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Proofgold Asset

asset id
780acb4c792a121590186659dfc014feb25ee4b9d5c01bdbbf78240670fc0c5b
asset hash
3a2ce838414d9fcd1b1ac02bceea84a0acd884365d65b08ac1e968dcefa4ab33
bday / block
3246
tx
bc03c..
preasset
doc published by PrCx1..
Known 46237..LoopI : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . binop_on x0 x1binop_on x0 x2binop_on x0 x3(∀ x5 . In x5 x0and (x1 x4 x5 = x5) (x1 x5 x4 = x5))(∀ x5 . In x5 x0∀ x6 . In x6 x0and (and (and (x2 x5 (x1 x5 x6) = x6) (x1 x5 (x2 x5 x6) = x6)) (x3 (x1 x5 x6) x6 = x5)) (x1 (x3 x5 x6) x6 = x5))Loop x0 x1 x2 x3 x4
Known 7c02f..andI : ∀ x0 x1 : ο . x0x1and x0 x1
Known 22d81..and4I : ∀ x0 x1 x2 x3 : ο . x0x1x2x3and (and (and x0 x1) x2) x3
Known 9ae18..SingE : ∀ x0 x1 . In x1 (Sing x0)x1 = x0
Known 86824..binop_on_I : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . In x2 x0∀ x3 . In x3 x0In (x1 x2 x3) x0)binop_on x0 x1
Known 1f15b..SingI : ∀ x0 . In x0 (Sing x0)
Theorem f3ad1.. : Loop (Sing 0) (λ x0 x1 . 0) (λ x0 x1 . 0) (λ x0 x1 . 0) (Sing 0) (proof)
Known 20499..Loop_with_defs_I : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop x0 x1 x2 x3 x4(∀ x14 . In x14 x0∀ x15 . In x15 x0x5 x14 x15 = x2 (x1 x15 x14) (x1 x14 x15))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x6 x14 x15 x16 = x2 (x1 x14 (x1 x15 x16)) (x1 (x1 x14 x15) x16))(∀ x14 . In x14 x0∀ x15 . In x15 x0and (and (and (and (x7 x14 x15 = x2 x14 (x1 x15 x14)) (x10 x14 x15 = x1 x14 (x1 x15 (x2 x14 x4)))) (x11 x14 x15 = x1 (x1 (x3 x4 x14) x15) x14)) (x12 x14 x15 = x1 (x2 x14 x15) (x2 (x2 x14 x4) x4))) (x13 x14 x15 = x1 (x3 x4 (x3 x4 x14)) (x3 x15 x14)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0and (x8 x14 x15 x16 = x2 (x1 x15 x14) (x1 x15 (x1 x14 x16))) (x9 x14 x15 x16 = x3 (x1 (x1 x16 x14) x15) (x1 x14 x15)))Loop_with_defs x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13
Known 1bd08..and5I : ∀ x0 x1 x2 x3 x4 : ο . x0x1x2x3x4and (and (and (and x0 x1) x2) x3) x4
Theorem 6e0b2.. : Loop_with_defs (Sing 0) (λ x0 x1 . 0) (λ x0 x1 . 0) (λ x0 x1 . 0) (Sing 0) (λ x0 x1 . 0) (λ x0 x1 x2 . 0) (λ x0 x1 . 0) (λ x0 x1 x2 . 0) (λ x0 x1 x2 . 0) (λ x0 x1 . 0) (λ x0 x1 . 0) (λ x0 x1 . 0) (λ x0 x1 . 0) (proof)
Known 9d0a4..Loop_with_defs_cex1_I : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . ∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0Loop_with_defs x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13not (x5 (x1 (x2 (x8 x15 x16 x14) x4) x14) x17 = x4)Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13
Known 8106d..notI : ∀ x0 : ο . (x0False)not x0
Known 2901c..EmptyE : ∀ x0 . In x0 0False
Theorem bc517.. : Loop_with_defs_cex1 (Sing 0) (λ x0 x1 . 0) (λ x0 x1 . 0) (λ x0 x1 . 0) (Sing 0) (λ x0 x1 . 0) (λ x0 x1 x2 . 0) (λ x0 x1 . 0) (λ x0 x1 x2 . 0) (λ x0 x1 x2 . 0) (λ x0 x1 . 0) (λ x0 x1 . 0) (λ x0 x1 . 0) (λ x0 x1 . 0) (proof)
Known 7054a..Loop_with_defs_cex2_I : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . ∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0Loop_with_defs x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13not (x6 x18 (x1 (x3 x4 x14) (x9 x15 x16 x14)) x17 = x4)Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13
Theorem 8fd72.. : Loop_with_defs_cex2 (Sing 0) (λ x0 x1 . 0) (λ x0 x1 . 0) (λ x0 x1 . 0) (Sing 0) (λ x0 x1 . 0) (λ x0 x1 x2 . 0) (λ x0 x1 . 0) (λ x0 x1 x2 . 0) (λ x0 x1 x2 . 0) (λ x0 x1 . 0) (λ x0 x1 . 0) (λ x0 x1 . 0) (λ x0 x1 . 0) (proof)
Known FalseEFalseE : False∀ x0 : ο . x0
Theorem 40262.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x10 x14 (x7 x15 (x7 x16 x17)) = x7 x15 (x7 x16 (x10 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x7 x16 (x10 x17 x18)) = x7 x16 (x10 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x13 x16 (x12 x17 x18))) = x13 x16 (x12 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x10 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x13 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x12 x17 (x7 x18 x19))) = x12 x17 (x7 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x7 x16 (x10 x17 (x7 x18 x19))) = x10 x17 (x7 x18 (x9 x14 x15 (x7 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x7 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x8 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x10 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x9 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x7 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x10 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x10 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x10 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x13 x16 (x10 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 68207.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x9 x14 x15 (x7 x14 (x13 x15 (x9 x14 x15 (x7 x14 (x13 x15 (x9 x14 x15 (x7 x14 (x13 x15 (x9 x14 x15 (x7 x14 (x13 x15 x16))))))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x9 x14 x15 (x12 x14 (x10 x15 (x9 x14 x15 (x12 x14 (x10 x15 (x9 x14 x15 (x12 x14 (x10 x15 x16)))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x12 x14 (x12 x15 (x12 x16 x17)) = x12 x15 (x12 x16 (x12 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x12 x16 (x13 x17 x18)) = x12 x16 (x13 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x13 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x12 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x13 x16 (x12 x17 x18))) = x13 x16 (x12 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x10 x16 (x13 x17 (x12 x18 x19))) = x13 x17 (x12 x18 (x8 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x12 x16 (x12 x17 (x10 x18 x19))) = x12 x17 (x10 x18 (x9 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x12 x16 (x8 x17 x18 (x13 x19 x20))) = x8 x17 x18 (x13 x19 (x8 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x13 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x7 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x13 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x7 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x13 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x8 x14 x15 (x10 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 26bea.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x12 x16 (x9 x14 x15 (x7 x16 (x9 x14 x15 (x12 x16 (x9 x14 x15 (x7 x16 x17))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x12 x14 (x12 x15 (x12 x16 x17)) = x12 x15 (x12 x16 (x12 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x7 x16 (x12 x17 x18)) = x7 x16 (x12 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x10 x15 (x7 x16 (x10 x17 x18))) = x7 x16 (x10 x17 (x7 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x7 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x10 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x12 x16 (x12 x17 (x10 x18 x19))) = x12 x17 (x10 x18 (x9 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x10 x16 (x13 x17 (x12 x18 x19))) = x13 x17 (x12 x18 (x8 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x7 x16 (x8 x17 x18 (x10 x19 x20))) = x8 x17 x18 (x10 x19 (x8 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x10 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x10 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x10 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x7 x16 (x10 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem fa6e4.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x7 x16 (x8 x14 x15 (x7 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x7 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x7 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x7 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x7 x16 x17))))))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x12 x16 (x10 x14 (x9 x15 x16 (x8 x14 x15 (x12 x16 (x10 x14 (x9 x15 x16 (x8 x14 x15 (x12 x16 (x10 x14 (x9 x15 x16 x17))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x10 x14 (x13 x15 (x7 x16 x17)) = x13 x15 (x7 x16 (x10 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x12 x16 (x12 x17 x18)) = x12 x16 (x12 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x12 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x7 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x12 x15 (x10 x16 (x7 x17 x18))) = x10 x16 (x7 x17 (x7 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x10 x16 (x12 x17 (x12 x18 x19))) = x12 x17 (x12 x18 (x8 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x7 x16 (x10 x17 (x7 x18 x19))) = x10 x17 (x7 x18 (x8 x14 x15 (x7 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x10 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x9 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x12 x16 (x8 x17 x18 (x10 x19 x20))) = x8 x17 x18 (x10 x19 (x8 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x13 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 27559.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x12 x14 (x12 x15 (x10 x16 x17)) = x12 x15 (x10 x16 (x12 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x10 x16 (x12 x17 x18)) = x10 x16 (x12 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x10 x16 (x13 x17 x18))) = x10 x16 (x13 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x7 x15 (x7 x16 (x10 x17 x18))) = x7 x16 (x10 x17 (x13 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x10 x16 (x10 x17 x18))) = x10 x16 (x10 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x7 x16 (x7 x17 x18))) = x7 x16 (x7 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x12 x15 (x7 x16 (x10 x17 x18))) = x7 x16 (x10 x17 (x7 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x7 x17 (x13 x18 x19))) = x7 x17 (x13 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x7 x16 (x13 x17 (x7 x18 x19))) = x13 x17 (x7 x18 (x9 x14 x15 (x7 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x10 x16 (x9 x17 x18 (x7 x19 x20))) = x9 x17 x18 (x7 x19 (x8 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x13 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x8 x14 x15 (x13 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x13 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x12 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x13 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem fa925.