(*#* #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 *)