+++ /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_upsilon pr0_refl ***************************************)
-
- Remark pr0_cong_upsilon_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_upsilon pr0_cong ***************************************)
-
- Remark pr0_cong_upsilon_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_upsilon pr0_delta **************************************)
-
- Remark pr0_cong_upsilon_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_upsilon pr0_zeta ***************************************)
-
- Remark pr0_cong_upsilon_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_upsilon pr0_upsilon *********************************************)
-
- Remark pr0_upsilon_upsilon: (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_upsilon_refl pr0_cong_upsilon_cong : ltlc.
- Hints Resolve pr0_cong_upsilon_delta pr0_cong_upsilon_zeta : ltlc.
- Hints Resolve pr0_cong_delta : ltlc.
- Hints Resolve pr0_upsilon_upsilon : ltlc.
- Hints Resolve pr0_delta_delta pr0_delta_epsilon : ltlc.
-
-(*#* #start file *)
-
-(*#* #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; XDEAuto 4.
- 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