Section APD.
Variable X V R : Ensf.
Variable S' : Elt.
Hypothesis Gram : isGram X V R S'.
apply isGram4 with S'.
exact Gram.
Qed.
Let P := union X V.
Let f_R_d (a : Elt) :=
couple (word (cons (first a) nil)) (couple (eps X) (second a)).
Let f_X_d (x : Elt) := couple (word (cons x nil)) (couple x (word nil)).
Let d := union (map f_R_d R) (map f_X_d X).
Let wd := cons S' nil.
Let wa := nil.
red in |- *.
intros x dans_x_d.
elimtype (dans x (map f_R_d R) \/ dans x (map f_X_d X)).
intros dans_x.
elimtype (exists y : Elt, dans y R /\ x = f_R_d y).
intros y temp; elim temp; clear temp.
unfold f_R_d in |- *.
intros dans_y_R eg_x_f_y.
exists (cons (first y) nil).
apply inmonoid_cons.
trivial.
elim (Regles_X_V_R y dans_y_R).
intros f_y dans_f_y_V temp; elim temp; clear temp.
intros u eg_y inmono_u.
rewrite eg_y.
unfold first in |- *.
unfold P in |- *.
auto.
exists (eps X).
auto.
exists (word_inv (second y)).
elim (Regles_X_V_R y dans_y_R).
intros f_y dans_f_y_V temp; elim temp; clear temp.
intros u eg_y inmono_u.
rewrite eg_y.
unfold second in |- *.
unfold word_inv in |- *.
assumption.
elim (Regles_X_V_R y dans_y_R).
intros f_y dans_f_y_V temp; elim temp; clear temp.
intros u eg_y inmono_u.
replace (word (word_inv (second y))) with (second y).
assumption.
replace (second y) with (word u).
apply refl_equal.
rewrite eg_y.
apply refl_equal.
apply dans_map.
assumption.
intros dans_x.
elimtype (exists y : Elt, dans y X /\ x = f_X_d y).
intros y temp; elim temp; clear temp.
unfold f_X_d in |- *.
intros dans_y_X eg_x_f_y.
exists (cons y nil).
apply inmonoid_cons.
trivial.
unfold P in |- *.
auto.
exists y.
auto.
exists nil.
trivial.
assumption.
apply dans_map.
assumption.
auto.
Qed.
red in |- *.
split.
unfold wd in |- *.
apply inmonoid_cons.
trivial.
unfold P in |- *.
apply union_d.
apply isGram3 with X R.
exact Gram.
split.
apply inmonoid_nil.
exact Trans.
Qed.
intro u.
pattern u in |- *.
apply Word_rec.
exists nil.
exists nil.
auto.
intros x w Hyp.
elim Hyp.
intros a' spec; elim spec.
intros b' temp; elim temp; clear temp.
intros inmonoid_a' temp; elim temp; clear temp.
intros spec_App spec_or.
elimtype ({dans x X} + {~ dans x X}).
intro dans_x_X.
exists (cons x a').
exists b'.
split.
apply inmonoid_cons; assumption.
split.
unfold Append in |- *.
simpl in |- *.
auto.
elim spec_or; intro; assumption.
intro N_dans_x_X.
exists nil.
exists (cons x w).
split.
auto.
split.
auto.
apply or_intror.
exists x.
assumption.
exists w.
apply refl_equal.
exact (Dans_spec x X).
Qed.
Definition cut (u : Word) :=
let (a, s) := cut_spec u in (a, let (b, s2) := s in b).
Definition cut1 (u : Word) := fst (cut u).
