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restrict2.v
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restrict2.v
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Require Import Utf8_core basic.
Require ILLVarInt. (* Don't want import it. *)
Import ILLVarInt.MILL. (* only this *)
Import FormulaMultiSet. (* and this *)
Open Scope ILL_scope.
(**
This module contains the proof that if a sequent e ⊢ f verifies property Ap
(see below) then any proof h:e ⊢ f is such that Ap holds also for each node
of h.
*)
Section Stability.
Set Implicit Arguments.
Variable pred: ∀ {e} {f} (h:e ⊢ f), Prop.
(** This property is true when pred holds for all nodes above the root
of h (it has not to hold for the root itself). *)
Inductive Istable: ∀ {e} {f} (h: e ⊢ f) , Prop :=
| IId: ∀ Γ p heq, Istable (Id Γ p heq)
| IImpl_R: ∀ Γ p q h, pred h → Istable h → Istable (Impl_R Γ p q h)
| IImpl_L: ∀ Γ Δ Δ' p q r hin heq h h',
pred h → pred h' → Istable h → Istable h' → Istable (Impl_L Γ Δ Δ' p q r hin heq h h')
| ITimes_R: ∀ Γ Δ Δ' p q heq h h', pred h → pred h' → Istable h → Istable h' → Istable (Times_R Γ Δ Δ' p q heq h h')
| ITimes_L: ∀ Γ p q r hin h, pred h → Istable h → Istable (Times_L Γ p q r hin h)
| IOne_R: ∀ Γ heq, Istable (One_R Γ heq)
| IOne_L: ∀ Γ p hin h, pred h → Istable h → Istable (One_L Γ p hin h)
| IAnd_R: ∀ Γ p q h h', pred h → pred h' → Istable h → Istable h' → Istable (And_R Γ p q h h')
| IAnd_L_2: ∀ Γ p q r hin h, pred h → Istable h → Istable (And_L_2 Γ p q r hin h)
| IAnd_L_1: ∀ Γ p q r hin h, pred h → Istable h → Istable (And_L_1 Γ p q r hin h)
| IOplus_L: ∀ Γ p q r hin h h', pred h → pred h' → Istable h → Istable h' → Istable (Oplus_L Γ p q r hin h h')
| IOplus_R_2: ∀ Γ p q h, Istable h → Istable (Oplus_R_2 Γ p q h)
| IOplus_R_1: ∀ Γ p q h, pred h → Istable h → Istable (Oplus_R_1 Γ p q h)
| IT_ : ∀ Γ, (Istable (T_ Γ)) → (Istable (T_ Γ))
| IZero_: ∀ Γ p truein, (Istable (Zero_ Γ p truein)) → (Istable (Zero_ Γ p truein))
| IBang_D: ∀ Γ p q hin h, pred h → Istable h → (Istable (Bang_D Γ p q hin h))
| IBang_C: ∀ Γ p q hin h, pred h → Istable h → (Istable (Bang_C Γ p q hin h))
| IBang_W: ∀ Γ p q hin h, pred h → Istable h → (Istable (Bang_W Γ p q hin h)).
End Stability.
Local Open Scope type_scope.
Local Open Scope ILL_scope.
(** The predicate that we want to check on all
nodes is that formula belong to the following
grammar. *)
Inductive Act : formula -> Prop := (* Act *)
| A1: Act 1
| A2:∀ φ₁ φ₂, Cres φ₁ → Context φ₂ → Act (φ₁ ⊸ φ₂)
| A3: ∀ φ₁ φ₂, Act φ₁ → Act φ₂ → Act (φ₁ ⊕ φ₂)
| A4: ∀ φ₁ φ₂, Act φ₁ → Act φ₂ → Act (φ₁ & φ₂)
| A5: ∀ φ, Act φ → Act (! φ)
with Cres: formula -> Prop:= (* CRes *)
| Cres1: Cres 1
| Cres2: ∀ n, Cres (Proposition n)
| Cres3: ∀ φ₁ φ₂, Cres φ₁ → Cres φ₂ → Cres (φ₁ ⊗ φ₂)
with Context: formula -> Prop:= (* Context *)
| Context1:∀ φ, Act φ → Context φ
| Context2:∀ φ, Res φ → Context φ
| Context3: ∀ φ₁ φ₂, Context φ₁ → Context φ₂ → Context (φ₁ ⊗ φ₂)
with Res: formula -> Prop := (* Res *)
R1: Res One
| R2: ∀ n, Res (Proposition n)
| R3: ∀ φ, Res φ → Res (!φ)
| R4: ∀ φ₁ φ₂, Res φ₁ → Res φ₂ → Res (φ₁ ⊗ φ₂)
| R5: ∀ φ₁ φ₂, Res φ₁ → Res φ₂ → Res (φ₁ & φ₂).
