contribution about \lambda-\delta

[helm.git] / helm / coq-contribs / LAMBDA-TYPES / pc3_subst0.v

diff --git a/helm/coq-contribs/LAMBDA-TYPES/pc3_subst0.v b/helm/coq-contribs/LAMBDA-TYPES/pc3_subst0.v
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+(*#* #stop file *)
+
+Require subst0_subst0.
+Require csubst0_defs.
+Require pr0_subst0.
+Require pc3_defs.
+
+   Section pc3_fsubst0. (****************************************************)
+
+      Theorem pc3_pr2_fsubst0: (c1:?; t1,t:?) (pr2 c1 t1 t) ->
+                               (i:?; u,c2,t2:?) (fsubst0 i u c1 t1 c2 t2) ->
+                               (e:?) (drop i (0) c1 (CTail e (Bind Abbr) u)) ->
+                               (pc3 c2 t2 t).
+      Intros until 1; XElim H.
+(* case 1: pr2_pr0 *)
+      Intros until 2; XElim H0; Intros.
+(* case 1.1: fsubst0_t *)
+      Pr0Subst0; [ XAuto | Apply (pc3_pr3_u c1 x); XEAuto ].
+(* case 1.2: fsubst0_c *)
+      XAuto.
+(* case 1.3: fsubst0_b *)
+      Pr0Subst0; CSubst0Drop; [ XAuto | Apply (pc3_pr3_u c0 x); XEAuto ].
+(* case 2 : pr2_delta *)
+      Intros until 3; XElim H1; Intros.
+(* case 2.1: fsubst0_t. *)
+      Apply (pc3_t t1); [ Apply pc3_s; XEAuto | XEAuto ].
+(* case 2.2: fsubst0_c. *)
+      Apply (lt_le_e i i0); Intros; CSubst0Drop.
+(* case 2.2.1: i < i0, none *)
+      XEAuto.
+(* case 2.2.2: i < i0, csubst0_fst *)
+      Inversion H; Rewrite <- H7 in H4; Rewrite <- H8 in H4; Rewrite <- H8 in H5; Rewrite <- H9 in H5; Clear H H7 H8 H9 c2 t2 x0 x1 x2.
+      Subst0Subst0; Rewrite <- lt_plus_minus_r in H6; [ CSubst0Drop | XAuto ].
+      Apply (pc3_pr3_u c0 x); XEAuto.
+(* case 2.2.3: i < i0, csubst0_snd *)
+      Inversion H; Rewrite <- H7 in H5; Rewrite <- H8 in H4; Rewrite <- H8 in H5; Rewrite <- H9 in H4; Clear H H7 H8 H9 c2 t2 x0 x1 x3.
+      Apply pc3_pr2_r; XEAuto.
+(* case 2.2.4: i < i0, csubst0_both *)
+      Inversion H; Rewrite <- H8 in H6; Rewrite <- H9 in H4; Rewrite <- H9 in H5; Rewrite <- H9 in H6; Rewrite <- H10 in H5; Clear H H8 H9 H10 c2 t2 x0 x1 x3.
+      Subst0Subst0; Rewrite <- lt_plus_minus_r in H7; [ CSubst0Drop | XAuto ].
+      Apply (pc3_pr3_u c0 x); XEAuto.
+(* case 2.2.5: i >= i0 *)
+      XEAuto.
+(* case 2.3: fsubst0_b *)
+      Apply (lt_le_e i i0); Intros; CSubst0Drop.
+(* case 2.3.1 : i < i0, none *)
+      CSubst0Drop; Apply pc3_pr3_u2 with t0 := t1; XEAuto.
+(* case 2.3.2 : i < i0, csubst0_fst *)
+      Inversion H; Rewrite <- H8 in H5; Rewrite <- H9 in H5; Rewrite <- H9 in H6; Rewrite <- H10 in H6; Clear H H8 H9 H10 c2 t2 x0 x1 x2.
