(**************************************************************************)
include "ground_2/relocation/rtmap_sle.ma".
+include "ground_2/relocation/rtmap_sdj.ma".
include "basic_2/notation/relations/relationstar_5.ma".
-include "basic_2/grammar/lenv.ma".
+include "basic_2/syntax/lenv.ma".
-(* GENERAL ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
+(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
-definition relation5 : Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0]
-≝ λA,B,C,D,E.A→B→C→D→E→Prop.
-
-definition relation6 : Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0]
-≝ λA,B,C,D,E,F.A→B→C→D→E→F→Prop.
-
-(* Basic_2A1: includes: lpx_sn_atom lpx_sn_pair *)
-inductive lexs (RN,RP:relation3 lenv term term): rtmap → relation lenv ≝
+inductive lexs (RN,RP:relation3 lenv bind bind): rtmap → relation lenv ≝
| lexs_atom: ∀f. lexs RN RP f (⋆) (⋆)
-| lexs_next: ∀I,L1,L2,V1,V2,f.
- lexs RN RP f L1 L2 → RN L1 V1 V2 →
- lexs RN RP (⫯f) (L1.â\93\91{I}V1) (L2.â\93\91{I}V2)
-| lexs_push: ∀I,L1,L2,V1,V2,f.
- lexs RN RP f L1 L2 → RP L1 V1 V2 →
- lexs RN RP (â\86\91f) (L1.â\93\91{I}V1) (L2.â\93\91{I}V2)
+| lexs_next: ∀f,I1,I2,L1,L2.
+ lexs RN RP f L1 L2 → RN L1 I1 I2 →
+ lexs RN RP (â\86\91f) (L1.â\93\98{I1}) (L2.â\93\98{I2})
+| lexs_push: ∀f,I1,I2,L1,L2.
+ lexs RN RP f L1 L2 → RP L1 I1 I2 →
+ lexs RN RP (⫯f) (L1.â\93\98{I1}) (L2.â\93\98{I2})
.
-interpretation "general entrywise extension (local environment)"
+interpretation "generic entrywise extension (local environment)"
'RelationStar RN RP f L1 L2 = (lexs RN RP f L1 L2).
-definition lpx_sn_confluent: relation6 (relation3 lenv term term)
- (relation3 lenv term term) … ≝
- λR1,R2,RN1,RP1,RN2,RP2.
- ∀f,L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
- ∀L1. L0 ⦻*[RN1, RP1, f] L1 → ∀L2. L0 ⦻*[RN2, RP2, f] L2 →
- ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
-
-definition lexs_transitive: relation4 (relation3 lenv term term)
- (relation3 lenv term term) … ≝
- λR1,R2,RN,RP.
- ∀f,L1,T1,T. R1 L1 T1 T → ∀L2. L1 ⦻*[RN, RP, f] L2 →
- ∀T2. R2 L2 T T2 → R1 L1 T1 T2.
+definition R_pw_confluent2_lexs: 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 rtmap lenv 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 →
+ ∃∃I. R2 L1 I1 I & R1 L2 I2 I.
+
+definition lexs_transitive: relation3 lenv bind bind → relation3 lenv bind bind →
+ relation3 lenv bind bind →
+ relation3 lenv bind bind → relation3 lenv bind bind →
+ relation3 rtmap lenv 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.
(* Basic inversion lemmas ***************************************************)
-fact lexs_inv_atom1_aux: ∀RN,RP,X,Y,f. X ⦻*[RN, RP, f] Y → X = ⋆ → Y = ⋆.
-#RN #RP #X #Y #f * -X -Y -f //
-#I #L1 #L2 #V1 #V2 #f #_ #_ #H destruct
+fact lexs_inv_atom1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → X = ⋆ → Y = ⋆.
+#RN #RP #f #X #Y * -f -X -Y //
+#f #I1 #I2 #L1 #L2 #_ #_ #H destruct
qed-.
(* Basic_2A1: includes lpx_sn_inv_atom1 *)
-lemma lexs_inv_atom1: ∀RN,RP,Y,f. ⋆ ⦻*[RN, RP, f] Y → Y = ⋆.
