(* File: In_NGamma.v  (last edited on 27/10/2000) (c) Klaus Weich  *)
(*******************************************************************)
(* The left hand side Gamma of a sequent consists of               *)
(*     work : a list of (arbitray) normalforms                     *)
(*            (to be inserted in the following structures)         *)
(*     ds: a list of disjunctions                                  *)
(*     ni: a list of decorated and undecorated nested implications *)
(*     ai: an AVL tree of atomic implications                      *)
(*     a: an AVL tree of atoms                                     *)
(*******************************************************************)
Require Export Kripke_Trees.
Require Export Normal_Forms.
(************************************************************)
(* disjunctions are stored as pairs of Int's                *)
Definition disj := (Int * Int)%type.
Definition disjs := list disj.
Definition DNil := nil (A:=disj).
Definition disj2form (x : disj) :=
match x with
| (i, j) => OrF (Atom i) (Atom j)
end.
Definition disj2nform (x : disj) := match x with
| (i, j) => NDisj i j
end.
Definition disjs2forms (ds : disjs) := map disj2form ds.
Definition disjs2nforms (ds : disjs) := map disj2nform ds.
(*****************************************************************)
(* Nested implications are stored either as a nimp or as a       *)
(* nimp and a counter-model                                      *)
Inductive nested_imp : Set :=
| Undecorated : nimp -> nested_imp
| Decorated : nimp -> kripke_tree -> nested_imp.
Definition nested_imp2nimp (ni : nested_imp) :=
match ni with
| Undecorated ni => ni
| Decorated ni _ => ni
end.
Definition nested_imp2form (x : nested_imp) := nimp2form (nested_imp2nimp x).
Definition nested_imp2nform (x : nested_imp) := NImp_NF (nested_imp2nimp x).
Definition nested_imps := list nested_imp.
Definition NNil := nil (A:=nested_imp).
Definition nested_imps2forms (ni : nested_imps) := map nested_imp2form ni.
Definition nested_imps2nforms (ni : nested_imps) := map nested_imp2nform ni.
Definition nested_imps2nimps (ni : nested_imps) := map nested_imp2nimp ni.
intros ni1 ni2.
elim ni1; clear ni1.
simpl in |- *; trivial.
intros x ni1 ih.
simpl in |- *.
rewrite ih; trivial.
Qed.
intros ni; elim ni; clear ni.
simpl in |- *; trivial.
intros x ni ih.
simpl in |- *.
rewrite ih; trivial.
Qed.
(************************************************************************)
(* Atomic implactions are stored in AVL Trees ai over lists of normal   *)
(* clauses.                                                             *)
(* For the node with the key field=i and with the data field=bs,        *)
(* we define that, for each b in bs, (Imp (Atom i) b) is in ai.         *)
Definition atomic_imps := AVL nf_list.
Definition AI_Nil := AVL_NIL nf_list.
(************************************************************************)
(* Atoms are stored in AVL Trees ai over unit.                          *)
(* For the node with the key field=i,                                   *)
(* we define that (Atom i) is in ai.                                    *)
(* (Definition in Kripke_Trees.v                                        *)
(************************************************************************)
(* The formulae on the left-hand side of the sequent are given by:      *)
Inductive in_ngamma (work : nf_list) (ds : disjs) (ni : nested_imps)
(ai : atomic_imps) (a : atoms) : normal_form -> Set :=
| In_Work :
forall (n : nat) (c : normal_form),
my_nth normal_form n work c -> in_ngamma work ds ni ai a c
| In_Disjs :
forall (n : nat) (i j : Int),
my_nth disj n ds (i, j) -> in_ngamma work ds ni ai a (NDisj i j)
| In_Nested_Imps :
forall (n : nat) (x : nimp),
my_nth nimp n (nested_imps2nimps ni) x ->
in_ngamma work ds ni ai a (NImp_NF x)
| In_Atomic_Imps :
forall (i : Int) (b : normal_form) (n : nat) (bs : nf_list),
LOOKUP nf_list i ai bs ->
my_nth normal_form n bs b -> in_ngamma work ds ni ai a (AImp i b)
| In_Atoms :
forall i : Int,
LOOKUP unit i a tt -> in_ngamma work ds ni ai a (NAtom i).
(********************************************************************)
intros c0 work ds ni ai a c in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n a0 nth; apply In_Work with (S n); apply My_NthS; assumption.
intros n i j nth; apply In_Disjs with n; assumption.
intros n x nth; apply In_Nested_Imps with n; assumption.
intros i b n bs lookup nth; apply In_Atomic_Imps with n bs; assumption.
intros i lookup; apply In_Atoms; assumption.
Qed.
intros c work ds ni ai a.
apply In_Work with 0.
apply My_NthO.
Qed.
intros bs work ds ni ai a c in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n c0 nth.
apply In_Work with (length bs + n); apply nth_app1; assumption.
intros n i j nth; apply In_Disjs with n; assumption.
intros n x nth; apply In_Nested_Imps with n; assumption.
intros i b n bs' lookup nth; apply In_Atomic_Imps with n bs'; assumption.
intros i lookup; apply In_Atoms; assumption.
Qed.
intros bs work ds ni ai a c in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n c0 nth.
elim (inv_nth_app normal_form n bs work c0 nth); clear nth.
intros nth0; right; exists n; assumption.
intros n' nth1; left; apply In_Work with n'; assumption.
intros n i j nth; left; apply In_Disjs with n; assumption.
intros n x nth; left; apply In_Nested_Imps with n; assumption.
intros i b n bs' lookup nth; left; apply In_Atomic_Imps with n bs';
assumption.
intros i lookup; left; apply In_Atoms; assumption.
Qed.
intros c0 work ds ni ai a c in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n; case n; clear n.
intros a0 nth; right; inversion_clear nth; trivial.
intros n c1 nth; left; apply In_Work with n; inversion_clear nth; assumption.
intros n i j nth; left; apply In_Disjs with n; assumption.
intros n x nth; left; apply In_Nested_Imps with n; assumption.
intros i b n bs lookup nth; left; apply In_Atomic_Imps with n bs; assumption.
intros i lookup; left; apply In_Atoms; assumption.
Qed.
intros work i j ds ni ai a c in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n c0 nth; apply In_Work with n; assumption.
intros n i' j' nth; apply In_Disjs with (S n); apply My_NthS; assumption.
intros n x nth; apply In_Nested_Imps with n; assumption.
intros i' b n bs lookup nth; apply In_Atomic_Imps with n bs; assumption.
intros i' lookup; apply In_Atoms; assumption.
Qed.
intros work i j ds ni ai a.
apply In_Disjs with 0.
apply My_NthO.
Qed.
intros work i j ds ni ai a c in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n c0 nth; left; apply In_Work with n; assumption.
intros n; case n; clear n.
intros i' j' nth; right; inversion_clear nth; trivial.
intros n i' j' nth; left; apply In_Disjs with n; inversion_clear nth;
assumption.
intros n x nth; left; apply In_Nested_Imps with n; assumption.
intros i' b n bs lookup nth; left; apply In_Atomic_Imps with n bs; assumption.
intros i' lookup; left; apply In_Atoms; assumption.
Qed.
intros work ds x ni ai a c in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n c0 nth; apply In_Work with n; assumption.
intros n i j nth; apply In_Disjs with n; assumption.
intros n x' nth; apply In_Nested_Imps with (S n).
simpl in |- *; apply My_NthS; assumption.
intros i' b n bs lookup nth; apply In_Atomic_Imps with n bs; assumption.
intros i' lookup; apply In_Atoms; assumption.
Qed.
intros work ds x ni ai a.
apply In_Nested_Imps with 0; simpl in |- *; apply My_NthO.
Qed.
intros work ds x ni ai a c in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n c1 nth; left; apply In_Work with n; assumption.
intros n i' j' nth; left; apply In_Disjs with n; assumption.
intros n; case n; clear n.
intros x' nth; right; inversion_clear nth; trivial.
intros n x' nth; left; apply In_Nested_Imps with n; inversion_clear nth;
assumption.
intros i' b n bs lookup nth; left; apply In_Atomic_Imps with n bs; assumption.
intros i' lookup; left; apply In_Atoms; assumption.
Qed.
intros work ds x ni1 ni2 ai a.
apply In_Nested_Imps with (n := length ni1) (x := nested_imp2nimp x).
rewrite (nested_imps2nimps_app ni1 (x :: ni2)).
rewrite (nested_imps2nimps_length ni1).
rewrite <- (plus_O (length (nested_imps2nimps ni1))).
apply nth_app1.
simpl in |- *; apply My_NthO.
Qed.
intros work ds x ni1 ni2 ai a c in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n c0 nth; apply In_Work with n; assumption.
intros n i j nth; apply In_Disjs with n; assumption.
intros n x0 nth.
elim (inv_nth_app nimp n (nested_imps2nimps ni1) (nested_imps2nimps ni2) x0).
intros nth'; apply In_Nested_Imps with n.
rewrite (nested_imps2nimps_app ni1 (x :: ni2)).
apply nth_app0; assumption.
intros n' nth'; apply In_Nested_Imps with (length ni1 + S n').
rewrite (nested_imps2nimps_app ni1 (x :: ni2)).
rewrite (nested_imps2nimps_length ni1).
apply nth_app1.
simpl in |- *; apply My_NthS; assumption.
rewrite <- (nested_imps2nimps_app ni1 ni2); assumption.
intros i b n bs lookup nth; apply In_Atomic_Imps with n bs; assumption.
intros i lookup; apply In_Atoms; assumption.
