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cl_beta_inv.v
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cl_beta_inv.v
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(**************************************************************)
(* Copyright Dominique Larchey-Wendling [*] *)
(* *)
(* [*] Affiliation LORIA -- CNRS *)
(**************************************************************)
(* This file is distributed under the terms of the *)
(* CeCILL v2 FREE SOFTWARE LICENSE AGREEMENT *)
(**************************************************************)
Require Import cl cl_eq cl_beta.
Set Implicit Arguments.
Section inversion_lemmas.
Lemma cl_beta_inv f g :
f -b-> g
-> f = I o g
\/ (exists y, f = K o g o y)
\/ (exists x y z, f = S o x o y o z /\ g = x o z o (y o z))
\/ (exists u v a, f = u o a /\ g = v o a /\ u -b-> v)
\/ (exists u a b, f = u o a /\ g = u o b /\ a -b-> b).
Proof.
induction 1.
do 0 right; left; auto.
do 1 right; left; exists y; auto.
do 2 right; left; exists x, y, z; auto.
do 3 right; left; exists f, g, a; auto.
do 4 right; exists f, a, b; auto.
Qed.
Fact cl_beta_app_inv f g v :
f o g -b-> v
-> (f = I /\ g = v)
\/ (f = K o v)
\/ (exists a b, f = S o a o b /\ v = a o g o (b o g))
\/ (exists f', v = f' o g /\ f -b-> f')
\/ (exists g', v = f o g' /\ g -b-> g').
Proof.
intros H.
apply cl_beta_inv in H.
destruct H as [ H
| [ (y & H)
| [ (x & y & z & H1 & H2)
| [ (x & y & z & H1 & H2 & H3)
| (x & y & z & H1 & H2 & H3) ] ] ] ].
apply cl_app_inj in H;
destruct H as [H0 H1]; rewrite H0; rewrite H1;
do 0 right; left; auto.
apply cl_app_inj in H;
destruct H as [H0 H1]; rewrite H0; rewrite H1;
do 1 right; left; auto.
apply cl_app_inj in H1;
destruct H1 as [H3 H4]; rewrite H3; rewrite H4;
do 2 right; left; exists x, y; split; auto.
apply cl_app_inj in H1;
destruct H1 as [H4 H5]; rewrite H4; rewrite H5;
do 3 right; left; exists y; split; auto.
apply cl_app_inj in H1;
destruct H1 as [H4 H5]; rewrite H4; rewrite H5;
do 4 right; exists z; split; auto.
Qed.
Fact cl_beta_var_0_inv p v : µ p -b-> v -> False.
Proof.
intros H.
apply cl_beta_inv in H.
destruct H as [ H
| [ (y & H)
| [ (x & y & z & H & _)
| [ (x & y & z & H & _ & _)
| (x & y & z & H & _ & _) ] ] ] ]; try discriminate H.
Qed.
Fact cl_beta_var_1_inv p a v : µ p o a -b-> v -> exists b, v = µ p o b /\ a -b-> b.
Proof.
intros H.
apply cl_beta_app_inv in H.
destruct H as [ (H & H2)
| [ H
| [ (f & g & H & _)
| [ (f & H1 & H2)
| (f & H1 & H2) ] ] ] ]; subst; try discriminate H.
exists a; split; apply cl_beta_var_0_inv in H2; contradiction.
exists f; split; auto.
Qed.
Fact cl_beta_var_2_inv p a b v : µ p o a o b -b-> v -> (exists a', v = µ p o a' o b /\ a -b-> a')
\/ (exists b', v = µ p o a o b' /\ b -b-> b').
Proof.
intros H.
apply cl_beta_app_inv in H.
destruct H as [ (H & H2)
| [ H
| [ (f & g & H & H2)
| [ (f & H & H2)
| (f & H & H2) ] ] ] ]; subst; try discriminate H.
apply cl_beta_var_1_inv in H2;
destruct H2 as [ c Hc ];
left; exists c; split.
Focus 2. destruct Hc; auto.
Focus 2.
right. exists f. split; auto.
destruct Hc.
rewrite H.
auto.
Qed.
Fact cl_beta_K_0_inv v : K -b-> v -> False.
Proof.
intros H.
apply cl_beta_inv in H.
destruct H as [ H
| [ (y & H)
| [ (x & y & z & H & _)
| [ (x & y & z & H & _ & _)
| (x & y & z & H & _ & _) ] ] ] ]; try discriminate H.
Qed.
Fact cl_beta_K_1_inv a v : K o a -b-> v -> exists b, v = K o b /\ a -b-> b.
Proof.
intro H.
apply cl_beta_inv in H.
destruct H as [ H
| [ (y & H)
| [ (x & y & z & H & _)
| [ (x & y & z & H & _ & _)
| (x & y & z & H & _ & _) ] ] ] ]; try discriminate H.
apply cl_app_inj in H.
Admitted.
Fact cl_beta_K_2 a b v : K o a o b -b-> v -> v = a
\/ (exists a', v = K o a' o b /\ a -b-> a')
\/ (exists b', v = K o a o b' /\ b -b-> b').
Proof.
intro H.
apply cl_beta_inv in H.
destruct H as [ H
| [ (y & H)
| [ (x & y & z & H & _)
| [ (x & y & z & H & _ & _)
| (x & y & z & H & _ & _) ] ] ] ]; try discriminate H.
apply cl_app_inj in H; destruct H as [H0 H1].
apply cl_app_inj in H0; destruct H0 as [H2 H3].
rewrite H3.
left; auto.
apply cl_app_inj in H; destruct H as [H0 H1].
right. left. exists a. inversion H0.
Focus 2.
do 2 right.
apply cl_app_inj in H. destruct H.
rewrite H.
(* apply cl_beta_K_1_inv .*)
Admitted.
Fact cl_beta_S_0_inv v : S -b-> v -> False.
Proof.
intros H.
apply cl_beta_inv in H.
destruct H as [ H
| [ (y & H)
| [ (x & y & z & H & _)
| [ (x & y & z & H & _ & _)
| (x & y & z & H & _ & _) ] ] ] ]; try discriminate H.
Qed.
Fact cl_beta_S_1_inv a v : S o a -b-> v -> exists b, v = S o b /\ a -b-> b.
Proof.
intros H.
apply cl_beta_inv in H.
destruct H as [ H
| [ (y & H)
| [ (x & y & z & H & _)
| [ (x & y & z & H & _ & _)
| (x & y & z & H & _ & _) ] ] ] ]; try discriminate H.
exists a.
apply cl_app_inj in H. destruct H as [H0 H1].
Admitted.
Fact cl_beta_S_2_inv a b v : S o a o b -b-> v
-> exists d, (v = S o d o b /\ a -b-> d)
\/ (v = S o a o d /\ b -b-> d).
Proof.
intro H.
apply cl_beta_inv in H.
destruct H as [ H
| [ (y & H)
| [ (x & y & z & H & _)
| [ (x & y & z & H & _ & _)
| (x & y & z & H & _ & _) ] ] ] ]; try discriminate H.
apply cl_app_inj in H. destruct H as [H0 H1].
exists b. left.
Admitted.
Fact cl_beta_I_0_inv v : I -b-> v -> False.
Proof.
intros H.
apply cl_beta_inv in H.
destruct H as [ H
| [ (y & H)
| [ (x & y & z & H & _)
| [ (x & y & z & H & _ & _)
| (x & y & z & H & _ & _) ] ] ] ]; try discriminate H.
Qed.
End inversion_lemmas.