Commit 3573759d by Samuel Ben Hamou

### Suite de la preuve de SuccRight.

parent 9ebd3ff8
 ... ... @@ -236,9 +236,26 @@ Proof. bsubst 0 (Succ (# 0)) (∀ Succ (# 1 + # 0) = # 1 + Succ (# 0))))). * apply R_Ax. unfold induction_schema. apply in_eq. * simpl. apply R_And_i. ++ assert ((∀ bsubst 0 Zero (∀ Succ (#1 + #0) = #1 + Succ (#0))) = (∀ Succ (#0 + Zero) = #0 + Succ (Zero))). { auto. (* EST-CE QUE C'EST AU MOINS VRAI ??? *) ++ compute. change (Fun "O" []) with Zero. apply R_All_i with (x := "x"); [ | compute; change (Fun "O" []) with Zero ]. -- unfold Names.In. compute. inversion 1. -- apply R_All_i with (x := "y"); [ compute; inversion 1 | ]. compute. change (Fun "O" []) with Zero. set (Γ := [(∀ ∀ Fun "S" [Zero + # 0] = Zero + Fun "S" [# 0]) /\ (∀ (∀ Fun "S" [# 1 + # 0] = # 1 + Fun "S" [# 0]) -> ∀ Fun "S" [Fun "S" [# 1] + # 0] = Fun "S" [# 1] + Fun "S" [# 0]) -> ∀ ∀ Fun "S" [# 1 + # 0] = # 1 + Fun "S" [# 0]; ∀ # 0 = # 0; ∀ ∀ # 1 = # 0 -> # 0 = # 1; ∀ ∀ ∀ # 2 = # 1 /\ # 1 = # 0 -> # 2 = # 0; ∀ ∀ # 1 = # 0 -> Fun "S" [# 1] = Fun "S" [# 0]; ∀ ∀ ∀ # 2 = # 1 -> # 2 + # 0 = # 1 + # 0; ∀ ∀ ∀ # 1 = # 0 -> # 2 + # 1 = # 2 + # 0; ∀ ∀ ∀ # 2 = # 1 -> # 2 * # 0 = # 1 * # 0; ∀ ∀ ∀ # 1 = # 0 -> # 2 * # 1 = # 2 * # 0; ∀ Zero + # 0 = # 0; ∀ ∀ Fun "S" [# 1] + # 0 = Fun "S" [# 1 + # 0]; ∀ Zero * # 0 = Zero; ∀ ∀ Fun "S" [# 1] * # 0 = # 1 * # 0 + # 0; ∀ ∀ Fun "S" [# 1] = Fun "S" [# 0] -> # 1 = # 0; ∀ ~ Fun "S" [# 0] = Zero]). assert (Pr Intuiti (Γ ⊢ Zero + FVar "y" = FVar "y")). { assert (AX9 : Pr Intuiti (Γ ⊢ ax9)). { apply R_Ax. compute; intuition. } unfold ax9 in AX9. apply R_All_e with (t := FVar "y") in AX9; [ | auto ]. compute in AX9. assumption. } Print bsubst. apply Lemma Comm : IsTheorem Intuiti PeanoTheory (∀∀ (#0 + #1 = #1 + #0)). Admitted. ... ...
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