+lemma lexs_inv_atom1: ∀RN,RP,f,Y. ⋆ ⪤*[RN, RP, f] Y → Y = ⋆.
/2 width=6 by lexs_inv_atom1_aux/ qed-.
-fact lexs_inv_next1_aux: ∀RN,RP,X,Y,f. X ⦻*[RN, RP, f] Y → ∀J,K1,W1,g. X = K1.ⓑ{J}W1 → f = ⫯g →
- ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2.
-#RN #RP #X #Y #f * -X -Y -f
-[ #f #J #K1 #W1 #g #H destruct
-| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K1 #W1 #g #H1 #H2 <(injective_next … H2) -g destruct
+fact lexs_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ↑g →
+ ∃∃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
+| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(injective_next … H2) -g destruct
/2 width=5 by ex3_2_intro/
-| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K1 #W1 #g #_ #H elim (discr_push_next … H)
+| #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (discr_push_next … H)
]
qed-.
(* Basic_2A1: includes lpx_sn_inv_pair1 *)
-lemma lexs_inv_next1: ∀RN,RP,J,K1,Y,W1,g. K1.ⓑ{J}W1 ⦻*[RN, RP, ⫯g] Y →
- ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2.
+lemma lexs_inv_next1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤*[RN, RP, ↑g] Y →
+ ∃∃J2,K2. K1 ⪤*[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}.
/2 width=7 by lexs_inv_next1_aux/ qed-.
-
-fact lexs_inv_push1_aux: ∀RN,RP,X,Y,f. X ⦻*[RN, RP, f] Y → ∀J,K1,W1,g. X = K1.ⓑ{J}W1 → f = ↑g →
- ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2.
-#RN #RP #X #Y #f * -X -Y -f
-[ #f #J #K1 #W1 #g #H destruct
-| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K1 #W1 #g #_ #H elim (discr_next_push … H)
-| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K1 #W1 #g #H1 #H2 <(injective_push … H2) -g destruct
+fact lexs_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ⫯g →
+ ∃∃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
/2 width=5 by ex3_2_intro/
]
qed-.
-lemma lexs_inv_push1: ∀RN,RP,J,K1,Y,W1,g. K1.ⓑ{J}W1 ⦻*[RN, RP, ↑g] Y →
- ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2.
+lemma lexs_inv_push1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤*[RN, RP, ⫯g] Y →
+ ∃∃J2,K2. K1 ⪤*[RN, RP, g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}.
/2 width=7 by lexs_inv_push1_aux/ qed-.
-fact lexs_inv_atom2_aux: ∀RN,RP,X,Y,f. X ⦻*[RN, RP, f] Y → Y = ⋆ → X = ⋆.
-#RN #RP #X #Y #f * -X -Y -f //
-#I #L1 #L2 #V1 #V2 #f #_ #_ #H destruct
+fact lexs_inv_atom2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → Y = ⋆ → X = ⋆.
+#RN #RP #f #X #Y * -f -X -Y //
+#f #I1 #I2 #L1 #L2 #_ #_ #H destruct
qed-.
(* Basic_2A1: includes lpx_sn_inv_atom2 *)
-lemma lexs_inv_atom2: ∀RN,RP,X,f. X ⦻*[RN, RP, f] ⋆ → X = ⋆.
+lemma lexs_inv_atom2: ∀RN,RP,f,X. X ⪤*[RN, RP, f] ⋆ → X = ⋆.
/2 width=6 by lexs_inv_atom2_aux/ qed-.
-fact lexs_inv_next2_aux: ∀RN,RP,X,Y,f. X ⦻*[RN, RP, f] Y → ∀J,K2,W2,g. Y = K2.ⓑ{J}W2 → f = ⫯g →
- ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1.
-#RN #RP #X #Y #f * -X -Y -f
-[ #f #J #K2 #W2 #g #H destruct
-| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K2 #W2 #g #H1 #H2 <(injective_next … H2) -g destruct
+fact lexs_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ↑g →
+ ∃∃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
+| #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(injective_next … H2) -g destruct
/2 width=5 by ex3_2_intro/
-| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K2 #W2 #g #_ #H elim (discr_push_next … H)
+| #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (discr_push_next … H)
]
qed-.
