*)
inductive lifts: rtmap → relation term ≝
| lifts_sort: ∀f,s. lifts f (⋆s) (⋆s)
-| lifts_lref: â\88\80f,i1,i2. @â¦\83i1,fâ¦\84 ≘ i2 → lifts f (#i1) (#i2)
+| lifts_lref: â\88\80f,i1,i2. @â\9dªi1,fâ\9d« ≘ i2 → lifts f (#i1) (#i2)
| lifts_gref: ∀f,l. lifts f (§l) (§l)
| lifts_bind: ∀f,p,I,V1,V2,T1,T2.
lifts f V1 V2 → lifts (⫯f) T1 T2 →
- lifts f (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
+ lifts f (ⓑ[p,I]V1.T1) (ⓑ[p,I]V2.T2)
| lifts_flat: ∀f,I,V1,V2,T1,T2.
lifts f V1 V2 → lifts f T1 T2 →
- lifts f (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+ lifts f (ⓕ[I]V1.T1) (ⓕ[I]V2.T2)
.
interpretation "uniform relocation (term)"
/2 width=4 by lifts_inv_sort1_aux/ qed-.
fact lifts_inv_lref1_aux: ∀f,X,Y. ⇧*[f] X ≘ Y → ∀i1. X = #i1 →
- â\88\83â\88\83i2. @â¦\83i1,fâ¦\84 ≘ i2 & Y = #i2.
+ â\88\83â\88\83i2. @â\9dªi1,fâ\9d« ≘ i2 & Y = #i2.
#f #X #Y * -f -X -Y
[ #f #s #x #H destruct
| #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
(* Basic_1: was: lift1_lref *)
(* Basic_2A1: includes: lift_inv_lref1 lift_inv_lref1_lt lift_inv_lref1_ge *)
lemma lifts_inv_lref1: ∀f,Y,i1. ⇧*[f] #i1 ≘ Y →
- â\88\83â\88\83i2. @â¦\83i1,fâ¦\84 ≘ i2 & Y = #i2.
+ â\88\83â\88\83i2. @â\9dªi1,fâ\9d« ≘ i2 & Y = #i2.
/2 width=3 by lifts_inv_lref1_aux/ qed-.
fact lifts_inv_gref1_aux: ∀f,X,Y. ⇧*[f] X ≘ Y → ∀l. X = §l → Y = §l.
/2 width=4 by lifts_inv_gref1_aux/ qed-.
fact lifts_inv_bind1_aux: ∀f,X,Y. ⇧*[f] X ≘ Y →
- ∀p,I,V1,T1. X = ⓑ{p,I}V1.T1 →
+ ∀p,I,V1,T1. X = ⓑ[p,I]V1.T1 →
∃∃V2,T2. ⇧*[f] V1 ≘ V2 & ⇧*[⫯f] T1 ≘ T2 &
- Y = ⓑ{p,I}V2.T2.
+ Y = ⓑ[p,I]V2.T2.
#f #X #Y * -f -X -Y
[ #f #s #q #J #W1 #U1 #H destruct
| #f #i1 #i2 #_ #q #J #W1 #U1 #H destruct
(* Basic_1: was: lift1_bind *)
(* Basic_2A1: includes: lift_inv_bind1 *)
-lemma lifts_inv_bind1: ∀f,p,I,V1,T1,Y. ⇧*[f] ⓑ{p,I}V1.T1 ≘ Y →
+lemma lifts_inv_bind1: ∀f,p,I,V1,T1,Y. ⇧*[f] ⓑ[p,I]V1.T1 ≘ Y →
∃∃V2,T2. ⇧*[f] V1 ≘ V2 & ⇧*[⫯f] T1 ≘ T2 &
- Y = ⓑ{p,I}V2.T2.
+ Y = ⓑ[p,I]V2.T2.
/2 width=3 by lifts_inv_bind1_aux/ qed-.
fact lifts_inv_flat1_aux: ∀f:rtmap. ∀X,Y. ⇧*[f] X ≘ Y →
- ∀I,V1,T1. X = ⓕ{I}V1.T1 →
+ ∀I,V1,T1. X = ⓕ[I]V1.T1 →
∃∃V2,T2. ⇧*[f] V1 ≘ V2 & ⇧*[f] T1 ≘ T2 &
- Y = ⓕ{I}V2.T2.
+ Y = ⓕ[I]V2.T2.
#f #X #Y * -f -X -Y
[ #f #s #J #W1 #U1 #H destruct
| #f #i1 #i2 #_ #J #W1 #U1 #H destruct
(* Basic_1: was: lift1_flat *)
(* Basic_2A1: includes: lift_inv_flat1 *)
-lemma lifts_inv_flat1: ∀f:rtmap. ∀I,V1,T1,Y. ⇧*[f] ⓕ{I}V1.T1 ≘ Y →
+lemma lifts_inv_flat1: ∀f:rtmap. ∀I,V1,T1,Y. ⇧*[f] ⓕ[I]V1.T1 ≘ Y →
∃∃V2,T2. ⇧*[f] V1 ≘ V2 & ⇧*[f] T1 ≘ T2 &
- Y = ⓕ{I}V2.T2.
+ Y = ⓕ[I]V2.T2.
/2 width=3 by lifts_inv_flat1_aux/ qed-.
fact lifts_inv_sort2_aux: ∀f,X,Y. ⇧*[f] X ≘ Y → ∀s. Y = ⋆s → X = ⋆s.
/2 width=4 by lifts_inv_sort2_aux/ qed-.
fact lifts_inv_lref2_aux: ∀f,X,Y. ⇧*[f] X ≘ Y → ∀i2. Y = #i2 →
- â\88\83â\88\83i1. @â¦\83i1,fâ¦\84 ≘ i2 & X = #i1.
+ â\88\83â\88\83i1. @â\9dªi1,fâ\9d« ≘ i2 & X = #i1.
#f #X #Y * -f -X -Y
[ #f #s #x #H destruct
| #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
(* Basic_1: includes: lift_gen_lref lift_gen_lref_lt lift_gen_lref_false lift_gen_lref_ge *)
(* Basic_2A1: includes: lift_inv_lref2 lift_inv_lref2_lt lift_inv_lref2_be lift_inv_lref2_ge lift_inv_lref2_plus *)
lemma lifts_inv_lref2: ∀f,X,i2. ⇧*[f] X ≘ #i2 →
- â\88\83â\88\83i1. @â¦\83i1,fâ¦\84 ≘ i2 & X = #i1.
+ â\88\83â\88\83i1. @â\9dªi1,fâ\9d« ≘ i2 & X = #i1.
/2 width=3 by lifts_inv_lref2_aux/ qed-.
fact lifts_inv_gref2_aux: ∀f,X,Y. ⇧*[f] X ≘ Y → ∀l. Y = §l → X = §l.
/2 width=4 by lifts_inv_gref2_aux/ qed-.
fact lifts_inv_bind2_aux: ∀f,X,Y. ⇧*[f] X ≘ Y →
- ∀p,I,V2,T2. Y = ⓑ{p,I}V2.T2 →
+ ∀p,I,V2,T2. Y = ⓑ[p,I]V2.T2 →
∃∃V1,T1. ⇧*[f] V1 ≘ V2 & ⇧*[⫯f] T1 ≘ T2 &
- X = ⓑ{p,I}V1.T1.
+ X = ⓑ[p,I]V1.T1.
#f #X #Y * -f -X -Y
[ #f #s #q #J #W2 #U2 #H destruct
| #f #i1 #i2 #_ #q #J #W2 #U2 #H destruct
(* Basic_1: includes: lift_gen_bind *)
(* Basic_2A1: includes: lift_inv_bind2 *)
-lemma lifts_inv_bind2: ∀f,p,I,V2,T2,X. ⇧*[f] X ≘ ⓑ{p,I}V2.T2 →
+lemma lifts_inv_bind2: ∀f,p,I,V2,T2,X. ⇧*[f] X ≘ ⓑ[p,I]V2.T2 →
∃∃V1,T1. ⇧*[f] V1 ≘ V2 & ⇧*[⫯f] T1 ≘ T2 &
- X = ⓑ{p,I}V1.T1.
+ X = ⓑ[p,I]V1.T1.
/2 width=3 by lifts_inv_bind2_aux/ qed-.
fact lifts_inv_flat2_aux: ∀f:rtmap. ∀X,Y. ⇧*[f] X ≘ Y →
- ∀I,V2,T2. Y = ⓕ{I}V2.T2 →
+ ∀I,V2,T2. Y = ⓕ[I]V2.T2 →
∃∃V1,T1. ⇧*[f] V1 ≘ V2 & ⇧*[f] T1 ≘ T2 &
- X = ⓕ{I}V1.T1.
+ X = ⓕ[I]V1.T1.
#f #X #Y * -f -X -Y
[ #f #s #J #W2 #U2 #H destruct
| #f #i1 #i2 #_ #J #W2 #U2 #H destruct
(* Basic_1: includes: lift_gen_flat *)
(* Basic_2A1: includes: lift_inv_flat2 *)
-lemma lifts_inv_flat2: ∀f:rtmap. ∀I,V2,T2,X. ⇧*[f] X ≘ ⓕ{I}V2.T2 →
+lemma lifts_inv_flat2: ∀f:rtmap. ∀I,V2,T2,X. ⇧*[f] X ≘ ⓕ[I]V2.T2 →
∃∃V1,T1. ⇧*[f] V1 ≘ V2 & ⇧*[f] T1 ≘ T2 &
- X = ⓕ{I}V1.T1.
+ X = ⓕ[I]V1.T1.
/2 width=3 by lifts_inv_flat2_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lifts_inv_atom1: ∀f,I,Y. ⇧*[f] ⓪{I} ≘ Y →
+lemma lifts_inv_atom1: ∀f,I,Y. ⇧*[f] ⓪[I] ≘ Y →
∨∨ ∃∃s. I = Sort s & Y = ⋆s
- | â\88\83â\88\83i,j. @â¦\83i,fâ¦\84 ≘ j & I = LRef i & Y = #j
+ | â\88\83â\88\83i,j. @â\9dªi,fâ\9d« ≘ j & I = LRef i & Y = #j
| ∃∃l. I = GRef l & Y = §l.
#f * #n #Y #H
[ lapply (lifts_inv_sort1 … H)
] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
qed-.
-lemma lifts_inv_atom2: ∀f,I,X. ⇧*[f] X ≘ ⓪{I} →
+lemma lifts_inv_atom2: ∀f,I,X. ⇧*[f] X ≘ ⓪[I] →
∨∨ ∃∃s. X = ⋆s & I = Sort s
- | â\88\83â\88\83i,j. @â¦\83i,fâ¦\84 ≘ j & X = #i & I = LRef j
+ | â\88\83â\88\83i,j. @â\9dªi,fâ\9d« ≘ j & X = #i & I = LRef j
| ∃∃l. X = §l & I = GRef l.
#f * #n #X #H
[ lapply (lifts_inv_sort2 … H)
qed-.
(* Basic_2A1: includes: lift_inv_pair_xy_x *)
-lemma lifts_inv_pair_xy_x: ∀f,I,V,T. ⇧*[f] ②{I}V.T ≘ V → ⊥.
+lemma lifts_inv_pair_xy_x: ∀f,I,V,T. ⇧*[f] ②[I]V.T ≘ V → ⊥.
#f #J #V elim V -V
[ * #i #U #H
[ lapply (lifts_inv_sort2 … H) -H #H destruct
(* Basic_1: includes: thead_x_lift_y_y *)
(* Basic_2A1: includes: lift_inv_pair_xy_y *)
-lemma lifts_inv_pair_xy_y: ∀I,T,V,f. ⇧*[f] ②{I}V.T ≘ T → ⊥.
+lemma lifts_inv_pair_xy_y: ∀I,T,V,f. ⇧*[f] ②[I]V.T ≘ T → ⊥.
#J #T elim T -T
[ * #i #W #f #H
[ lapply (lifts_inv_sort2 … H) -H #H destruct
(* Basic forward lemmas *****************************************************)
(* Basic_2A1: includes: lift_inv_O2 *)
-lemma lifts_fwd_isid: â\88\80f,T1,T2. â\87§*[f] T1 â\89\98 T2 â\86\92 ð\9d\90\88â¦\83fâ¦\84 → T1 = T2.
+lemma lifts_fwd_isid: â\88\80f,T1,T2. â\87§*[f] T1 â\89\98 T2 â\86\92 ð\9d\90\88â\9dªfâ\9d« → T1 = T2.
#f #T1 #T2 #H elim H -f -T1 -T2
/4 width=3 by isid_inv_at_mono, isid_push, eq_f2, eq_f/
qed-.
(* Basic_2A1: includes: lift_fwd_pair1 *)
-lemma lifts_fwd_pair1: ∀f:rtmap. ∀I,V1,T1,Y. ⇧*[f] ②{I}V1.T1 ≘ Y →
- ∃∃V2,T2. ⇧*[f] V1 ≘ V2 & Y = ②{I}V2.T2.
+lemma lifts_fwd_pair1: ∀f:rtmap. ∀I,V1,T1,Y. ⇧*[f] ②[I]V1.T1 ≘ Y →
+ ∃∃V2,T2. ⇧*[f] V1 ≘ V2 & Y = ②[I]V2.T2.
#f * [ #p ] #I #V1 #T1 #Y #H
[ elim (lifts_inv_bind1 … H) -H /2 width=4 by ex2_2_intro/
| elim (lifts_inv_flat1 … H) -H /2 width=4 by ex2_2_intro/
qed-.
(* Basic_2A1: includes: lift_fwd_pair2 *)
-lemma lifts_fwd_pair2: ∀f:rtmap. ∀I,V2,T2,X. ⇧*[f] X ≘ ②{I}V2.T2 →
- ∃∃V1,T1. ⇧*[f] V1 ≘ V2 & X = ②{I}V1.T1.
+lemma lifts_fwd_pair2: ∀f:rtmap. ∀I,V2,T2,X. ⇧*[f] X ≘ ②[I]V2.T2 →
+ ∃∃V1,T1. ⇧*[f] V1 ≘ V2 & X = ②[I]V1.T1.
#f * [ #p ] #I #V2 #T2 #X #H
[ elim (lifts_inv_bind2 … H) -H /2 width=4 by ex2_2_intro/
| elim (lifts_inv_flat2 … H) -H /2 width=4 by ex2_2_intro/
(* Basic_1: includes: lift_r *)
(* Basic_2A1: includes: lift_refl *)
-lemma lifts_refl: â\88\80T,f. ð\9d\90\88â¦\83fâ¦\84 → ⇧*[f] T ≘ T.
+lemma lifts_refl: â\88\80T,f. ð\9d\90\88â\9dªfâ\9d« → ⇧*[f] T ≘ T.
#T elim T -T *
/4 width=3 by lifts_flat, lifts_bind, lifts_lref, isid_inv_at, isid_push/
qed.
(* Properties with uniform relocation ***************************************)
-lemma lifts_uni: â\88\80n1,n2,T,U. â\87§*[ð\9d\90\94â\9d´n1â\9dµâ\88\98ð\9d\90\94â\9d´n2â\9dµ] T ≘ U → ⇧*[n1+n2] T ≘ U.
+lemma lifts_uni: â\88\80n1,n2,T,U. â\87§*[ð\9d\90\94â\9d¨n1â\9d©â\88\98ð\9d\90\94â\9d¨n2â\9d©] T ≘ U → ⇧*[n1+n2] T ≘ U.
/3 width=4 by lifts_eq_repl_back, after_inv_total/ qed.
(* Basic_2A1: removed theorems 14:
projects / helm.git / blobdiff