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ex_meta2.v
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ex_meta2.v
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Require Import Utf8_core.
Require ILLVarInt. (* Don't want import it. *)
Import ILLVarInt.MILL. (* only this *)
Import ILLVarInt.Tacs. (* only this *)
Require Import unprove.
Import FormulaMultiSet. (* and this *)
Require Import ILL_equiv.
Require Import emma_orig.
Open Scope ILL_scope.
Open Scope Emma.
(* Ltac up x := repeat progress setoid_rewrite (add_comm _ x). *)
Inductive boolP : Prop := trueP:boolP | falseP:boolP.
Program Fixpoint exists_AtheseA_on_formula
(cont: forall (e1:env) (f1:formula) (h1: e1 ⊢ f1),boolP)
(φl φr:formula) (e:env) (f:formula) (h: e ⊢ f) {struct h}: boolP :=
match h with
| Oplus_R_1 _ p q x =>
if FormulaOrdered.eq_dec p φl
then if FormulaOrdered.eq_dec q φr
then cont _ _ x
else exists_AtheseA_on_formula cont φl φr _ _ x
else exists_AtheseA_on_formula cont φl φr _ _ x
| Oplus_R_2 _ q p x =>
if FormulaOrdered.eq_dec p φl
then if FormulaOrdered.eq_dec q φr
then cont _ _ x
else exists_AtheseA_on_formula cont φl φr _ _ x
else exists_AtheseA_on_formula cont φl φr _ _ x
| Id _ _ p => falseP
| Impl_R _ p q x => exists_AtheseA_on_formula cont φl φr _ _ x
| Impl_L _ Δ Δ' p q r _ _ x x0 => if exists_AtheseA_on_formula cont φl φr _ _ x then trueP else exists_AtheseA_on_formula cont φl φr _ _ x0
| Times_R _ Δ p q _ _ x x0 => if exists_AtheseA_on_formula cont φl φr _ _ x then trueP else exists_AtheseA_on_formula cont φl φr _ _ x0
| Times_L _ p q r _ x => exists_AtheseA_on_formula cont φl φr _ _ x
| One_R _ _ => falseP
| One_L _ p _ x => exists_AtheseA_on_formula cont φl φr _ _ x
| And_R _ p q x x0 => if exists_AtheseA_on_formula cont φl φr _ _ x then trueP else exists_AtheseA_on_formula cont φl φr _ _ x0
| And_L_1 _ p q r _ x => exists_AtheseA_on_formula cont φl φr _ _ x
| And_L_2 _ p q r _ x => exists_AtheseA_on_formula cont φl φr _ _ x
| Oplus_L _ p q r _ x x0 => if exists_AtheseA_on_formula cont φl φr _ _ x then trueP else exists_AtheseA_on_formula cont φl φr _ _ x0
| T_ _ => falseP
| Zero_ _ p x => falseP
| Bang_D _ p q _ x => exists_AtheseA_on_formula cont φl φr _ _ x
| Bang_C _ p q _ x => exists_AtheseA_on_formula cont φl φr _ _ x
| Bang_W _ p q _ x => exists_AtheseA_on_formula cont φl φr _ _ x
end.
Eval vm_compute in exists_AtheseA_on_formula (fun _ _ _ => trueP) A M _ _ emma_orig.originelle.
Eval vm_compute in exists_AtheseA_on_formula (fun _ _ _ => trueP) A M _ _ emma_orig.simpl_ex.
Lemma exists_AtheseA_on_formula_proof_eq_compat :
∀ f1 f2 Γ Γ' φ (h1:Γ⊢φ) (h2:Γ'⊢φ),
Proof_eq.eq h1 h2 ->
exists_AtheseA_on_formula (fun _ _ _ => trueP) f1 f2 _ _ h1 = exists_AtheseA_on_formula (fun _ _ _ => trueP) f1 f2 _ _ h2.
Proof.
intros f1 f2 Γ Γ' φ h1 h2 H.
induction H;simpl.
- reflexivity.
- auto.
- rewrite IHeq1;rewrite IHeq2;reflexivity.
- rewrite IHeq1;rewrite IHeq2;reflexivity.
- rewrite IHeq;reflexivity.
- reflexivity.
- rewrite IHeq;reflexivity.
- rewrite IHeq1;rewrite IHeq2;reflexivity.
- rewrite IHeq;reflexivity.
- rewrite IHeq;reflexivity.
- rewrite IHeq1;rewrite IHeq2;reflexivity.
- rewrite IHeq;reflexivity.
- rewrite IHeq;reflexivity.
- reflexivity.
- reflexivity.
- rewrite IHeq;reflexivity.
- rewrite IHeq;reflexivity.
- rewrite IHeq;reflexivity.
Qed.
