'RLiftStar f T1 T2 = (lifts f T1 T2).
definition liftable2_sn: predicate (relation term) ≝
- λR. â\88\80T1,T2. R T1 T2 â\86\92 â\88\80f,U1. â¬\86*[f] T1 ≘ U1 →
- â\88\83â\88\83U2. â¬\86*[f] T2 ≘ U2 & R U1 U2.
+ λR. â\88\80T1,T2. R T1 T2 â\86\92 â\88\80f,U1. â\87§*[f] T1 ≘ U1 →
+ â\88\83â\88\83U2. â\87§*[f] T2 ≘ U2 & R U1 U2.
definition deliftable2_sn: predicate (relation term) ≝
- λR. â\88\80U1,U2. R U1 U2 â\86\92 â\88\80f,T1. â¬\86*[f] T1 ≘ U1 →
- â\88\83â\88\83T2. â¬\86*[f] T2 ≘ U2 & R T1 T2.
+ λR. â\88\80U1,U2. R U1 U2 â\86\92 â\88\80f,T1. â\87§*[f] T1 ≘ U1 →
+ â\88\83â\88\83T2. â\87§*[f] T2 ≘ U2 & R T1 T2.
definition liftable2_bi: predicate (relation term) ≝
- λR. â\88\80T1,T2. R T1 T2 â\86\92 â\88\80f,U1. â¬\86*[f] T1 ≘ U1 →
- â\88\80U2. â¬\86*[f] T2 ≘ U2 → R U1 U2.
+ λR. â\88\80T1,T2. R T1 T2 â\86\92 â\88\80f,U1. â\87§*[f] T1 ≘ U1 →
+ â\88\80U2. â\87§*[f] T2 ≘ U2 → R U1 U2.
definition deliftable2_bi: predicate (relation term) ≝
- λR. â\88\80U1,U2. R U1 U2 â\86\92 â\88\80f,T1. â¬\86*[f] T1 ≘ U1 →
- â\88\80T2. â¬\86*[f] T2 ≘ U2 → R T1 T2.
+ λR. â\88\80U1,U2. R U1 U2 â\86\92 â\88\80f,T1. â\87§*[f] T1 ≘ U1 →
+ â\88\80T2. â\87§*[f] T2 ≘ U2 → R T1 T2.
definition liftable2_dx: predicate (relation term) ≝
- λR. â\88\80T1,T2. R T1 T2 â\86\92 â\88\80f,U2. â¬\86*[f] T2 ≘ U2 →
- â\88\83â\88\83U1. â¬\86*[f] T1 ≘ U1 & R U1 U2.
+ λR. â\88\80T1,T2. R T1 T2 â\86\92 â\88\80f,U2. â\87§*[f] T2 ≘ U2 →
+ â\88\83â\88\83U1. â\87§*[f] T1 ≘ U1 & R U1 U2.
definition deliftable2_dx: predicate (relation term) ≝
- λR. â\88\80U1,U2. R U1 U2 â\86\92 â\88\80f,T2. â¬\86*[f] T2 ≘ U2 →
- â\88\83â\88\83T1. â¬\86*[f] T1 ≘ U1 & R T1 T2.
+ λR. â\88\80U1,U2. R U1 U2 â\86\92 â\88\80f,T2. â\87§*[f] T2 ≘ U2 →
+ â\88\83â\88\83T1. â\87§*[f] T1 ≘ U1 & R T1 T2.
(* Basic inversion lemmas ***************************************************)
-fact lifts_inv_sort1_aux: â\88\80f,X,Y. â¬\86*[f] X ≘ Y → ∀s. X = ⋆s → Y = ⋆s.
+fact lifts_inv_sort1_aux: â\88\80f,X,Y. â\87§*[f] X ≘ Y → ∀s. X = ⋆s → Y = ⋆s.
#f #X #Y * -f -X -Y //
[ #f #i1 #i2 #_ #x #H destruct
| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
(* Basic_1: was: lift1_sort *)
(* Basic_2A1: includes: lift_inv_sort1 *)
-lemma lifts_inv_sort1: â\88\80f,Y,s. â¬\86*[f] ⋆s ≘ Y → Y = ⋆s.