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x10 x14 (x13 x15 (x13 x16 x17)) = x13 x15 (x13 x16 (x10 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x12 x16 (x10 x17 x18)) = x12 x16 (x10 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x10 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x7 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x7 x16 (x10 x17 x18))) = x7 x16 (x10 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x7 x16 (x10 x17 x18))) = x7 x16 (x10 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x13 x17 (x10 x18 x19))) = x13 x17 (x10 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x10 x16 (x7 x17 (x12 x18 x19))) = x7 x17 (x12 x18 (x9 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x8 x17 x18 (x10 x19 x20))) = x8 x17 x18 (x10 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x13 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x13 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x13 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem ab8e0.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x12 x14 (x13 x15 (x7 x16 x17)) = x13 x15 (x7 x16 (x12 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x10 x16 (x10 x17 x18)) = x10 x16 (x10 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x13 x16 (x7 x17 x18))) = x13 x16 (x7 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x10 x16 (x12 x17 (x10 x18 x19))) = x12 x17 (x10 x18 (x9 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x12 x17 (x10 x18 x19))) = x12 x17 (x10 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x13 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x8 x14 x15 (x13 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x10 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x8 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x7 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x7 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x13 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x7 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 7af9d.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x9 x14 x15 (x10 x14 (x12 x15 (x9 x14 x15 (x10 x14 (x12 x15 (x9 x14 x15 (x10 x14 (x12 x15 (x9 x14 x15 (x10 x14 (x12 x15 x16))))))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x7 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x7 x16 (x9 x14 x15 (x7 x16 x17))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x13 x14 (x13 x15 (x12 x16 x17)) = x13 x15 (x12 x16 (x13 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x10 x16 (x12 x17 x18)) = x10 x16 (x12 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x10 x16 (x7 x17 x18))) = x10 x16 (x7 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x7 x16 (x10 x17 x18))) = x7 x16 (x10 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x13 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x7 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x7 x17 (x10 x18 x19))) = x7 x17 (x10 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x10 x16 (x13 x17 (x7 x18 x19))) = x13 x17 (x7 x18 (x8 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x12 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x8 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x12 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x8 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x10 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x7 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x7 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x13 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x10 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x10 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x7 x16 (x10 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem ba19f.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x12 x16 (x7 x14 (x8 x15 x16 (x8 x14 x15 (x12 x16 (x7 x14 (x8 x15 x16 (x8 x14 x15 (x12 x16 (x7 x14 (x8 x15 x16 x17))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x9 x14 x15 (x7 x14 (x10 x15 (x9 x14 x15 (x7 x14 (x10 x15 (x9 x14 x15 (x7 x14 (x10 x15 (x9 x14 x15 (x7 x14 (x10 x15 x16))))))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x12 x14 (x12 x15 (x7 x16 x17)) = x12 x15 (x7 x16 (x12 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x13 x16 (x7 x17 x18)) = x13 x16 (x7 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x10 x16 (x10 x17 x18))) = x10 x16 (x10 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x12 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x13 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x12 x16 (x12 x17 (x10 x18 x19))) = x12 x17 (x10 x18 (x9 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x12 x17 (x7 x18 x19))) = x12 x17 (x7 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x13 x16 (x8 x17 x18 (x13 x19 x20))) = x8 x17 x18 (x13 x19 (x9 x14 x15 (x13 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x10 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x7 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x10 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x10 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x7 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x12 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x9 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem a7002.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x8 x14 x15 (x10 x14 (x13 x15 (x8 x14 x15 (x10 x14 (x13 x15 (x8 x14 x15 (x10 x14 (x13 x15 x16)))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x7 x14 (x10 x15 (x13 x16 x17)) = x10 x15 (x13 x16 (x7 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x12 x16 (x10 x17 x18)) = x12 x16 (x10 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x13 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x12 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x12 x15 (x13 x16 (x7 x17 x18))) = x13 x16 (x7 x17 (x13 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x7 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x13 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x10 x16 (x7 x17 (x7 x18 x19))) = x7 x17 (x7 x18 (x8 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x12 x16 (x10 x17 (x7 x18 x19))) = x10 x17 (x7 x18 (x9 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x9 x17 x18 (x10 x19 x20))) = x9 x17 x18 (x10 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x9 x17 x18 (x10 x19 x20))) = x9 x17 x18 (x10 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x10 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x7 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x10 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x8 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x10 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x12 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x9 x14 x15 (x13 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem fefaa.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x12 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x12 x16 (x9 x14 x15 (x7 x16 x17))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x7 x16 (x7 x14 (x8 x15 x16 (x9 x14 x15 (x7 x16 (x7 x14 (x8 x15 x16 (x9 x14 x15 (x7 x16 (x7 x14 (x8 x15 x16 x17))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x13 x14 (x10 x15 (x13 x16 x17)) = x10 x15 (x13 x16 (x13 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x12 x16 (x7 x17 x18)) = x12 x16 (x7 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x13 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x12 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x7 x16 (x7 x17 x18))) = x7 x16 (x7 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x13 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x7 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x13 x16 (x12 x17 x18))) = x13 x16 (x12 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x10 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x13 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x12 x17 (x12 x18 x19))) = x12 x17 (x12 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x7 x16 (x12 x17 (x7 x18 x19))) = x12 x17 (x7 x18 (x8 x14 x15 (x7 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x10 x16 (x9 x17 x18 (x10 x19 x20))) = x9 x17 x18 (x10 x19 (x9 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x9 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x13 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x13 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x10 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x8 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x10 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x9 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x9 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 9e713.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x12 x16 (x7 x14 (x9 x15 x16 (x8 x14 x15 (x12 x16 (x7 x14 (x9 x15 x16 (x8 x14 x15 (x12 x16 (x7 x14 (x9 x15 x16 (x8 x14 x15 (x12 x16 (x7 x14 (x9 x15 x16 (x8 x14 x15 (x12 x16 (x7 x14 (x9 x15 x16 x17))))))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x13 x16 (x7 x14 (x8 x15 x16 (x9 x14 x15 (x13 x16 (x7 x14 (x8 x15 x16 (x9 x14 x15 (x13 x16 (x7 x14 (x8 x15 x16 x17))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x10 x14 (x12 x15 (x13 x16 x17)) = x12 x15 (x13 x16 (x10 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x7 x16 (x10 x17 x18)) = x7 x16 (x10 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x12 x15 (x10 x16 (x10 x17 x18))) = x10 x16 (x10 x17 (x7 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x10 x15 (x13 x16 (x10 x17 x18))) = x13 x16 (x10 x17 (x10 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x12 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x7 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x12 x16 (x12 x17 (x13 x18 x19))) = x12 x17 (x13 x18 (x9 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x13 x16 (x13 x17 (x7 x18 x19))) = x13 x17 (x7 x18 (x9 x14 x15 (x13 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x7 x16 (x9 x17 x18 (x7 x19 x20))) = x9 x17 x18 (x7 x19 (x8 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x13 x16 (x8 x17 x18 (x7 x19 x20))) = x8 x17 x18 (x7 x19 (x9 x14 x15 (x13 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x13 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x8 x14 x15 (x13 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x13 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x7 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x8 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x9 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x10 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x10 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x10 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x10 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 11192.