(* (c) Copyright 2006-2015 Microsoft Corporation and Inria. *) (* Distributed under the terms of CeCILL-B. *) Require Import mathcomp.ssreflect.ssreflect. From mathcomp Require Import ssrbool ssrfun eqtype ssrnat seq. From mathcomp Require Import fintype. Lemma test15: forall (y : nat) (x : 'I_2), y < 1 -> val x = y -> Some x = insub y. move=> y x le_1 defx; rewrite insubT ?(leq_trans le_1) // => ?. by congr (Some _); apply: val_inj=> /=; exact: defx. Qed. Axiom P : nat -> Prop. Axiom Q : forall n, P n -> Prop. Definition R := fun (x : nat) (p : P x) m (q : P (x+1)) => m > 0. Inductive myEx : Type := ExI : forall n (pn : P n) pn', Q n pn -> R n pn n pn' -> myEx. Variable P1 : P 1. Variable P11 : P (1 + 1). Variable Q1 : forall P1, Q 1 P1. Lemma testmE1 : myEx. Proof. apply: ExI 1 _ _ _ _. match goal with |- P 1 => exact: P1 | _ => fail end. match goal with |- P (1+1) => exact: P11 | _ => fail end. match goal with |- forall p : P 1, Q 1 p => move=>*; exact: Q1 | _ => fail end. match goal with |- forall (p : P 1) (q : P (1+1)), is_true (R 1 p 1 q) => done | _ => fail end. Qed. Lemma testE2 : exists y : { x | P x }, sval y = 1. Proof. apply: ex_intro (exist _ 1 _) _. match goal with |- P 1 => exact: P1 | _ => fail end. match goal with |- forall p : P 1, @sval _ _ (@exist _ _ 1 p) = 1 => done | _ => fail end. Qed. Lemma testE3 : exists y : { x | P x }, sval y = 1. Proof. have := (ex_intro _ (exist _ 1 _) _); apply. match goal with |- P 1 => exact: P1 | _ => fail end. match goal with |- forall p : P 1, @sval _ _ (@exist _ _ 1 p) = 1 => done | _ => fail end. Qed. Lemma testE4 : P 2 -> exists y : { x | P x }, sval y = 2. Proof. move=> P2; apply: ex_intro (exist _ 2 _) _. match goal with |- @sval _ _ (@exist _ _ 2 P2) = 2 => done | _ => fail end. Qed. Hint Resolve P1. Lemma testmE12 : myEx. Proof. apply: ExI 1 _ _ _ _. match goal with |- P (1+1) => exact: P11 | _ => fail end. match goal with |- Q 1 P1 => exact: Q1 | _ => fail end. match goal with |- forall (q : P (1+1)), is_true (R 1 P1 1 q) => done | _ => fail end. Qed. Create HintDb SSR. Hint Resolve P11 : SSR. Ltac ssrautoprop := trivial with SSR. Lemma testmE13 : myEx. Proof. apply: ExI 1 _ _ _ _. match goal with |- Q 1 P1 => exact: Q1 | _ => fail end. match goal with |- is_true (R 1 P1 1 P11) => done | _ => fail end. Qed. Definition R1 := fun (x : nat) (p : P x) m (q : P (x+1)) (r : Q x p) => m > 0. Inductive myEx1 : Type := ExI1 : forall n (pn : P n) pn' (q : Q n pn), R1 n pn n pn' q -> myEx1. Hint Resolve (Q1 P1) : SSR. (* tests that goals in prop are solved in the right order, propagating instantiations, thus the goal Q 1 ?p1 is faced by trivial after ?p1, and is thus evar free *) Lemma testmE14 : myEx1. Proof. apply: ExI1 1 _ _ _ _. match goal with |- is_true (R1 1 P1 1 P11 (Q1 P1)) => done | _ => fail end. Qed.