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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *)
Require Import ssreflect.
Require Import ssrbool ssrfun TestSuite.ssr_mini_mathcomp.
Axiom daemon : False. Ltac myadmit := case: daemon.
(* Ltac debugging feature: recursive elim + eq generation *)
Lemma testL1 : forall A (s : seq A), s = s.
Proof.
move=> A s; elim branch: s => [|x xs _].
match goal with _ : _ = [::] |- [::] = [::] => move: branch => // | _ => fail end.
match goal with _ : _ = _ :: _ |- _ :: _ = _ :: _ => move: branch => // | _ => fail end.
Qed.
(* The same but with explicit eliminator and a conflict in the intro pattern *)
Lemma testL2 : forall A (s : seq A), s = s.
Proof.
move=> A s; elim/last_ind branch: s => [|x s _].
match goal with _ : _ = [::] |- [::] = [::] => move: branch => // | _ => fail end.
match goal with _ : _ = rcons _ _ |- rcons _ _ = rcons _ _ => move: branch => // | _ => fail end.
Qed.
(* The same but without names for variables involved in the generated eq *)
Lemma testL3 : forall A (s : seq A), s = s.
Proof.
move=> A s; elim branch: s; move: (s) => _.
match goal with _ : _ = [::] |- [::] = [::] => move: branch => // | _ => fail end.
move=> _; match goal with _ : _ = _ :: _ |- _ :: _ = _ :: _ => move: branch => // | _ => fail end.
Qed.
Inductive foo : Type := K1 : foo | K2 : foo -> foo -> foo | K3 : (nat -> foo) -> foo.
(* The same but with more intros to be done *)
Lemma testL4 : forall (o : foo), o = o.
Proof.
move=> o; elim branch: o.
match goal with _ : _ = K1 |- K1 = K1 => move: branch => // | _ => fail end.
move=> _; match goal with _ : _ = K2 _ _ |- K2 _ _ = K2 _ _ => move: branch => // | _ => fail end.
move=> _; match goal with _ : _ = K3 _ |- K3 _ = K3 _ => move: branch => // | _ => fail end.
Qed.
(* Occurrence counting *)
Lemma testO1: forall (b : bool), b = b.
Proof.
move=> b; case: (b) / idP.
match goal with |- is_true b -> true = true => done | _ => fail end.
match goal with |- ~ is_true b -> false = false => done | _ => fail end.
Qed.
(* The same but only the second occ *)
Lemma testO2: forall (b : bool), b = b.
Proof.
move=> b; case: {2}(b) / idP.
match goal with |- is_true b -> b = true => done | _ => fail end.
match goal with |- ~ is_true b -> b = false => move/(introF idP) => // | _ => fail end.
Qed.
(* The same but with eq generation *)
Lemma testO3: forall (b : bool), b = b.
Proof.
move=> b; case E: {2}(b) / idP.
match goal with _ : is_true b, _ : b = true |- b = true => move: E => _; done | _ => fail end.
match goal with H : ~ is_true b, _ : b = false |- b = false => move: E => _; move/(introF idP): H => // | _ => fail end.
Qed.
(* Views *)
Lemma testV1 : forall A (s : seq A), s = s.
Proof.
move=> A s; case/lastP E: {1}s => [| x xs].
match goal with _ : s = [::] |- [::] = s => symmetry; exact E | _ => fail end.
match goal with _ : s = rcons x xs |- rcons _ _ = s => symmetry; exact E | _ => fail end.
Qed.
Lemma testV2 : forall A (s : seq A), s = s.
Proof.
move=> A s; case/lastP E: s => [| x xs].
match goal with _ : s = [::] |- [::] = [::] => done | _ => fail end.
match goal with _ : s = rcons x xs |- rcons _ _ = rcons _ _ => done | _ => fail end.
Qed.
Lemma testV3 : forall A (s : seq A), s = s.
Proof.
move=> A s; case/lastP: s => [| x xs].
match goal with |- [::] = [::] => done | _ => fail end.
match goal with |- rcons _ _ = rcons _ _ => done | _ => fail end.
Qed.