Lemma exists_AtheseA_on_formula_proof_eq_pre_morph_compat :
∀ f1 f2 Γ Γ' φ (h1:Γ⊢φ) (h2:Γ==Γ'),
exists_AtheseA_on_formula (fun _ _ _ => trueP) f1 f2 _ _ h1 = exists_AtheseA_on_formula (fun _ _ _ => trueP) f1 f2 _ _ (ILL_proof_pre_morph φ Γ Γ' h2 h1).
Proof.
intros f1 f2 Γ Γ' φ h1 h2.
apply exists_AtheseA_on_formula_proof_eq_compat.
apply Proof_eq.sym;apply Proof_eq.eq_pre_morph.
Qed.
Lemma simple: {P&1, B&1, (V⊸A)&1, (E⊸A)&1,(P⊸M)&1,B ⊸ 1,V}⊢A⊕M.
Proof with try solve [ apply Id;reflexivity | prove_multiset_eq].
and_l_1 B 1.
weak_impl_l B 1...
one_l.
and_l_2 P 1.
and_l_2 (E ⊸ A) 1.
and_l_2 (P ⊸ M) 1.
repeat one_l.
and_l_1 (V ⊸ A) 1... (* !D au lieu de WL *)
weak_impl_l V A...
apply Oplus_R_1...
Qed.
(*
Notation "'all_proofs_of' x" := (forall (p:x), exists_AtheseA_on_formula (fun _ _ _ => trueP) A M _ _ p =trueP) (at level 80,only parsing).
Notation "'no_proof_for' x" := (forall (p:x), False) (at level 80,only parsing).
*)
Definition all_proofs_of env gamma := (forall (p:env⊢gamma), exists_AtheseA_on_formula (fun _ _ _ => trueP) A M _ _ p =trueP).
Definition no_proof_for env gamma := (forall (p:env⊢gamma), False).
#[local] Hint Unfold all_proofs_of no_proof_for : proof.
Lemma all_proofs_of_pre_morph : forall φ Γ Γ',
all_proofs_of Γ φ -> eq_bool Γ Γ' = true -> all_proofs_of Γ' φ.
Proof.
unfold all_proofs_of.
intros φ Γ Γ' Hall Heq p.
apply eq_bool_correct in Heq.
assert (h:exists p': Γ⊢φ, Proof_eq.eq p p').
symmetry in Heq; exists (ILL_proof_pre_morph _ _ _ Heq p).
apply Proof_eq.sym;apply Proof_eq.eq_pre_morph.
destruct h as [p' h].
rewrite exists_AtheseA_on_formula_proof_eq_compat with (h2:=p') (1:=h); auto.
Qed.
#[local] Hint Resolve all_proofs_of_pre_morph : proof.
Lemma all_proofs_of_pre_morph' :
forall φ Γ Γ', all_proofs_of Γ φ -> eq_bool Γ Γ' = true ->
forall (p:Γ'⊢φ), exists_AtheseA_on_formula (fun _ _ _ => trueP) A M _ _ p =trueP.
Proof.
intros φ Γ Γ' H H0 p.
eapply all_proofs_of_pre_morph;eassumption.
Qed.
#[local] Hint Resolve all_proofs_of_pre_morph' : proof.
#[local] Hint Rewrite all_proofs_of_pre_morph' : proof.
Require Import Setoid.
Add Morphism all_proofs_of with signature (eq ==> Logic.eq ==> iff) as
all_proof_of_morph.
Proof.
intros x y H y0.
split;intros; eapply all_proofs_of_pre_morph;try eassumption.
apply eq_bool_complete;assumption.
apply eq_bool_complete;symmetry;assumption.
Qed.
#[local] Hint Extern 0 ( _ == _ ) => apply eq_bool_correct;vm_compute;reflexivity : proof.
Tactic Notation "complete" tactic1(t) := t; fail.
Ltac clean p :=
try complete eauto 2 with proof;
(dependent simple inversion p||inversion p);clear p;subst;try discriminate;simpl.
Ltac decompose_add :=
repeat (match goal with
| H : _ ∈ (_ :: _) |- _ =>
destruct (mem_destruct _ _ _ H);clear H
| H : _ ∈ ∅ |- _ =>
rewrite empty_no_mem in H;discriminate
| H : ILLVarInt.MILL.eq _ _ |- _ => apply eq_is_eq in H;subst
end).
Ltac var_not_in_env_tac_simple n' H :=
elim unusable_var_in_env with (n:=n') (1:=H);
[
vm_compute;reflexivity |
vm_compute;reflexivity |
vm_compute;reflexivity |
vm_compute;reflexivity |
intros;decompose_add;simpl in *;repeat split;try discriminate;reflexivity
]
.
Ltac var_not_in_env_tac_aux H env :=
match env with
| Proposition ?n::?env' =>
(complete (var_not_in_env_tac_simple n H)) ||
var_not_in_env_tac_aux H env'
| _ :: ?env' => var_not_in_env_tac_aux H env'
end.
Ltac var_not_in_env_tac H :=
match type of H with
| ?env ⊢ _ =>
var_not_in_env_tac_aux H env
end.