Definition cut2 (u : Word) := snd (cut u).
intros u a b a' b' temp1 temp2; elim temp1; elim temp2; clear temp1;
clear temp2.
intros inmon temp inmon' temp'; elim temp; elim temp'; clear temp;
clear temp'.
intros App spec_or App' spec_or'.
elimtype
(exists w : Word,
a = Append a' w /\ b' = Append w b \/ a' = Append a w /\ b = Append w b').
intros x temp; elim temp; clear temp.
pattern a' at 2 in |- *.
replace a' with (Append a' nil).
generalize x; clear x; simple induction x.
intro temp; elim temp.
auto.
intros x0 w Hyp temp; elim temp; clear temp.
intros a_eg b'_eg.
absurd (dans x0 X).
elim spec_or'.
rewrite b'_eg.
intro cons_nil.
absurd (cons x0 (Append w b) = nil).
discriminate.
assumption.
intro temp; elim temp; clear temp.
intros x1 N_dans_x1 temp; elim temp; clear temp.
intros w' b'_eg_2.
replace x0 with x1.
assumption.
apply cons_cons_inv1 with w' (Append w b).
rewrite <- b'_eg_2.
assumption.
apply inmonoid_cons_inv2 with w.
apply inmonoid_Append_inv2 with a'.
rewrite <- a_eg.
assumption.
apply Append_w_nil.
pattern a at 2 in |- *.
replace a with (Append a nil).
generalize x; clear x; simple induction x.
intro temp; elim temp.
auto.
intros x0 w Hyp temp; elim temp; clear temp.
intros a'_eg b_eg.
absurd (dans x0 X).
elim spec_or.
rewrite b_eg.
intro cons_nil.
absurd (cons x0 (Append w b') = nil).
discriminate.
assumption.
intro temp; elim temp; clear temp.
intros x1 N_dans_x1 temp; elim temp; clear temp.
intros w' b_eg_2.
replace x0 with x1.
assumption.
apply cons_cons_inv1 with w' (Append w b').
rewrite <- b_eg_2.
assumption.
apply inmonoid_cons_inv2 with w.
apply inmonoid_Append_inv2 with a.
rewrite <- a'_eg.
assumption.
apply Append_w_nil.
apply Append_Append.
rewrite App'; assumption.
Qed.
intro w.
unfold cut1, cut in |- *.
elim (cut_spec w).
intros a temp; elim temp; clear temp.
intros b temp; elim temp; clear temp.
simpl in |- *; auto.
Qed.
intro w.
unfold cut1, cut2, cut in |- *.
elim (cut_spec w).
intros a temp; elim temp; clear temp.
intros b temp; elim temp; clear temp.
intros inm temp; elim temp.
simpl in |- *; auto.
Qed.
intros a w dans_a_X.
unfold cut1, cut in |- *.
elim (cut_spec w).
intros x temp; elim temp; clear temp.
intros b temp; elim temp; clear temp.
intros inmon_X_x temp; elim temp; clear temp.
elim (cut_spec (cons a w)).
intros x' temp; elim temp; clear temp.
intros b' temp; elim temp; clear temp.
intros inmon_X_x' temp; elim temp; clear temp.
intros App_eg' spec' App_eg spec.
simpl in |- *.
cut (inmonoid X x').
cut (Append x' b' = cons a w).
clear inmon_X_x' App_eg'.
generalize x'; clear x'; simple induction x'.
intro App_nil_b'_eg_cons_a_w.
absurd (dans a X).
elim spec'.
intro b'_nil.
absurd (cons a w = nil).
discriminate.
rewrite <- b'_nil.
rewrite <- App_nil_b'_eg_cons_a_w.
trivial.
intro temp; elim temp; clear temp.
intros x0 N_dans_x0_X temp; elim temp; clear temp.
intros w1 eg_b'_cons.
replace a with x0.
assumption.
apply cons_cons_inv1 with w1 w.
rewrite <- eg_b'_cons.
assumption.
assumption.
intros x0 w0 Hyp App_eg_cons inmon_cons.
apply cons_cons.
apply cons_cons_inv1 with (Append w0 b') w.
assumption.
apply cut_unicite with w b' b.
split.
apply inmonoid_cons_inv with x0; assumption.
split.
apply cons_cons_inv2 with x0 a.
assumption.
assumption.
split; auto.
assumption.
assumption.
Qed.
intros u v.
generalize u; clear u; simple induction u.
auto.
intros x w Hyp inmon_x_w.
replace (cut1 (Append (cons x w) v)) with (cons x (cut1 (Append w v))).
replace (Append (cons x w) (cut1 v)) with (cons x (Append w (cut1 v))).
apply cons_cons.
trivial.
apply Hyp.
apply inmonoid_cons_inv with x; assumption.
trivial.
apply sym_equal.
unfold Append in |- *.
simpl in |- *.
apply cut1_cons.
apply inmonoid_cons_inv2 with w.
trivial.