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x10 x16 (x12 x14 (x8 x15 x16 (x8 x14 x15 (x10 x16 (x12 x14 (x8 x15 x16 (x8 x14 x15 (x10 x16 (x12 x14 (x8 x15 x16 (x8 x14 x15 (x10 x16 (x12 x14 (x8 x15 x16 (x8 x14 x15 (x10 x16 (x12 x14 (x8 x15 x16 x17))))))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x8 x14 x15 (x12 x14 (x10 x15 (x8 x14 x15 (x12 x14 (x10 x15 (x8 x14 x15 (x12 x14 (x10 x15 (x8 x14 x15 (x12 x14 (x10 x15 (x8 x14 x15 (x12 x14 (x10 x15 x16)))))))))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x13 x14 (x10 x15 (x12 x16 x17)) = x10 x15 (x12 x16 (x13 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x7 x16 (x7 x17 x18)) = x7 x16 (x7 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x10 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x7 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x13 x15 (x10 x16 (x7 x17 x18))) = x10 x16 (x7 x17 (x12 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x10 x16 (x10 x17 x18))) = x10 x16 (x10 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x13 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x10 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x7 x16 (x10 x17 x18))) = x7 x16 (x10 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x10 x16 (x12 x17 (x12 x18 x19))) = x12 x17 (x12 x18 (x8 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x10 x16 (x10 x17 (x10 x18 x19))) = x10 x17 (x10 x18 (x8 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x7 x16 (x8 x17 x18 (x7 x19 x20))) = x8 x17 x18 (x7 x19 (x8 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x9 x17 x18 (x10 x19 x20))) = x9 x17 x18 (x10 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x7 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x10 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x10 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x8 x14 x15 (x7 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 6c1c6.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x9 x14 x15 (x12 x14 (x10 x15 (x9 x14 x15 (x12 x14 (x10 x15 x16))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x13 x16 (x10 x14 (x8 x15 x16 (x8 x14 x15 (x13 x16 (x10 x14 (x8 x15 x16 (x8 x14 x15 (x13 x16 (x10 x14 (x8 x15 x16 (x8 x14 x15 (x13 x16 (x10 x14 (x8 x15 x16 x17))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x13 x14 (x12 x15 (x13 x16 x17)) = x12 x15 (x13 x16 (x13 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x12 x16 (x12 x17 x18)) = x12 x16 (x12 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x10 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x7 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x10 x16 (x7 x17 x18))) = x10 x16 (x7 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x7 x15 (x13 x16 (x13 x17 x18))) = x13 x16 (x13 x17 (x10 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x12 x15 (x7 x16 (x10 x17 x18))) = x7 x16 (x10 x17 (x7 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x13 x16 (x12 x17 (x10 x18 x19))) = x12 x17 (x10 x18 (x9 x14 x15 (x13 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x7 x16 (x7 x17 (x13 x18 x19))) = x7 x17 (x13 x18 (x8 x14 x15 (x7 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x10 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x8 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x8 x17 x18 (x7 x19 x20))) = x8 x17 x18 (x7 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x10 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x9 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x7 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x10 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x7 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem d55c5.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x12 x14 (x10 x15 (x7 x16 x17)) = x10 x15 (x7 x16 (x12 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x12 x16 (x10 x17 x18)) = x12 x16 (x10 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x12 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x7 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x7 x15 (x13 x16 (x13 x17 x18))) = x13 x16 (x13 x17 (x13 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x10 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x10 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x10 x16 (x10 x17 (x13 x18 x19))) = x10 x17 (x13 x18 (x8 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x7 x16 (x13 x17 (x12 x18 x19))) = x13 x17 (x12 x18 (x8 x14 x15 (x7 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x10 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x9 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x10 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x8 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x7 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x9 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x10 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 498c1.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x9 x14 x15 (x13 x14 (x10 x15 (x9 x14 x15 (x13 x14 (x10 x15 (x9 x14 x15 (x13 x14 (x10 x15 (x9 x14 x15 (x13 x14 (x10 x15 (x9 x14 x15 (x13 x14 (x10 x15 x16)))))))))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x10 x16 (x8 x14 x15 (x13 x16 (x8 x14 x15 (x10 x16 (x8 x14 x15 (x13 x16 (x8 x14 x15 (x10 x16 (x8 x14 x15 (x13 x16 x17))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x13 x14 (x7 x15 (x7 x16 x17)) = x7 x15 (x7 x16 (x13 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x10 x16 (x12 x17 x18)) = x10 x16 (x12 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x7 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x13 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x7 x16 (x7 x17 x18))) = x7 x16 (x7 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x10 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x7 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x10 x15 (x10 x16 (x10 x17 x18))) = x10 x16 (x10 x17 (x10 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x13 x15 (x10 x16 (x10 x17 x18))) = x10 x16 (x10 x17 (x12 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x13 x16 (x12 x17 (x12 x18 x19))) = x12 x17 (x12 x18 (x9 x14 x15 (x13 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x12 x17 (x12 x18 x19))) = x12 x17 (x12 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x10 x16 (x8 x17 x18 (x7 x19 x20))) = x8 x17 x18 (x7 x19 (x8 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x9 x17 x18 (x7 x19 x20))) = x9 x17 x18 (x7 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x10 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x13 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x13 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x9 x14 x15 (x12 x16 (x13 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem a3ce2.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 x17))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 x17))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x7 x14 (x12 x15 (x10 x16 x17)) = x12 x15 (x10 x16 (x7 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x12 x16 (x12 x17 x18)) = x12 x16 (x12 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x13 x15 (x10 x16 (x13 x17 x18))) = x10 x16 (x13 x17 (x12 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x7 x16 (x13 x17 x18))) = x7 x16 (x13 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x7 x16 (x7 x17 x18))) = x7 x16 (x7 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x12 x15 (x7 x16 (x7 x17 x18))) = x7 x16 (x7 x17 (x7 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x10 x17 (x12 x18 x19))) = x10 x17 (x12 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x12 x16 (x7 x17 (x13 x18 x19))) = x7 x17 (x13 x18 (x9 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x13 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x9 x14 x15 (x13 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x12 x16 (x9 x17 x18 (x7 x19 x20))) = x9 x17 x18 (x7 x19 (x8 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x10 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x7 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x9 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x7 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem c7a16.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x12 x16 (x12 x14 (x9 x15 x16 (x8 x14 x15 (x12 x16 (x12 x14 (x9 x15 x16 (x8 x14 x15 (x12 x16 (x12 x14 (x9 x15 x16 (x8 x14 x15 (x12 x16 (x12 x14 (x9 x15 x16 (x8 x14 x15 (x12 x16 (x12 x14 (x9 x15 x16 x17))))))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 x17))))))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x13 x14 (x12 x15 (x13 x16 x17)) = x12 x15 (x13 x16 (x13 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x7 x16 (x13 x17 x18)) = x7 x16 (x13 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x13 x16 (x7 x17 x18))) = x13 x16 (x7 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x10 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x10 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x13 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x12 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x10 x16 (x10 x17 (x13 x18 x19))) = x10 x17 (x13 x18 (x9 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x7 x16 (x7 x17 (x12 x18 x19))) = x7 x17 (x12 x18 (x8 x14 x15 (x7 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x10 x16 (x9 x17 x18 (x13 x19 x20))) = x9 x17 x18 (x13 x19 (x8 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x12 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x8 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x10 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x8 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x10 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x7 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x10 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem a23be.