(* Basic_2A1: includes lpx_sn_inv_pair2 *)
-lemma lexs_inv_next2: ∀RN,RP,J,X,K2,W2,g. X ⦻*[RN, RP, ⫯g] K2.ⓑ{J}W2 →
- ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1.
+lemma lexs_inv_next2: ∀RN,RP,g,J2,X,K2. X ⪤*[RN, RP, ↑g] K2.ⓘ{J2} →
+ ∃∃J1,K1. K1 ⪤*[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}.
/2 width=7 by lexs_inv_next2_aux/ qed-.
-fact lexs_inv_push2_aux: ∀RN,RP,X,Y,f. X ⦻*[RN, RP, f] Y → ∀J,K2,W2,g. Y = K2.ⓑ{J}W2 → f = ↑g →
- ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1.
-#RN #RP #X #Y #f * -X -Y -f
-[ #f #J #K2 #W2 #g #H destruct
-| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K2 #W2 #g #_ #H elim (discr_next_push … H)
-| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K2 #W2 #g #H1 #H2 <(injective_push … H2) -g destruct
+fact lexs_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ⫯g →
+ ∃∃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
/2 width=5 by ex3_2_intro/
]
qed-.
-lemma lexs_inv_push2: ∀RN,RP,J,X,K2,W2,g. X ⦻*[RN, RP, ↑g] K2.ⓑ{J}W2 →
- ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1.
+lemma lexs_inv_push2: ∀RN,RP,g,J2,X,K2. X ⪤*[RN, RP, ⫯g] K2.ⓘ{J2} →
+ ∃∃J1,K1. K1 ⪤*[RN, RP, g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}.
/2 width=7 by lexs_inv_push2_aux/ qed-.
(* Basic_2A1: includes lpx_sn_inv_pair *)
-lemma lexs_inv_next: ∀RN,RP,I1,I2,L1,L2,V1,V2,f.
- L1.â\93\91{I1}V1 ⦻*[RN, RP, ⫯f] (L2.â\93\91{I2}V2) →
- ∧∧ L1 ⦻*[RN, RP, f] L2 & RN L1 V1 V2 & I1 = I2.
-#RN #RP #I1 #I2 #L1 #L2 #V1 #V2 #f #H elim (lexs_inv_next1 … H) -H
-#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
+lemma lexs_inv_next: ∀RN,RP,f,I1,I2,L1,L2.
+ L1.â\93\98{I1} ⪤*[RN, RP, â\86\91f] L2.â\93\98{I2} →
+ L1 ⪤*[RN, RP, f] L2 ∧ RN L1 I1 I2.
+#RN #RP #f #I1 #I2 #L1 #L2 #H elim (lexs_inv_next1 … H) -H
+#I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/
qed-.
-lemma lexs_inv_push: ∀RN,RP,I1,I2,L1,L2,V1,V2,f.
- L1.â\93\91{I1}V1 ⦻*[RN, RP, â\86\91f] (L2.â\93\91{I2}V2) →
- ∧∧ L1 ⦻*[RN, RP, f] L2 & RP L1 V1 V2 & I1 = I2.
-#RN #RP #I1 #I2 #L1 #L2 #V1 #V2 #f #H elim (lexs_inv_push1 … H) -H
-#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
+lemma lexs_inv_push: ∀RN,RP,f,I1,I2,L1,L2.
+ L1.â\93\98{I1} ⪤*[RN, RP, ⫯f] L2.â\93\98{I2} →
+ L1 ⪤*[RN, RP, f] L2 ∧ RP L1 I1 I2.
+#RN #RP #f #I1 #I2 #L1 #L2 #H elim (lexs_inv_push1 … H) -H
+#I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/
qed-.
-lemma lexs_inv_tl: ∀RN,RP,I,L1,L2,V1,V2,f. L1 ⦻*[RN, RP, ⫱f] L2 →
- RN L1 V1 V2 → RP L1 V1 V2 →
- L1.â\93\91{I}V1 ⦻*[RN, RP, f] L2.â\93\91{I}V2.
