--- /dev/null
+(*#* #stop file *)
+
+Require tlt_defs.
+Require lift_gen.
+Require lift_tlt.
+Require subst0_gen.
+Require subst0_confluence.
+Require pr0_defs.
+Require pr0_lift.
+Require pr0_gen.
+Require pr0_subst0.
+
+ Section pr0_confluence. (*************************************************)
+
+ Tactic Definition SSubstInv :=
+ Match Context With
+ | [ H0: (TTail ? ? ?) = (TTail ? ? ?) |- ? ] ->
+ Inversion H0; Clear H0
+ | [ H0: (pr0 (TTail (Bind ?) ? ?) ?) |- ? ] ->
+ Inversion H0; Clear H0
+ | _ -> EqFalse Orelse LiftGen Orelse Pr0Gen.
+
+ Tactic Definition SSubstBack :=
+ Match Context With
+ | [ H0: Abst = ?1; H1:? |- ? ] ->
+ Rewrite <- H0 in H1 Orelse Rewrite <- H0 Orelse Clear H0 ?1
+ | [ H0: Abbr = ?1; H1:? |- ? ] ->
+ Rewrite <- H0 in H1 Orelse Rewrite <- H0 Orelse Clear H0 ?1
+ | [ H0: (? ?) = ?1; H1:? |- ? ] ->
+ Rewrite <- H0 in H1 Orelse Rewrite <- H0 Orelse Clear H0 ?1
+ | [ H0: (? ? ? ?) = ?1; H1:? |- ? ] ->
+ Rewrite <- H0 in H1 Orelse Rewrite <- H0 Orelse Clear H0 ?1.
+
+ Tactic Definition SSubst :=
+ Match Context With
+ [ H0: ?1 = ?; H1:? |- ? ] ->
+ Rewrite H0 in H1 Orelse Rewrite H0 Orelse Clear H0 ?1.
+
+ Tactic Definition XSubst :=
+ Repeat (SSubstInv Orelse SSubstBack Orelse SSubst).
+
+ Tactic Definition IH :=
+ Match Context With
+ | [ H0: (pr0 ?1 ?2); H1: (pr0 ?1 ?3) |- ? ] ->
+ LApply (H ?1); [ Intros H_x | XEAuto ];
+ LApply (H_x ?2); [ Clear H_x; Intros H_x | XAuto ];
+ LApply (H_x ?3); [ Clear H_x; Intros H_x | XAuto ];
+ XElim H_x; Clear H0 H1; Intros.
+
+(* case pr0_cong pr0_gamma pr0_refl *****************************************)
+
+ Remark pr0_cong_gamma_refl: (b:?) ~ b = Abst ->
+ (u0,u3:?) (pr0 u0 u3) ->
+ (t4,t5:?) (pr0 t4 t5) ->
+ (u2,v2,x:?) (pr0 u2 x) -> (pr0 v2 x) ->
+ (EX t:T | (pr0 (TTail (Flat Appl) u2 (TTail (Bind b) u0 t4)) t) &
+ (pr0 (TTail (Bind b) u3 (TTail (Flat Appl) (lift (1) (0) v2) t5)) t)).
+ Intros.
+ Apply ex2_intro with x:=(TTail (Bind b) u3 (TTail (Flat Appl) (lift (1) (0) x) t5)); XAuto.
+ Qed.
+
+(* case pr0_cong pr0_gamma pr0_cong *****************************************)
+
+ Remark pr0_cong_gamma_cong: (b:?) ~ b = Abst ->
+ (u2,v2,x:?) (pr0 u2 x) -> (pr0 v2 x) ->
+ (t2,t5,x0:?) (pr0 t2 x0) -> (pr0 t5 x0) ->
+ (u5,u3,x1:?) (pr0 u5 x1) -> (pr0 u3 x1) ->
+ (EX t:T | (pr0 (TTail (Flat Appl) u2 (TTail (Bind b) u5 t2)) t) &
+ (pr0 (TTail (Bind b) u3 (TTail (Flat Appl) (lift (1) (0) v2) t5)) t)).
+ Intros.
+ Apply ex2_intro with x:=(TTail (Bind b) x1 (TTail (Flat Appl) (lift (1) (0) x) x0)); XAuto.
+ Qed.
+
+(* case pr0_cong pr0_gamma pr0_delta *****************************************)
+
+ Remark pr0_cong_gamma_delta: ~ Abbr = Abst ->
+ (u5,t2,w:?) (subst0 (0) u5 t2 w) ->
+ (u2,v2,x:?) (pr0 u2 x) -> (pr0 v2 x) ->
+ (t5,x0:?) (pr0 t2 x0) -> (pr0 t5 x0) ->
+ (u3,x1:?) (pr0 u5 x1) -> (pr0 u3 x1) ->
+ (EX t:T | (pr0 (TTail (Flat Appl) u2 (TTail (Bind Abbr) u5 w)) t) &
+ (pr0 (TTail (Bind Abbr) u3 (TTail (Flat Appl) (lift (1) (0) v2) t5)) t)).
+ Intros; Pr0Subst0.
+(* case 1: x0 is a lift *)
+ Apply ex2_intro with x:=(TTail (Bind Abbr) x1 (TTail (Flat Appl) (lift (1) (0) x) x0)); XAuto.