Ltac is_var_env gamma :=
match gamma with
| empty => fail 1
| _ :: _ => fail 1
| _ \ _ => fail 1
| _ ∪ _ => fail 1
| _ => idtac
end.
Ltac finish :=
simpl;try reflexivity;
try discriminate;
try complete auto with proof;
try autorewrite with proof;
try complete (apply False_ind;auto with proof);
match goal with
|- (if ?e then trueP else trueP ) = trueP =>
case e;reflexivity
| i:?e⊢Proposition ?n' |- _ =>
elim var_in_env with (n:=n') (3:=i);vm_compute;reflexivity
| H: ?env⊢?g |- _ =>
complete var_not_in_env_tac H
| H : ?s == ?t |- _ =>
(complete (apply eq_bool_complete in H;vm_compute in H;
discriminate))|| (progress repeat (rewrite H in * ))
end.
Ltac tutu :=
simpl;try reflexivity;
try discriminate;
(* try complete auto with proof; *)
(* try autorewrite with proof; *)
match goal with
| H : ?t = ?t |- _ => clear H
| H : ?t == ?t |- _ => clear H
| H:_ :: _ == _ ∪ _ |- _ =>
symmetry in H
| H: _ ∪ _ == _ :: _ |- _ =>
let delta := fresh "Δ" in
let h1 := fresh "H" in
let h2 := fresh "H" in
destruct (union_decompose _ _ _ _ H)
as [[delta [h1 h2]]|[delta [h1 h2]]];clear H
| H: empty == _ ∪ _ |- _ =>
symmetry in H
| H: _ ∪ _ == empty |- _ =>
let h1 := fresh "H" in
let h2 := fresh "H" in
destruct (union_empty_decompose _ _ H) as [h1 h2];
clear H
| H : ?t == _ |- _ =>
is_var_env t;
match goal with
| H': context [t] |- _ =>
match H' with
| H => fail 1
| _ => fail 2
end
| _ => clear H
end
| H: ILLVarInt.MILL.eq _ _ |- _ => apply eq_is_eq in H; try (injection H;clear H;intros;subst)
| H: _ ∈ _ |- _ => complete (vm_compute in H;discriminate)
| H: _ ∈ (add _ _) |- _ =>
destruct (mem_destruct _ _ _ H);clear H
(* | H : ?s == ?t |- _ => *)
(* (complete (apply eq_bool_complete in H;vm_compute in H; *)
(* discriminate))|| (progress repeat (rewrite H in * )) *)
| H : ?s == ?t |- _ =>
(progress repeat (rewrite H in * ))
| H: context C [ remove ?f ?env ] |- _ =>
match env with
context C' [ add f ?env' ] =>
let e := context C' [ env' ] in
setoid_replace (remove f env) with e in H by (apply eq_bool_correct;vm_compute;reflexivity)
end
| H:(?x ⊸ ?y) = _ |- _ =>
try discriminate H;injection H;clear H;intros;subst
| H: (_ ⊕ _) = _ |- _ =>
try discriminate H;injection H;clear H;intros;subst
| H:(_ ⊗ _) = _ |- _ =>
try discriminate H;injection H;clear H;intros;subst
| H: _ & _ = _ |- _ =>
try discriminate H;injection H;clear H;intros;subst
| H: _ = (?x ⊸ ?y) |- _ =>
try discriminate H;injection H;clear H;intros;subst
| H: _ = (_ ⊕ _) |- _ =>
try discriminate H;injection H;clear H;intros;subst
| H: _ = (_ ⊗ _) |- _ =>
try discriminate H;injection H;clear H;intros;subst
| H: _ = _ & _ |- _ =>
try discriminate H;injection H;clear H;intros;subst
| H: ?delta ⊢ _, H' : ?delta == ∅ |- _ =>
apply False_ind;rewrite H' in H;clear - H;clean H;finish
| H: ?env⊢?g |- _ =>
(* try complete var_not_in_env_tac H; *)
match env with
| context C' [?env' \ ?f] =>
match env' with
| context C [add f ?env''] =>
let e' := context C [ env'' ] in
let e := context C' [ e' ] in
assert (heq: e == env) by (apply eq_bool_correct;vm_compute;reflexivity);
symmetry in heq;
let h := fresh "H" in
let i' := fresh "i" in
assert (h:(exists i':ILL_proof e g, Proof_eq.eq H i')) by (exists (ILL_proof_pre_morph _ _ _ heq H);
apply Proof_eq.sym;
apply Proof_eq.eq_pre_morph);
destruct h as [i' h];
rewrite exists_AtheseA_on_formula_proof_eq_compat with (h2:=i') (1:=h);
clear H h heq
end
end
| H: ?t == ?t', i: ?env⊢?f |- _ =>
match env with
| context [ t ] =>
let f_env :=
match eval pattern t in env with
| ?f _ => f
end
in
let env'0 := constr:(f_env t') in
let env' := eval cbv beta iota in env'0 in
let h := fresh "H" in
let i' := fresh "i" in
let heq := fresh "heq" in
assert (h:exists i': env'⊢f, Proof_eq.eq i i');[
assert (heq:env'==env) by (rewrite H;reflexivity);
symmetry in heq;
exists (ILL_proof_pre_morph _ _ _ heq i);
apply Proof_eq.sym;apply Proof_eq.eq_pre_morph
|
destruct h as [i' h];
rewrite exists_AtheseA_on_formula_proof_eq_compat with (h2:=i') (1:=h);clear i h;try (rewrite H in *;clear H)
]
end
end.