+lemma lifts_inv_sort1: â\88\80f,Y,s. â\87§*[f] ⋆s ≘ Y → Y = ⋆s.
/2 width=4 by lifts_inv_sort1_aux/ qed-.
-fact lifts_inv_lref1_aux: â\88\80f,X,Y. â¬\86*[f] X ≘ Y → ∀i1. X = #i1 →
+fact lifts_inv_lref1_aux: â\88\80f,X,Y. â\87§*[f] X ≘ Y → ∀i1. X = #i1 →
∃∃i2. @⦃i1,f⦄ ≘ i2 & Y = #i2.
#f #X #Y * -f -X -Y
[ #f #s #x #H destruct
(* 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 ≘ Y →
+lemma lifts_inv_lref1: â\88\80f,Y,i1. â\87§*[f] #i1 ≘ Y →
∃∃i2. @⦃i1,f⦄ ≘ i2 & Y = #i2.
/2 width=3 by lifts_inv_lref1_aux/ qed-.
-fact lifts_inv_gref1_aux: â\88\80f,X,Y. â¬\86*[f] X ≘ Y → ∀l. X = §l → Y = §l.
+fact lifts_inv_gref1_aux: â\88\80f,X,Y. â\87§*[f] X ≘ Y → ∀l. X = §l → Y = §l.
#f #X #Y * -f -X -Y //
[ #f #i1 #i2 #_ #x #H destruct
| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
qed-.
(* Basic_2A1: includes: lift_inv_gref1 *)
-lemma lifts_inv_gref1: â\88\80f,Y,l. â¬\86*[f] §l ≘ Y → Y = §l.
+lemma lifts_inv_gref1: â\88\80f,Y,l. â\87§*[f] §l ≘ Y → Y = §l.
/2 width=4 by lifts_inv_gref1_aux/ qed-.
-fact lifts_inv_bind1_aux: â\88\80f,X,Y. â¬\86*[f] X ≘ Y →
+fact lifts_inv_bind1_aux: â\88\80f,X,Y. â\87§*[f] X ≘ Y →
∀p,I,V1,T1. X = ⓑ{p,I}V1.T1 →
- â\88\83â\88\83V2,T2. â¬\86*[f] V1 â\89\98 V2 & â¬\86*[⫯f] T1 ≘ T2 &
+ â\88\83â\88\83V2,T2. â\87§*[f] V1 â\89\98 V2 & â\87§*[⫯f] T1 ≘ T2 &
Y = ⓑ{p,I}V2.T2.
#f #X #Y * -f -X -Y
[ #f #s #q #J #W1 #U1 #H destruct
(* Basic_1: was: lift1_bind *)
(* Basic_2A1: includes: lift_inv_bind1 *)
-lemma lifts_inv_bind1: â\88\80f,p,I,V1,T1,Y. â¬\86*[f] ⓑ{p,I}V1.T1 ≘ Y →
- â\88\83â\88\83V2,T2. â¬\86*[f] V1 â\89\98 V2 & â¬\86*[⫯f] T1 ≘ T2 &
+lemma lifts_inv_bind1: â\88\80f,p,I,V1,T1,Y. â\87§*[f] ⓑ{p,I}V1.T1 ≘ Y →
+ â\88\83â\88\83V2,T2. â\87§*[f] V1 â\89\98 V2 & â\87§*[⫯f] T1 ≘ T2 &
Y = ⓑ{p,I}V2.T2.
/2 width=3 by lifts_inv_bind1_aux/ qed-.
-fact lifts_inv_flat1_aux: â\88\80f:rtmap. â\88\80X,Y. â¬\86*[f] X ≘ Y →
+fact lifts_inv_flat1_aux: â\88\80f:rtmap. â\88\80X,Y. â\87§*[f] X ≘ Y →
∀I,V1,T1. X = ⓕ{I}V1.T1 →
- â\88\83â\88\83V2,T2. â¬\86*[f] V1 â\89\98 V2 & â¬\86*[f] T1 ≘ T2 &
+ â\88\83â\88\83V2,T2. â\87§*[f] V1 â\89\98 V2 & â\87§*[f] T1 ≘ T2 &
Y = ⓕ{I}V2.T2.