+      Subst0Subst0; Rewrite <- lt_plus_minus_r in H7; [ CSubst0Drop | XAuto ].
+      Apply (pc3_pr3_u2 c0 t1); [ Idtac | Apply (pc3_pr3_u c0 x) ]; XEAuto.
+(* case 2.3.3: i < i0, csubst0_snd *)
+      Inversion H; Rewrite <- H8 in H6; Rewrite <- H9 in H5; Rewrite <- H9 in H6; Rewrite <- H10 in H5; Clear H H8 H9 H10 c2 t2 x0 x1 x3.
+      CSubst0Drop; Apply (pc3_pr3_u2 c0 t1); [ Idtac | Apply pc3_pr2_r ]; XEAuto.
+(* case 2.3.4: i < i0, csubst0_both *)
+      Inversion H; Rewrite <- H9 in H7; Rewrite <- H10 in H5; Rewrite <- H10 in H6; Rewrite <- H10 in H7; Rewrite <- H11 in H6; Clear H H9 H10 H11 c2 t2 x0 x1 x3.
+      Subst0Subst0; Rewrite <- lt_plus_minus_r in H8; [ CSubst0Drop | XAuto ].
+      Apply (pc3_pr3_u2 c0 t1); [ Idtac | Apply (pc3_pr3_u c0 x) ]; XEAuto.
+(* case 2.3.5: i >= i0 *)
+      CSubst0Drop; Apply (pc3_pr3_u2 c0 t1); XEAuto.
+      Qed.
+
+      Theorem pc3_pr2_fsubst0_back: (c1:?; t,t1:?) (pr2 c1 t t1) ->
+                                    (i:?; u,c2,t2:?) (fsubst0 i u c1 t1 c2 t2) ->
+                                    (e:?) (drop i (0) c1 (CTail e (Bind Abbr) u)) ->
+                                    (pc3 c2 t t2).
+      Intros until 1; XElim H.
+(* case 1: pr2_pr0 *)
+      Intros until 2; XElim H0; Intros.
+(* case 1.1: fsubst0_t. *)
+      Apply (pc3_pr3_u c1 t1); XEAuto.
+(* case 1.2: fsubst0_c. *)
+      XAuto.
+(* case 1.3: fsubst0_c. *)
+      CSubst0Drop; Apply (pc3_pr3_u c0 t1); XEAuto.
+(* case 2: pr2_delta *)
+      Intros until 3; XElim H1; Intros.
+(* case 2.1: fsubst0_t. *)
+      Apply (pc3_t t1); XEAuto.
+(* case 2.2: fsubst0_c. *)
+      Apply (lt_le_e i i0); Intros; CSubst0Drop.
+(* case 2.2.1: i < i0, none *)
+      XEAuto.
+(* case 2.2.2: i < i0, csubst0_bind *)
+      Inversion H; Rewrite <- H7 in H4; Rewrite <- H8 in H4; Rewrite <- H8 in H5; Rewrite <- H9 in H5; Clear H H7 H8 H9 c2 t2 x0 x1 x2.
+      Subst0Subst0; Rewrite <- lt_plus_minus_r in H6; [ CSubst0Drop | XAuto ].
+      Apply (pc3_pr3_u c0 x); XEAuto.
+(* case 2.2.3: i < i0, csubst0_snd *)
+      Inversion H; Rewrite <- H7 in H5; Rewrite <- H8 in H4; Rewrite <- H8 in H5; Rewrite <- H9 in H4; Clear H H7 H8 H9 c2 t2 x0 x1 x3.
+      Apply pc3_pr2_r; XEAuto.
+(* case 2.2.4: i < i0, csubst0_both *)
+      Inversion H; Rewrite <- H8 in H6; Rewrite <- H9 in H4; Rewrite <- H9 in H5; Rewrite <- H9 in H6; Rewrite <- H10 in H5; Clear H H8 H9 H10 c2 t2 x0 x1 x3.