Ltac one_step p := clean p; (repeat tutu);try finish;auto with proof;eauto 3 with proof.
Ltac unusable_implies_tac n' f H :=
apply unusable_implies with (1:=H) (n:=n') (φ:=f);
[
vm_compute;reflexivity |
vm_compute;reflexivity |
vm_compute;reflexivity |
intros;decompose_add;repeat split;simpl in *;try discriminate;reflexivity].
Ltac unusable_var_strong_tac n1 n2 H :=
apply unusable_var_in_env_strong with (1:=H) (n:=n1);[
vm_compute;reflexivity|
vm_compute;reflexivity|
vm_compute;reflexivity|
vm_compute;reflexivity|
intros;decompose_add;simpl in *; repeat split;try discriminate;try reflexivity;auto;
intros _;right;
exists n2;
split;[((left;reflexivity)||(right;reflexivity))|split;[
intros;decompose_add;simpl;reflexivity|vm_compute;reflexivity]]].
(*
Lemma aux_10 : no_proof_for ({E ⊸ A, P, (P ⊸ M) & 1, V}) V.
Proof.
intro p;unusable_implies_tac 7%nat A p.
Qed.
Hint Resolve aux_10 : proof.
Lemma aux_12 : no_proof_for ({P ⊸ M, P, V}) V.
Proof.
intro p.
unusable_var_strong_tac 1%nat 8%nat p.
Qed.
Hint Resolve aux_12 : proof.
Lemma aux_13 : no_proof_for ({1, P ⊸ M, P, V}) V.
Proof.
intro p.
unusable_var_strong_tac 1%nat 8%nat p.
Qed.
Hint Resolve aux_13 : proof.
Lemma aux_14 : no_proof_for ({P ⊸ M, P, (E ⊸ A) & 1, V}) V.
Proof.
intro p.
unusable_var_strong_tac 1%nat 8%nat p.
Qed.
Hint Resolve aux_14 : proof.
Lemma aux_16 : no_proof_for ({1, P, (P ⊸ M) & 1, V}) V.
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_16 : proof.
Lemma aux_18 : no_proof_for ({E ⊸ A, A, P, (P ⊸ M) & 1}) (A ⊕ M).
Proof.
intro p;unusable_implies_tac 7%nat A p.
Qed.
Hint Resolve aux_18 : proof.
Lemma aux_19 : no_proof_for ({M, A}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_19 : proof.
Lemma aux_20 : no_proof_for ({E ⊸ A, M, A}) (A ⊕ M).
Proof.
intro p;unusable_implies_tac 7%nat A p.
Qed.
Hint Resolve aux_20 : proof.
Lemma aux_21 : no_proof_for ({1, M, A})(A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_21 : proof.
Lemma aux_22 : no_proof_for ({M, A, (E ⊸ A) & 1}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_22 : proof.
Lemma aux_23 : no_proof_for ({E ⊸ A, P ⊸ M, A, P}) (A ⊕ M).
Proof.
intro p;unusable_implies_tac 7%nat A p.
Qed.
Hint Resolve aux_23 : proof.
Lemma aux_24 : no_proof_for ({P ⊸ M, A, P}) A.
Proof.
intro p.
unusable_var_strong_tac 1%nat 8%nat p.
Qed.
Hint Resolve aux_24 : proof.
Lemma aux_25 : no_proof_for ({P ⊸ M, A, P}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_25 : proof.
Lemma aux_26 : no_proof_for ({1, P ⊸ M, A, P}) A.
Proof.
intro p.
unusable_var_strong_tac 1%nat 8%nat p.
Qed.
Hint Resolve aux_26 : proof.
Lemma aux_27 : no_proof_for ({1, P ⊸ M, A, P}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(*
{P ⊸ M, A, P} ⊢ A ⊕ M
{1, P ⊸ M, A, P} ⊢ A
*)
Qed.
Hint Resolve aux_27 : proof.
Lemma aux_29 : no_proof_for ({P ⊸ M, A, P, (E ⊸ A) & 1}) A.
Proof.
intro p.
unusable_var_strong_tac 1%nat 8%nat p.
Qed.
Hint Resolve aux_29 : proof.