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x10 x16 (x10 x14 (x8 x15 x16 (x8 x14 x15 (x10 x16 (x10 x14 (x8 x15 x16 (x8 x14 x15 (x10 x16 (x10 x14 (x8 x15 x16 (x8 x14 x15 (x10 x16 (x10 x14 (x8 x15 x16 (x8 x14 x15 (x10 x16 (x10 x14 (x8 x15 x16 x17))))))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x12 x16 (x12 x14 (x8 x15 x16 (x8 x14 x15 (x12 x16 (x12 x14 (x8 x15 x16 (x8 x14 x15 (x12 x16 (x12 x14 (x8 x15 x16 (x8 x14 x15 (x12 x16 (x12 x14 (x8 x15 x16 x17))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x7 x14 (x10 x15 (x12 x16 x17)) = x10 x15 (x12 x16 (x7 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x12 x16 (x10 x17 x18)) = x12 x16 (x10 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x10 x16 (x7 x17 x18))) = x10 x16 (x7 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x7 x16 (x7 x17 x18))) = x7 x16 (x7 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x10 x16 (x12 x17 (x7 x18 x19))) = x12 x17 (x7 x18 (x9 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x13 x16 (x7 x17 (x12 x18 x19))) = x7 x17 (x12 x18 (x9 x14 x15 (x13 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x7 x16 (x8 x17 x18 (x10 x19 x20))) = x8 x17 x18 (x10 x19 (x8 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x12 x16 (x9 x17 x18 (x13 x19 x20))) = x9 x17 x18 (x13 x19 (x8 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x13 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x9 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x13 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x13 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x9 x14 x15 (x10 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 837e6.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x10 x14 (x12 x15 (x10 x16 x17)) = x12 x15 (x10 x16 (x10 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x10 x16 (x10 x17 x18)) = x10 x16 (x10 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x7 x16 (x7 x17 x18))) = x7 x16 (x7 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x7 x16 (x10 x17 x18))) = x7 x16 (x10 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x13 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x10 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x13 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x12 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x13 x15 (x7 x16 (x10 x17 x18))) = x7 x16 (x10 x17 (x13 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x13 x16 (x7 x17 (x7 x18 x19))) = x7 x17 (x7 x18 (x8 x14 x15 (x13 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x10 x16 (x7 x17 (x7 x18 x19))) = x7 x17 (x7 x18 (x8 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x10 x16 (x8 x17 x18 (x7 x19 x20))) = x8 x17 x18 (x7 x19 (x8 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x7 x16 (x8 x17 x18 (x7 x19 x20))) = x8 x17 x18 (x7 x19 (x9 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x8 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x7 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x13 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x13 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem b130f.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x8 x14 x15 (x12 x14 (x12 x15 (x8 x14 x15 (x12 x14 (x12 x15 x16))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x8 x14 x15 (x7 x14 (x12 x15 (x8 x14 x15 (x7 x14 (x12 x15 (x8 x14 x15 (x7 x14 (x12 x15 x16)))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x12 x14 (x10 x15 (x12 x16 x17)) = x10 x15 (x12 x16 (x12 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x7 x16 (x7 x17 x18)) = x7 x16 (x7 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x10 x15 (x13 x16 (x12 x17 x18))) = x13 x16 (x12 x17 (x10 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x10 x16 (x13 x17 x18))) = x10 x16 (x13 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x10 x16 (x13 x17 x18))) = x10 x16 (x13 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x13 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x13 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x13 x16 (x7 x17 (x10 x18 x19))) = x7 x17 (x10 x18 (x9 x14 x15 (x13 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x12 x17 (x7 x18 x19))) = x12 x17 (x7 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x7 x16 (x9 x17 x18 (x7 x19 x20))) = x9 x17 x18 (x7 x19 (x8 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x12 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x8 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x10 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x12 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x7 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x10 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x9 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 7f910.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x8 x14 x15 (x10 x14 (x7 x15 (x8 x14 x15 (x10 x14 (x7 x15 x16))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x8 x14 x15 (x12 x14 (x12 x15 (x8 x14 x15 (x12 x14 (x12 x15 (x8 x14 x15 (x12 x14 (x12 x15 x16)))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x7 x14 (x10 x15 (x12 x16 x17)) = x10 x15 (x12 x16 (x7 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x13 x16 (x7 x17 x18)) = x13 x16 (x7 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x7 x15 (x10 x16 (x7 x17 x18))) = x10 x16 (x7 x17 (x10 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x13 x16 (x7 x17 x18))) = x13 x16 (x7 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x13 x15 (x10 x16 (x10 x17 x18))) = x10 x16 (x10 x17 (x12 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x7 x15 (x7 x16 (x7 x17 x18))) = x7 x16 (x7 x17 (x10 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x10 x16 (x13 x17 (x12 x18 x19))) = x13 x17 (x12 x18 (x8 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x13 x16 (x10 x17 (x12 x18 x19))) = x10 x17 (x12 x18 (x9 x14 x15 (x13 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x10 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x9 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x10 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x9 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x13 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x7 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 63e2b.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x10 x16 (x9 x14 x15 (x12 x16 (x9 x14 x15 (x10 x16 (x9 x14 x15 (x12 x16 (x9 x14 x15 (x10 x16 (x9 x14 x15 (x12 x16 (x9 x14 x15 (x10 x16 (x9 x14 x15 (x12 x16 x17))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x9 x14 x15 (x7 x14 (x12 x15 (x9 x14 x15 (x7 x14 (x12 x15 (x9 x14 x15 (x7 x14 (x12 x15 (x9 x14 x15 (x7 x14 (x12 x15 (x9 x14 x15 (x7 x14 (x12 x15 x16)))))))))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x12 x14 (x7 x15 (x12 x16 x17)) = x7 x15 (x12 x16 (x12 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x7 x16 (x10 x17 x18)) = x7 x16 (x10 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x10 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x10 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x10 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x7 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x13 x16 (x12 x17 x18))) = x13 x16 (x12 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x13 x16 (x12 x17 x18))) = x13 x16 (x12 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x12 x16 (x13 x17 x18))) = x12 x16 (x13 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x12 x16 (x10 x17 (x7 x18 x19))) = x10 x17 (x7 x18 (x9 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x13 x16 (x13 x17 (x12 x18 x19))) = x13 x17 (x12 x18 (x8 x14 x15 (x13 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x13 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x9 x14 x15 (x13 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x10 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x8 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x7 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem b0997.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x7 x16 (x10 x14 (x8 x15 x16 (x8 x14 x15 (x7 x16 (x10 x14 (x8 x15 x16 x17))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x10 x16 (x7 x14 (x8 x15 x16 (x9 x14 x15 (x10 x16 (x7 x14 (x8 x15 x16 (x9 x14 x15 (x10 x16 (x7 x14 (x8 x15 x16 (x9 x14 x15 (x10 x16 (x7 x14 (x8 x15 x16 (x9 x14 x15 (x10 x16 (x7 x14 (x8 x15 x16 x17))))))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x12 x14 (x12 x15 (x7 x16 x17)) = x12 x15 (x7 x16 (x12 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x12 x16 (x13 x17 x18)) = x12 x16 (x13 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x10 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x7 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x10 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x13 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x12 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x7 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x12 x15 (x12 x16 (x13 x17 x18))) = x12 x16 (x13 x17 (x7 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x10 x16 (x10 x17 x18))) = x10 x16 (x10 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x7 x16 (x10 x17 (x7 x18 x19))) = x10 x17 (x7 x18 (x9 x14 x15 (x7 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x12 x17 (x10 x18 x19))) = x12 x17 (x10 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x12 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x8 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x12 x16 (x9 x17 x18 (x10 x19 x20))) = x9 x17 x18 (x10 x19 (x8 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x13 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x7 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x8 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem db474.