Qed.
intros work ds x ni1 ni2 ai a c in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n c0 nth; left; apply In_Work with n; assumption.
intros n i j nth; left; apply In_Disjs with n; assumption.
intros n x0 nth.
elim
(inv_nth_app nimp n (nested_imps2nimps ni1) (nested_imps2nimps (x :: ni2))
x0).
intros n0; left; apply In_Nested_Imps with n.
rewrite (nested_imps2nimps_app ni1 ni2).
apply nth_app0; assumption.
intros n'; case n'; clear n'.
intros nth'; right; inversion_clear nth'; trivial.
intros n' nth'; left; apply In_Nested_Imps with (length ni1 + n').
rewrite (nested_imps2nimps_length ni1).
rewrite (nested_imps2nimps_app ni1 ni2).
apply nth_app1.
inversion_clear nth'; assumption.
rewrite <- (nested_imps2nimps_app ni1 (x :: ni2)); assumption.
intros i b n bs lookup nth; left; apply In_Atomic_Imps with n bs; assumption.
intros i lookup; left; apply In_Atoms; assumption.
Qed.
intros work ds ni ni' ai a c eq in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n c0 nth; apply In_Work with n; assumption.
intros n i j nth; apply In_Disjs with n; assumption.
intros n x' nth; apply In_Nested_Imps with n.
rewrite <- eq; assumption.
intros i' b n bs lookup nth; apply In_Atomic_Imps with n bs; assumption.
intros i' lookup; apply In_Atoms; assumption.
Qed.
intros work ds ni i b ai ai' a c equiv_ins in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n c0 nth; apply In_Work with n; assumption.
intros n i' j' nth; apply In_Disjs with n; assumption.
intros n x nth; apply In_Nested_Imps with n; assumption.
intros i0 b0 n bs lookup nth.
case (equal_dec i0 i).
intro equal_i0_i.
apply In_Atomic_Imps with (S n) (b :: bs).
generalize equiv_ins lookup; clear equiv_ins lookup.
elim ai; clear ai; intros t avl_t.
elim ai'; clear ai'; intros t' avl_t'.
simpl in |- *.
intros equiv_ins lookup.
inversion_clear equiv_ins.
apply H; assumption.
apply My_NthS; assumption.
intros not_equal_i0_i.
apply In_Atomic_Imps with n bs.
generalize equiv_ins lookup; clear equiv_ins lookup.
elim ai; clear ai; intros t avl_t.
elim ai'; clear ai'; intros t' avl_t'.
simpl in |- *.
intros equiv_ins lookup.
inversion_clear equiv_ins.
apply H1; assumption.
assumption.
intros i' lookup; apply In_Atoms; assumption.
Qed.
intros work ds ni i b ai ai' a notlookup equiv_ins.
apply In_Atomic_Imps with 0 (b :: nf_nil).
generalize equiv_ins notlookup; clear equiv_ins notlookup.
elim ai; clear ai; intros t avl_t.
elim ai'; clear ai'; intros t' avl_t'.
simpl in |- *.
intros.
inversion_clear equiv_ins.
clear H1 H2.
apply H0.
apply equal_refl.
assumption.
apply My_NthO.
Qed.
intros work ds ni i b bs ai ai' a lookup equiv_ins.
apply In_Atomic_Imps with 0 (b :: bs).
generalize equiv_ins lookup; clear equiv_ins lookup.
elim ai; clear ai; intros t avl_t.
elim ai'; clear ai'; intros t' avl_t'.
simpl in |- *.
intros.
inversion_clear equiv_ins.
clear H1 H2.
apply H.
apply equal_refl.
assumption.
apply My_NthO.
Qed.
intros work ds ni i b ai ai' a c equiv_ins in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n c0; left; apply In_Work with n; assumption.
intros n i' j' nth; left; apply In_Disjs with n; assumption.
intros n x nth; left; apply In_Nested_Imps with n; assumption.
intros i0.
elim (equal_dec i0 i).
intro equal.
intros b0 n.
case n; clear n.
intros bs lookup nth.
right.
generalize equiv_ins lookup; clear equiv_ins lookup.
elim ai; clear ai; intros t avl_t.
elim ai'; clear ai'; intros t' avl_t'.
simpl in |- *.
intros equiv_ins lookup.
inversion_clear equiv_ins.
clear H1 H2.
elim (lookup_dec nf_list i0 t avl_t).
intros bs' lookup0.
generalize (H i0 bs' equal lookup0); clear H H0.
intros.
rewrite (lookup_avl_equal nf_list i0 i0 t' bs (b :: bs')) in nth;
try assumption.