Lemma aux_30 : no_proof_for ({P ⊸ M, A, P, (E ⊸ A) & 1}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(*
{M, A} ⊢ A ⊕ M
{M, A, (E ⊸ A) & 1} ⊢ A ⊕ M
{E ⊸ A, P ⊸ M, A, P} ⊢ A ⊕ M
{1, P ⊸ M, A, P} ⊢ A ⊕ M
{P ⊸ M, A, P, (E ⊸ A) & 1} ⊢ A e
*)
Qed.
Hint Resolve aux_30 : proof.
Lemma aux_31 : no_proof_for ({A, P, (P ⊸ M) & 1}) A.
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_31 : proof.
Lemma aux_32 : no_proof_for ({A, P, (P ⊸ M) & 1}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(* ({A, P, (P ⊸ M) & 1}) A *)
Qed.
Hint Resolve aux_32 : proof.
Lemma aux_33 : no_proof_for ({1, A, P, (P ⊸ M) & 1}) A.
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_33 : proof.
Lemma aux_34 : no_proof_for ({1, A, P, (P ⊸ M) & 1}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(*
{A, P, (P ⊸ M) & 1} ⊢ A ⊕ M
{1, A, P, (P ⊸ M) & 1} ⊢ A
*)
Qed.
Hint Resolve aux_34 : proof.
Lemma aux_35 : no_proof_for ({E ⊸ A, A, P, (P ⊸ M) & 1}) A.
Proof.
intro p;unusable_implies_tac 7%nat A p.
Qed.
Hint Resolve aux_35 : proof.
Lemma aux_36 : no_proof_for ({A, P, (E ⊸ A) & 1, (P ⊸ M) & 1}) A.
Proof.
intros p; one_step p;eauto with proof.
(*
{E ⊸ A, A, P, (P ⊸ M) & 1} ⊢ A
*)
Qed.
Hint Resolve aux_36 : proof.
Lemma aux_38 : no_proof_for ({E ⊸ A, V ⊸ A, P, (P ⊸ M) & 1, V}) (A ⊕ M).
Proof.
intro p;unusable_implies_tac 7%nat A p.
Qed.
Hint Resolve aux_38 : proof.
Lemma aux_40 : no_proof_for ({V ⊸ A, P, V}) P.
Proof.
intro p.
unusable_var_strong_tac 5%nat 6%nat p.
Qed.
Hint Resolve aux_40 : proof.
Lemma aux_42 : no_proof_for ({V ⊸ A, P, (E ⊸ A) & 1, V}) P.
Proof.
intro p.
unusable_var_strong_tac 5%nat 6%nat p.
Qed.
Hint Resolve aux_42 : proof.
Lemma aux_43 : no_proof_for ({M, V ⊸ A, V}) M.
Proof.
intro p.
unusable_var_strong_tac 5%nat 6%nat p.
Qed.
Hint Resolve aux_43 : proof.
Lemma aux_44 : no_proof_for ({M, V ⊸ A, V}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(*
{M, V ⊸ A, V} ⊢ M
*)
Qed.
Hint Resolve aux_44 : proof.
Lemma aux_45 : no_proof_for ({E ⊸ A, M, V ⊸ A, V}) (A ⊕ M).
Proof.
intro p;unusable_implies_tac 7%nat A p.
Qed.
Hint Resolve aux_45 : proof.
Lemma aux_46 : no_proof_for ({1, M, V ⊸ A, V}) (M).
Proof.
intro p.
unusable_var_strong_tac 5%nat 6%nat p.
Qed.
Hint Resolve aux_46 : proof.
Lemma aux_47 : no_proof_for ({1, M, V ⊸ A, V}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_47 : proof.
Lemma aux_49 : no_proof_for ({M, V ⊸ A, (E ⊸ A) & 1, V}) M.
Proof.
intro p.
unusable_var_strong_tac 5%nat 6%nat p.
Qed.
Hint Resolve aux_49 : proof.
Lemma aux_50 : no_proof_for ({M, V ⊸ A, (E ⊸ A) & 1, V}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(*
{E ⊸ A, M, V ⊸ A, V} ⊢ A ⊕ M
{1, M, V ⊸ A, V} ⊢ A ⊕ M
{M, V ⊸ A, (E ⊸ A) & 1, V} ⊢ M
*)
Qed.
Hint Resolve aux_50 : proof.
Lemma aux_41 : no_proof_for ({1, V ⊸ A, P, V}) P.
Proof.
intro p.
unusable_var_strong_tac 5%nat 6%nat p.
Qed.
Hint Resolve aux_41 : proof.
Lemma aux_52 : no_proof_for ({P ⊸ M, V ⊸ A, P, V}) A.
Proof.
intro p.
unusable_var_strong_tac 1%nat 8%nat p.
Qed.
Hint Resolve aux_52 : proof.
Lemma aux_53 : no_proof_for ({P ⊸ M, V ⊸ A, P, V}) M.
Proof.
intro p.
unusable_var_strong_tac 5%nat 6%nat p.
Qed.
Hint Resolve aux_53 : proof.