+      Subst0Subst0; Rewrite <- lt_plus_minus_r in H7; [ CSubst0Drop | XAuto ].
+      Apply (pc3_pr3_u c0 x); XEAuto.
+(* case 2.2.5: i >= i0 *)
+      XEAuto.
+(* case 2.3: fsubst0_b *)
+      Apply (lt_le_e i i0); Intros; CSubst0Drop.
+(* case 2.3.1 : i < i0, none *)
+      CSubst0Drop; Apply pc3_pr3_u with t2 := t1; XEAuto.
+(* case 2.3.2 : i < i0, csubst0_fst *)
+      Inversion H; Rewrite <- H8 in H5; Rewrite <- H9 in H5; Rewrite <- H9 in H6; Rewrite <- H10 in H6; Clear H H8 H9 H10 c2 t2 x0 x1 x2.
+      Subst0Subst0; Rewrite <- lt_plus_minus_r in H7; [ CSubst0Drop | XAuto ].
+      Apply (pc3_pr3_u c0 x); [ Idtac | Apply (pc3_pr3_u2 c0 t1) ]; XEAuto.
+(* case 2.3.3: i < i0, csubst0_snd *)
+      Inversion H; Rewrite <- H8 in H6; Rewrite <- H9 in H5; Rewrite <- H9 in H6; Rewrite <- H10 in H5; Clear H H8 H9 H10 c2 t2 x0 x1 x3.
+      CSubst0Drop; Apply (pc3_pr3_u c0 t1); [ Idtac | Apply pc3_pr2_r ]; XEAuto.
+(* case 2.3.4: i < i0, csubst0_both *)
+      Inversion H; Rewrite <- H9 in H7; Rewrite <- H10 in H5; Rewrite <- H10 in H6; Rewrite <- H10 in H7; Rewrite <- H11 in H6; Clear H H9 H10 H11 c2 t2 x0 x1 x3.
+      Subst0Subst0; Rewrite <- lt_plus_minus_r in H8; [ CSubst0Drop | XAuto ].
+      Apply (pc3_pr3_u c0 x); [ Idtac | Apply (pc3_pr3_u2 c0 t1) ]; XEAuto.
+(* case 2.3.5: i >= i0 *)
+      CSubst0Drop; Apply (pc3_pr3_u c0 t1); XEAuto.
+      Qed.
+
+      Theorem pc3_pc2_fsubst0: (c1:?; t1,t:?) (pc2 c1 t1 t) ->
+                               (i:?; u,c2,t2:?) (fsubst0 i u c1 t1 c2 t2) ->
+                               (e:?) (drop i (0) c1 (CTail e (Bind Abbr) u)) ->
+                               (pc3 c2 t2 t).
+      Intros until 1; XElim H; Intros.
+(* case 1: pc2_r *)
+      EApply pc3_pr2_fsubst0; XEAuto.
+(* case 2: pc2_x *)
+      Apply pc3_s; EApply pc3_pr2_fsubst0_back; XEAuto.
+      Qed.
+
+      Theorem pc3_fsubst0: (c1:?; t1,t:?) (pc3 c1 t1 t) ->
+                           (i:?; u,c2,t2:?) (fsubst0 i u c1 t1 c2 t2) ->
+                           (e:?) (drop i (0) c1 (CTail e (Bind Abbr) u)) ->
+                           (pc3 c2 t2 t).
+      Intros until 1; XElim H.
+(* case 1: pc3_r *)
+      Intros until 1; XElim H; Intros; Try CSubst0Drop; XEAuto.
+(* case 2: pc3_u *)
+      Intros until 4; XElim H2; Intros;
+      (Apply (pc3_t t2); [ EApply pc3_pc2_fsubst0; XEAuto | XEAuto ]).
+      Qed.
+
+   End pc3_fsubst0.
+
+      Hints Resolve pc3_fsubst0.