Qed.
Axiom
cut2_cons :
forall (A : Elt) (v : Word), dans A X -> cut2 (cons A v) = cut2 v.
intros A v N_dans_A_X.
unfold cut2, cut in |- *.
elim (cut_spec (cons A v)).
intros a temp; elim temp; clear temp.
intros b temp; elim temp; clear temp.
intros inmon_a temp; elim temp; clear temp.
intros App_eg spec_or.
cut (inmonoid X a).
cut (Append a b = cons A v).
clear App_eg inmon_a.
generalize a; clear a; simple induction a.
auto.
intros x w Hyp App_eg_x_w inmon.
absurd (dans A X).
assumption.
replace A with x.
apply inmonoid_cons_inv2 with w.
assumption.
apply cons_cons_inv1 with (Append w b) v.
assumption.
assumption.
assumption.
Qed.
intros x y Der.
elim Der.
intros.
replace (cut1 (cons A v)) with nil.
exists (Append u v).
trivial.
replace (cut2 (cons A v)) with (cons A v).
apply Deriveg1.
assumption.
apply sym_equal.
apply cut2_cons_N.
apply inter_dans with V.
apply sym_inter.
apply isGram2 with R S'.
exact Gram.
apply Regles_inv1 with X R (word u).
apply isGram4 with S'.
exact Gram.
assumption.
unfold cut1, cut in |- *.
elim (cut_spec (cons A v)).
intros a p.
elim p.
intros b temp; elim temp; clear temp.
intros inmon_a temp; elim temp; clear temp.
intros App_eg spec_or.
cut (inmonoid X a). (**)(*pour l'induction*)
cut (Append a b = cons A v). (***)(*pour l'induction*)
clear App_eg inmon_a p.
generalize a; clear a; simple induction a.
auto.
intros x0 w Hyp App inmon.
absurd (dans A X).
intros.
apply inter_dans with V.
apply sym_inter.
apply isGram2 with R S'.
exact Gram.
apply Regles_inv1 with X R (word u).
apply isGram4 with S'.
exact Gram.
assumption.
apply inmonoid_cons_inv2 with w.
replace A with x0.
assumption.
apply cons_cons_inv1 with (Append w b) v.
assumption.
assumption. (**)
assumption. (***)
intros u v x0 dans_x0_X Deriveg_X_R_u_v temp.
elim temp; clear temp.
intros w eg_App_cut Der_g_cut2.
exists w.
replace (cut1 (cons x0 u)) with (cons x0 (cut1 u)).
unfold Append in |- *.
simpl in |- *.
apply cons_cons; trivial.
apply sym_equal.
apply cut1_cons.
assumption.
replace (cut2 (cons x0 u)) with (cut2 u).
assumption.
apply sym_equal.
apply cut2_cons.
assumption.
Qed.
intros x y Derg.
elim Derg.
intros u0 v0 A dans_A_u0_R u v eg_App.
elimtype (u = nil \/ (exists w : Word, (exists x : Elt, u = cons x w))).
intros u_eg.
exists (Append u0 v0).
rewrite u_eg.
trivial.
replace v with (cons A v0).
apply Deriveg1; assumption.
replace v with (Append u v).
auto.
rewrite u_eg.
trivial.
intro temp; elim temp; clear temp.
intros w temp; elim temp; clear temp.
intros x0 u_eg inmon_u.
absurd (dans x0 V).
apply inter_dans with X.
apply isGram2 with R S'.
exact Gram.
apply inmonoid_cons_inv2 with w.
rewrite <- u_eg.
assumption.
replace x0 with A.
apply Regles_inv1 with X R (word u0).
exact Regles_X_V_R.
assumption.
apply cons_cons_inv1 with v0 (Append w v).
change (cons A v0 = Append (cons x0 w) v) in |- *.
rewrite <- u_eg.
auto.
clear eg_App.
generalize u; clear u; simple induction u.
auto.
intros x0 w hyp.
apply or_intror.
exists w.