+(* case 2: x0 is not a lift *)
+ Apply ex2_intro with x:=(TTail (Bind Abbr) x1 (TTail (Flat Appl) (lift (1) (0) x) x2)); XEAuto.
+ Qed.
+
+(* case pr0_cong pr0_gamma pr0_zeta *****************************************)
+
+ Remark pr0_cong_gamma_zeta: (b:?) ~ b = Abst ->
+ (u0,u3:?) (pr0 u0 u3) ->
+ (u2,v2,x0:?) (pr0 u2 x0) -> (pr0 v2 x0) ->
+ (x,t3,x1:?) (pr0 x x1) -> (pr0 t3 x1) ->
+ (EX t:T | (pr0 (TTail (Flat Appl) u2 t3) t) &
+ (pr0 (TTail (Bind b) u3 (TTail (Flat Appl) (lift (1) (0) v2) (lift (1) (0) x))) t)).
+ Intros; LiftTailRwBack; XEAuto.
+ Qed.
+
+(* case pr0_cong pr0_delta **************************************************)
+
+ Remark pr0_cong_delta: (u3,t5,w:?) (subst0 (0) u3 t5 w) ->
+ (u2,x:?) (pr0 u2 x) -> (pr0 u3 x) ->
+ (t3,x0:?) (pr0 t3 x0) -> (pr0 t5 x0) ->
+ (EX t:T | (pr0 (TTail (Bind Abbr) u2 t3) t) &
+ (pr0 (TTail (Bind Abbr) u3 w) t)).
+ Intros; Pr0Subst0; XEAuto.
+ Qed.
+
+(* case pr0_gamma pr0_gamma *************************************************)
+
+ Remark pr0_gamma_gamma: (b:?) ~ b = Abst ->
+ (v1,v2,x0:?) (pr0 v1 x0) -> (pr0 v2 x0) ->
+ (u1,u2,x1:?) (pr0 u1 x1) -> (pr0 u2 x1) ->
+ (t1,t2,x2:?) (pr0 t1 x2) -> (pr0 t2 x2) ->
+ (EX t:T | (pr0 (TTail (Bind b) u1 (TTail (Flat Appl) (lift (1) (0) v1) t1)) t) &
+ (pr0 (TTail (Bind b) u2 (TTail (Flat Appl) (lift (1) (0) v2) t2)) t)).
+ Intros.
+ Apply ex2_intro with x:=(TTail (Bind b) x1 (TTail (Flat Appl) (lift (1) (0) x0) x2)); XAuto.
+ Qed.
+
+(* case pr0_delta pr0_delta *************************************************)
+
+ Remark pr0_delta_delta: (u2,t3,w:?) (subst0 (0) u2 t3 w) ->
+ (u3,t5,w0:?) (subst0 (0) u3 t5 w0) ->
+ (x:?) (pr0 u2 x) -> (pr0 u3 x) ->
+ (x0:?) (pr0 t3 x0) -> (pr0 t5 x0) ->
+ (EX t:T | (pr0 (TTail (Bind Abbr) u2 w) t) &
+ (pr0 (TTail (Bind Abbr) u3 w0) t)).
+ Intros; Pr0Subst0; Pr0Subst0; Try Subst0Confluence; XSubst; XEAuto.
+ Qed.
+
+(* case pr0_delta pr0_epsilon ***********************************************)
+
+ Remark pr0_delta_epsilon: (u2,t3,w:?) (subst0 (0) u2 t3 w) ->
+ (t4:?) (pr0 (lift (1) (0) t4) t3) ->
+ (t2:?) (EX t:T | (pr0 (TTail (Bind Abbr) u2 w) t) & (pr0 t2 t)).
+ Intros; Pr0Gen; XSubst; Subst0Gen.
+ Qed.
+
+(* main *********************************************************************)
+
+ Hints Resolve pr0_cong_gamma_refl pr0_cong_gamma_cong : ltlc.
+ Hints Resolve pr0_cong_gamma_delta pr0_cong_gamma_zeta : ltlc.
+ Hints Resolve pr0_cong_delta : ltlc.
+ Hints Resolve pr0_gamma_gamma : ltlc.
+ Hints Resolve pr0_delta_delta pr0_delta_epsilon : ltlc.
+
+(*#* #start proof *)
+
+(*#* #caption "confluence with itself: Church-Rosser property" *)
+(*#* #cap #cap t0, t1, t2, t *)
+
+ Theorem pr0_confluence : (t0,t1:?) (pr0 t0 t1) -> (t2:?) (pr0 t0 t2) ->
+ (EX t | (pr0 t1 t) & (pr0 t2 t)).
+
+(*#* #stop file *)
+
+ XElimUsing tlt_wf_ind t0; Intros.
+ Inversion H0; Inversion H1; Clear H0 H1;
+ XSubst; Repeat IH; XEAuto.
+ Qed.
+
+ End pr0_confluence.
+
+ Tactic Definition Pr0Confluence :=
+ Match Context With
+ [ H1: (pr0 ?1 ?2); H2: (pr0 ?1 ?3) |-? ] ->
+ LApply (pr0_confluence ?1 ?2); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?3); [ Clear H1 H2; Intros H1 | XAuto ];
+ XElim H1; Intros.
+
+(*#* #start file *)
+
+(*#* #single *)
projects / helm.git / blobdiff