*)
inductive lifts: rtmap → relation term ≝
| lifts_sort: ∀f,s. lifts f (⋆s) (⋆s)
*)
inductive lifts: rtmap → relation term ≝
| lifts_sort: ∀f,s. lifts f (⋆s) (⋆s)
-| lifts_lref: â\88\80f,i1,i2. @â¦\83i1, fâ¦\84 â\89¡ i2 → lifts f (#i1) (#i2)
+| lifts_lref: â\88\80f,i1,i2. @â¦\83i1, fâ¦\84 â\89\98 i2 → lifts f (#i1) (#i2)
| lifts_gref: ∀f,l. lifts f (§l) (§l)
| lifts_bind: ∀f,p,I,V1,V2,T1,T2.
| lifts_gref: ∀f,l. lifts f (§l) (§l)
| lifts_bind: ∀f,p,I,V1,V2,T1,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 (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
| lifts_flat: ∀f,I,V1,V2,T1,T2.
lifts f V1 V2 → lifts f T1 T2 →
- λR. â\88\80T1,T2. R T1 T2 â\86\92 â\88\80f,U1. â¬\86*[f] T1 â\89¡ U1 →
- â\88\83â\88\83U2. â¬\86*[f] T2 â\89¡ U2 & R U1 U2.
+ λR. â\88\80T1,T2. R T1 T2 â\86\92 â\88\80f,U1. â¬\86*[f] T1 â\89\98 U1 →
+ â\88\83â\88\83U2. â¬\86*[f] T2 â\89\98 U2 & R U1 U2.
- λR. â\88\80U1,U2. R U1 U2 â\86\92 â\88\80f,T1. â¬\86*[f] T1 â\89¡ U1 →
- â\88\83â\88\83T2. â¬\86*[f] T2 â\89¡ U2 & R T1 T2.
+ λR. â\88\80U1,U2. R U1 U2 â\86\92 â\88\80f,T1. â¬\86*[f] T1 â\89\98 U1 →
+ â\88\83â\88\83T2. â¬\86*[f] T2 â\89\98 U2 & R T1 T2.
- λR. â\88\80T1,T2. R T1 T2 â\86\92 â\88\80f,U1. â¬\86*[f] T1 â\89¡ U1 →
- â\88\80U2. â¬\86*[f] T2 â\89¡ U2 → R U1 U2.
+ λR. â\88\80T1,T2. R T1 T2 â\86\92 â\88\80f,U1. â¬\86*[f] T1 â\89\98 U1 →
+ â\88\80U2. â¬\86*[f] T2 â\89\98 U2 → R U1 U2.
- λR. â\88\80U1,U2. R U1 U2 â\86\92 â\88\80f,T1. â¬\86*[f] T1 â\89¡ U1 →
- â\88\80T2. â¬\86*[f] T2 â\89¡ U2 → R T1 T2.
+ λR. â\88\80U1,U2. R U1 U2 â\86\92 â\88\80f,T1. â¬\86*[f] T1 â\89\98 U1 →
+ â\88\80T2. â¬\86*[f] T2 â\89\98 U2 → R T1 T2.
-fact lifts_inv_lref1_aux: â\88\80f,X,Y. â¬\86*[f] X â\89¡ Y → ∀i1. X = #i1 →
- â\88\83â\88\83i2. @â¦\83i1, fâ¦\84 â\89¡ i2 & Y = #i2.
+fact lifts_inv_lref1_aux: â\88\80f,X,Y. â¬\86*[f] X â\89\98 Y → ∀i1. X = #i1 →
+ â\88\83â\88\83i2. @â¦\83i1, fâ¦\84 â\89\98 i2 & Y = #i2.
(* Basic_1: was: lift1_lref *)
(* Basic_2A1: includes: lift_inv_lref1 lift_inv_lref1_lt lift_inv_lref1_ge *)
(* Basic_1: was: lift1_lref *)
(* Basic_2A1: includes: lift_inv_lref1 lift_inv_lref1_lt lift_inv_lref1_ge *)
-lemma lifts_inv_lref1: â\88\80f,Y,i1. â¬\86*[f] #i1 â\89¡ Y →
- â\88\83â\88\83i2. @â¦\83i1, fâ¦\84 â\89¡ i2 & Y = #i2.
+lemma lifts_inv_lref1: â\88\80f,Y,i1. â¬\86*[f] #i1 â\89\98 Y →
+ â\88\83â\88\83i2. @â¦\83i1, fâ¦\84 â\89\98 i2 & Y = #i2.
∀p,I,V1,T1. X = ⓑ{p,I}V1.T1 →
∀p,I,V1,T1. X = ⓑ{p,I}V1.T1 →
-lemma lifts_inv_bind1: â\88\80f,p,I,V1,T1,Y. â¬\86*[f] â\93\91{p,I}V1.T1 â\89¡ Y →
- â\88\83â\88\83V2,T2. â¬\86*[f] V1 â\89¡ V2 & â¬\86*[â\86\91f] T1 â\89¡ T2 &
+lemma lifts_inv_bind1: â\88\80f,p,I,V1,T1,Y. â¬\86*[f] â\93\91{p,I}V1.T1 â\89\98 Y →
+ â\88\83â\88\83V2,T2. â¬\86*[f] V1 â\89\98 V2 & â¬\86*[⫯f] T1 â\89\98 T2 &
-lemma lifts_inv_flat1: â\88\80f:rtmap. â\88\80I,V1,T1,Y. â¬\86*[f] â\93\95{I}V1.T1 â\89¡ Y →
- â\88\83â\88\83V2,T2. â¬\86*[f] V1 â\89¡ V2 & â¬\86*[f] T1 â\89¡ T2 &
+lemma lifts_inv_flat1: â\88\80f:rtmap. â\88\80I,V1,T1,Y. â¬\86*[f] â\93\95{I}V1.T1 â\89\98 Y →
+ â\88\83â\88\83V2,T2. â¬\86*[f] V1 â\89\98 V2 & â¬\86*[f] T1 â\89\98 T2 &
-fact lifts_inv_lref2_aux: â\88\80f,X,Y. â¬\86*[f] X â\89¡ Y → ∀i2. Y = #i2 →
- â\88\83â\88\83i1. @â¦\83i1, fâ¦\84 â\89¡ i2 & X = #i1.
+fact lifts_inv_lref2_aux: â\88\80f,X,Y. â¬\86*[f] X â\89\98 Y → ∀i2. Y = #i2 →
+ â\88\83â\88\83i1. @â¦\83i1, fâ¦\84 â\89\98 i2 & X = #i1.
(* 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 *)
(* 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: â\88\80f,X,i2. â¬\86*[f] X â\89¡ #i2 →
- â\88\83â\88\83i1. @â¦\83i1, fâ¦\84 â\89¡ i2 & X = #i1.
+lemma lifts_inv_lref2: â\88\80f,X,i2. â¬\86*[f] X â\89\98 #i2 →
+ â\88\83â\88\83i1. @â¦\83i1, fâ¦\84 â\89\98 i2 & X = #i1.
∀p,I,V2,T2. Y = ⓑ{p,I}V2.T2 →
∀p,I,V2,T2. Y = ⓑ{p,I}V2.T2 →
-lemma lifts_inv_bind2: â\88\80f,p,I,V2,T2,X. â¬\86*[f] X â\89¡ ⓑ{p,I}V2.T2 →
- â\88\83â\88\83V1,T1. â¬\86*[f] V1 â\89¡ V2 & â¬\86*[â\86\91f] T1 â\89¡ T2 &
+lemma lifts_inv_bind2: â\88\80f,p,I,V2,T2,X. â¬\86*[f] X â\89\98 ⓑ{p,I}V2.T2 →
+ â\88\83â\88\83V1,T1. â¬\86*[f] V1 â\89\98 V2 & â¬\86*[⫯f] T1 â\89\98 T2 &
-lemma lifts_inv_flat2: â\88\80f:rtmap. â\88\80I,V2,T2,X. â¬\86*[f] X â\89¡ ⓕ{I}V2.T2 →
- â\88\83â\88\83V1,T1. â¬\86*[f] V1 â\89¡ V2 & â¬\86*[f] T1 â\89¡ T2 &
+lemma lifts_inv_flat2: â\88\80f:rtmap. â\88\80I,V2,T2,X. â¬\86*[f] X â\89\98 ⓕ{I}V2.T2 →
+ â\88\83â\88\83V1,T1. â¬\86*[f] V1 â\89\98 V2 & â¬\86*[f] T1 â\89\98 T2 &
X = ⓕ{I}V1.T1.
/2 width=3 by lifts_inv_flat2_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
X = ⓕ{I}V1.T1.
/2 width=3 by lifts_inv_flat2_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
qed-.
] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
qed-.
-lemma lifts_inv_pair_xy_x: â\88\80f,I,V,T. â¬\86*[f] â\91¡{I}V.T â\89¡ V → ⊥.
+lemma lifts_inv_pair_xy_x: â\88\80f,I,V,T. â¬\86*[f] â\91¡{I}V.T â\89\98 V → ⊥.
-lemma lifts_inv_pair_xy_y: â\88\80I,T,V,f. â¬\86*[f] â\91¡{I}V.T â\89¡ T → ⊥.
+lemma lifts_inv_pair_xy_y: â\88\80I,T,V,f. â¬\86*[f] â\91¡{I}V.T â\89\98 T → ⊥.
∃∃i1. X = #i1 & i2 = l + i1.
#l #X #i2 #H elim (lifts_inv_lref2 … H) -H
/3 width=3 by at_inv_uni, ex2_intro/
qed-.
∃∃i1. X = #i1 & i2 = l + i1.
#l #X #i2 #H elim (lifts_inv_lref2 … H) -H
/3 width=3 by at_inv_uni, ex2_intro/
qed-.
#l #X #i2 #H elim (lifts_inv_lref2_uni … H) -H
#i1 #H1 #H2 destruct /4 width=2 by injective_plus_r, eq_f, sym_eq/
qed-.
#l #X #i2 #H elim (lifts_inv_lref2_uni … H) -H
#i1 #H1 #H2 destruct /4 width=2 by injective_plus_r, eq_f, sym_eq/
qed-.
(* Basic forward lemmas *****************************************************)
(* Basic_2A1: includes: lift_inv_O2 *)
(* Basic forward lemmas *****************************************************)
(* Basic_2A1: includes: lift_inv_O2 *)
#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 *)
#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: â\88\80f:rtmap. â\88\80I,V1,T1,Y. â¬\86*[f] â\91¡{I}V1.T1 â\89¡ Y →
- â\88\83â\88\83V2,T2. â¬\86*[f] V1 â\89¡ V2 & Y = ②{I}V2.T2.
+lemma lifts_fwd_pair1: â\88\80f:rtmap. â\88\80I,V1,T1,Y. â¬\86*[f] â\91¡{I}V1.T1 â\89\98 Y →
+ â\88\83â\88\83V2,T2. â¬\86*[f] V1 â\89\98 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/
#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/
-lemma lifts_fwd_pair2: â\88\80f:rtmap. â\88\80I,V2,T2,X. â¬\86*[f] X â\89¡ ②{I}V2.T2 →
- â\88\83â\88\83V1,T1. â¬\86*[f] V1 â\89¡ V2 & X = ②{I}V1.T1.
+lemma lifts_fwd_pair2: â\88\80f:rtmap. â\88\80I,V2,T2,X. â¬\86*[f] X â\89\98 ②{I}V2.T2 →
+ â\88\83â\88\83V1,T1. â¬\86*[f] V1 â\89\98 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/
#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/
#T1 #T2 #f1 #H elim H -T1 -T2 -f1
/4 width=5 by lifts_flat, lifts_bind, lifts_lref, at_eq_repl_back, eq_push/
qed-.
#T1 #T2 #f1 #H elim H -T1 -T2 -f1
/4 width=5 by lifts_flat, lifts_bind, lifts_lref, at_eq_repl_back, eq_push/
qed-.
#T1 #T2 @eq_repl_sym /2 width=3 by lifts_eq_repl_back/ (**) (* full auto fails *)
qed-.
(* Basic_1: includes: lift_r *)
(* Basic_2A1: includes: lift_refl *)
#T1 #T2 @eq_repl_sym /2 width=3 by lifts_eq_repl_back/ (**) (* full auto fails *)
qed-.
(* Basic_1: includes: lift_r *)
(* Basic_2A1: includes: lift_refl *)
#T elim T -T *
/4 width=3 by lifts_flat, lifts_bind, lifts_lref, isid_inv_at, isid_push/
qed.
(* Basic_2A1: includes: lift_total *)
#T elim T -T *
/4 width=3 by lifts_flat, lifts_bind, lifts_lref, isid_inv_at, isid_push/
qed.
(* Basic_2A1: includes: lift_total *)
#T1 elim T1 -T1 *
/3 width=2 by lifts_lref, lifts_sort, lifts_gref, ex_intro/
[ #p ] #I #V1 #T1 #IHV1 #IHT1 #f
elim (IHV1 f) -IHV1 #V2 #HV12
#T1 elim T1 -T1 *
/3 width=2 by lifts_lref, lifts_sort, lifts_gref, ex_intro/
[ #p ] #I #V1 #T1 #IHV1 #IHT1 #f
elim (IHV1 f) -IHV1 #V2 #HV12
#l elim l -l /2 width=1 by lifts_lref/
qed.
(* Basic_1: includes: lift_free (right to left) *)
(* Basic_2A1: includes: lift_split *)
#l elim l -l /2 width=1 by lifts_lref/
qed.
(* Basic_1: includes: lift_free (right to left) *)
(* Basic_2A1: includes: lift_split *)
-lemma lifts_split_trans: â\88\80f,T1,T2. â¬\86*[f] T1 â\89¡ T2 →
- â\88\80f1,f2. f2 â\8a\9a f1 â\89¡ f →
- â\88\83â\88\83T. â¬\86*[f1] T1 â\89¡ T & â¬\86*[f2] T â\89¡ T2.
+lemma lifts_split_trans: â\88\80f,T1,T2. â¬\86*[f] T1 â\89\98 T2 →
+ â\88\80f1,f2. f2 â\8a\9a f1 â\89\98 f →
+ â\88\83â\88\83T. â¬\86*[f1] T1 â\89\98 T & â¬\86*[f2] T â\89\98 T2.
#f #T1 #T2 #H elim H -f -T1 -T2
[ /3 width=3 by lifts_sort, ex2_intro/
| #f #i1 #i2 #Hi #f1 #f2 #Ht elim (after_at_fwd … Hi … Ht) -Hi -Ht
/3 width=3 by lifts_lref, ex2_intro/
| /3 width=3 by lifts_gref, ex2_intro/
| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f1 #f2 #Ht
#f #T1 #T2 #H elim H -f -T1 -T2
[ /3 width=3 by lifts_sort, ex2_intro/
| #f #i1 #i2 #Hi #f1 #f2 #Ht elim (after_at_fwd … Hi … Ht) -Hi -Ht
/3 width=3 by lifts_lref, ex2_intro/
| /3 width=3 by lifts_gref, ex2_intro/
| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f1 #f2 #Ht
/3 width=5 by lifts_bind, after_O2, ex2_intro/
| #f #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f1 #f2 #Ht
elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht
/3 width=5 by lifts_bind, after_O2, ex2_intro/
| #f #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f1 #f2 #Ht
elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht
-lemma lifts_split_div: â\88\80f1,T1,T2. â¬\86*[f1] T1 â\89¡ T2 →
- â\88\80f2,f. f2 â\8a\9a f1 â\89¡ f →
- â\88\83â\88\83T. â¬\86*[f2] T2 â\89¡ T & â¬\86*[f] T1 â\89¡ T.
+lemma lifts_split_div: â\88\80f1,T1,T2. â¬\86*[f1] T1 â\89\98 T2 →
+ â\88\80f2,f. f2 â\8a\9a f1 â\89\98 f →
+ â\88\83â\88\83T. â¬\86*[f2] T2 â\89\98 T & â¬\86*[f] T1 â\89\98 T.
#f1 #T1 #T2 #H elim H -f1 -T1 -T2
[ /3 width=3 by lifts_sort, ex2_intro/
| #f1 #i1 #i2 #Hi #f2 #f #Ht elim (after_at1_fwd … Hi … Ht) -Hi -Ht
/3 width=3 by lifts_lref, ex2_intro/
| /3 width=3 by lifts_gref, ex2_intro/
| #f1 #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #f #Ht
#f1 #T1 #T2 #H elim H -f1 -T1 -T2
[ /3 width=3 by lifts_sort, ex2_intro/
| #f1 #i1 #i2 #Hi #f2 #f #Ht elim (after_at1_fwd … Hi … Ht) -Hi -Ht
/3 width=3 by lifts_lref, ex2_intro/
| /3 width=3 by lifts_gref, ex2_intro/
| #f1 #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #f #Ht
/3 width=5 by lifts_bind, after_O2, ex2_intro/
| #f1 #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #f #Ht
elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht
/3 width=5 by lifts_bind, after_O2, ex2_intro/
| #f1 #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #f #Ht
elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht
#T1 elim T1 -T1
[ * [1,3: /3 width=2 by lifts_sort, lifts_gref, ex_intro, or_introl/ ]
#i2 #f elim (is_at_dec f i2) //
#T1 elim T1 -T1
[ * [1,3: /3 width=2 by lifts_sort, lifts_gref, ex_intro, or_introl/ ]
#i2 #f elim (is_at_dec f i2) //
[ * #T1 #HT12 @or_introl /3 width=2 by lifts_bind, ex_intro/
| -V1 #HT2 @or_intror * #X #H
elim (lifts_inv_bind2 … H) -H /3 width=2 by ex_intro/
[ * #T1 #HT12 @or_introl /3 width=2 by lifts_bind, ex_intro/
| -V1 #HT2 @or_intror * #X #H
elim (lifts_inv_bind2 … H) -H /3 width=2 by ex_intro/
-lemma lifts_uni: â\88\80n1,n2,T,U. â¬\86*[ð\9d\90\94â\9d´n1â\9dµâ\88\98ð\9d\90\94â\9d´n2â\9dµ] T â\89¡ U â\86\92 â¬\86*[n1+n2] T â\89¡ U.
+lemma lifts_uni: â\88\80n1,n2,T,U. â¬\86*[ð\9d\90\94â\9d´n1â\9dµâ\88\98ð\9d\90\94â\9d´n2â\9dµ] T â\89\98 U â\86\92 â¬\86*[n1+n2] T â\89\98 U.
/3 width=4 by lifts_eq_repl_back, after_inv_total/ qed.
(* Basic_2A1: removed theorems 14:
/3 width=4 by lifts_eq_repl_back, after_inv_total/ qed.
(* Basic_2A1: removed theorems 14: