From a0cfa4f118023d35b767a999d5a2ac4b082857b4 Mon Sep 17 00:00:00 2001 From: Samuel Mimram Date: Fri, 25 Jul 2008 15:12:53 +0200 Subject: Imported Upstream version 8.2~beta3+dfsg --- contrib/funind/Recdef.v | 48 ++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 48 insertions(+) create mode 100644 contrib/funind/Recdef.v (limited to 'contrib/funind/Recdef.v') diff --git a/contrib/funind/Recdef.v b/contrib/funind/Recdef.v new file mode 100644 index 00000000..2d206220 --- /dev/null +++ b/contrib/funind/Recdef.v @@ -0,0 +1,48 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* A) -> A -> A := + fun (fl : A -> A) (def : A) => + match n with + | O => def + | S m => fl (iter m fl def) + end. +End Iter. + +Theorem SSplus_lt : forall p p' : nat, p < S (S (p + p')). + intro p; intro p'; change (S p <= S (S (p + p'))); + apply le_S; apply Gt.gt_le_S; change (p < S (p + p')); + apply Lt.le_lt_n_Sm; apply Plus.le_plus_l. +Qed. + + +Theorem Splus_lt : forall p p' : nat, p' < S (p + p'). + intro p; intro p'; change (S p' <= S (p + p')); + apply Gt.gt_le_S; change (p' < S (p + p')); apply Lt.le_lt_n_Sm; + apply Plus.le_plus_r. +Qed. + +Theorem le_lt_SS : forall x y, x <= y -> x < S (S y). +intro x; intro y; intro H; change (S x <= S (S y)); + apply le_S; apply Gt.gt_le_S; change (x < S y); + apply Lt.le_lt_n_Sm; exact H. +Qed. + +Inductive max_type (m n:nat) : Set := + cmt : forall v, m <= v -> n <= v -> max_type m n. + +Definition max : forall m n:nat, max_type m n. +intros m n; case (Compare_dec.le_gt_dec m n). +intros h; exists n; [exact h | apply le_n]. +intros h; exists m; [apply le_n | apply Lt.lt_le_weak; exact h]. +Defined. -- cgit v1.2.3