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x7 x16 (x9 x14 x15 (x12 x16 (x9 x14 x15 (x7 x16 (x9 x14 x15 (x12 x16 (x9 x14 x15 (x7 x16 (x9 x14 x15 (x12 x16 (x9 x14 x15 (x7 x16 (x9 x14 x15 (x12 x16 x17))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x12 x16 (x8 x14 x15 (x13 x16 (x9 x14 x15 (x12 x16 (x8 x14 x15 (x13 x16 (x9 x14 x15 (x12 x16 (x8 x14 x15 (x13 x16 (x9 x14 x15 (x12 x16 (x8 x14 x15 (x13 x16 x17))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x10 x14 (x12 x15 (x12 x16 x17)) = x12 x15 (x12 x16 (x10 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x10 x16 (x12 x17 x18)) = x10 x16 (x12 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x7 x15 (x12 x16 (x13 x17 x18))) = x12 x16 (x13 x17 (x10 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x12 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x13 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x13 x15 (x13 x16 (x10 x17 x18))) = x13 x16 (x10 x17 (x10 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x7 x16 (x12 x17 (x12 x18 x19))) = x12 x17 (x12 x18 (x9 x14 x15 (x7 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x13 x17 (x13 x18 x19))) = x13 x17 (x13 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x7 x16 (x8 x17 x18 (x10 x19 x20))) = x8 x17 x18 (x10 x19 (x8 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x10 x16 (x8 x17 x18 (x7 x19 x20))) = x8 x17 x18 (x7 x19 (x9 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x10 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x13 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x13 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x7 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x10 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 8cc14.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x10 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x10 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x10 x16 (x9 x14 x15 (x7 x16 x17))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x7 x14 (x12 x15 (x7 x16 x17)) = x12 x15 (x7 x16 (x7 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x12 x16 (x12 x17 x18)) = x12 x16 (x12 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x10 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x7 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x10 x15 (x7 x16 (x7 x17 x18))) = x7 x16 (x7 x17 (x13 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x10 x16 (x12 x17 (x12 x18 x19))) = x12 x17 (x12 x18 (x9 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x12 x16 (x10 x17 (x7 x18 x19))) = x10 x17 (x7 x18 (x9 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x7 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x8 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x13 x16 (x9 x17 x18 (x7 x19 x20))) = x9 x17 x18 (x7 x19 (x8 x14 x15 (x13 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x7 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x9 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x10 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x7 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x10 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x10 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x13 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x13 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 955ae.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x9 x14 x15 (x7 x14 (x13 x15 (x9 x14 x15 (x7 x14 (x13 x15 x16))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x10 x14 (x10 x15 (x12 x16 x17)) = x10 x15 (x12 x16 (x10 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x7 x16 (x12 x17 x18)) = x7 x16 (x12 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x10 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x10 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x10 x16 (x13 x17 x18))) = x10 x16 (x13 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x10 x15 (x7 x16 (x7 x17 x18))) = x7 x16 (x7 x17 (x10 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x12 x17 (x10 x18 x19))) = x12 x17 (x10 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x10 x16 (x12 x17 (x12 x18 x19))) = x12 x17 (x12 x18 (x9 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x10 x16 (x8 x17 x18 (x13 x19 x20))) = x8 x17 x18 (x13 x19 (x9 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x12 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x8 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x13 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x9 x14 x15 (x10 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x7 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x10 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x9 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem d7c6d.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x7 x16 (x8 x14 x15 (x12 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x12 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x12 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x12 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x12 x16 x17))))))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x7 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x7 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x7 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x7 x16 (x9 x14 x15 (x7 x16 x17))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x7 x14 (x13 x15 (x12 x16 x17)) = x13 x15 (x12 x16 (x7 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x10 x16 (x10 x17 x18)) = x10 x16 (x10 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x10 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x7 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x12 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x13 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x13 x15 (x12 x16 (x13 x17 x18))) = x12 x16 (x13 x17 (x12 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x10 x16 (x7 x17 x18))) = x10 x16 (x7 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x13 x16 (x10 x17 (x12 x18 x19))) = x10 x17 (x12 x18 (x9 x14 x15 (x13 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x10 x16 (x13 x17 (x13 x18 x19))) = x13 x17 (x13 x18 (x8 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x10 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x9 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x10 x16 (x9 x17 x18 (x7 x19 x20))) = x9 x17 x18 (x7 x19 (x8 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x13 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x7 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x10 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x9 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x13 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem bb2ef.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x9 x14 x15 (x10 x14 (x12 x15 (x9 x14 x15 (x10 x14 (x12 x15 x16))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x10 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x10 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x10 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x10 x16 (x9 x14 x15 (x7 x16 (x8 x14 x15 (x10 x16 (x9 x14 x15 (x7 x16 x17))))))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x7 x14 (x10 x15 (x10 x16 x17)) = x10 x15 (x10 x16 (x7 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x7 x16 (x12 x17 x18)) = x7 x16 (x12 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x13 x16 (x12 x17 x18))) = x13 x16 (x12 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x13 x16 (x10 x17 x18))) = x13 x16 (x10 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x13 x15 (x13 x16 (x10 x17 x18))) = x13 x16 (x10 x17 (x7 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x13 x16 (x12 x17 (x12 x18 x19))) = x12 x17 (x12 x18 (x8 x14 x15 (x13 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x12 x17 (x10 x18 x19))) = x12 x17 (x10 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x7 x16 (x9 x17 x18 (x13 x19 x20))) = x9 x17 x18 (x13 x19 (x9 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x12 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x8 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 000b7.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x9 x14 x15 (x7 x14 (x7 x15 (x9 x14 x15 (x7 x14 (x7 x15 (x9 x14 x15 (x7 x14 (x7 x15 (x9 x14 x15 (x7 x14 (x7 x15 (x9 x14 x15 (x7 x14 (x7 x15 x16)))))))))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x13 x16 (x12 x14 (x8 x15 x16 (x9 x14 x15 (x13 x16 (x12 x14 (x8 x15 x16 (x9 x14 x15 (x13 x16 (x12 x14 (x8 x15 x16 (x9 x14 x15 (x13 x16 (x12 x14 (x8 x15 x16 (x9 x14 x15 (x13 x16 (x12 x14 (x8 x15 x16 x17))))))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x7 x14 (x12 x15 (x12 x16 x17)) = x12 x15 (x12 x16 (x7 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x10 x16 (x13 x17 x18)) = x10 x16 (x13 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x13 x15 (x10 x16 (x7 x17 x18))) = x10 x16 (x7 x17 (x7 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x10 x15 (x7 x16 (x7 x17 x18))) = x7 x16 (x7 x17 (x10 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x12 x15 (x10 x16 (x13 x17 x18))) = x10 x16 (x13 x17 (x13 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x13 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x12 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x12 x16 (x7 x17 (x7 x18 x19))) = x7 x17 (x7 x18 (x9 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x12 x16 (x13 x17 (x7 x18 x19))) = x13 x17 (x7 x18 (x9 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x9 x17 x18 (x10 x19 x20))) = x9 x17 x18 (x10 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x13 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x9 x14 x15 (x13 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x13 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x7 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x9 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x10 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x13 x16 (x10 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 9e722.