#f #X #Y * -f -X -Y
[ #f #s #J #W1 #U1 #H destruct
(* Basic_1: was: lift1_flat *)
(* Basic_2A1: includes: lift_inv_flat1 *)
-lemma lifts_inv_flat1: â\88\80f:rtmap. â\88\80I,V1,T1,Y. â¬\86*[f] ⓕ{I}V1.T1 ≘ Y →
- â\88\83â\88\83V2,T2. â¬\86*[f] V1 â\89\98 V2 & â¬\86*[f] T1 ≘ T2 &
+lemma lifts_inv_flat1: â\88\80f:rtmap. â\88\80I,V1,T1,Y. â\87§*[f] ⓕ{I}V1.T1 ≘ Y →
+ â\88\83â\88\83V2,T2. â\87§*[f] V1 â\89\98 V2 & â\87§*[f] T1 ≘ T2 &
Y = ⓕ{I}V2.T2.
/2 width=3 by lifts_inv_flat1_aux/ qed-.
-fact lifts_inv_sort2_aux: â\88\80f,X,Y. â¬\86*[f] X ≘ Y → ∀s. Y = ⋆s → X = ⋆s.
+fact lifts_inv_sort2_aux: â\88\80f,X,Y. â\87§*[f] X ≘ Y → ∀s. Y = ⋆s → X = ⋆s.
#f #X #Y * -f -X -Y //
[ #f #i1 #i2 #_ #x #H destruct
| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
(* Basic_1: includes: lift_gen_sort *)
(* Basic_2A1: includes: lift_inv_sort2 *)
-lemma lifts_inv_sort2: â\88\80f,X,s. â¬\86*[f] X ≘ ⋆s → X = ⋆s.
+lemma lifts_inv_sort2: â\88\80f,X,s. â\87§*[f] X ≘ ⋆s → X = ⋆s.
/2 width=4 by lifts_inv_sort2_aux/ qed-.
-fact lifts_inv_lref2_aux: â\88\80f,X,Y. â¬\86*[f] X ≘ Y → ∀i2. Y = #i2 →
+fact lifts_inv_lref2_aux: â\88\80f,X,Y. â\87§*[f] X ≘ Y → ∀i2. Y = #i2 →
∃∃i1. @⦃i1,f⦄ ≘ i2 & X = #i1.
#f #X #Y * -f -X -Y
[ #f #s #x #H destruct
(* 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 ≘ #i2 →
+lemma lifts_inv_lref2: â\88\80f,X,i2. â\87§*[f] X ≘ #i2 →
∃∃i1. @⦃i1,f⦄ ≘ i2 & X = #i1.
/2 width=3 by lifts_inv_lref2_aux/ qed-.
-fact lifts_inv_gref2_aux: â\88\80f,X,Y. â¬\86*[f] X ≘ Y → ∀l. Y = §l → X = §l.
+fact lifts_inv_gref2_aux: â\88\80f,X,Y. â\87§*[f] X ≘ Y → ∀l. Y = §l → X = §l.
#f #X #Y * -f -X -Y //
[ #f #i1 #i2 #_ #x #H destruct
| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
qed-.
(* Basic_2A1: includes: lift_inv_gref1 *)
-lemma lifts_inv_gref2: â\88\80f,X,l. â¬\86*[f] X ≘ §l → X = §l.
+lemma lifts_inv_gref2: â\88\80f,X,l. â\87§*[f] X ≘ §l → X = §l.
/2 width=4 by lifts_inv_gref2_aux/ qed-.
-fact lifts_inv_bind2_aux: â\88\80f,X,Y. â¬\86*[f] X ≘ Y →
+fact lifts_inv_bind2_aux: â\88\80f,X,Y. â\87§*[f] X ≘ Y →
∀p,I,V2,T2. Y = ⓑ{p,I}V2.T2 →
- â\88\83â\88\83V1,T1. â¬\86*[f] V1 â\89\98 V2 & â¬\86*[⫯f] T1 ≘ T2 &
+ â\88\83â\88\83V1,T1. â\87§*[f] V1 â\89\98 V2 & â\87§*[⫯f] T1 ≘ T2 &
X = ⓑ{p,I}V1.T1.