Lemma aux_54 : no_proof_for ({P ⊸ M, V ⊸ A, P, V}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(*
{P ⊸ M, V ⊸ A, P, V} ⊢ A
{P ⊸ M, V ⊸ A, P, V} ⊢ M
*)
Qed.
Hint Resolve aux_54 : proof.
Lemma aux_55 : no_proof_for ({1, P ⊸ M, V ⊸ A, P, V}) A.
Proof.
intro p.
unusable_var_strong_tac 1%nat 8%nat p.
Qed.
Hint Resolve aux_55 : proof.
Lemma aux_56 : no_proof_for ({1, P ⊸ M, V ⊸ A, P, V}) M.
Proof.
intro p.
unusable_var_strong_tac 5%nat 6%nat p.
Qed.
Hint Resolve aux_56 : proof.
Lemma aux_57 : no_proof_for ({1, P ⊸ M, V ⊸ A, P, V}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(*
{1, V ⊸ A, P, V} ⊢ P
{P ⊸ M, V ⊸ A, P, V} ⊢ A ⊕ M
{1, P ⊸ M, V ⊸ A, P, V} ⊢ A
{1, P ⊸ M, V ⊸ A, P, V} ⊢ M
*)
Qed.
Hint Resolve aux_57 : proof.
Lemma aux_59 : no_proof_for ({P ⊸ M, V ⊸ A, P, (E ⊸ A) & 1, V}) A.
Proof.
intro p.
unusable_var_strong_tac 1%nat 8%nat p.
Qed.
Hint Resolve aux_59 : proof.
Lemma aux_60 : no_proof_for ({E ⊸ A, P ⊸ M, V ⊸ A, P, V}) M.
Proof.
intro p;unusable_implies_tac 7%nat A p.
Qed.
Hint Resolve aux_60 : proof.
Lemma aux_61 : no_proof_for ({P ⊸ M, V ⊸ A, P, (E ⊸ A) & 1, V}) M.
Proof.
intro p.
unusable_var_strong_tac 5%nat 6%nat p.
Qed.
Hint Resolve aux_61 : proof.
Lemma aux_51 : no_proof_for ({E ⊸ A, P ⊸ M, V ⊸ A, P, V}) (A ⊕ M).
Proof.
intro p;unusable_implies_tac 7%nat A p.
Qed.
Hint Resolve aux_51 : proof.
Lemma aux_62 : no_proof_for ({P ⊸ M, V ⊸ A, P, (E ⊸ A) & 1, V}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(*
{V ⊸ A, P, (E ⊸ A) & 1, V} ⊢ P e
{V ⊸ A, P, V} ⊢ P e
{M, V ⊸ A, V} ⊢ A ⊕ M
{M, V ⊸ A, (E ⊸ A) & 1, V} ⊢ A ⊕ M
{E ⊸ A, P ⊸ M, V ⊸ A, P, V} ⊢ A ⊕ M
{1, P ⊸ M, V ⊸ A, P, V} ⊢ A ⊕ M
{P ⊸ M, V ⊸ A, P, (E ⊸ A) & 1, V} ⊢ A e
{P ⊸ M, V ⊸ A, P, (E ⊸ A) & 1, V} ⊢ M e
*)
Qed.
Hint Resolve aux_62 : proof.
Lemma aux_63 : no_proof_for ({V ⊸ A, P, (P ⊸ M) & 1, V}) A.
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_63 : proof.
Lemma aux_64 : no_proof_for ({V ⊸ A, P, (P ⊸ M) & 1, V}) M.
Proof.
intro p.
unusable_var_strong_tac 5%nat 6%nat p.
Qed.
Hint Resolve aux_64 : proof.
Lemma aux_65 : no_proof_for ({V ⊸ A, P, (P ⊸ M) & 1, V}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(*
{V ⊸ A, P, (P ⊸ M) & 1, V} ⊢ A
{V ⊸ A, P, (P ⊸ M) & 1, V} ⊢ M e
*)
Qed.
Hint Resolve aux_65 : proof.
Lemma aux_66 : no_proof_for ({1, V ⊸ A, P, (P ⊸ M) & 1, V}) A.
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_66 : proof.
Lemma aux_67 : no_proof_for ({1, V ⊸ A, P, (P ⊸ M) & 1, V}) M.
Proof.
intro p.
unusable_var_strong_tac 5%nat 6%nat p.
Qed.
Hint Resolve aux_67 : proof.
Lemma aux_68 : no_proof_for ({1, V ⊸ A, P, (P ⊸ M) & 1, V}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(*
{V ⊸ A, P, (P ⊸ M) & 1, V} ⊢ A ⊕ M
{1, V ⊸ A, P, (P ⊸ M) & 1, V} ⊢ A
{1, V ⊸ A, P, (P ⊸ M) & 1, V} ⊢ M e
*)
Qed.
Hint Resolve aux_68 : proof.
