Known bfa6d..^{Loop_with_defs_E} : ∀ 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 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 ⟶ ∀ x14 : ο . (Loop x0 x1 x2 x3 x4 ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x5 x15 x16 = x2 (x1 x16 x15) (x1 x15 x16)) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ ∀ x17 . In x17 x0 ⟶ x6 x15 x16 x17 = x2 (x1 x15 (x1 x16 x17)) (x1 (x1 x15 x16) x17)) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ and (and (and (and (x7 x15 x16 = x2 x15 (x1 x16 x15)) (x10 x15 x16 = x1 x15 (x1 x16 (x2 x15 x4)))) (x11 x15 x16 = x1 (x1 (x3 x4 x15) x16) x15)) (x12 x15 x16 = x1 (x2 x15 x16) (x2 (x2 x15 x4) x4))) (x13 x15 x16 = x1 (x3 x4 (x3 x4 x15)) (x3 x16 x15))) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ ∀ x17 . In x17 x0 ⟶ and (x8 x15 x16 x17 = x2 (x1 x16 x15) (x1 x16 (x1 x15 x17))) (x9 x15 x16 x17 = x3 (x1 (x1 x17 x15) x16) (x1 x15 x16))) ⟶ x14) ⟶ x14
Known 21d45..^Loop_def : Loop = λ x1 . λ x2 x3 x4 : ι → ι → ι . λ x5 . and (and (and (and (binop_on x1 x2) (binop_on x1 x3)) (binop_on x1 x4)) (∀ x6 . In x6 x1 ⟶ and (x2 x5 x6 = x6) (x2 x6 x5 = x6))) (∀ x6 . In x6 x1 ⟶ ∀ x7 . In x7 x1 ⟶ and (and (and (x3 x6 (x2 x6 x7) = x7) (x2 x6 (x3 x6 x7) = x7)) (x4 (x2 x6 x7) x7 = x6)) (x2 (x4 x6 x7) x7 = x6))
Known andE^andE : ∀ x0 x1 : ο . and x0 x1 ⟶ ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known 8b9e9..^binop_on_E : ∀ x0 . ∀ x1 : ι → ι → ι . binop_on x0 x1 ⟶ ∀ x2 . In x2 x0 ⟶ ∀ x3 . In x3 x0 ⟶ In (x1 x2 x3) x0
Theorem 683ad.. : ∀ 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 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 ⟶ In x4 x0 ⟶ ∀ x14 : ο . ((∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ In (x1 x15 x16) x0) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ In (x3 x15 x16) x0) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ In (x2 x15 x16) x0) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x5 x15 x16 = x2 (x1 x16 x15) (x1 x15 x16)) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ In (x5 x15 x16) x0) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ ∀ x17 . In x17 x0 ⟶ x6 x15 x16 x17 = x2 (x1 x15 (x1 x16 x17)) (x1 (x1 x15 x16) x17)) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ ∀ x17 . In x17 x0 ⟶ In (x6 x15 x16 x17) x0) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x7 x15 x16 = x2 x15 (x1 x16 x15)) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ In (x7 x15 x16) x0) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ ∀ x17 . In x17 x0 ⟶ x8 x15 x16 x17 = x2 (x1 x16 x15) (x1 x16 (x1 x15 x17))) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ ∀ x17 . In x17 x0 ⟶ In (x8 x15 x16 x17) x0) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ ∀ x17 . In x17 x0 ⟶ x9 x15 x16 x17 = x3 (x1 (x1 x17 x15) x16) (x1 x15 x16)) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ ∀ x17 . In x17 x0 ⟶ In (x9 x15 x16 x17) x0) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x10 x15 x16 = x1 x15 (x1 x16 (x2 x15 x4))) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ In (x10 x15 x16) x0) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x12 x15 x16 = x1 (x2 x15 x16) (x2 (x2 x15 x4) x4)) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ In (x12 x15 x16) x0) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x11 x15 x16 = x1 (x1 (x3 x4 x15) x16) x15) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ In (x11 x15 x16) x0) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x13 x15 x16 = x1 (x3 x4 (x3 x4 x15)) (x3 x16 x15)) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ In (x13 x15 x16) x0) ⟶ (∀ x15 . In x15 x0 ⟶ x1 x4 x15 = x15) ⟶ (∀ x15 . In x15 x0 ⟶ x1 x15 x4 = x15) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x2 x15 (x1 x15 x16) = x16) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x1 x15 (x2 x15 x16) = x16) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x3 (x1 x15 x16) x16 = x15) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x1 (x3 x15 x16) x16 = x15) ⟶ x14) ⟶ x14 (proof)
Known 1449b..^{Loop_with_defs_cex1_E} : ∀ 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 : ο . (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ ∀ x17 . In x17 x0 ⟶ ∀ x18 . In x18 x0 ⟶ Loop_with_defs x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 ⟶ not (x5 (x1 (x2 (x8 x16 x17 x15) x4) x15) x18 = x4) ⟶ x14) ⟶ x14
Theorem 5d9b3.. : ∀ 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 ⟶ In x4 x0 ⟶ ∀ x14 : ο . (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ ∀ x17 . In x17 x0 ⟶ ∀ x18 . In x18 x0 ⟶ not (x5 (x1 (x2 (x8 x15 x16 x17) x4) x17) x18 = x4) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ In (x1 x19 x20) x0) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ In (x3 x19 x20) x0) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ In (x2 x19 x20) x0) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ x5 x19 x20 = x2 (x1 x20 x19) (x1 x19 x20)) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ In (x5 x19 x20) x0) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ x6 x19 x20 x21 = x2 (x1 x19 (x1 x20 x21)) (x1 (x1 x19 x20) x21)) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ In (x6 x19 x20 x21) x0) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ x7 x19 x20 = x2 x19 (x1 x20 x19)) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ In (x7 x19 x20) x0) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ x8 x19 x20 x21 = x2 (x1 x20 x19) (x1 x20 (x1 x19 x21))) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ In (x8 x19 x20 x21) x0) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ x9 x19 x20 x21 = x3 (x1 (x1 x21 x19) x20) (x1 x19 x20)) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ In (x9 x19 x20 x21) x0) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ x10 x19 x20 = x1 x19 (x1 x20 (x2 x19 x4))) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ In (x10 x19 x20) x0) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ x12 x19 x20 = x1 (x2 x19 x20) (x2 (x2 x19 x4) x4)) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ In (x12 x19 x20) x0) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ x11 x19 x20 = x1 (x1 (x3 x4 x19) x20) x19) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ In (x11 x19 x20) x0) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ x13 x19 x20 = x1 (x3 x4 (x3 x4 x19)) (x3 x20 x19)) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ In (x13 x19 x20) x0) ⟶ (∀ x19 . In x19 x0 ⟶ x1 x4 x19 = x19) ⟶ (∀ x19 . In x19 x0 ⟶ x1 x19 x4 = x19) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ x2 x19 (x1 x19 x20) = x20) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ x1 x19 (x2 x19 x20) = x20) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ x3 (x1 x19 x20) x20 = x19) ⟶ (∀ x19 . In x19 x0 ⟶ ∀ x20 . In x20 x0 ⟶ x1 (x3 x19 x20) x20 = x19) ⟶ x14) ⟶ x14 (proof)
Known 9c580..^{Loop_with_defs_cex2_E} : ∀ 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 : ο . (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ ∀ x17 . In x17 x0 ⟶ ∀ x18 . In x18 x0 ⟶ ∀ x19 . In x19 x0 ⟶ Loop_with_defs x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 ⟶ not (x6 x19 (x1 (x3 x4 x15) (x9 x16 x17 x15)) x18 = x4) ⟶ x14) ⟶ x14
Theorem 02f51.. : ∀ 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 ⟶ In x4 x0 ⟶ ∀ x14 : ο . (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ ∀ x17 . In x17 x0 ⟶ ∀ x18 . In x18 x0 ⟶ ∀ x19 . In x19 x0 ⟶ not (x6 x15 (x1 (x3 x4 x16) (x9 x17 x18 x16)) x19 = x4) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ In (x1 x20 x21) x0) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ In (x3 x20 x21) x0) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ In (x2 x20 x21) x0) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ x5 x20 x21 = x2 (x1 x21 x20) (x1 x20 x21)) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ In (x5 x20 x21) x0) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ ∀ x22 . In x22 x0 ⟶ x6 x20 x21 x22 = x2 (x1 x20 (x1 x21 x22)) (x1 (x1 x20 x21) x22)) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ ∀ x22 . In x22 x0 ⟶ In (x6 x20 x21 x22) x0) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ x7 x20 x21 = x2 x20 (x1 x21 x20)) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ In (x7 x20 x21) x0) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ ∀ x22 . In x22 x0 ⟶ x8 x20 x21 x22 = x2 (x1 x21 x20) (x1 x21 (x1 x20 x22))) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ ∀ x22 . In x22 x0 ⟶ In (x8 x20 x21 x22) x0) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ ∀ x22 . In x22 x0 ⟶ x9 x20 x21 x22 = x3 (x1 (x1 x22 x20) x21) (x1 x20 x21)) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ ∀ x22 . In x22 x0 ⟶ In (x9 x20 x21 x22) x0) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ x10 x20 x21 = x1 x20 (x1 x21 (x2 x20 x4))) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ In (x10 x20 x21) x0) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ x12 x20 x21 = x1 (x2 x20 x21) (x2 (x2 x20 x4) x4)) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ In (x12 x20 x21) x0) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ x11 x20 x21 = x1 (x1 (x3 x4 x20) x21) x20) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ In (x11 x20 x21) x0) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ x13 x20 x21 = x1 (x3 x4 (x3 x4 x20)) (x3 x21 x20)) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ In (x13 x20 x21) x0) ⟶ (∀ x20 . In x20 x0 ⟶ x1 x4 x20 = x20) ⟶ (∀ x20 . In x20 x0 ⟶ x1 x20 x4 = x20) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ x2 x20 (x1 x20 x21) = x21) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ x1 x20 (x2 x20 x21) = x21) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ x3 (x1 x20 x21) x21 = x20) ⟶ (∀ x20 . In x20 x0 ⟶ ∀ x21 . In x21 x0 ⟶ x1 (x3 x20 x21) x21 = x20) ⟶ x14) ⟶ x14 (proof)

Proofgold Address