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)).
+(* 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_gamma pr0_cong *****************************************)
+(* case pr0_cong pr0_upsilon 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)).
+ 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_gamma pr0_delta *****************************************)
+(* case pr0_cong pr0_upsilon 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)).
+ 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.
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 *****************************************)
+(* case pr0_cong pr0_upsilon 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)).
+ 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.
Intros; Pr0Subst0; XEAuto.
Qed.
-(* case pr0_gamma pr0_gamma *************************************************)
+(* case pr0_upsilon pr0_upsilon *********************************************)
- 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)).
+ 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.
(* 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_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_gamma_gamma : ltlc.
+ Hints Resolve pr0_upsilon_upsilon : ltlc.
Hints Resolve pr0_delta_delta pr0_delta_epsilon : ltlc.
-(*#* #start proof *)
+(*#* #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)).
-
+ 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.
+ XSubst; Repeat IH; XDEAuto 4.
Qed.
End pr0_confluence.
projects / helm.git / blobdiff