#f #X #Y * -f -X -Y
[ #f #s #q #J #W2 #U2 #H destruct
(* Basic_1: includes: lift_gen_bind *)
(* Basic_2A1: includes: lift_inv_bind2 *)
-lemma lifts_inv_bind2: â\88\80f,p,I,V2,T2,X. â¬\86*[f] X ≘ ⓑ{p,I}V2.T2 →
- â\88\83â\88\83V1,T1. â¬\86*[f] V1 â\89\98 V2 & â¬\86*[⫯f] T1 ≘ T2 &
+lemma lifts_inv_bind2: â\88\80f,p,I,V2,T2,X. â\87§*[f] X ≘ ⓑ{p,I}V2.T2 →
+ â\88\83â\88\83V1,T1. â\87§*[f] V1 â\89\98 V2 & â\87§*[⫯f] T1 ≘ T2 &
X = ⓑ{p,I}V1.T1.
/2 width=3 by lifts_inv_bind2_aux/ qed-.
-fact lifts_inv_flat2_aux: â\88\80f:rtmap. â\88\80X,Y. â¬\86*[f] X ≘ Y →
+fact lifts_inv_flat2_aux: â\88\80f:rtmap. â\88\80X,Y. â\87§*[f] X ≘ Y →
∀I,V2,T2. Y = ⓕ{I}V2.T2 →
- â\88\83â\88\83V1,T1. â¬\86*[f] V1 â\89\98 V2 & â¬\86*[f] T1 ≘ T2 &
+ â\88\83â\88\83V1,T1. â\87§*[f] V1 â\89\98 V2 & â\87§*[f] T1 ≘ T2 &
X = ⓕ{I}V1.T1.
#f #X #Y * -f -X -Y
[ #f #s #J #W2 #U2 #H destruct
(* Basic_1: includes: lift_gen_flat *)
(* Basic_2A1: includes: lift_inv_flat2 *)
-lemma lifts_inv_flat2: â\88\80f:rtmap. â\88\80I,V2,T2,X. â¬\86*[f] X ≘ ⓕ{I}V2.T2 →
- â\88\83â\88\83V1,T1. â¬\86*[f] V1 â\89\98 V2 & â¬\86*[f] T1 ≘ T2 &
+lemma lifts_inv_flat2: â\88\80f:rtmap. â\88\80I,V2,T2,X. â\87§*[f] X ≘ ⓕ{I}V2.T2 →
+ â\88\83â\88\83V1,T1. â\87§*[f] V1 â\89\98 V2 & â\87§*[f] T1 ≘ T2 &
X = ⓕ{I}V1.T1.
/2 width=3 by lifts_inv_flat2_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lifts_inv_atom1: â\88\80f,I,Y. â¬\86*[f] ⓪{I} ≘ Y →
+lemma lifts_inv_atom1: â\88\80f,I,Y. â\87§*[f] ⓪{I} ≘ Y →
∨∨ ∃∃s. I = Sort s & Y = ⋆s
| ∃∃i,j. @⦃i,f⦄ ≘ j & I = LRef i & Y = #j
| ∃∃l. I = GRef l & Y = §l.
] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
qed-.
-lemma lifts_inv_atom2: â\88\80f,I,X. â¬\86*[f] X ≘ ⓪{I} →
+lemma lifts_inv_atom2: â\88\80f,I,X. â\87§*[f] X ≘ ⓪{I} →
∨∨ ∃∃s. X = ⋆s & I = Sort s
| ∃∃i,j. @⦃i,f⦄ ≘ j & X = #i & I = LRef j
| ∃∃l. X = §l & I = GRef l.
qed-.
(* Basic_2A1: includes: lift_inv_pair_xy_x *)
-lemma lifts_inv_pair_xy_x: â\88\80f,I,V,T. â¬\86*[f] ②{I}V.T ≘ V → ⊥.
+lemma lifts_inv_pair_xy_x: â\88\80f,I,V,T. â\87§*[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: â\88\80I,T,V,f. â¬\86*[f] ②{I}V.T ≘ T → ⊥.
+lemma lifts_inv_pair_xy_y: â\88\80I,T,V,f. â\87§*[f] ②{I}V.T ≘ T → ⊥.
#J #T elim T -T
[ * #i #W #f #H
[ lapply (lifts_inv_sort2 … H) -H #H destruct
]
qed-.