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x8 x14 x15 (x10 x14 (x12 x15 (x8 x14 x15 (x10 x14 (x12 x15 (x8 x14 x15 (x10 x14 (x12 x15 (x8 x14 x15 (x10 x14 (x12 x15 (x8 x14 x15 (x10 x14 (x12 x15 x16)))))))))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x7 x16 (x7 x14 (x9 x15 x16 (x9 x14 x15 (x7 x16 (x7 x14 (x9 x15 x16 (x9 x14 x15 (x7 x16 (x7 x14 (x9 x15 x16 (x9 x14 x15 (x7 x16 (x7 x14 (x9 x15 x16 (x9 x14 x15 (x7 x16 (x7 x14 (x9 x15 x16 x17))))))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x7 x14 (x12 x15 (x12 x16 x17)) = x12 x15 (x12 x16 (x7 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x13 x16 (x13 x17 x18)) = x13 x16 (x13 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x12 x15 (x7 x16 (x7 x17 x18))) = x7 x16 (x7 x17 (x13 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x7 x16 (x13 x17 x18))) = x7 x16 (x13 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x7 x16 (x13 x17 x18))) = x7 x16 (x13 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x13 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x7 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x12 x16 (x12 x17 (x10 x18 x19))) = x12 x17 (x10 x18 (x9 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x10 x16 (x10 x17 (x12 x18 x19))) = x10 x17 (x12 x18 (x9 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x10 x16 (x9 x17 x18 (x7 x19 x20))) = x9 x17 x18 (x7 x19 (x8 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x10 x16 (x8 x17 x18 (x7 x19 x20))) = x8 x17 x18 (x7 x19 (x9 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x10 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x10 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x9 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x13 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x10 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x7 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 48eba.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x7 x16 (x9 x14 x15 (x10 x16 (x8 x14 x15 (x7 x16 (x9 x14 x15 (x10 x16 x17))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x12 x14 (x7 x15 (x13 x16 x17)) = x7 x15 (x13 x16 (x12 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x7 x16 (x12 x17 x18)) = x7 x16 (x12 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x13 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x10 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x13 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x12 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x13 x16 (x12 x17 x18))) = x13 x16 (x12 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x13 x16 (x12 x17 x18))) = x13 x16 (x12 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x7 x16 (x12 x17 (x12 x18 x19))) = x12 x17 (x12 x18 (x9 x14 x15 (x7 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x10 x16 (x10 x17 (x12 x18 x19))) = x10 x17 (x12 x18 (x8 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x7 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x9 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x10 x16 (x8 x17 x18 (x7 x19 x20))) = x8 x17 x18 (x7 x19 (x8 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x7 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x13 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x9 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x7 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x10 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem dc95c.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 x17))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x10 x14 (x10 x15 (x12 x16 x17)) = x10 x15 (x12 x16 (x10 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x7 x16 (x12 x17 x18)) = x7 x16 (x12 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x10 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x13 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x10 x16 (x13 x17 x18))) = x10 x16 (x13 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x7 x15 (x10 x16 (x7 x17 x18))) = x10 x16 (x7 x17 (x10 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x13 x15 (x13 x16 (x7 x17 x18))) = x13 x16 (x7 x17 (x12 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x7 x15 (x7 x16 (x10 x17 x18))) = x7 x16 (x10 x17 (x13 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x7 x16 (x13 x17 (x12 x18 x19))) = x13 x17 (x12 x18 (x8 x14 x15 (x7 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x10 x17 (x7 x18 x19))) = x10 x17 (x7 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x10 x16 (x8 x17 x18 (x10 x19 x20))) = x8 x17 x18 (x10 x19 (x9 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x13 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x12 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x10 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x12 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x8 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x12 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x8 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem f3491.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x8 x14 x15 (x13 x14 (x13 x15 (x8 x14 x15 (x13 x14 (x13 x15 (x8 x14 x15 (x13 x14 (x13 x15 (x8 x14 x15 (x13 x14 (x13 x15 x16))))))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x10 x16 (x7 x14 (x9 x15 x16 (x8 x14 x15 (x10 x16 (x7 x14 (x9 x15 x16 (x8 x14 x15 (x10 x16 (x7 x14 (x9 x15 x16 (x8 x14 x15 (x10 x16 (x7 x14 (x9 x15 x16 x17))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x7 x14 (x13 x15 (x12 x16 x17)) = x13 x15 (x12 x16 (x7 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x7 x16 (x13 x17 x18)) = x7 x16 (x13 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x13 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x12 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x7 x16 (x10 x17 x18))) = x7 x16 (x10 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x10 x15 (x13 x16 (x10 x17 x18))) = x13 x16 (x10 x17 (x7 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x7 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x10 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x10 x16 (x10 x17 (x12 x18 x19))) = x10 x17 (x12 x18 (x8 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x13 x16 (x10 x17 (x7 x18 x19))) = x10 x17 (x7 x18 (x8 x14 x15 (x13 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x9 x17 x18 (x13 x19 x20))) = x9 x17 x18 (x13 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x8 x17 x18 (x13 x19 x20))) = x8 x17 x18 (x13 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x13 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x13 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x13 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x7 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x10 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem b45b9.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x10 x16 (x12 x14 (x8 x15 x16 (x8 x14 x15 (x10 x16 (x12 x14 (x8 x15 x16 (x8 x14 x15 (x10 x16 (x12 x14 (x8 x15 x16 x17))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x10 x16 (x9 x14 x15 (x7 x16 (x9 x14 x15 (x10 x16 (x9 x14 x15 (x7 x16 (x9 x14 x15 (x10 x16 (x9 x14 x15 (x7 x16 x17))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x7 x14 (x12 x15 (x12 x16 x17)) = x12 x15 (x12 x16 (x7 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x12 x16 (x10 x17 x18)) = x12 x16 (x10 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x7 x15 (x10 x16 (x10 x17 x18))) = x10 x16 (x10 x17 (x13 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x10 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x10 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x10 x16 (x7 x17 x18))) = x10 x16 (x7 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x10 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x13 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x13 x16 (x13 x17 x18))) = x13 x16 (x13 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x13 x17 (x7 x18 x19))) = x13 x17 (x7 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x12 x17 (x7 x18 x19))) = x12 x17 (x7 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x9 x17 x18 (x7 x19 x20))) = x9 x17 x18 (x7 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x7 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x13 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x7 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x7 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x7 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 3e28e.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x8 x14 x15 (x10 x14 (x12 x15 (x8 x14 x15 (x10 x14 (x12 x15 x16))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 x17))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x12 x14 (x12 x15 (x12 x16 x17)) = x12 x15 (x12 x16 (x12 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x7 x16 (x12 x17 x18)) = x7 x16 (x12 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x10 x15 (x7 x16 (x13 x17 x18))) = x7 x16 (x13 x17 (x7 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x10 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x7 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x10 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x7 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x10 x16 (x7 x17 (x12 x18 x19))) = x7 x17 (x12 x18 (x8 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x12 x16 (x7 x17 (x12 x18 x19))) = x7 x17 (x12 x18 (x9 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x12 x16 (x8 x17 x18 (x10 x19 x20))) = x8 x17 x18 (x10 x19 (x8 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x12 x16 (x9 x17 x18 (x10 x19 x20))) = x9 x17 x18 (x10 x19 (x8 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x10 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x7 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x10 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x10 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x9 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem bd24a.