-#RN #RP #I #L2 #L2 #V1 #V2 #f elim (pn_split f) *
+lemma lexs_inv_tl: ∀RN,RP,f,I1,I2,L1,L2. L1 ⪤*[RN, RP, ⫱f] L2 →
+ RN L1 I1 I2 → RP L1 I1 I2 →
+ L1.â\93\98{I1} ⪤*[RN, RP, f] L2.â\93\98{I2}.
+#RN #RP #f #I1 #I2 #L2 #L2 elim (pn_split f) *
/2 width=1 by lexs_next, lexs_push/
qed-.
+(* Basic forward lemmas *****************************************************)
+
+lemma lexs_fwd_bind: ∀RN,RP,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 (pn_split f) * #g #H destruct
+[ elim (lexs_inv_push … Hf) | elim (lexs_inv_next … Hf) ] -Hf //
+qed-.
+
(* Basic properties *********************************************************)
-lemma lexs_eq_repl_back: â\88\80RN,RP,L1,L2. eq_repl_back â\80¦ (λf. L1 ⦻*[RN, RP, f] L2).
-#RN #RP #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
-#I #L1 #L2 #V1 #v2 #f1 #_ #HV #IH #f2 #H
+lemma lexs_eq_repl_back: â\88\80RN,RP,L1,L2. eq_repl_back â\80¦ (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 lexs_next/
| elim (eq_inv_px … H) -H /3 width=3 by lexs_push/
]
qed-.
-lemma lexs_eq_repl_fwd: â\88\80RN,RP,L1,L2. eq_repl_fwd â\80¦ (λf. L1 ⦻*[RN, RP, f] L2).
+lemma lexs_eq_repl_fwd: â\88\80RN,RP,L1,L2. eq_repl_fwd â\80¦ (λf. L1 ⪤*[RN, RP, f] L2).
#RN #RP #L1 #L2 @eq_repl_sym /2 width=3 by lexs_eq_repl_back/ (**) (* full auto fails *)
qed-.
-(* Note: fexs_sym and fexs_trans hold, but lexs_sym and lexs_trans do not *)
-(* Basic_2A1: includes: lpx_sn_refl *)
-lemma lexs_refl: ∀RN,RP,f.
- (∀L. reflexive … (RN L)) →
- (∀L. reflexive … (RP L)) →
- reflexive … (lexs RN RP f).
-#RN #RP #f #HRN #HRP #L generalize in match f; -f elim L -L //
-#L #I #V #IH * * /2 width=1 by lexs_next, lexs_push/
+lemma lexs_refl: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
+ ∀f.reflexive … (lexs RN RP f).
+#RN #RP #HRN #HRP #f #L generalize in match f; -f elim L -L //
+#L #I #IH #f elim (pn_split f) *
+#g #H destruct /2 width=1 by lexs_next, lexs_push/
qed.
-lemma sle_lexs_trans: ∀RN,RP. (∀L,T1,T2. RN L T1 T2 → RP L T1 T2) →
- ∀L1,L2,f2. L1 ⦻*[RN, RP, f2] L2 →
- ∀f1. f1 ⊆ f2 → L1 ⦻*[RN, RP, f1] L2.
-#RN #RP #HR #L1 #L2 #f2 #H elim H -L1 -L2 -f2 //
-#I #L1 #L2 #V1 #V2 #f2 #_ #HV12 #IH
-[ * * [2: #n1 ] ] #f1 #H
-[ /4 width=5 by lexs_next, sle_inv_nn/
-| /4 width=5 by lexs_push, sle_inv_pn/
-| elim (sle_inv_xp … H) -H [2,3: // ]
+lemma lexs_sym: ∀RN,RP.
+ (∀L1,L2,I1,I2. RN L1 I1 I2 → RN L2 I2 I1) →
+ (∀L1,L2,I1,I2. RP L1 I1 I2 → RP L2 I2 I1) →
+ ∀f. symmetric … (lexs RN RP f).