exists x0.
trivial.
intros u0 v0 x0 dans_x0_X Derg_u_v Hyp u v.
elimtype (u = nil \/ (exists w : Word, (exists x : Elt, u = cons x w))).
intro eg_u.
rewrite eg_u.
intros App_eg inmon_nil.
exists (cons x0 v0).
trivial.
replace v with (cons x0 u0).
apply Deriveg2.
assumption.
assumption.
intro temp; elim temp; clear temp.
intros w temp; elim temp; clear temp.
intros x1 u_eg.
rewrite u_eg.
intros App_cons_eg inmonoid_X_cons_x1_w.
elimtype
(ex2 (fun w1 : Word => Append w w1 = v0) (fun w1 : Word => Deriveg X R v w1)).
intros w0 App_eg Der_v_w0.
exists w0.
change (cons x1 (Append w w0) = cons x0 v0) in |- *.
apply cons_cons.
apply cons_cons_inv1 with (Append w v) u0.
assumption.
assumption.
assumption.
apply Hyp.
apply cons_cons_inv2 with x1 x0.
assumption.
apply inmonoid_cons_inv with x1.
assumption.
generalize u; clear u; simple induction u.
auto.
intros x2 w hyp.
apply or_intror.
exists w.
exists x2.
trivial.
Qed.
unfold Derivegstar in |- *.
intros x y Der_star.
unfold Rstar in Der_star.
pattern x, y in |- *.
apply Der_star.
intros; exists v.
assumption.
apply Rstar_reflexive.
intros u v w Der_u_v Hyp u0 v0 App_eg inmon_u0.
elimtype
(ex2 (fun w1 : Word => Append u0 w1 = v)
(fun w1 : Word => Deriveg X R v0 w1)).
intros w0 App_eg_u0_w0 Deriveg_v0_x0.
elimtype
(ex2 (fun w1 : Word => Append u0 w1 = w)
(fun w1 : Word => Rstar Word (Deriveg X R) w0 w1)).
intros w1 App_eg_u0_w1_w Rstar_Der.
exists w1.
assumption.
apply Rstar_R with w0.
assumption.
assumption.
apply Hyp; assumption.
apply Deriveg_imp_Deriveg_App with u; assumption.
Qed.
Inductive Derive_P_A_2 : Conf -> Conf -> Prop :=
| Derive_X :
forall (w u : Word) (x : Elt),
dans x X -> Derive_P_A_2 (cons x w, cons x u) (w, u)
| Derive_V :
forall (v w u : Word) (x : Elt),
dans (couple x (word u)) R ->
Derive_P_A_2 (cons x w, v) (Append u w, v).
Definition Derive_P_A_2_nind (x y : Conf) :=
ex2 (fun a : Elt => dans a X)
(fun a : Elt => fst x = cons a (fst y) /\ snd x = cons a (snd y)) \/
(exists a : Elt,
ex2 (fun w : Word => cons a w = fst x)
(fun w : Word =>
ex2 (fun u : Word => dans (couple a (word u)) R)
(fun u : Word => Append u w = fst y))).
intros x y Der.
unfold Derive_P_A_2_nind in |- *.
elim Der.
intros w u x0 dans_x0_X.
apply or_introl.
exists x0; auto.
intros v w u x0 dans_couple_Der.
apply or_intror.
exists x0.
exists w.
auto.
exists u; auto.
Qed.
Definition Derivestar_P_A_2 := Rstar Conf Derive_P_A_2.
intros x u v dans_x_X Der_star.
elimtype
((cons x u, v) = (nil, nil) \/
ex2 (fun z : Conf => Derive_P_A_2 (cons x u, v) z)
(fun z : Conf => Rstar (Word * Word) Derive_P_A_2 z (nil, nil))).
intro eg.
absurd (cons x u = nil).
discriminate.
change (fst (cons x u, v) = fst (nil, nil)) in |- *.
apply (f_equal (fst (A:=Word) (B:=Word))).
assumption.
intro temp; elim temp; clear temp; intros z Der Der_star_2.
cut (Derive_P_A_2_nind (cons x u, v) z).
unfold Derive_P_A_2_nind in |- *.
