Proofgold Address

Param ap^ap : ι → ι → ι
Param lam^Sigma : ι → (ι → ι) → ι
Param ordsucc^ordsucc : ι → ι
Param If_i^If_i : ο → ι → ι → ι
Definition u17_to_Church17 := λ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . ap (lam 17 (λ x18 . If_i (x18 = 0) x1 (If_i (x18 = 1) x2 (If_i (x18 = 2) x3 (If_i (x18 = 3) x4 (If_i (x18 = 4) x5 (If_i (x18 = 5) x6 (If_i (x18 = 6) x7 (If_i (x18 = 7) x8 (If_i (x18 = 8) x9 (If_i (x18 = 9) x10 (If_i (x18 = 10) x11 (If_i (x18 = 11) x12 (If_i (x18 = 12) x13 (If_i (x18 = 13) x14 (If_i (x18 = 14) x15 (If_i (x18 = 15) x16 x17))))))))))))))))) x0
Param u3 : ι
Known 2a0b0.. : (∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 . x3 ∈ x1 ⟶ ap (lam x1 (λ x5 . If_i (x5 = x3) x0 (x2 (ordsucc x3) x5))) x3 = x0) ⟶ (∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 x4 . (x4 = x3 ⟶ ∀ x5 : ο . x5) ⟶ ap (lam x1 (λ x6 . If_i (x6 = x3) x0 (x2 (ordsucc x3) x6))) x4 = ap (lam x1 (x2 (ordsucc x3))) x4) ⟶ ∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . ap (lam 17 (λ x18 . If_i (x18 = 0) x0 (If_i (x18 = 1) x1 (If_i (x18 = 2) x2 (If_i (x18 = 3) x3 (If_i (x18 = 4) x4 (If_i (x18 = 5) x5 (If_i (x18 = 6) x6 (If_i (x18 = 7) x7 (If_i (x18 = 8) x8 (If_i (x18 = 9) x9 (If_i (x18 = 10) x10 (If_i (x18 = 11) x11 (If_i (x18 = 12) x12 (If_i (x18 = 13) x13 (If_i (x18 = 14) x14 (If_i (x18 = 15) x15 x16))))))))))))))))) u3 = x3
Known 48efb.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 . x3 ∈ x1 ⟶ ap (lam x1 (λ x5 . If_i (x5 = x3) x0 (x2 (ordsucc x3) x5))) x3 = x0
Known d21a1.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 x4 . (x4 = x3 ⟶ ∀ x5 : ο . x5) ⟶ ap (lam x1 (λ x6 . If_i (x6 = x3) x0 (x2 (ordsucc x3) x6))) x4 = ap (lam x1 (x2 (ordsucc x3))) x4
Theorem fbcaf.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . ap (lam 17 (λ x18 . If_i (x18 = 0) x0 (If_i (x18 = 1) x1 (If_i (x18 = 2) x2 (If_i (x18 = 3) x3 (If_i (x18 = 4) x4 (If_i (x18 = 5) x5 (If_i (x18 = 6) x6 (If_i (x18 = 7) x7 (If_i (x18 = 8) x8 (If_i (x18 = 9) x9 (If_i (x18 = 10) x10 (If_i (x18 = 11) x11 (If_i (x18 = 12) x12 (If_i (x18 = 13) x13 (If_i (x18 = 14) x14 (If_i (x18 = 15) x15 x16))))))))))))))))) u3 = x3 (proof)
Param u4 : ι
Known b09cb.. : (∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 . x3 ∈ x1 ⟶ ap (lam x1 (λ x5 . If_i (x5 = x3) x0 (x2 (ordsucc x3) x5))) x3 = x0) ⟶ (∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 x4 . (x4 = x3 ⟶ ∀ x5 : ο . x5) ⟶ ap (lam x1 (λ x6 . If_i (x6 = x3) x0 (x2 (ordsucc x3) x6))) x4 = ap (lam x1 (x2 (ordsucc x3))) x4) ⟶ ∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . ap (lam 17 (λ x18 . If_i (x18 = 0) x0 (If_i (x18 = 1) x1 (If_i (x18 = 2) x2 (If_i (x18 = 3) x3 (If_i (x18 = 4) x4 (If_i (x18 = 5) x5 (If_i (x18 = 6) x6 (If_i (x18 = 7) x7 (If_i (x18 = 8) x8 (If_i (x18 = 9) x9 (If_i (x18 = 10) x10 (If_i (x18 = 11) x11 (If_i (x18 = 12) x12 (If_i (x18 = 13) x13 (If_i (x18 = 14) x14 (If_i (x18 = 15) x15 x16))))))))))))))))) u4 = x4