+lemma lifts_inv_push_zero_sn (f):
+ ∀X. ⇧*[⫯f]#0 ≘ X → #0 = X.
+#f #X #H
+elim (lifts_inv_lref1 … H) -H #i #Hi #H destruct
+lapply (at_inv_ppx … Hi ???) -Hi //
+qed-.
+
+lemma lifts_inv_push_succ_sn (f) (i1):
+ ∀X. ⇧*[⫯f]#(↑i1) ≘ X →
+ ∃∃i2. ⇧*[f]#i1 ≘ #i2 & #(↑i2) = X.
+#f #i1 #X #H
+elim (lifts_inv_lref1 … H) -H #j #Hij #H destruct
+elim (at_inv_npx … Hij) -Hij [|*: // ] #i2 #Hi12 #H destruct
+/3 width=3 by lifts_lref, ex2_intro/
+qed-.
+
(* Inversion lemmas with uniform relocations ********************************)
-lemma lifts_inv_lref1_uni: â\88\80l,Y,i. â¬\86*[l] #i ≘ Y → Y = #(l+i).
+lemma lifts_inv_lref1_uni: â\88\80l,Y,i. â\87§*[l] #i ≘ Y → Y = #(l+i).
#l #Y #i1 #H elim (lifts_inv_lref1 … H) -H /4 width=4 by at_mono, eq_f/
qed-.
-lemma lifts_inv_lref2_uni: â\88\80l,X,i2. â¬\86*[l] X ≘ #i2 →
+lemma lifts_inv_lref2_uni: â\88\80l,X,i2. â\87§*[l] X ≘ #i2 →
∃∃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-.
-lemma lifts_inv_lref2_uni_ge: â\88\80l,X,i. â¬\86*[l] X ≘ #(l + i) → X = #i.
+lemma lifts_inv_lref2_uni_ge: â\88\80l,X,i. â\87§*[l] X ≘ #(l + i) → X = #i.
#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-.
-lemma lifts_inv_lref2_uni_lt: â\88\80l,X,i. â¬\86*[l] X ≘ #i → i < l → ⊥.
+lemma lifts_inv_lref2_uni_lt: â\88\80l,X,i. â\87§*[l] X ≘ #i → i < l → ⊥.
#l #X #i2 #H elim (lifts_inv_lref2_uni … H) -H
#i1 #_ #H1 #H2 destruct /2 width=4 by lt_le_false/
qed-.
(* Basic forward lemmas *****************************************************)
(* Basic_2A1: includes: lift_inv_O2 *)
-lemma lifts_fwd_isid: â\88\80f,T1,T2. â¬\86*[f] T1 ≘ T2 → 𝐈⦃f⦄ → T1 = T2.
+lemma lifts_fwd_isid: â\88\80f,T1,T2. â\87§*[f] T1 ≘ T2 → 𝐈⦃f⦄ → 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: â\88\80f:rtmap. â\88\80I,V1,T1,Y. â¬\86*[f] ②{I}V1.T1 ≘ Y →
- â\88\83â\88\83V2,T2. â¬\86*[f] V1 ≘ V2 & Y = ②{I}V2.T2.
+lemma lifts_fwd_pair1: â\88\80f:rtmap. â\88\80I,V1,T1,Y. â\87§*[f] ②{I}V1.T1 ≘ Y →
+ â\88\83â\88\83V2,T2. â\87§*[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: â\88\80f:rtmap. â\88\80I,V2,T2,X. â¬\86*[f] X ≘ ②{I}V2.T2 →
- â\88\83â\88\83V1,T1. â¬\86*[f] V1 ≘ V2 & X = ②{I}V1.T1.
+lemma lifts_fwd_pair2: â\88\80f:rtmap. â\88\80I,V2,T2,X. â\87§*[f] X ≘ ②{I}V2.T2 →
+ â\88\83â\88\83V1,T1. â\87§*[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/
elim (H1R … U1 … HTU2) -H1R /3 width=3 by ex2_intro/
qed-.
-lemma lifts_eq_repl_back: â\88\80T1,T2. eq_repl_back â\80¦ (λf. â¬\86*[f] T1 ≘ T2).