intro temp; elim temp; clear temp; intro temp; elim temp; clear temp.
simpl in |- *.
intros x0 dans_x0_X temp; elim temp; clear temp.
intros eg_cons_x_x0 eg_v_cons.
exists (snd z).
replace x with x0.
trivial.
apply cons_cons_inv1 with (fst z) u.
auto.
unfold Derivestar_P_A_2 in |- *.
replace u with (fst z).
cut (Rstar (Word * Word) Derive_P_A_2 z (nil, nil)).
elim z.
auto.
assumption.
apply cons_cons_inv2 with x0 x.
auto.
simpl in |- *.
intros x0 temp; elim temp; clear temp; intros w eg temp; elim temp;
clear temp.
intros u0 dans_couple.
absurd (dans x0 V).
apply inter_dans with X.
exact (isGram2 X V R S' Gram).
replace x0 with x.
trivial.
apply cons_cons_inv1 with u w.
auto.
apply Regles_inv1 with X R (word u0).
exact Regles_X_V_R.
assumption.
apply Derive_P_A_2_inv.
assumption.
apply Rstar_inv.
assumption.
Qed.
unfold Derivestar_P_A_2 in |- *.
simple induction x.
intros y Der_P_A_2 u v.
elimtype (u = nil \/ (exists w : Word, (exists x : Elt, u = cons x w))).
intros u_eg.
rewrite u_eg.
intros App_eg inmon_nil.
exists y; auto.
replace v with nil; trivial.
intro temp; elim temp; clear temp; intros w temp; elim temp; clear temp.
intros x0 eg_u.
rewrite eg_u.
intros cons_eg_nil.
absurd (cons x0 (Append w v) = nil).
discriminate.
trivial.
generalize u; clear u; simple induction u.
auto.
intros x1 w1 Hyp; apply or_intror; exists w1; exists x1; trivial.
intros x0 w Hyp y Der_star_cons u v App_eg inmon_X.
elimtype (u = nil \/ (exists w : Word, (exists x : Elt, u = cons x w))).
intros u_eg.
rewrite u_eg.
replace v with (cons x0 w).
exists y; trivial.
replace v with (Append u v).
auto.
rewrite u_eg.
trivial.
intro temp; elim temp; clear temp; intros w1 temp; elim temp; clear temp.
intros x1 eg_u.
rewrite eg_u.
elimtype
(ex2 (fun v2 : Word => y = cons x0 v2)
(fun v2 : Word => Derivestar_P_A_2 (w, v2) (nil, nil))).
intros v2 y_eg Der_w_v2.
elimtype
(ex2 (fun w : Word => Append w1 w = v2)
(fun w : Word => Rstar Conf Derive_P_A_2 (v, w) (nil, nil))).
intros we App_w1_we_eg_v2 Rstar_Der.
exists we.
rewrite y_eg.
replace (Append (cons x1 w1) we) with (cons x1 (Append w1 we)).
apply cons_cons.
apply cons_cons_inv1 with (Append w1 v) w.
rewrite <- App_eg.
rewrite eg_u.
trivial.
trivial.
trivial.
trivial.
apply Hyp.
trivial.
apply cons_cons_inv2 with x1 x0.
rewrite <- App_eg.
rewrite eg_u.
trivial.
apply inmonoid_cons_inv with x1.
rewrite <- eg_u.
trivial.
apply Der_cons_inv.
replace x0 with x1.
apply inmonoid_cons_inv2 with w1.
rewrite <- eg_u.
trivial.
apply cons_cons_inv1 with (Append w1 v) w.
rewrite <- App_eg.
rewrite eg_u.
trivial.
assumption.
clear inmon_X App_eg.
generalize u; clear u; simple induction u.
auto.
intros x1 w1 tmp; apply or_intror; exists w1; exists x1; trivial.
Qed.

Lemma Derive_P_A_2_imp_Der_P_A_2_cons :
forall (u : Elt) (x y : Word),
Derivestar_P_A_2 (cons u x, y) (nil, nil) ->
dans u X ->
ex2 (fun w : Word => cons u w = y)
(fun w : Word => Derivestar_P_A_2 (x, w) (nil, nil)).