Lemma aux_69 : no_proof_for ({E ⊸ A, V ⊸ A, P, (P ⊸ M) & 1, V}) A.
Proof.
intro p;unusable_implies_tac 7%nat A p.
Qed.
Hint Resolve aux_69 : proof.
Lemma aux_70 : no_proof_for ({V ⊸ A, P, (E ⊸ A) & 1, (P ⊸ M) & 1, V}) A.
Proof.
intros p; one_step p;eauto with proof.
(*
{E ⊸ A, V ⊸ A, P, (P ⊸ M) & 1, V} ⊢ A
*)
Qed.
Hint Resolve aux_70 : proof.
Lemma aux_72 : no_proof_for ({V ⊸ A, P, (E ⊸ A) & 1, (P ⊸ M) & 1, V}) M.
Proof.
intro p.
unusable_var_strong_tac 5%nat 6%nat p.
Qed.
Hint Resolve aux_72 : proof.
Lemma aux_74 : no_proof_for ({E ⊸ A, P, (V ⊸ A) & 1, (P ⊸ M) & 1, V}) (A ⊕ M).
Proof.
intro p;unusable_implies_tac 7%nat A p.
Qed.
Hint Resolve aux_74 : proof.
Lemma aux_75 : no_proof_for ({E ⊸ A, P, (V ⊸ A) & 1, V}) P.
Proof.
intro p;unusable_implies_tac 7%nat A p.
Qed.
Hint Resolve aux_75 : proof.
Lemma aux_76 : no_proof_for ({P, (V ⊸ A) & 1, V}) P.
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_76 : proof.
Lemma aux_77 : no_proof_for ({1, P, (V ⊸ A) & 1, V}) P.
Proof.
intros p; one_step p;eauto with proof.
(*
{P, (V ⊸ A) & 1, V} ⊢ P
*)
Qed.
Hint Resolve aux_77 : proof.
Lemma aux_78 : no_proof_for ({P, (V ⊸ A) & 1, (E ⊸ A) & 1, V}) P.
Proof.
intros p; one_step p;eauto with proof.
(*
{E ⊸ A, P, (V ⊸ A) & 1, V} ⊢ P
{1, P, (V ⊸ A) & 1, V} ⊢ P
*)
Qed.
Hint Resolve aux_78 : proof.
Lemma aux_79 : no_proof_for ({E ⊸ A, M, (V ⊸ A) & 1, V}) (A ⊕ M).
Proof.
intro p;unusable_implies_tac 7%nat A p.
Qed.
Hint Resolve aux_79 : proof.
Lemma aux_80 : no_proof_for ({M, (V ⊸ A) & 1, V}) (M).
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_80 : proof.
Lemma aux_88 : no_proof_for ({M, (V ⊸ A) & 1, V}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(*
{M, (V ⊸ A) & 1, V} ⊢ M
*)
Qed.
Hint Resolve aux_88 : proof.
Lemma aux_82 : no_proof_for ({1, M, (V ⊸ A) & 1, V}) M.
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_82 : proof.
Lemma aux_83 : no_proof_for ({1, M, (V ⊸ A) & 1, V}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(*
{M, (V ⊸ A) & 1, V} ⊢ A ⊕ M
{1, M, (V ⊸ A) & 1, V} ⊢ M
*)
Qed.
Hint Resolve aux_83 : proof.
Lemma aux_87 : no_proof_for ({E ⊸ A, M, (V ⊸ A) & 1, V}) M.
Proof.
intro p;unusable_implies_tac 7%nat A p.
Qed.
Hint Resolve aux_87 : proof.
Lemma aux_85 : no_proof_for ({M, (V ⊸ A) & 1, (E ⊸ A) & 1, V}) M.
Proof.
intros p; one_step p;eauto with proof.
(*
{E ⊸ A, M, (V ⊸ A) & 1, V} ⊢ M
*)
Qed.
Hint Resolve aux_85 : proof.
Lemma aux_86 : no_proof_for ({M, (V ⊸ A) & 1, (E ⊸ A) & 1, V}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(*
{E ⊸ A, M, (V ⊸ A) & 1, V} ⊢ A ⊕M
{1, M, (V ⊸ A) & 1, V} ⊢ A ⊕ M
{M, (V ⊸ A) & 1, (E ⊸ A) & 1, V} ⊢ M
*)
Qed.
Hint Resolve aux_86 : proof.
Lemma aux_89 : no_proof_for ({E ⊸ A, P ⊸ M, P, (V ⊸ A) & 1, V}) (A ⊕ M).
Proof.
intro p;unusable_implies_tac 7%nat A p.
Qed.
Hint Resolve aux_89 : proof.
Lemma aux_90 : no_proof_for ({P ⊸ M, P, (V ⊸ A) & 1, V}) A.
Proof.
intro p.
unusable_var_strong_tac 1%nat 8%nat p.
Qed.
Hint Resolve aux_90 : proof.