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x8 x14 x15 (x7 x14 (x7 x15 (x8 x14 x15 (x7 x14 (x7 x15 (x8 x14 x15 (x7 x14 (x7 x15 (x8 x14 x15 (x7 x14 (x7 x15 x16))))))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x12 x16 (x9 x14 x15 (x10 x16 (x9 x14 x15 (x12 x16 (x9 x14 x15 (x10 x16 (x9 x14 x15 (x12 x16 (x9 x14 x15 (x10 x16 (x9 x14 x15 (x12 x16 (x9 x14 x15 (x10 x16 x17))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x7 x14 (x13 x15 (x10 x16 x17)) = x13 x15 (x10 x16 (x7 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x13 x16 (x10 x17 x18)) = x13 x16 (x10 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x13 x16 (x7 x17 x18))) = x13 x16 (x7 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x10 x16 (x7 x17 x18))) = x10 x16 (x7 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x10 x15 (x7 x16 (x10 x17 x18))) = x7 x16 (x10 x17 (x13 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x13 x16 (x12 x17 x18))) = x13 x16 (x12 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x10 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x7 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x12 x16 (x7 x17 (x7 x18 x19))) = x7 x17 (x7 x18 (x9 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x10 x17 (x12 x18 x19))) = x10 x17 (x12 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x10 x16 (x9 x17 x18 (x7 x19 x20))) = x9 x17 x18 (x7 x19 (x9 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x13 x16 (x8 x17 x18 (x7 x19 x20))) = x8 x17 x18 (x7 x19 (x8 x14 x15 (x13 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x10 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x12 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x7 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x13 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x7 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 618ea.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 x17))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x10 x14 (x12 x15 (x12 x16 x17)) = x12 x15 (x12 x16 (x10 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x7 x16 (x7 x17 x18)) = x7 x16 (x7 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x7 x16 (x10 x17 x18))) = x7 x16 (x10 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x12 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x7 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x12 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x7 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x13 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x12 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x7 x16 (x12 x17 (x12 x18 x19))) = x12 x17 (x12 x18 (x9 x14 x15 (x7 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x12 x16 (x13 x17 (x10 x18 x19))) = x13 x17 (x10 x18 (x9 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x12 x16 (x8 x17 x18 (x7 x19 x20))) = x8 x17 x18 (x7 x19 (x8 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x7 x16 (x9 x17 x18 (x13 x19 x20))) = x9 x17 x18 (x13 x19 (x8 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x13 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x7 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x7 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x10 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x10 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem b8327.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x9 x14 x15 (x7 x14 (x7 x15 (x9 x14 x15 (x7 x14 (x7 x15 (x9 x14 x15 (x7 x14 (x7 x15 (x9 x14 x15 (x7 x14 (x7 x15 x16))))))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x12 x16 (x12 x14 (x8 x15 x16 (x8 x14 x15 (x12 x16 (x12 x14 (x8 x15 x16 (x8 x14 x15 (x12 x16 (x12 x14 (x8 x15 x16 (x8 x14 x15 (x12 x16 (x12 x14 (x8 x15 x16 (x8 x14 x15 (x12 x16 (x12 x14 (x8 x15 x16 x17))))))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x12 x14 (x7 x15 (x7 x16 x17)) = x7 x15 (x7 x16 (x12 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x12 x16 (x12 x17 x18)) = x12 x16 (x12 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x10 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x10 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x7 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x10 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x10 x15 (x7 x16 (x10 x17 x18))) = x7 x16 (x10 x17 (x13 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x7 x17 (x7 x18 x19))) = x7 x17 (x7 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x12 x16 (x10 x17 (x7 x18 x19))) = x10 x17 (x7 x18 (x9 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x8 x17 x18 (x10 x19 x20))) = x8 x17 x18 (x10 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x7 x16 (x9 x17 x18 (x7 x19 x20))) = x9 x17 x18 (x7 x19 (x9 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x13 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x10 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x13 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x10 x16 (x13 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem ac910.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x13 x16 (x10 x14 (x9 x15 x16 (x8 x14 x15 (x13 x16 (x10 x14 (x9 x15 x16 (x8 x14 x15 (x13 x16 (x10 x14 (x9 x15 x16 (x8 x14 x15 (x13 x16 (x10 x14 (x9 x15 x16 (x8 x14 x15 (x13 x16 (x10 x14 (x9 x15 x16 x17))))))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x10 x16 (x9 x14 x15 (x10 x16 (x9 x14 x15 (x10 x16 (x9 x14 x15 (x10 x16 (x9 x14 x15 (x10 x16 (x9 x14 x15 (x10 x16 x17))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x7 x14 (x7 x15 (x12 x16 x17)) = x7 x15 (x12 x16 (x7 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x13 x16 (x12 x17 x18)) = x13 x16 (x12 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x10 x16 (x7 x17 x18))) = x10 x16 (x7 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x12 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x13 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x10 x15 (x7 x16 (x13 x17 x18))) = x7 x16 (x13 x17 (x7 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x10 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x7 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x13 x16 (x7 x17 (x10 x18 x19))) = x7 x17 (x10 x18 (x8 x14 x15 (x13 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x7 x16 (x12 x17 (x12 x18 x19))) = x12 x17 (x12 x18 (x8 x14 x15 (x7 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x13 x16 (x8 x17 x18 (x10 x19 x20))) = x8 x17 x18 (x10 x19 (x9 x14 x15 (x13 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x7 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x8 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x10 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x10 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x10 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x13 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x13 x16 (x13 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 8389e.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x9 x14 x15 (x7 x14 (x10 x15 (x9 x14 x15 (x7 x14 (x10 x15 (x9 x14 x15 (x7 x14 (x10 x15 x16)))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x12 x16 (x10 x14 (x8 x15 x16 (x9 x14 x15 (x12 x16 (x10 x14 (x8 x15 x16 (x9 x14 x15 (x12 x16 (x10 x14 (x8 x15 x16 (x9 x14 x15 (x12 x16 (x10 x14 (x8 x15 x16 x17))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x10 x14 (x10 x15 (x10 x16 x17)) = x10 x15 (x10 x16 (x10 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x7 x16 (x10 x17 x18)) = x7 x16 (x10 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x12 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x7 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x7 x16 (x13 x17 x18))) = x7 x16 (x13 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x7 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x13 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x7 x16 (x7 x17 (x12 x18 x19))) = x7 x17 (x12 x18 (x9 x14 x15 (x7 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x12 x16 (x7 x17 (x7 x18 x19))) = x7 x17 (x7 x18 (x9 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x7 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x8 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x7 x16 (x8 x17 x18 (x7 x19 x20))) = x8 x17 x18 (x7 x19 (x8 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x13 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x13 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 0bd48.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x10 x14 (x12 x15 (x12 x16 x17)) = x12 x15 (x12 x16 (x10 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x12 x16 (x12 x17 x18)) = x12 x16 (x12 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x13 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x10 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x12 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x7 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x13 x15 (x10 x16 (x7 x17 x18))) = x10 x16 (x7 x17 (x12 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x12 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x7 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x7 x16 (x12 x17 (x13 x18 x19))) = x12 x17 (x13 x18 (x8 x14 x15 (x7 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x7 x16 (x7 x17 (x12 x18 x19))) = x7 x17 (x12 x18 (x8 x14 x15 (x7 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x9 x17 x18 (x7 x19 x20))) = x9 x17 x18 (x7 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x7 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x8 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x10 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x13 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x7 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 2c066.