| sex_atom: ∀f. sex RN RP f (⋆) (⋆)
| sex_next: ∀f,I1,I2,L1,L2.
sex RN RP f L1 L2 → RN L1 I1 I2 →
| sex_atom: ∀f. sex RN RP f (⋆) (⋆)
| sex_next: ∀f,I1,I2,L1,L2.
sex RN RP f L1 L2 → RN L1 I1 I2 →
relation3 lenv bind bind → relation3 lenv bind bind →
relation3 lenv bind bind →
relation3 lenv bind bind → relation3 lenv bind bind →
relation3 lenv bind bind → relation3 lenv bind bind →
relation3 lenv bind bind →
relation3 lenv bind bind → relation3 lenv bind bind →
λR1,R2,R3,RN,RP,f,L1,I1.
∀I. R1 L1 I1 I → ∀L2. L1 ⪤[RN,RP,f] L2 →
∀I2. R2 L2 I I2 → R3 L1 I1 I2.
λR1,R2,R3,RN,RP,f,L1,I1.
∀I. R1 L1 I1 I → ∀L2. L1 ⪤[RN,RP,f] L2 →
∀I2. R2 L2 I I2 → R3 L1 I1 I2.
definition R_pw_confluent1_sex:
relation3 lenv bind bind → relation3 lenv bind bind →
relation3 lenv bind bind → relation3 lenv bind bind →
definition R_pw_confluent1_sex:
relation3 lenv bind bind → relation3 lenv bind bind →
relation3 lenv bind bind → relation3 lenv bind bind →
relation3 lenv bind bind → relation3 lenv bind bind →
relation3 lenv bind bind → relation3 lenv bind bind →
relation3 lenv bind bind → relation3 lenv bind bind →
relation3 lenv bind bind → relation3 lenv bind bind →
relation3 lenv bind bind → relation3 lenv bind bind →
relation3 lenv bind bind → relation3 lenv bind bind →
λR1,R2,RN1,RP1,RN2,RP2,f,L0,I0.
∀I1. R1 L0 I0 I1 → ∀I2. R2 L0 I0 I2 →
∀L1. L0 ⪤[RN1,RP1,f] L1 → ∀L2. L0 ⪤[RN2,RP2,f] L2 →
λR1,R2,RN1,RP1,RN2,RP2,f,L0,I0.
∀I1. R1 L0 I0 I1 → ∀I2. R2 L0 I0 I2 →
∀L1. L0 ⪤[RN1,RP1,f] L1 → ∀L2. L0 ⪤[RN2,RP2,f] L2 →
relation3 lenv bind bind → relation3 lenv bind bind →
relation3 lenv bind bind → relation3 lenv bind bind →
relation3 lenv bind bind → relation3 lenv bind bind →
relation3 lenv bind bind → relation3 lenv bind bind →
relation3 lenv bind bind → relation3 lenv bind bind →
relation3 lenv bind bind → relation3 lenv bind bind →
λR1,R2,RN1,RP1,RN2,RP2,f,L0,I0.
∀I1. R1 L0 I0 I1 → ∀I2. R2 L0 I0 I2 →
∀L1. L0 ⪤[RN1,RP1,f] L1 → ∀L2. L0 ⪤[RN2,RP2,f] L2 →
λR1,R2,RN1,RP1,RN2,RP2,f,L0,I0.
∀I1. R1 L0 I0 I1 → ∀I2. R2 L0 I0 I2 →
∀L1. L0 ⪤[RN1,RP1,f] L1 → ∀L2. L0 ⪤[RN2,RP2,f] L2 →
∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & Y = K2.ⓘ[J2].
#RN #RP #f #X #Y * -f -X -Y
[ #f #g #J1 #K1 #H destruct
∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & Y = K2.ⓘ[J2].
#RN #RP #f #X #Y * -f -X -Y
[ #f #g #J1 #K1 #H destruct
∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & Y = K2.ⓘ[J2].
