| 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 RN RP (↑f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
+ sex RN RP (↑f) (L1.ⓘ[I1]) (L2.ⓘ[I2])
| sex_push: ∀f,I1,I2,L1,L2.
sex RN RP f L1 L2 → RP L1 I1 I2 →
- sex RN RP (⫯f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
+ sex RN RP (⫯f) (L1.ⓘ[I1]) (L2.ⓘ[I2])
.
interpretation "generic entrywise extension (local environment)"
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 →
+ ∀L1. L0 ⪤[RN1,RP1,f] L1 → ∀L2. L0 ⪤[RN2,RP2,f] L2 →
∃∃I. R2 L1 I1 I & R1 L2 I2 I.
definition sex_transitive: 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 →
+ ∀I. R1 L1 I1 I → ∀L2. L1 ⪤[RN,RP,f] L2 →
∀I2. R2 L2 I I2 → R3 L1 I1 I2.
(* Basic inversion lemmas ***************************************************)
-fact sex_inv_atom1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → X = ⋆ → Y = ⋆.
+fact sex_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 sex_inv_atom1: ∀RN,RP,f,Y. ⋆ ⪤[RN, RP, f] Y → Y = ⋆.
+lemma sex_inv_atom1: ∀RN,RP,f,Y. ⋆ ⪤[RN,RP,f] Y → Y = ⋆.
/2 width=6 by sex_inv_atom1_aux/ qed-.
-fact sex_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}.
+fact sex_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
qed-.
(* Basic_2A1: includes lpx_sn_inv_pair1 *)
-lemma sex_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}.
+lemma sex_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 sex_inv_next1_aux/ qed-.
-fact sex_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}.
+fact sex_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)
]
qed-.
-lemma sex_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}.
+lemma sex_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 sex_inv_push1_aux/ qed-.
-fact sex_inv_atom2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → Y = ⋆ → X = ⋆.
+fact sex_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 sex_inv_atom2: ∀RN,RP,f,X. X ⪤[RN, RP, f] ⋆ → X = ⋆.
+lemma sex_inv_atom2: ∀RN,RP,f,X. X ⪤[RN,RP,f] ⋆ → X = ⋆.
/2 width=6 by sex_inv_atom2_aux/ qed-.
-fact sex_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}.
+fact sex_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
qed-.
(* Basic_2A1: includes lpx_sn_inv_pair2 *)
-lemma sex_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}.
+lemma sex_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 sex_inv_next2_aux/ qed-.
-fact sex_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}.
+fact sex_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)
]
qed-.
-lemma sex_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}.
+lemma sex_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 sex_inv_push2_aux/ qed-.
(* Basic_2A1: includes lpx_sn_inv_pair *)
lemma sex_inv_next: ∀RN,RP,f,I1,I2,L1,L2.
- L1.ⓘ{I1} ⪤[RN, RP, ↑f] L2.ⓘ{I2} →
- L1 ⪤[RN, RP, f] L2 ∧ RN L1 I1 I2.
+ L1.ⓘ[I1] ⪤[RN,RP,↑f] L2.ⓘ[I2] →
+ L1 ⪤[RN,RP,f] L2 ∧ RN L1 I1 I2.
#RN #RP #f #I1 #I2 #L1 #L2 #H elim (sex_inv_next1 … H) -H
#I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/
qed-.
lemma sex_inv_push: ∀RN,RP,f,I1,I2,L1,L2.
- L1.ⓘ{I1} ⪤[RN, RP, ⫯f] L2.ⓘ{I2} →
- L1 ⪤[RN, RP, f] L2 ∧ RP L1 I1 I2.
+ L1.ⓘ[I1] ⪤[RN,RP,⫯f] L2.ⓘ[I2] →
+ L1 ⪤[RN,RP,f] L2 ∧ RP L1 I1 I2.
#RN #RP #f #I1 #I2 #L1 #L2 #H elim (sex_inv_push1 … H) -H
#I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/
qed-.
-lemma sex_inv_tl: ∀RN,RP,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}.
+lemma sex_inv_tl: ∀RN,RP,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].
#RN #RP #f #I1 #I2 #L2 #L2 elim (pn_split f) *
/2 width=1 by sex_next, sex_push/
qed-.
(* Basic forward lemmas *****************************************************)
-lemma sex_fwd_bind: ∀RN,RP,f,I1,I2,L1,L2.
- L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2} →
- L1 ⪤[RN, RP, ⫱f] L2.
+lemma sex_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 (sex_inv_push … Hf) | elim (sex_inv_next … Hf) ] -Hf //
(* Basic properties *********************************************************)
-lemma sex_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⪤[RN, RP, f] L2).
+lemma sex_eq_repl_back: ∀RN,RP,L1,L2. 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/
]
qed-.
-lemma sex_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⪤[RN, RP, f] L2).
+lemma sex_eq_repl_fwd: ∀RN,RP,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 *)
qed-.
qed-.
lemma sex_pair_repl: ∀RN,RP,f,I1,I2,L1,L2.
- L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2} →
+ L1.ⓘ[I1] ⪤[RN,RP,f] L2.ⓘ[I2] →
∀J1,J2. RN L1 J1 J2 → RP L1 J1 J2 →
- L1.ⓘ{J1} ⪤[RN, RP, f] L2.ⓘ{J2}.
+ L1.ⓘ[J1] ⪤[RN,RP,f] L2.ⓘ[J2].
/3 width=3 by sex_inv_tl, sex_fwd_bind/ qed-.
lemma sex_co: ∀RN1,RP1,RN2,RP2. RN1 ⊆ RN2 → RP1 ⊆ RP2 →
- ∀f,L1,L2. L1 ⪤[RN1, RP1, f] L2 → L1 ⪤[RN2, RP2, f] L2.
+ ∀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 sex_atom, sex_next, sex_push/
qed-.
lemma sex_co_isid: ∀RN1,RP1,RN2,RP2. RP1 ⊆ RP2 →
- ∀f,L1,L2. L1 ⪤[RN1, RP1, f] L2 → 𝐈⦃f⦄ →
- L1 ⪤[RN2, RP2, f] L2.
+ ∀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 //
qed-.
lemma sex_sdj: ∀RN,RP. RP ⊆ RN →
- ∀f1,L1,L2. L1 ⪤[RN, RP, f1] L2 →
- ∀f2. f1 ∥ f2 → L1 ⪤[RP, RN, f2] L2.
+ ∀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: // ]
qed-.
lemma sle_sex_trans: ∀RN,RP. RN ⊆ RP →
- ∀f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
- ∀f1. f1 ⊆ f2 → L1 ⪤[RN, RP, f1] L2.
+ ∀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) * ]
qed-.
lemma sle_sex_conf: ∀RN,RP. RP ⊆ RN →
- ∀f1,L1,L2. L1 ⪤[RN, RP, f1] L2 →
- ∀f2. f1 ⊆ f2 → L1 ⪤[RN, RP, f2] L2.
+ ∀f1,L1,L2. L1 ⪤[RN,RP,f1] L2 →
+ ∀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) * ]
qed-.
lemma sex_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.
+ ∀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 sex_atom, ex2_intro/ ]
#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
qed-.
lemma sex_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.
+ ∀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 sex_atom, ex2_intro/ ]
#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
lemma sex_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).
+ ∀L1,L2,f. Decidable (L1 ⪤[RN,RP,f] L2).
#RN #RP #HRN #HRP #L1 elim L1 -L1 [ * | #L1 #I1 #IH * ]
[ /2 width=1 by sex_atom, or_introl/
| #L2 #I2 #f @or_intror #H