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(* A bit complex but realistic example whose last fixpoint definition
used to fail in 8.1 because of wrong environment in conversion
problems (see revision 9664) *)
Require Import List.
Require Import Arith.
Parameter predicate : Set.
Parameter function : Set.
Definition variable := nat.
Definition x0 := 0.
Definition var_eq_dec := eq_nat_dec.
Inductive term : Set :=
| App : function -> term -> term
| Var : variable -> term.
Definition atom := (predicate * term)%type.
Inductive formula : Set :=
| Atom : atom -> formula
| Imply : formula -> formula -> formula
| Forall : variable -> formula -> formula.
Notation "A --> B" := (Imply A B) (at level 40).
Definition substitution range := list (variable * range).
Fixpoint remove_assoc (A:Set)(x:variable)(rho: substitution A){struct rho}
: substitution A :=
match rho with
| nil => rho
| (y,t) :: rho => if var_eq_dec x y then remove_assoc A x rho
else (y,t) :: remove_assoc A x rho
end.
Fixpoint assoc (A:Set)(x:variable)(rho:substitution A){struct rho}
: option A :=
match rho with
| nil => None
| (y,t) :: rho => if var_eq_dec x y then Some t
else assoc A x rho
end.
Fixpoint subst_term (rho:substitution term)(t:term){struct t} : term :=
match t with
| Var x => match assoc _ x rho with
| Some a => a
| None => Var x
end
| App f t' => App f (subst_term rho t')
end.
Fixpoint subst_formula (rho:substitution term)(A:formula){struct A}:formula :=
match A with
| Atom (p,t) => Atom (p, subst_term rho t)
| A --> B => subst_formula rho A --> subst_formula rho B
| Forall y A => Forall y (subst_formula (remove_assoc _ y rho) A)
(* assume t closed *)
end.
Definition subst A x t := subst_formula ((x,t):: nil) A.
Record Kripke : Type := {
worlds: Set;
wle : worlds -> worlds -> Type;
wle_refl : forall w, wle w w ;
wle_trans : forall w w' w'', wle w w' -> wle w' w'' -> wle w w'';
domain : Set;
vars : variable -> domain;
funs : function -> domain -> domain;
atoms : worlds -> predicate * domain -> Type;
atoms_mon : forall w w', wle w w' -> forall P, atoms w P -> atoms w' P
}.
Section Sem.
Variable K : Kripke.
Fixpoint sem (rho: substitution (domain K))(t:term){struct t} : domain K :=
match t with
| Var x => match assoc _ x rho with
| Some a => a
| None => vars K x
end
| App f t' => funs K f (sem rho t')
end.
End Sem.
Notation "w <= w'" := (wle _ w w').
Set Implicit Arguments.
Reserved Notation "w ||- A" (at level 70).
Definition context := list formula.
Variable fresh : variable -> context -> Prop.
Variable fresh_out : context -> variable.
Axiom fresh_out_spec : forall Gamma, fresh (fresh_out Gamma) Gamma.
Axiom fresh_peel : forall x A Gamma, fresh x (A::Gamma) -> fresh x Gamma.
Fixpoint force (K:Kripke)(rho: substitution (domain K))(w:worlds K)(A:formula)
{struct A} : Type :=
match A with
| Atom (p,t) => atoms K w (p, sem K rho t)
| A --> B => forall w', w <= w' -> force K rho w' A -> force K rho w' B
| Forall x A => forall w', w <= w' -> forall t, force K ((x,t)::remove_assoc _ x rho) w' A
end.
Notation "w ||- A" := (force _ nil w A).
Reserved Notation "Gamma |- A" (at level 70).
Reserved Notation "Gamma ; A |- C" (at level 70, A at next level).
Inductive context_prefix (Gamma:context) : context -> Type :=
| CtxPrefixRefl : context_prefix Gamma Gamma
| CtxPrefixTrans : forall A Gamma', context_prefix Gamma Gamma' -> context_prefix Gamma (cons A Gamma').
Inductive in_context (A:formula) : list formula -> Prop :=
| InAxiom : forall Gamma, in_context A (cons A Gamma)
| OmWeak : forall Gamma B, in_context A Gamma -> in_context A (cons B Gamma).
Inductive prove : list formula -> formula -> Type :=
| ProofImplyR : forall A B Gamma, prove (cons A Gamma) B
-> prove Gamma (A --> B)
| ProofForallR : forall x A Gamma, (forall y, fresh y (A::Gamma)
-> prove Gamma (subst A x (Var y))) -> prove Gamma (Forall x A)
| ProofCont : forall A Gamma Gamma' C, context_prefix (A::Gamma) Gamma'
-> (prove_stoup Gamma' A C) -> (Gamma' |- C)
where "Gamma |- A" := (prove Gamma A)
with prove_stoup : list formula -> formula -> formula -> Type :=
| ProofAxiom Gamma C: Gamma ; C |- C
| ProofImplyL Gamma C : forall A B, (Gamma |- A)
-> (prove_stoup Gamma B C) -> (prove_stoup Gamma (A --> B) C)
| ProofForallL Gamma C : forall x t A, (prove_stoup Gamma (subst A x t) C)
-> (prove_stoup Gamma (Forall x A) C)
where " Gamma ; B |- A " := (prove_stoup Gamma B A).
Axiom context_prefix_trans :
forall Gamma Gamma' Gamma'',
context_prefix Gamma Gamma'
-> context_prefix Gamma' Gamma''
-> context_prefix Gamma Gamma''.
Axiom Weakening :
forall Gamma Gamma' A,
context_prefix Gamma Gamma' -> Gamma |- A -> Gamma' |- A.
Axiom universal_weakening :
forall Gamma Gamma', context_prefix Gamma Gamma'
-> forall P, Gamma |- Atom P -> Gamma' |- Atom P.
Canonical Structure Universal := Build_Kripke
context
context_prefix
CtxPrefixRefl
context_prefix_trans
term
Var
App
(fun Gamma P => Gamma |- Atom P)
universal_weakening.
Axiom subst_commute :
forall A rho x t,
subst_formula ((x,t)::rho) A = subst (subst_formula rho A) x t.
Axiom subst_formula_atom :
forall rho p t,
Atom (p, sem _ rho t) = subst_formula rho (Atom (p,t)).
Fixpoint universal_completeness (Gamma:context)(A:formula){struct A}
: forall rho:substitution term,
force _ rho Gamma A -> Gamma |- subst_formula rho A
:=
match A
return forall rho, force _ rho Gamma A
-> Gamma |- subst_formula rho A
with
| Atom (p,t) => fun rho H => eq_rect _ (fun A => Gamma |- A) H _ (subst_formula_atom rho p t)
| A --> B => fun rho HImplyAB =>
let A' := subst_formula rho A in
ProofImplyR (universal_completeness (A'::Gamma) B rho
(HImplyAB (A'::Gamma)(CtxPrefixTrans A' (CtxPrefixRefl Gamma))
(universal_completeness_stoup A rho (fun C Gamma' Hle p
=> ProofCont Hle p))))
| Forall x A => fun rho HForallA
=> ProofForallR x (fun y Hfresh
=> eq_rect _ _ (universal_completeness Gamma A _
(HForallA Gamma (CtxPrefixRefl Gamma)(Var y))) _ (subst_commute _ _ _ _ ))
end
with universal_completeness_stoup (Gamma:context)(A:formula){struct A}
: forall rho, (forall C Gamma', context_prefix Gamma Gamma'
-> Gamma' ; subst_formula rho A |- C -> Gamma' |- C)
-> force _ rho Gamma A
:=
match A return forall rho,
(forall C Gamma', context_prefix Gamma Gamma'
-> Gamma' ; subst_formula rho A |- C
-> Gamma' |- C)
-> force _ rho Gamma A
with
| Atom (p,t) as C => fun rho H
=> H _ Gamma (CtxPrefixRefl Gamma)(ProofAxiom _ _)
| A --> B => fun rho H => fun Gamma' Hle HA
=> universal_completeness_stoup B rho (fun C Gamma'' Hle' p
=> H C Gamma'' (context_prefix_trans Hle Hle')
(ProofImplyL (Weakening Hle' (universal_completeness Gamma' A rho HA)) p))
| Forall x A => fun rho H => fun Gamma' Hle t
=> (universal_completeness_stoup A ((x,t)::remove_assoc _ x rho)
(fun C Gamma'' Hle' p =>
H C Gamma'' (context_prefix_trans Hle Hle')
(ProofForallL x t (subst_formula (remove_assoc _ x rho) A)
(eq_rect _ (fun D => Gamma'' ; D |- C) p _ (subst_commute _ _ _ _)))))
end.
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