inversion_clear nth.
rewrite (equal_eq i0 i).
trivial.
assumption.
apply equal_refl.
intros notlookup0.
generalize (H0 i0 equal notlookup0); clear H H0.
intros.
rewrite (lookup_avl_equal nf_list i0 i0 t' bs (b :: nf_nil)) in nth;
try assumption.
inversion_clear nth.
rewrite (equal_eq i0 i).
trivial.
assumption.
apply equal_refl.
intros n bs lookup nth.
left; apply In_Atomic_Imps with n (tail bs).
generalize equiv_ins lookup; clear equiv_ins lookup.
elim ai; clear ai; intros t avl_t.
elim ai'; clear ai'; intros t' avl_t'.
simpl in |- *.
intros equiv_ins lookup.
inversion_clear equiv_ins.
clear H1 H2.
elim (lookup_dec nf_list i0 t avl_t).
intros bs' lookup0.
generalize (H i0 bs' equal lookup0); clear H H0.
intros.
rewrite (lookup_avl_equal nf_list i0 i0 t' bs (b :: bs')); try assumption.
apply equal_refl.
intros notlookup0.
generalize (H0 i0 equal notlookup0); clear H H0.
intros.
rewrite (lookup_avl_equal nf_list i0 i0 t' bs (b :: nf_nil)) in nth;
try assumption.
inversion_clear nth.
inversion_clear H0.
apply equal_refl.
inversion_clear nth.
assumption.
intros notequal.
intros b0 n bs lookup nth.
left; apply In_Atomic_Imps with n bs.
generalize equiv_ins lookup; clear equiv_ins lookup.
elim ai; clear ai; intros t avl_t.
elim ai'; clear ai'; intros t' avl_t'.
simpl in |- *.
intros equiv_ins lookup.
inversion_clear equiv_ins.
clear H H0.
apply H2; assumption.
assumption.
intros i' lookup; left; apply In_Atoms; assumption.
Qed.
intros work ds ni i ai ai' a c equiv_del in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n c0 nth; apply In_Work with n; assumption.
intros n i' j' nth; apply In_Disjs with n; assumption.
intros n x' nth; apply In_Nested_Imps with n; assumption.
intros i0 b0 n bs lookup nth.
apply In_Atomic_Imps with n bs.
generalize equiv_del lookup; clear equiv_del lookup.
elim ai; clear ai; intros t avl_t.
elim ai'; clear ai'; intros t' avl_t'.
simpl in |- *.
intros equiv_del lookup.
inversion_clear equiv_del.
elim (equal_dec i0 i).
intro equal.
elimtype False.
apply (H i0 equal bs lookup).
intros not_equal.
apply H1; assumption.
assumption.
intros i' lookup; apply In_Atoms; assumption.
Qed.
Inductive in_ngamma_del_ai_rev_spec (i : Int) (bs : nf_list)
(c : normal_form) : Set :=
In_NGamma_Del_AI_Rev_Spec_Intro :
forall (b : normal_form) (n : nat),
my_nth normal_form n bs b ->
c = AImp i b -> in_ngamma_del_ai_rev_spec i bs c.
intros work ds ni i bs ai ai' a c lookup equiv_del in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n c0 nth; left; apply In_Work with n; assumption.
intros n i' j' nth; left; apply In_Disjs with n; assumption.
intros n x' nth; left; apply In_Nested_Imps with n; assumption.
intros i0 b0 n bs0 lookup0 nth.
elim (equal_dec i0 i).
intro equal.
right.
rewrite (equal_eq i0 i equal).
exists b0 n.
generalize equiv_del lookup0 lookup; clear equiv_del lookup0 lookup.
elim ai; clear ai; intros t avl_t.
elim ai'; clear ai'; intros t' avl_t'.
simpl in |- *.
intros.
rewrite <- (lookup_avl_equal nf_list i0 i t' bs0 bs); try assumption.
trivial.
intros notequal.
left.
apply In_Atomic_Imps with n bs0.
generalize lookup equiv_del lookup0; clear lookup equiv_del lookup0.
elim ai; clear ai; intros t avl_t.
elim ai'; clear ai'; intros t' avl_t'.
simpl in |- *; intros.
inversion_clear equiv_del.
apply H0; assumption.
assumption.
intros i' lookup'; left; apply In_Atoms; assumption.
Qed.

Lemma in_ngamma_ins_a_tail :
forall (work : nf_list) (ds : disjs) (ni : nested_imps)
(ai : atomic_imps) (i : Int) (a a' : atoms) (c : normal_form),
EQUIV_INS unit i (fun _ : unit => tt) tt a a' ->
in_ngamma work ds ni ai a c -> in_ngamma work ds ni ai a' c.