(* Copyright (C) 2012  G. Gonthier, B. Ziliani, A. Nanevski, D. Dreyer  This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.  This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License for more details.  You should have received a copy of the GNU General Public License along with this program.  If not, see <http://www.gnu.org/licenses/>. *)
From mathcomp
From LemmaOverloading
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
(* Automated proving of a proposition in a logic with binders *)
(* adapted from VeriML paper of Stampoulist and Shao. *)
(* first another searching structure; can probably be reused from some *)
(* other file, but I don't bother now *)
(* simply check if a proposition x is in the context g, represented as a list *)
Structure tagged_seq := TagS {untags :> seq Prop}.
Definition recurse := TagS.
Canonical Structure found (g : seq Prop) := recurse g.
Structure find (x : Prop) :=
Find {seq_of :> tagged_seq;
_ : x \In untags seq_of}.
Program Canonical Structure
found_struct x g := @Find x (found (x :: g)) _.
Next Obligation. by rewrite InE; left. Qed.
Program Canonical Structure
recurse_struct x y (g : find x) := @Find x (recurse (y :: g)) _.
Next Obligation. by rewrite InE /=; right; case: g. Qed.
(* then a helper structure for controlling the information flow *)
(* it is like the hoisting pattern *)
Structure equate_to (x : Prop) := Equate {assign :> Prop}.
Canonical Structure singleton x := Equate x x.
Structure check (x : Prop) (g : seq Prop) :=
Check {x_of :> equate_to x;
_ : assign x_of \In g}.
Program  Canonical Structure
start x (f : find x) := @Check x f (singleton x) _.
Next Obligation. by case: f=>[[]]. Qed.
(**************************************************************)
(* Now the main body -- branches on the structure of the prop *)
(**************************************************************)
(* if p is a conjunction, prove both sides *)
(* if p is a disjunction, try to prove left then right side *)
(* if p is an implication, put the hypothesis into the context g and recurse *)
(* if p is a universal, abstract over the bound variable *)
(* if neither, check if p is in the context g *)
Structure tagged_prop := Tag {untag :> Prop}.
Definition var_tag := Tag.
Definition all_tag := var_tag.
Definition imp_tag := all_tag.
Definition orL_tag := imp_tag.
Definition orR_tag := orL_tag.
Canonical Structure and_tag p := orR_tag p.
Structure form (g : seq Prop) :=
Form {prop_of :> tagged_prop;
_ : foldr and True g -> untag prop_of}.
Program Canonical Structure
and_struct g (p1 p2 : form g) :=
@Form g (@and_tag (p1 /\ p2)) _.
Next Obligation.
case: p1 p2=>[[p1]] H1 [[p2]] H2.
by split; [apply: H1 | apply: H2]; apply: H.
Qed.
Program Canonical Structure
orL_struct g (p1 : form g) (p2 : Prop) :=
@Form g (@orL_tag (p1 \/ p2)) _.
Next Obligation. by case: p1=>[[p1]] H1; left; apply: H1 H. Qed.
Program Canonical Structure
orR_struct g (p1 : Prop) (p2 : form g) :=
@Form g (@orR_tag (p1 \/ p2)) _.
Next Obligation. by case: p2=>[[p2]] H2; right; apply: H2 H. Qed.
Program Canonical Structure
imp_struct g (p : Prop) (q : form (p :: g)) :=
@Form g (@imp_tag (p -> q)) _.
Next Obligation. by case: q=>[[q]] H1; apply: H1. Qed.
Program Canonical Structure
all_struct A g (p : A -> form g) :=
@Form g (@all_tag (forall x, p x)) _.
Next Obligation. by case: (p x)=>[[q]]; apply. Qed.
Program Canonical Structure
var_struct x g (c : check x g) :=
@Form g (@var_tag c) _ .
Next Obligation.
case: c=>[[p]] /=; elim: g H=>[//|t s IH] /=.
case=>H1 H2; rewrite InE /=.
by case; [move=>-> | apply: IH H2].
Qed.
(* main lemma *)

Lemma auto (p : form [::]) : untag p.