Theorem 12883.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . ap (lam 17 (λ x18 . If_i (x18 = 0) x0 (If_i (x18 = 1) x1 (If_i (x18 = 2) x2 (If_i (x18 = 3) x3 (If_i (x18 = 4) x4 (If_i (x18 = 5) x5 (If_i (x18 = 6) x6 (If_i (x18 = 7) x7 (If_i (x18 = 8) x8 (If_i (x18 = 9) x9 (If_i (x18 = 10) x10 (If_i (x18 = 11) x11 (If_i (x18 = 12) x12 (If_i (x18 = 13) x13 (If_i (x18 = 14) x14 (If_i (x18 = 15) x15 x16))))))))))))))))) u4 = x4 (proof)
Param u5 : ι
Known 497c7.. : (∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 . x3 ∈ x1 ⟶ ap (lam x1 (λ x5 . If_i (x5 = x3) x0 (x2 (ordsucc x3) x5))) x3 = x0) ⟶ (∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 x4 . (x4 = x3 ⟶ ∀ x5 : ο . x5) ⟶ ap (lam x1 (λ x6 . If_i (x6 = x3) x0 (x2 (ordsucc x3) x6))) x4 = ap (lam x1 (x2 (ordsucc x3))) x4) ⟶ ∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . ap (lam 17 (λ x18 . If_i (x18 = 0) x0 (If_i (x18 = 1) x1 (If_i (x18 = 2) x2 (If_i (x18 = 3) x3 (If_i (x18 = 4) x4 (If_i (x18 = 5) x5 (If_i (x18 = 6) x6 (If_i (x18 = 7) x7 (If_i (x18 = 8) x8 (If_i (x18 = 9) x9 (If_i (x18 = 10) x10 (If_i (x18 = 11) x11 (If_i (x18 = 12) x12 (If_i (x18 = 13) x13 (If_i (x18 = 14) x14 (If_i (x18 = 15) x15 x16))))))))))))))))) u5 = x5
Theorem 8021a.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . ap (lam 17 (λ x18 . If_i (x18 = 0) x0 (If_i (x18 = 1) x1 (If_i (x18 = 2) x2 (If_i (x18 = 3) x3 (If_i (x18 = 4) x4 (If_i (x18 = 5) x5 (If_i (x18 = 6) x6 (If_i (x18 = 7) x7 (If_i (x18 = 8) x8 (If_i (x18 = 9) x9 (If_i (x18 = 10) x10 (If_i (x18 = 11) x11 (If_i (x18 = 12) x12 (If_i (x18 = 13) x13 (If_i (x18 = 14) x14 (If_i (x18 = 15) x15 x16))))))))))))))))) u5 = x5 (proof)
Known aa7c9.. : ∀ x0 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . ∀ x1 . ∀ x2 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . (∀ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x0 x1 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 = x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19) ⟶ x0 x1 = x2
Theorem 1ef08.. : u17_to_Church17 u3 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x4 (proof)
Theorem 05513.. : u17_to_Church17 u4 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x5 (proof)
Theorem 22977.. : u17_to_Church17 u5 = λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x6 (proof)