Inductive Goal : formula → Prop :=
| G1: Goal 1
| G2: ∀ n, Goal (Proposition n)
| G3: ∀ φ₁ φ₂, Goal φ₁ → Goal φ₂ → Goal (φ₁ ⊗ φ₂)
| G4: ∀ φ₁ φ₂, Goal φ₁ → Goal φ₂ → Goal (φ₁ ⊕ φ₂)
| G5: ∀ φ₁ φ₂, Goal φ₁ → Goal φ₂ → Goal (φ₁ & φ₂).
(** Predicate [Contextall Γ f _] is true if all
formulas of a [Γ] are in grammar [Context] and [f]
is in Goal. *)
Definition Contextall Γ f (_:Γ⊢f):Prop := Goal f /\ ∀g:formula, g ∈ Γ → Context g.
Lemma inEnv:∀ g p Γ, g ∈ (p :: Γ) → g = p ∨ g ∈ Γ .
Proof.
intros g p Γ H.
generalize (mem_destruct _ _ _ H).
intros [H1| H1];auto.
apply eq_is_eq in H1.
auto.
Qed.
Lemma inAdd1: ∀ f Γ, f ∈ (f :: Γ) .
Proof.
intros f Γ.
apply add_is_mem.
apply FormulaOrdered.eq_refl.
Qed.
Lemma inAdd2: ∀ f g Γ, f ∈ Γ → f ∈ (g :: Γ).
Proof.
apply mem_add_is_mem.
Qed.
Lemma inUnion1: ∀ g Δ Γ, g ∈ Δ -> g ∈ (Δ ∪ Γ).
apply mem_union_l.
Qed.
Lemma inUnion2: ∀ g Δ Γ, g ∈ Γ -> g ∈ (Δ ∪ Γ).
apply mem_union_r.
Qed.
Lemma inAddUnion1: ∀ g Δ Γ f, g ∈ Δ -> g ∈ (f :: (Δ ∪ Γ)).
Proof.
intros g Δ Γ f H.
apply inAdd2.
apply inUnion1;assumption.
Qed.
Lemma inAddUnion2: ∀ g Δ Γ f, g ∈ Γ -> g ∈ (f :: (Δ ∪ Γ)).
Proof.
intros g Δ Γ f H.
apply inAdd2.
apply inUnion2;assumption.
Qed.
Lemma mem_remove: ∀ Γ f Δ g, Γ \ f == Δ → g ∈ Δ → g ∈ Γ.
Proof.
intros Γ f Δ g H H0.
apply mem_remove_2 with (b := f).
rewrite <- mem_morph_eq with (Γ:=Δ);auto.
apply eq_sym.
assumption.
Qed.
Lemma Cres_Context: ∀x, Cres x → Context x.
Proof.
intros x H.
induction H.
constructor 1;constructor 1.
constructor 2;constructor.
constructor 3;assumption.
Qed.
Lemma Context_Goal: ∀x, Cres x → Goal x.
Proof.
intros x H.
induction H.
constructor.
constructor.
constructor;assumption.
Qed.