+#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -L1 -L2 -f
+/3 width=2 by lexs_next, lexs_push/
+qed-.
+
+lemma lexs_pair_repl: ∀RN,RP,f,I1,I2,L1,L2.
+ L1.ⓘ{I1} ⪤*[RN, RP, f] L2.ⓘ{I2} →
+ ∀J1,J2. RN L1 J1 J2 → RP L1 J1 J2 →
+ L1.ⓘ{J1} ⪤*[RN, RP, f] L2.ⓘ{J2}.
+/3 width=3 by lexs_inv_tl, lexs_fwd_bind/ qed-.
+
+lemma lexs_co: ∀RN1,RP1,RN2,RP2. RN1 ⊆ RN2 → RP1 ⊆ RP2 →
+ ∀f,L1,L2. L1 ⪤*[RN1, RP1, f] L2 → L1 ⪤*[RN2, RP2, f] L2.
+#RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
+/3 width=1 by lexs_atom, lexs_next, lexs_push/
+qed-.
+
+lemma lexs_co_isid: ∀RN1,RP1,RN2,RP2. RP1 ⊆ RP2 →
+ ∀f,L1,L2. L1 ⪤*[RN1, RP1, f] L2 → 𝐈⦃f⦄ →
+ L1 ⪤*[RN2, RP2, f] L2.
+#RN1 #RP1 #RN2 #RP2 #HR #f #L1 #L2 #H elim H -f -L1 -L2 //
+#f #I1 #I2 #K1 #K2 #_ #HI12 #IH #H
+[ elim (isid_inv_next … H) -H //
+| /4 width=3 by lexs_push, isid_inv_push/
+]
+qed-.
+
+lemma lexs_sdj: ∀RN,RP. RP ⊆ RN →
+ ∀f1,L1,L2. L1 ⪤*[RN, RP, f1] L2 →
+ ∀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
+[ elim (sdj_inv_nx … H12) -H12 [2,3: // ]
+ #g2 #H #H2 destruct /3 width=1 by lexs_push/
+| elim (sdj_inv_px … H12) -H12 [2,4: // ] *
+ #g2 #H #H2 destruct /3 width=1 by lexs_next, lexs_push/
+]
+qed-.
+
+lemma sle_lexs_trans: ∀RN,RP. RN ⊆ RP →
+ ∀f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
+ ∀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 lexs_push, sle_inv_pn/
+| /4 width=5 by lexs_next, sle_inv_nn/
+| elim (sle_inv_xp … H12) -H12 [2,3: // ]
#g1 #H #H1 destruct /3 width=5 by lexs_push/
]
qed-.
-lemma sle_lexs_conf: ∀RN,RP. (∀L,T1,T2. RP L T1 T2 → RN L T1 T2) →
- ∀L1,L2,f1. L1 ⦻*[RN, RP, f1] L2 →
- â\88\80f2. f1 â\8a\86 f2 â\86\92 L1 ⦻*[RN, RP, f2] L2.
-#RN #RP #HR #L1 #L2 #f2 #H elim H -L1 -L2 -f2 //
-#I #L1 #L2 #V1 #V2 #f1 #_ #HV12 #IH
-[2: * * [2: #n2 ] ] #f2 #H
-[ /4 width=5 by lexs_next, sle_inv_pn/
-| /4 width=5 by lexs_push, sle_inv_pp/
-| elim (sle_inv_nx … H) -H [2,3: // ]
+lemma sle_lexs_conf: ∀RN,RP. RP ⊆ RN →
+ ∀f1,L1,L2. L1 ⪤*[RN, RP, f1] L2 →
+ â\88\80f2. f1 â\8a\86 f2 â\86\92 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 lexs_push, sle_inv_pp/
+| /4 width=5 by lexs_next, sle_inv_pn/
+| elim (sle_inv_nx … H12) -H12 [2,3: // ]
#g2 #H #H2 destruct /3 width=5 by lexs_next/
]
qed-.
-lemma lexs_co: ∀RN1,RP1,RN2,RP2.
- (∀L1,T1,T2. RN1 L1 T1 T2 → RN2 L1 T1 T2) →
- (∀L1,T1,T2. RP1 L1 T1 T2 → RP2 L1 T1 T2) →
- ∀L1,L2,f. L1 ⦻*[RN1, RP1, f] L2 → L1 ⦻*[RN2, RP2, f] L2.
-#RN1 #RP1 #RN2 #RP2 #HRN #HRP #L1 #L2 #f #H elim H -L1 -L2 -f
-/3 width=1 by lexs_atom, lexs_next, lexs_push/
+lemma lexs_sle_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 →
+ ∀f,L1,L2. L1 ⪤*[R1, RP, f] L2 → ∀g. f ⊆ g →
+ ∃∃L. L1 ⪤*[R1, RP, g] L & L ⪤*[R2, cfull, f] L2.
+#R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
+[ /2 width=3 by lexs_atom, ex2_intro/ ]
+#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
+[ elim (sle_inv_nx … H ??) -H [ |*: // ] #g #Hfg #H destruct
+ elim (IH … Hfg) -IH -Hfg /3 width=5 by lexs_next, ex2_intro/
+| elim (sle_inv_px … H ??) -H [1,3: * |*: // ] #g #Hfg #H destruct
+ elim (IH … Hfg) -IH -Hfg /3 width=5 by lexs_next, lexs_push, ex2_intro/
+]
+qed-.
+
+lemma lexs_sdj_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 →
+ ∀f,L1,L2. L1 ⪤*[R1, RP, f] L2 → ∀g. f ∥ g →
+ ∃∃L. L1 ⪤*[RP, R1, g] L & L ⪤*[R2, cfull, f] L2.
+#R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
+[ /2 width=3 by lexs_atom, ex2_intro/ ]
+#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
+[ elim (sdj_inv_nx … H ??) -H [ |*: // ] #g #Hfg #H destruct
+ elim (IH … Hfg) -IH -Hfg /3 width=5 by lexs_next, lexs_push, ex2_intro/
+| elim (sdj_inv_px … H ??) -H [1,3: * |*: // ] #g #Hfg #H destruct
+ elim (IH … Hfg) -IH -Hfg /3 width=5 by lexs_next, lexs_push, ex2_intro/
+]
qed-.
-(* Basic_2A1: removed theorems 17:
- llpx_sn_inv_bind llpx_sn_inv_flat
- llpx_sn_fwd_lref llpx_sn_fwd_pair_sn llpx_sn_fwd_length
- llpx_sn_fwd_bind_sn llpx_sn_fwd_bind_dx llpx_sn_fwd_flat_sn llpx_sn_fwd_flat_dx
- llpx_sn_refl llpx_sn_Y llpx_sn_bind_O llpx_sn_ge_up llpx_sn_ge llpx_sn_co
- llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx
-*)
+lemma lexs_dec: ∀RN,RP.
+ (∀L,I1,I2. Decidable (RN L I1 I2)) →
+ (∀L,I1,I2. Decidable (RP L I1 I2)) →
+ ∀L1,L2,f. Decidable (L1 ⪤*[RN, RP, f] L2).
+#RN #RP #HRN #HRP #L1 elim L1 -L1 [ * | #L1 #I1 #IH * ]
+[ /2 width=1 by lexs_atom, or_introl/
+| #L2 #I2 #f @or_intror #H
+ lapply (lexs_inv_atom1 … H) -H #H destruct
+| #f @or_intror #H
+ lapply (lexs_inv_atom2 … H) -H #H destruct
+| #L2 #I2 #f elim (IH L2 (⫱f)) -IH #HL12
+ [2: /4 width=3 by lexs_fwd_bind, or_intror/ ]
+ elim (pn_split f) * #g #H destruct
+ [ elim (HRP L1 I1 I2) | elim (HRN L1 I1 I2) ] -HRP -HRN #HV12
+ [1,3: /3 width=1 by lexs_push, lexs_next, or_introl/ ]
+ @or_intror #H
+ [ elim (lexs_inv_push … H) | elim (lexs_inv_next … H) ] -H
+ /2 width=1 by/
+]
+qed-.
projects / helm.git / blobdiff