(* Patterns *)
Lemma testP1: forall (x y : nat), (y == x) && (y == x) -> y == x.
move=> x y; elim: {2}(_ == _) / eqP.
match goal with |- (y = x -> is_true ((y == x) && true) -> is_true (y == x)) => move=> -> // | _ => fail end.
match goal with |- (y <> x -> is_true ((y == x) && false) -> is_true (y == x)) => move=> _; rewrite andbC // | _ => fail end.
Qed.
(* The same but with an implicit pattern *)
Lemma testP2 : forall (x y : nat), (y == x) && (y == x) -> y == x.
move=> x y; elim: {2}_ / eqP.
match goal with |- (y = x -> is_true ((y == x) && true) -> is_true (y == x)) => move=> -> // | _ => fail end.
match goal with |- (y <> x -> is_true ((y == x) && false) -> is_true (y == x)) => move=> _; rewrite andbC // | _ => fail end.
Qed.
(* The same but with an eq generation switch *)
Lemma testP3 : forall (x y : nat), (y == x) && (y == x) -> y == x.
move=> x y; elim E: {2}_ / eqP.
match goal with _ : y = x |- (is_true ((y == x) && true) -> is_true (y == x)) => rewrite E; reflexivity | _ => fail end.
match goal with _ : y <> x |- (is_true ((y == x) && false) -> is_true (y == x)) => rewrite E => /= H; exact H | _ => fail end.
Qed.
Inductive spec : nat -> nat -> nat -> Prop :=
| specK : forall a b c, a = 0 -> b = 2 -> c = 4 -> spec a b c.
Lemma specP : spec 0 2 4. Proof. by constructor. Qed.
Lemma testP4 : (1+1) * 4 = 2 + (1+1) + (2 + 2).
Proof.
case: specP => a b c defa defb defc.
match goal with |- (a.+1 + a.+1) * c = b + (a.+1 + a.+1) + (b + b) => subst; done | _ => fail end.
Qed.
Lemma testP5 : (1+1) * 4 = 2 + (1+1) + (2 + 2).
Proof.
case: (1 + 1) _ / specP => a b c defa defb defc.
match goal with |- b * c = a.+2 + b + (a.+2 + a.+2) => subst; done | _ => fail end.
Qed.
Lemma testP6 : (1+1) * 4 = 2 + (1+1) + (2 + 2).
Proof.
case: {2}(1 + 1) _ / specP => a b c defa defb defc.
match goal with |- (a.+1 + a.+1) * c = a.+2 + b + (a.+2 + a.+2) => subst; done | _ => fail end.
Qed.
Lemma testP7 : (1+1) * 4 = 2 + (1+1) + (2 + 2).
Proof.
case: _ (1 + 1) (2 + _) / specP => a b c defa defb defc.
match goal with |- b * a.+4 = c + c => subst; done | _ => fail end.
Qed.
Lemma testP8 : (1+1) * 4 = 2 + (1+1) + (2 + 2).
Proof.
case E: (1 + 1) (2 + _) / specP=> [a b c defa defb defc].
match goal with |- b * a.+4 = c + c => subst; done | _ => fail end.
Qed.
Variables (T : Type) (tr : T -> T).
Inductive exec (cf0 cf1 : T) : seq T -> Prop :=
| exec_step : tr cf0 = cf1 -> exec cf0 cf1 [::]
| exec_star : forall cf2 t, tr cf0 = cf2 ->
exec cf2 cf1 t -> exec cf0 cf1 (cf2 :: t).
Inductive execr (cf0 cf1 : T) : seq T -> Prop :=
| execr_step : tr cf0 = cf1 -> execr cf0 cf1 [::]
| execr_star : forall cf2 t, execr cf0 cf2 t ->
tr cf2 = cf1 -> execr cf0 cf1 (t ++ [:: cf2]).
Lemma execP : forall cf0 cf1 t, exec cf0 cf1 t <-> execr cf0 cf1 t.