#RN #RP #f #X #Y * -f -X -Y
[ #f #g #J1 #K1 #H destruct
∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & Y = K2.ⓘ[J2].
#RN #RP #f #X #Y * -f -X -Y
[ #f #g #J1 #K1 #H destruct
-| #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (discr_next_push … H)
-| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(injective_push … H2) -g destruct
+| #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (eq_inv_pr_next_push … H)
+| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(eq_inv_pr_push_bi … H2) -g destruct
∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & X = K1.ⓘ[J1].
#RN #RP #f #X #Y * -f -X -Y
[ #f #g #J2 #K2 #H destruct
∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & X = K1.ⓘ[J1].
#RN #RP #f #X #Y * -f -X -Y
[ #f #g #J2 #K2 #H destruct
∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & X = K1.ⓘ[J1].
#RN #RP #f #X #Y * -f -X -Y
[ #f #J2 #K2 #g #H destruct
∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & X = K1.ⓘ[J1].
#RN #RP #f #X #Y * -f -X -Y
[ #f #J2 #K2 #g #H destruct
-| #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (discr_next_push … H)
-| #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(injective_push … H2) -g destruct
+| #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (eq_inv_pr_next_push … H)
+| #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(eq_inv_pr_push_bi … H2) -g destruct
∀f,I1,I2,L1,L2. L1 ⪤[RN,RP,⫰f] L2 →
RN L1 I1 I2 → RP L1 I1 I2 →
L1.ⓘ[I1] ⪤[RN,RP,f] L2.ⓘ[I2].
∀f,I1,I2,L1,L2. L1 ⪤[RN,RP,⫰f] L2 →
RN L1 I1 I2 → RP L1 I1 I2 →
L1.ⓘ[I1] ⪤[RN,RP,f] L2.ⓘ[I2].
∀f,I1,I2,L1,L2.
L1.ⓘ[I1] ⪤[RN,RP,f] L2.ⓘ[I2] → L1 ⪤[RN,RP,⫰f] L2.
#RN #RP #f #I1 #I2 #L1 #L2 #Hf
∀f,I1,I2,L1,L2.
L1.ⓘ[I1] ⪤[RN,RP,f] L2.ⓘ[I2] → L1 ⪤[RN,RP,⫰f] L2.
#RN #RP #f #I1 #I2 #L1 #L2 #Hf
[ elim (sex_inv_push … Hf) | elim (sex_inv_next … Hf) ] -Hf //
qed-.
(* Basic properties *********************************************************)
lemma sex_eq_repl_back (RN) (RP):
[ elim (sex_inv_push … Hf) | elim (sex_inv_next … Hf) ] -Hf //
qed-.
(* Basic properties *********************************************************)
lemma sex_eq_repl_back (RN) (RP):
- ∀L1,L2. eq_repl_back … (λf. L1 ⪤[RN,RP,f] L2).
+ ∀L1,L2. pr_eq_repl_back … (λf. L1 ⪤[RN,RP,f] L2).
#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
#f1 #I1 #I2 #L1 #L2 #_ #HI #IH #f2 #H
[ elim (eq_inv_nx … H) -H /3 width=3 by sex_next/
#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
#f1 #I1 #I2 #L1 #L2 #_ #HI #IH #f2 #H
[ elim (eq_inv_nx … H) -H /3 width=3 by sex_next/
- ∀L1,L2. eq_repl_fwd … (λf. L1 ⪤[RN,RP,f] L2).
-#RN #RP #L1 #L2 @eq_repl_sym /2 width=3 by sex_eq_repl_back/ (**) (* full auto fails *)
+ ∀L1,L2. pr_eq_repl_fwd … (λf. L1 ⪤[RN,RP,f] L2).
+#RN #RP #L1 #L2 @pr_eq_repl_sym /2 width=3 by sex_eq_repl_back/ (**) (* full auto fails *)
qed-.
lemma sex_refl (RN) (RP):
c_reflexive … RN → c_reflexive … RP →
∀f.reflexive … (sex RN RP f).