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x8 x14 x15 (x7 x16 (x12 x14 (x8 x15 x16 (x8 x14 x15 (x7 x16 (x12 x14 (x8 x15 x16 (x8 x14 x15 (x7 x16 (x12 x14 (x8 x15 x16 x17))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x9 x14 x15 (x12 x14 (x7 x15 (x9 x14 x15 (x12 x14 (x7 x15 (x9 x14 x15 (x12 x14 (x7 x15 x16)))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x12 x14 (x13 x15 (x10 x16 x17)) = x13 x15 (x10 x16 (x12 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x8 x14 x15 (x13 x16 (x12 x17 x18)) = x13 x16 (x12 x17 (x8 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x12 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x7 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x12 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x7 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x7 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x10 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x13 x15 (x10 x16 (x7 x17 x18))) = x10 x16 (x7 x17 (x13 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x13 x16 (x10 x17 (x12 x18 x19))) = x10 x17 (x12 x18 (x8 x14 x15 (x13 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x10 x16 (x12 x17 (x13 x18 x19))) = x12 x17 (x13 x18 (x8 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x7 x16 (x9 x17 x18 (x7 x19 x20))) = x9 x17 x18 (x7 x19 (x9 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x10 x16 (x8 x17 x18 (x7 x19 x20))) = x8 x17 x18 (x7 x19 (x9 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x7 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x10 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x7 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x10 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x10 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x7 x16 (x10 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 2bd7e.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 (x9 x14 x15 (x12 x16 (x8 x14 x15 (x12 x16 x17))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x8 x14 x15 (x7 x14 (x10 x15 (x8 x14 x15 (x7 x14 (x10 x15 x16))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x12 x14 (x12 x15 (x12 x16 x17)) = x12 x15 (x12 x16 (x12 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x7 x16 (x12 x17 x18)) = x7 x16 (x12 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x10 x16 (x13 x17 x18))) = x10 x16 (x13 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x13 x16 (x13 x17 x18))) = x13 x16 (x13 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x13 x15 (x10 x16 (x13 x17 x18))) = x10 x16 (x13 x17 (x12 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x10 x15 (x10 x16 (x12 x17 x18))) = x10 x16 (x12 x17 (x13 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x10 x16 (x12 x17 (x7 x18 x19))) = x12 x17 (x7 x18 (x8 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x7 x17 (x10 x18 x19))) = x7 x17 (x10 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x12 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x9 x14 x15 (x12 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x10 x16 (x8 x17 x18 (x12 x19 x20))) = x8 x17 x18 (x12 x19 (x8 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x13 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x9 x14 x15 (x10 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x12 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x9 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x13 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x10 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x10 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x12 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x8 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 84b88.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x12 x16 (x12 x14 (x8 x15 x16 (x9 x14 x15 (x12 x16 (x12 x14 (x8 x15 x16 x17))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x9 x14 x15 (x12 x16 (x12 x14 (x9 x15 x16 (x9 x14 x15 (x12 x16 (x12 x14 (x9 x15 x16 (x9 x14 x15 (x12 x16 (x12 x14 (x9 x15 x16 (x9 x14 x15 (x12 x16 (x12 x14 (x9 x15 x16 (x9 x14 x15 (x12 x16 (x12 x14 (x9 x15 x16 x17))))))))))))))))))) = x17)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x12 x14 (x12 x15 (x12 x16 x17)) = x12 x15 (x12 x16 (x12 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x7 x16 (x7 x17 x18)) = x7 x16 (x7 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x13 x14 (x13 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x13 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x13 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x7 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x13 x16 (x12 x17 x18))) = x13 x16 (x12 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x12 x17 (x13 x18 x19))) = x12 x17 (x13 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x10 x16 (x12 x17 (x13 x18 x19))) = x12 x17 (x13 x18 (x9 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x7 x16 (x9 x17 x18 (x12 x19 x20))) = x9 x17 x18 (x12 x19 (x8 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x13 x16 (x9 x17 x18 (x10 x19 x20))) = x9 x17 x18 (x10 x19 (x9 x14 x15 (x13 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x13 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x10 x16 (x13 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x7 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x8 x14 x15 (x7 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x10 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x7 x16 (x10 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x7 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x7 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x13 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 7048a.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x13 x14 (x7 x15 (x13 x16 x17)) = x7 x15 (x13 x16 (x13 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x12 x16 (x12 x17 x18)) = x12 x16 (x12 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x7 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x7 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x7 x14 (x13 x15 (x7 x16 (x10 x17 x18))) = x7 x16 (x10 x17 (x7 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x13 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x10 x14 (x13 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x10 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x10 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x7 x16 (x12 x17 x18))) = x7 x16 (x12 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x9 x14 x15 (x12 x16 (x7 x17 (x13 x18 x19))) = x7 x17 (x13 x18 (x9 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x12 x16 (x13 x17 (x7 x18 x19))) = x13 x17 (x7 x18 (x8 x14 x15 (x12 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x10 x16 (x8 x17 x18 (x10 x19 x20))) = x8 x17 x18 (x10 x19 (x9 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x9 x14 x15 (x13 x16 (x9 x17 x18 (x7 x19 x20))) = x9 x17 x18 (x7 x19 (x9 x14 x15 (x13 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x7 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x7 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x13 x16 (x12 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x9 x14 x15 (x13 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x13 x16 (x10 x17 (x9 x18 x19 (x10 x20 x21)))) = x9 x18 x19 (x10 x20 (x8 x14 x15 (x13 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x7 x17 (x9 x18 x19 (x12 x20 x21)))) = x9 x18 x19 (x12 x20 (x8 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)
Theorem 85af4.. : (∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ι → ι . ∀ x10 x11 x12 x13 : ι → ι → ι . Loop_with_defs_cex1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x9 x14 x15 (x10 x14 (x7 x15 (x9 x14 x15 (x10 x14 (x7 x15 (x9 x14 x15 (x10 x14 (x7 x15 (x9 x14 x15 (x10 x14 (x7 x15 (x9 x14 x15 (x10 x14 (x7 x15 x16)))))))))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0x8 x14 x15 (x10 x14 (x12 x15 (x8 x14 x15 (x10 x14 (x12 x15 (x8 x14 x15 (x10 x14 (x12 x15 x16)))))))) = x16)(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0x12 x14 (x7 x15 (x12 x16 x17)) = x7 x15 (x12 x16 (x12 x14 x17)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x9 x14 x15 (x12 x16 (x13 x17 x18)) = x12 x16 (x13 x17 (x9 x14 x15 x18)))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x12 x15 (x13 x16 (x7 x17 x18))) = x13 x16 (x7 x17 (x12 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x10 x15 (x13 x16 (x7 x17 x18))) = x13 x16 (x7 x17 (x12 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x12 x14 (x7 x15 (x12 x16 (x7 x17 x18))) = x12 x16 (x7 x17 (x12 x14 (x7 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x12 x15 (x12 x16 (x12 x17 x18))) = x12 x16 (x12 x17 (x10 x14 (x12 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0x10 x14 (x10 x15 (x12 x16 (x10 x17 x18))) = x12 x16 (x10 x17 (x10 x14 (x10 x15 x18))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x7 x16 (x7 x17 (x12 x18 x19))) = x7 x17 (x12 x18 (x8 x14 x15 (x7 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0x8 x14 x15 (x10 x16 (x12 x17 (x10 x18 x19))) = x12 x17 (x10 x18 (x8 x14 x15 (x10 x16 x19))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x7 x16 (x9 x17 x18 (x7 x19 x20))) = x9 x17 x18 (x7 x19 (x8 x14 x15 (x7 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0x8 x14 x15 (x10 x16 (x9 x17 x18 (x7 x19 x20))) = x9 x17 x18 (x7 x19 (x8 x14 x15 (x10 x16 x20))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x10 x16 (x7 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x9 x14 x15 (x10 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x8 x18 x19 (x10 x20 x21)))) = x8 x18 x19 (x10 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x10 x16 (x12 x17 (x9 x18 x19 (x7 x20 x21)))) = x9 x18 x19 (x7 x20 (x8 x14 x15 (x10 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x10 x17 (x8 x18 x19 (x13 x20 x21)))) = x8 x18 x19 (x13 x20 (x9 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x7 x17 (x8 x18 x19 (x12 x20 x21)))) = x8 x18 x19 (x12 x20 (x8 x14 x15 (x12 x16 (x7 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x9 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x9 x14 x15 (x12 x16 (x12 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x10 x17 (x8 x18 x19 (x7 x20 x21)))) = x8 x18 x19 (x7 x20 (x8 x14 x15 (x12 x16 (x10 x17 x21)))))(∀ x14 . In x14 x0∀ x15 . In x15 x0∀ x16 . In x16 x0∀ x17 . In x17 x0∀ x18 . In x18 x0∀ x19 . In x19 x0∀ x20 . In x20 x0∀ x21 . In x21 x0x8 x14 x15 (x12 x16 (x12 x17 (x9 x18 x19 (x13 x20 x21)))) = x9 x18 x19 (x13 x20 (x8 x14 x15 (x12 x16 (x12 x17 x21)))))False)∀ x0 : ο . x0 (proof)