Proof.
move=> cf0 cf1 t; split => [] Ecf.
elim: Ecf.
match goal with |- forall cf2 cf3 : T, tr cf2 = cf3 ->
execr cf2 cf3 [::] => myadmit | _ => fail end.
match goal with |- forall (cf2 cf3 cf4 : T) (t0 : seq T),
tr cf2 = cf4 -> exec cf4 cf3 t0 -> execr cf4 cf3 t0 ->
execr cf2 cf3 (cf4 :: t0) => myadmit | _ => fail end.
elim: Ecf.
match goal with |- forall cf2 : T,
tr cf0 = cf2 -> exec cf0 cf2 [::] => myadmit | _ => fail end.
match goal with |- forall (cf2 cf3 : T) (t0 : seq T),
execr cf0 cf3 t0 -> exec cf0 cf3 t0 -> tr cf3 = cf2 ->
exec cf0 cf2 (t0 ++ [:: cf3]) => myadmit | _ => fail end.
Qed.
Fixpoint plus (m n : nat) {struct n} : nat :=
match n with
| 0 => m
| S p => S (plus m p)
end.
Definition plus_equation :
forall m n : nat,
plus m n =
match n with
| 0 => m
| p.+1 => (plus m p).+1
end
:=
fun m n : nat =>
match
n as n0
return
(forall m0 : nat,
plus m0 n0 =
match n0 with
| 0 => m0
| p.+1 => (plus m0 p).+1
end)
with
| 0 => @erefl nat
| n0.+1 => fun m0 : nat => erefl (plus m0 n0).+1
end m.
Definition plus_rect :
forall (m : nat) (P : nat -> nat -> Type),
(forall n : nat, n = 0 -> P 0 m) ->
(forall n p : nat,
n = p.+1 -> P p (plus m p) -> P p.+1 (plus m p).+1) ->
forall n : nat, P n (plus m n)
:=
fun (m : nat) (P : nat -> nat -> Type)
(f0 : forall n : nat, n = 0 -> P 0 m)
(f : forall n p : nat,
n = p.+1 -> P p (plus m p) -> P p.+1 (plus m p).+1) =>
fix plus0 (n : nat) : P n (plus m n) :=
eq_rect_r [eta P n]
(let f1 := f0 n in
let f2 := f n in
match
n as n0
return
(n = n0 ->
(forall p : nat,
n0 = p.+1 -> P p (plus m p) -> P p.+1 (plus m p).+1) ->
(n0 = 0 -> P 0 m) ->
P n0 match n0 with
| 0 => m
| p.+1 => (plus m p).+1
end)
with
| 0 =>
fun (_ : n = 0)
(_ : forall p : nat,
0 = p.+1 ->
P p (plus m p) -> P p.+1 (plus m p).+1)
(f4 : 0 = 0 -> P 0 m) => unkeyed (f4 (erefl 0))
| n0.+1 =>
fun (_ : n = n0.+1)
(f3 : forall p : nat,
n0.+1 = p.+1 ->
P p (plus m p) -> P p.+1 (plus m p).+1)
(_ : n0.+1 = 0 -> P 0 m) =>
let f5 :=
let p := n0 in
let H := erefl n0.+1 : n0.+1 = p.+1 in f3 p H in
unkeyed (let Hrec := plus0 n0 in f5 Hrec)
end (erefl n) f2 f1) (plus_equation m n).
Definition plus_ind := plus_rect.
Lemma exF x y z: plus (plus x y) z = plus x (plus y z).
elim/plus_ind: z / (plus _ z).
match goal with |- forall n : nat, n = 0 -> plus x y = plus x (plus y 0) => idtac end.
Undo 2.
elim/plus_ind: (plus _ z).
match goal with |- forall n : nat, n = 0 -> plus x y = plus x (plus y 0) => idtac end.
Undo 2.
elim/plus_ind: {z}(plus _ z).
match goal with |- forall n : nat, n = 0 -> plus x y = plus x (plus y 0) => idtac end.
Undo 2.
elim/plus_ind: {z}_.
match goal with |- forall n : nat, n = 0 -> plus x y = plus x (plus y 0) => idtac end.
Undo 2.
elim/plus_ind: z / _.
match goal with |- forall n : nat, n = 0 -> plus x y = plus x (plus y 0) => idtac end.
done.
by move=> _ p _ ->.
Qed.
(* BUG elim-False *)
Lemma testeF : False -> 1 = 0.
Proof. by elim. Qed.
|