#RN #RP #HRN #HRP #f #L generalize in match f; -f elim L -L //
qed-.
lemma sex_refl (RN) (RP):
c_reflexive … RN → c_reflexive … RP →
∀f.reflexive … (sex RN RP f).
#RN #RP #HRN #HRP #f #L generalize in match f; -f elim L -L //
-[ elim (isid_inv_next … H) -H //
-| /4 width=3 by sex_push, isid_inv_push/
+[ elim (pr_isi_inv_next … H) -H //
+| /4 width=3 by sex_push, pr_isi_inv_push/
∀f2. f1 ∥ f2 → L1 ⪤[RP,RN,f2] L2.
#RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 //
#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12
∀f2. f1 ∥ f2 → L1 ⪤[RP,RN,f2] L2.
#RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 //
#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12
∀f1. f1 ⊆ f2 → L1 ⪤[RN,RP,f1] L2.
#RN #RP #HR #f2 #L1 #L2 #H elim H -f2 -L1 -L2 //
#f2 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f1 #H12
∀f1. f1 ⊆ f2 → L1 ⪤[RN,RP,f1] L2.
#RN #RP #HR #f2 #L1 #L2 #H elim H -f2 -L1 -L2 //
#f2 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f1 #H12
-[ elim (pn_split f1) * ]
-[ /4 width=5 by sex_push, sle_inv_pn/
-| /4 width=5 by sex_next, sle_inv_nn/
-| elim (sle_inv_xp … H12) -H12 [2,3: // ]
+[ elim (pr_map_split_tl f1) * ]
+[ /4 width=5 by sex_push, pr_sle_inv_push_next/
+| /4 width=5 by sex_next, pr_sle_inv_next_bi/
+| elim (pr_sle_inv_push_dx … H12) -H12 [2,3: // ]
∀f2. f1 ⊆ f2 → L1 ⪤[RN,RP,f2] L2.
#RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 //
#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12
∀f2. f1 ⊆ f2 → L1 ⪤[RN,RP,f2] L2.
#RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 //
#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12
-[2: elim (pn_split f2) * ]
-[ /4 width=5 by sex_push, sle_inv_pp/
-| /4 width=5 by sex_next, sle_inv_pn/
-| elim (sle_inv_nx … H12) -H12 [2,3: // ]
+[2: elim (pr_map_split_tl f2) * ]
+[ /4 width=5 by sex_push, pr_sle_inv_push_bi/
+| /4 width=5 by sex_next, pr_sle_inv_push_next/
+| elim (pr_sle_inv_next_sn … H12) -H12 [2,3: // ]
#R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
[ /2 width=3 by sex_atom, ex2_intro/ ]
#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
#R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
[ /2 width=3 by sex_atom, ex2_intro/ ]
#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
#R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
[ /2 width=3 by sex_atom, ex2_intro/ ]
#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
#R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
[ /2 width=3 by sex_atom, ex2_intro/ ]
#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
lapply (sex_inv_atom2 … H) -H #H destruct
| #L2 #I2 #f elim (IH L2 (⫰f)) -IH #HL12
[2: /4 width=3 by sex_fwd_bind, or_intror/ ]
lapply (sex_inv_atom2 … H) -H #H destruct
| #L2 #I2 #f elim (IH L2 (⫰f)) -IH #HL12
[2: /4 width=3 by sex_fwd_bind, or_intror/ ]
[ elim (HRP L1 I1 I2) | elim (HRN L1 I1 I2) ] -HRP -HRN #HV12
[1,3: /3 width=1 by sex_push, sex_next, or_introl/ ]
@or_intror #H
[ elim (HRP L1 I1 I2) | elim (HRN L1 I1 I2) ] -HRP -HRN #HV12
[1,3: /3 width=1 by sex_push, sex_next, or_introl/ ]
@or_intror #H