# Library Coq.rtauto.Rtauto

Require Export List.
Require Export Bintree.
Require Import Bool.
Ltac caseq t := generalize (refl_equal t); pattern t at -1; case t.
Ltac clean:=try (simpl;congruence).
Inductive form:Set:=
Atom : positive -> form
| Arrow : form -> form -> form
| Bot
| Conjunct : form -> form -> form
| Disjunct : form -> form -> form.
Notation "[ n ]":=(Atom n).
Notation "A =>> B":= (Arrow A B) (at level 59, right associativity).
Notation "#" := Bot.
Notation "A //\\ B" := (Conjunct A B) (at level 57, left associativity).
Notation "A \\// B" := (Disjunct A B) (at level 58, left associativity).
Definition ctx := Store form.
Fixpoint pos_eq (m n:positive) {struct m} :bool :=
match m with
xI mm => match n with xI nn => pos_eq mm nn | _ => false end
| xO mm => match n with xO nn => pos_eq mm nn | _ => false end
| xH => match n with xH => true | _ => false end
end.
Theorem pos_eq_refl : forall m n, pos_eq m n = true -> m = n.
Fixpoint form_eq (p q:form) {struct p} :bool :=
match p with
Atom m => match q with Atom n => pos_eq m n | _ => false end
| Arrow p1 p2 =>
match q with
Arrow q1 q2 => form_eq p1 q1 && form_eq p2 q2
| _ => false end
| Bot => match q with Bot => true | _ => false end
| Conjunct p1 p2 =>
match q with
Conjunct q1 q2 => form_eq p1 q1 && form_eq p2 q2
| _ => false
end
| Disjunct p1 p2 =>
match q with
Disjunct q1 q2 => form_eq p1 q1 && form_eq p2 q2
| _ => false
end
end.
Theorem form_eq_refl: forall p q, form_eq p q = true -> p = q.
Implicit Arguments form_eq_refl [p q].
Section with_env.
Variable env:Store Prop.
Fixpoint interp_form (f:form): Prop :=
match f with
[n]=> match get n env with PNone => True | PSome P => P end
| A =>> B => (interp_form A) -> (interp_form B)
| # => False
| A //\\ B => (interp_form A) /\ (interp_form B)
| A \\// B => (interp_form A) \/ (interp_form B)
end.
Notation "[[ A ]]" := (interp_form A).
Fixpoint interp_ctx (hyps:ctx) (F:Full hyps) (G:Prop) {struct F} : Prop :=
match F with
F_empty => G
| F_push H hyps0 F0 => interp_ctx hyps0 F0 ([[H]] -> G)
end.
Require Export BinPos.
Ltac wipe := intros;simpl;constructor.
Lemma compose0 :
forall hyps F (A:Prop),
A ->
(interp_ctx hyps F A).
Lemma compose1 :
forall hyps F (A B:Prop),
(A -> B) ->
(interp_ctx hyps F A) ->
(interp_ctx hyps F B).
Theorem compose2 :
forall hyps F (A B C:Prop),
(A -> B -> C) ->
(interp_ctx hyps F A) ->
(interp_ctx hyps F B) ->
(interp_ctx hyps F C).
Theorem compose3 :
forall hyps F (A B C D:Prop),
(A -> B -> C -> D) ->
(interp_ctx hyps F A) ->
(interp_ctx hyps F B) ->
(interp_ctx hyps F C) ->
(interp_ctx hyps F D).
Lemma weaken : forall hyps F f G,
(interp_ctx hyps F G) ->
(interp_ctx (hyps\f) (F_push f hyps F) G).
Theorem project_In : forall hyps F g,
In g hyps F ->
interp_ctx hyps F [[g]].
Theorem project : forall hyps F p g,
get p hyps = PSome g->
interp_ctx hyps F [[g]].
Implicit Arguments project [hyps p g].
Inductive proof:Set :=
Ax : positive -> proof
| I_Arrow : proof -> proof
| E_Arrow : positive -> positive -> proof -> proof
| D_Arrow : positive -> proof -> proof -> proof
| E_False : positive -> proof
| I_And: proof -> proof -> proof
| E_And: positive -> proof -> proof
| D_And: positive -> proof -> proof
| I_Or_l: proof -> proof
| I_Or_r: proof -> proof
| E_Or: positive -> proof -> proof -> proof
| D_Or: positive -> proof -> proof
| Cut: form -> proof -> proof -> proof.
Notation "hyps \ A" := (push A hyps) (at level 72,left associativity).
Fixpoint check_proof (hyps:ctx) (gl:form) (P:proof) {struct P}: bool :=
match P with
Ax i =>
match get i hyps with
PSome F => form_eq F gl
| _ => false
end
| I_Arrow p =>
match gl with
A =>> B => check_proof (hyps \ A) B p
| _ => false
end
| E_Arrow i j p =>
match get i hyps,get j hyps with
PSome A,PSome (B =>>C) =>
form_eq A B && check_proof (hyps \ C) (gl) p
| _,_ => false
end
| D_Arrow i p1 p2 =>
match get i hyps with
PSome ((A =>>B)=>>C) =>
(check_proof ( hyps \ B =>> C \ A) B p1) && (check_proof (hyps \ C) gl p2)
| _ => false
end
| E_False i =>
match get i hyps with
PSome # => true
| _ => false
end
| I_And p1 p2 =>
match gl with
A //\\ B =>
check_proof hyps A p1 && check_proof hyps B p2
| _ => false
end
| E_And i p =>
match get i hyps with
PSome (A //\\ B) => check_proof (hyps \ A \ B) gl p
| _=> false
end
| D_And i p =>
match get i hyps with
PSome (A //\\ B =>> C) => check_proof (hyps \ A=>>B=>>C) gl p
| _=> false
end
| I_Or_l p =>
match gl with
(A \\// B) => check_proof hyps A p
| _ => false
end
| I_Or_r p =>
match gl with
(A \\// B) => check_proof hyps B p
| _ => false
end
| E_Or i p1 p2 =>
match get i hyps with
PSome (A \\// B) =>
check_proof (hyps \ A) gl p1 && check_proof (hyps \ B) gl p2
| _=> false
end
| D_Or i p =>
match get i hyps with
PSome (A \\// B =>> C) =>
(check_proof (hyps \ A=>>C \ B=>>C) gl p)
| _=> false
end
| Cut A p1 p2 =>
check_proof hyps A p1 && check_proof (hyps \ A) gl p2
end.
Theorem interp_proof:
forall p hyps F gl,
check_proof hyps gl p = true -> interp_ctx hyps F [[gl]].
Theorem Reflect: forall gl prf, if check_proof empty gl prf then [[gl]] else True.
End with_env.