Ltac zap :=
match goal with
| |- ?x == ?x => apply eq_refl
| H: ILLVarInt.MILL.eq ?g ?q |- _ => apply eq_is_eq in H;subst
| H:?g ∈ (?q :: ?D') |- _ => destruct (mem_destruct _ _ _ H);clear H
| H: ?p ∈ ?Γ , H':∀ g : formula, g ∈ ?Γ → Context g |- _ => assert (Context (p)) by auto;clear H
| H: Cres ?p |- Goal ?p => apply Context_Goal;assumption
| H:Context (?p ⊸ ?q) |- _ => inversion H;clear H
| H:Act (?p ⊸ ?q) |- _ => inversion H;clear H
| H:Res (?p ⊸ ?q) |- _ => inversion H;clear H
| H:Cres (?p ⊸ ?q) |- _ => inversion H;clear H
| H:Goal (?p ⊸ ?q) |- _ => inversion H;clear H
| H:Context (?p ⊕ ?q) |- _ => inversion H;clear H
| H:Act (?p ⊕ ?q) |- _ => inversion H;clear H
| H:Res (?p ⊕ ?q) |- _ => inversion H;clear H
| H:Cres (?p ⊕ ?q) |- _ => inversion H;clear H
| H:Goal (?p ⊕ ?q) |- _ => inversion H;clear H
| H:Context (?p ⊗ ?q) |- _ => inversion H;clear H
| H:Act (?p ⊗ ?q) |- _ => inversion H;clear H
| H:Res (?p ⊗ ?q) |- _ => inversion H;clear H
| H:Cres (?p ⊗ ?q) |- _ => inversion H;clear H
| H:Goal (?p ⊗ ?q) |- _ => inversion H;clear H
| H:Context (?p & ?q) |- _ => inversion H;clear H
| H:Act (?p & ?q) |- _ => inversion H;clear H
| H:Res (?p & ?q) |- _ => inversion H;clear H
| H:Cres (?p & ?q) |- _ => inversion H;clear H
| H:Goal (?p & ?q) |- _ => inversion H;clear H
| H: ?g ∈ ?Δ |- ?g ∈ (?Δ ∪ ?Δ') => apply mem_union_l;assumption
| H: ?g ∈ ?Δ |- ?g ∈ (?Δ' ∪ ?Δ) => apply mem_union_r;assumption
| H: ILLVarInt.MILL.eq ?g ?q |- _ => apply eq_is_eq in H;subst
| H: (?Γ \ ?p) == ?D |- ?g ∈ ?Γ => apply mem_remove with (Δ:=D) (f:=p);[auto|]
| H: ?g ∈ (?Γ \ ?p) |- ?g ∈ ?Γ => apply mem_remove with (Δ:=(Γ \ p)) (f:=p);[auto|]
| H:?Γ == ?Δ ∪ ?Δ' |- ?g ∈ ?Γ => rewrite (mem_morph_eq _ _ (Δ ∪ Δ'));[|auto]
| H:Res ?p |- Context ?p => constructor 2;assumption
| H: Act ?p |- Context ?p => constructor 1;assumption
end
;try assumption.
Lemma Grammar_Stable : ∀ Γ φ (h:Γ ⊢ φ), Contextall h → Istable Contextall h.
Proof.
fix Grammar_Stable 3.
destruct h; intros; try constructor;try (unfold Contextall in *;decompose [and] H;clear H); try split;intros; try solve [ repeat (zap;subst);apply H1;repeat zap
].
Guarded.
apply Grammar_Stable;split;intros; repeat (zap;subst);apply H1;repeat zap.
Guarded.
apply Grammar_Stable;split;intros;repeat (zap;subst);apply H1;repeat zap.
apply Grammar_Stable;split;intros;repeat (zap;subst);apply H1;repeat zap.
Guarded.
apply Grammar_Stable;split;intros;repeat (zap;subst);apply H1;repeat zap.
Guarded.
apply Grammar_Stable;split;intros;repeat (zap;subst);apply H1;repeat zap.
Guarded.
apply Grammar_Stable;split;intros;repeat (zap;subst);apply H1;repeat zap.
Guarded.
apply Grammar_Stable;split;intros;repeat (zap;subst);apply H1;repeat zap.
Guarded.
apply Grammar_Stable;split;intros;repeat (zap;subst);apply H1;repeat zap.
Guarded.
apply Grammar_Stable;split;intros;repeat (zap;subst);apply H1;repeat zap.
Guarded.
apply Grammar_Stable;split;intros;repeat (zap;subst);apply H1;repeat zap.
Guarded.
apply Grammar_Stable;split;intros;repeat (zap;subst);apply H1;repeat zap.
Guarded.
apply Grammar_Stable;split;intros;repeat (zap;subst);apply H1;repeat zap.
Guarded.
apply Grammar_Stable;split;intros;repeat (zap;subst);apply H1;repeat zap.
Guarded.
apply Grammar_Stable;split;intros;repeat (zap;subst);apply H1;repeat zap.
Guarded.
inversion H0.
Guarded.
assert ( h : Context 0) by auto.
inversion h.
inversion H.
inversion H.
repeat zap.
inversion H;subst.
inversion H2.
constructor 1;assumption.
inversion H2.
constructor 2;assumption.
repeat (zap;subst);apply H1;repeat zap.
Guarded.
apply Grammar_Stable;split;intros.
assumption.
repeat zap.
inversion H;subst.
inversion H2.
constructor 1;assumption.
inversion H2.
constructor 2;assumption.
repeat (zap;subst);apply H1;repeat zap.
Guarded.
apply Grammar_Stable;split;intros;repeat (zap;subst);apply H1;repeat zap.
Guarded.
apply Grammar_Stable;split;intros;repeat (zap;subst);apply H1;repeat zap.
Guarded.
Qed.