Lemma aux_91 : no_proof_for ({P ⊸ M, P, (V ⊸ A) & 1, V}) M.
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_91 : proof.
Lemma aux_92 : no_proof_for ({P ⊸ M, P, (V ⊸ A) & 1, V}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(*
{P ⊸ M, P, (V ⊸ A) & 1, V} ⊢ A e
{P ⊸ M, P, (V ⊸ A) & 1, V} ⊢ M
*)
Qed.
Hint Resolve aux_92 : proof.
Lemma aux_93 : no_proof_for ({1, P ⊸ M, P, (V ⊸ A) & 1, V}) A.
Proof.
intro p.
unusable_var_strong_tac 1%nat 8%nat p.
Qed.
Hint Resolve aux_93 : proof.
Lemma aux_94 : no_proof_for ({1, P ⊸ M, P, (V ⊸ A) & 1, V}) M.
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_94 : proof.
Lemma aux_95 : no_proof_for ({1, P ⊸ M, P, (V ⊸ A) & 1, V}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(*
{P ⊸ M, P, (V ⊸ A) & 1, V} ⊢ A ⊕ M
{1, P ⊸ M, P, (V ⊸ A) & 1, V} ⊢ A e
{1, P ⊸ M, P, (V ⊸ A) & 1, V} ⊢ M
*)
Qed.
Hint Resolve aux_95 : proof.
Lemma aux_97 : no_proof_for ({P ⊸ M, P, (V ⊸ A) & 1, (E ⊸ A) & 1, V}) A.
Proof.
intro p.
unusable_var_strong_tac 1%nat 8%nat p.
Qed.
Hint Resolve aux_97 : proof.
Lemma aux_98 : no_proof_for ({E ⊸ A, P ⊸ M, P, (V ⊸ A) & 1, V}) M.
Proof.
intro p;unusable_implies_tac 7%nat A p.
Qed.
Hint Resolve aux_98 : proof.
Lemma aux_99 : no_proof_for ({P ⊸ M, P, (V ⊸ A) & 1, (E ⊸ A) & 1, V}) M.
Proof.
intros p; one_step p;eauto with proof.
(*
{E ⊸ A, P ⊸ M, P, (V ⊸ A) & 1, V} ⊢ M
*)
Qed.
Hint Resolve aux_99 : proof.
Lemma aux_100 : no_proof_for ({P ⊸ M, P, (V ⊸ A) & 1, (E ⊸ A) & 1, V}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(*
{P, (V ⊸ A) & 1, (E ⊸ A) & 1, V} ⊢ P
{P, (V ⊸ A) & 1, V} ⊢ P
{M, (V ⊸ A) & 1, (E ⊸ A) & 1, V} ⊢ A ⊕ M
{M, (V ⊸ A) & 1, V} ⊢ A ⊕ M
{E ⊸ A, P ⊸ M, P, (V ⊸ A) & 1, V} ⊢ A ⊕ M
{1, P ⊸ M, P, (V ⊸ A) & 1, V} ⊢ A ⊕ M
{P ⊸ M, P, (V ⊸ A) & 1, (E ⊸ A) & 1, V} ⊢ A e
{P ⊸ M, P, (V ⊸ A) & 1, (E ⊸ A) & 1, V} ⊢ M
*)
Qed.
Hint Resolve aux_100 : proof.
Lemma aux_101 : no_proof_for ({P, (V ⊸ A) & 1, (P ⊸ M) & 1, V}) A.
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_101 : proof.
Lemma aux_102 : no_proof_for ({P, (V ⊸ A) & 1, (P ⊸ M) & 1, V}) M.
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_102 : proof.
Lemma aux_103 : no_proof_for ({P, (V ⊸ A) & 1, (P ⊸ M) & 1, V}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(*
{P, (V ⊸ A) & 1, (P ⊸ M) & 1, V} ⊢ A
{P, (V ⊸ A) & 1, (P ⊸ M) & 1, V} ⊢ M
*)
Qed.
Hint Resolve aux_103 : proof.
Lemma aux_104 : no_proof_for ({1, P, (V ⊸ A) & 1, (P ⊸ M) & 1, V}) A.
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_104 : proof.
Lemma aux_105 : no_proof_for ({1, P, (V ⊸ A) & 1, (P ⊸ M) & 1, V}) M.
Proof.
intros p; one_step p;eauto with proof.
Qed.
Hint Resolve aux_105 : proof.
Lemma aux_106 : no_proof_for ({1, P, (V ⊸ A) & 1, (P ⊸ M) & 1, V}) (A ⊕ M).
Proof.
intros p; one_step p;eauto with proof.
(*
{P, (V ⊸ A) & 1, (P ⊸ M) & 1, V} ⊢ A ⊕ M
{1, P, (V ⊸ A) & 1, (P ⊸ M) & 1, V} ⊢ A
{1, P, (V ⊸ A) & 1, (P ⊸ M) & 1, V} ⊢ M
*)
Qed.