+lemma lifts_eq_repl_back: â\88\80T1,T2. eq_repl_back â\80¦ (λf. â\87§*[f] T1 ≘ T2).
#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-.
-lemma lifts_eq_repl_fwd: â\88\80T1,T2. eq_repl_fwd â\80¦ (λf. â¬\86*[f] T1 ≘ T2).
+lemma lifts_eq_repl_fwd: â\88\80T1,T2. eq_repl_fwd â\80¦ (λf. â\87§*[f] T1 ≘ T2).
#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 *)
-lemma lifts_refl: â\88\80T,f. ð\9d\90\88â¦\83fâ¦\84 â\86\92 â¬\86*[f] T ≘ T.
+lemma lifts_refl: â\88\80T,f. ð\9d\90\88â¦\83fâ¦\84 â\86\92 â\87§*[f] T ≘ T.
#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 *)
-lemma lifts_total: â\88\80T1,f. â\88\83T2. â¬\86*[f] T1 ≘ T2.
+lemma lifts_total: â\88\80T1,f. â\88\83T2. â\87§*[f] T1 ≘ T2.
#T1 elim T1 -T1 *
/3 width=2 by lifts_lref, lifts_sort, lifts_gref, ex_intro/
[ #p ] #I #V1 #T1 #IHV1 #IHT1 #f
]
qed-.
-lemma lifts_push_zero (f): â¬\86*[⫯f]#0 ≘ #0.
+lemma lifts_push_zero (f): â\87§*[⫯f]#0 ≘ #0.
/2 width=1 by lifts_lref/ qed.
-lemma lifts_push_lref (f) (i1) (i2): â¬\86*[f]#i1 â\89\98 #i2 â\86\92 â¬\86*[⫯f]#(↑i1) ≘ #(↑i2).
+lemma lifts_push_lref (f) (i1) (i2): â\87§*[f]#i1 â\89\98 #i2 â\86\92 â\87§*[⫯f]#(↑i1) ≘ #(↑i2).
#f1 #i1 #i2 #H
elim (lifts_inv_lref1 … H) -H #j #Hij #H destruct
/3 width=7 by lifts_lref, at_push/
qed.
-lemma lifts_lref_uni: â\88\80l,i. â¬\86*[l] #i ≘ #(l+i).
+lemma lifts_lref_uni: â\88\80l,i. â\87§*[l] #i ≘ #(l+i).
#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 ≘ T2 →
+lemma lifts_split_trans: â\88\80f,T1,T2. â\87§*[f] T1 ≘ T2 →
∀f1,f2. f2 ⊚ f1 ≘ f →
- â\88\83â\88\83T. â¬\86*[f1] T1 â\89\98 T & â¬\86*[f2] T ≘ T2.
+ â\88\83â\88\83T. â\87§*[f1] T1 â\89\98 T & â\87§*[f2] T ≘ 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
qed-.
(* Note: apparently, this was missing in Basic_2A1 *)
-lemma lifts_split_div: â\88\80f1,T1,T2. â¬\86*[f1] T1 ≘ T2 →
+lemma lifts_split_div: â\88\80f1,T1,T2. â\87§*[f1] T1 ≘ T2 →
∀f2,f. f2 ⊚ f1 ≘ f →
- â\88\83â\88\83T. â¬\86*[f2] T2 â\89\98 T & â¬\86*[f] T1 ≘ T.
+ â\88\83â\88\83T. â\87§*[f2] T2 â\89\98 T & â\87§*[f] T1 ≘ 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
(* Basic_1: includes: dnf_dec2 dnf_dec *)
(* Basic_2A1: includes: is_lift_dec *)
-lemma is_lifts_dec: â\88\80T2,f. Decidable (â\88\83T1. â¬\86*[f] T1 ≘ T2).
+lemma is_lifts_dec: â\88\80T2,f. Decidable (â\88\83T1. â\87§*[f] T1 ≘ T2).
#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) //
(* Properties with uniform relocation ***************************************)
-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 ≘ U.
+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 â\89\98 U â\86\92 â\87§*[n1+n2] T ≘ U.
/3 width=4 by lifts_eq_repl_back, after_inv_total/ qed.
(* Basic_2A1: removed theorems 14: