(* *)
(**************************************************************************)
-include "ground_2/relocation/nstream_after.ma".
+include "ground/arith/nat_le_plus.ma".
+include "ground/relocation/pr_compose.ma".
+include "ground/relocation/pr_nat_uni.ma".
+include "ground/relocation/pr_isi_nat.ma".
+include "ground/relocation/pr_ist_ist.ma".
+include "ground/relocation/pr_after_uni.ma".
+include "ground/relocation/pr_after_nat.ma".
+include "ground/relocation/pr_after_ist.ma".
include "static_2/notation/relations/rliftstar_3.ma".
+include "static_2/notation/relations/rlift_3.ma".
include "static_2/syntax/term.ma".
(* GENERIC RELOCATION FOR TERMS *********************************************)
lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
lifts_nil lifts_cons
*)
-inductive lifts: rtmap → relation term ≝
+inductive lifts: pr_map → relation term ≝
| lifts_sort: ∀f,s. lifts f (⋆s) (⋆s)
-| lifts_lref: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → lifts f (#i1) (#i2)
+| lifts_lref: ∀f,i1,i2. ＠§❨i1,f❩ ≘ 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)"
- 'RLiftStar i T1 T2 = (lifts (uni i) T1 T2).
-
interpretation "generic relocation (term)"
'RLiftStar f T1 T2 = (lifts f T1 T2).
+interpretation "uniform relocation (term)"
+ 'RLift i T1 T2 = (lifts (pr_uni i) T1 T2).
+
definition liftable2_sn: predicate (relation term) ≝
- λR. ∀T1,T2. R T1 T2 → ∀f,U1. ⇧*[f] T1 ≘ U1 →
+ λR. ∀T1,T2. R T1 T2 → ∀f,U1. ⇧*[f] T1 ≘ U1 →
∃∃U2. ⇧*[f] T2 ≘ U2 & R U1 U2.
definition deliftable2_sn: predicate (relation term) ≝
∃∃T2. ⇧*[f] T2 ≘ U2 & R T1 T2.
definition liftable2_bi: predicate (relation term) ≝
- λR. ∀T1,T2. R T1 T2 → ∀f,U1. ⇧*[f] T1 ≘ U1 →
+ λR. ∀T1,T2. R T1 T2 → ∀f,U1. ⇧*[f] T1 ≘ U1 →
∀U2. ⇧*[f] T2 ≘ U2 → R U1 U2.
definition deliftable2_bi: predicate (relation term) ≝
/2 width=4 by lifts_inv_sort1_aux/ qed-.
fact lifts_inv_lref1_aux: ∀f,X,Y. ⇧*[f] X ≘ Y → ∀i1. X = #i1 →
- ∃∃i2. @⦃i1,f⦄ ≘ i2 & Y = #i2.
+ ∃∃i2. ＠§❨i1,f❩ ≘ 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 →
- ∃∃i2. @⦃i1,f⦄ ≘ i2 & Y = #i2.
+ ∃∃i2. ＠§❨i1,f❩ ≘ 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 →
+fact lifts_inv_flat1_aux: ∀f,X,Y. ⇧*[f] X ≘ Y →
+ ∀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,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 →
- ∃∃i1. @⦃i1,f⦄ ≘ i2 & X = #i1.
+ ∃∃i1. ＠§❨i1,f❩ ≘ 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 →
- ∃∃i1. @⦃i1,f⦄ ≘ i2 & X = #i1.
+ ∃∃i1. ＠§❨i1,f❩ ≘ 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 →
+fact lifts_inv_flat2_aux: ∀f,X,Y. ⇧*[f] X ≘ Y →
+ ∀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,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
- | ∃∃i,j. @⦃i,f⦄ ≘ j & I = LRef i & Y = #j
+ | ∃∃i,j. ＠§❨i,f❩ ≘ 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
- | ∃∃i,j. @⦃i,f⦄ ≘ j & X = #i & I = LRef j
+ | ∃∃i,j. ＠§❨i,f❩ ≘ 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
qed-.
lemma lifts_inv_push_zero_sn (f):
- ∀X. ⇧*[⫯f]#0 ≘ X → #0 = X.
+ ∀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 //
+lapply (pr_pat_inv_unit_push … Hi ???) -Hi //
qed-.
lemma lifts_inv_push_succ_sn (f) (i1):
∃∃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
+elim (pr_nat_inv_succ_push … 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: ∀l,Y,i. ⇧*[l] #i ≘ Y → Y = #(l+i).
-#l #Y #i1 #H elim (lifts_inv_lref1 … H) -H /4 width=4 by at_mono, eq_f/
+lemma lifts_inv_lref1_uni: ∀l,Y,i. ⇧[l] #i ≘ Y → Y = #(l+i).
+#l #Y #i1 #H elim (lifts_inv_lref1 … H) -H
+#i2 #H #H2 destruct
+/4 width=4 by pr_nat_mono, eq_f/
qed-.
-lemma lifts_inv_lref2_uni: ∀l,X,i2. ⇧*[l] X ≘ #i2 →
- ∃∃i1. X = #i1 & i2 = l + i1.
+lemma lifts_inv_lref2_uni: ∀l,X,i2. ⇧[l] X ≘ #i2 →
+ ∃∃i1. X = #i1 & i1 + l = i2.
#l #X #i2 #H elim (lifts_inv_lref2 … H) -H
-/3 width=3 by at_inv_uni, ex2_intro/
+/3 width=3 by pr_nat_inv_uni, ex2_intro/
qed-.
-lemma lifts_inv_lref2_uni_ge: ∀l,X,i. ⇧*[l] X ≘ #(l + i) → X = #i.
+lemma lifts_inv_lref2_uni_ge: ∀l,X,i. ⇧[l] X ≘ #(i+l) → 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/
+#i1 #H1 #H2 destruct
+/4 width=2 by eq_inv_nplus_bi_dx, eq_f/
qed-.
-lemma lifts_inv_lref2_uni_lt: ∀l,X,i. ⇧*[l] X ≘ #i → i < l → ⊥.
+lemma lifts_inv_lref2_uni_lt: ∀l,X,i. ⇧[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/
+#i1 #_ #H1 #H2 destruct
+/2 width=4 by nlt_ge_false/
qed-.
(* 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\9d¨fâ\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/
+/4 width=3 by pr_isi_nat_des, pr_isi_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,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,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/
elim (H1R … U1 … HTU2) -H1R /3 width=3 by ex2_intro/
qed-.
-lemma lifts_eq_repl_back: ∀T1,T2. eq_repl_back … (λf. ⇧*[f] T1 ≘ T2).
+lemma lifts_eq_repl_back: ∀T1,T2. pr_eq_repl_back … (λf. ⇧*[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/
+/4 width=5 by lifts_flat, lifts_bind, lifts_lref, pr_pat_eq_repl_back, pr_eq_push/
qed-.
-lemma lifts_eq_repl_fwd: ∀T1,T2. eq_repl_fwd … (λf. ⇧*[f] T1 ≘ T2).
-#T1 #T2 @eq_repl_sym /2 width=3 by lifts_eq_repl_back/ (**) (* full auto fails *)
+lemma lifts_eq_repl_fwd: ∀T1,T2. pr_eq_repl_fwd … (λf. ⇧*[f] T1 ≘ T2).
+#T1 #T2 @pr_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 → ⇧*[f] T ≘ T.
+lemma lifts_refl: â\88\80T,f. ð\9d\90\88â\9d¨fâ\9d© → ⇧*[f] T ≘ T.
#T elim T -T *
-/4 width=3 by lifts_flat, lifts_bind, lifts_lref, isid_inv_at, isid_push/
+/4 width=3 by lifts_flat, lifts_bind, lifts_lref, pr_isi_inv_pat, pr_isi_push/
qed.
(* Basic_2A1: includes: lift_total *)
-lemma lifts_total: ∀T1,f. ∃T2. ⇧*[f] T1 ≘ T2.
+lemma lifts_total: ∀T1,f. 𝐓❨f❩ → ∃T2. ⇧*[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
-elim (IHV1 f) -IHV1 #V2 #HV12
-[ elim (IHT1 (⫯f)) -IHT1 /3 width=2 by lifts_bind, ex_intro/
+/3 width=2 by lifts_sort, lifts_gref, ex_intro/
+[ #i #f #Hf elim (Hf (↑i)) -Hf /3 width=2 by ex_intro, lifts_lref/ ]
+[ #p ] #I #V1 #T1 #IHV1 #IHT1 #f #Hf
+elim (IHV1 f) -IHV1 // #V2 #HV12
+[ elim (IHT1 (⫯f)) -IHT1 /3 width=2 by pr_ist_push, ex_intro, lifts_bind/
| elim (IHT1 f) -IHT1 /3 width=2 by lifts_flat, ex_intro/
]
qed-.
-lemma lifts_push_zero (f): ⇧*[⫯f]#0 ≘ #0.
-/2 width=1 by lifts_lref/ qed.
+lemma lifts_push_zero (f): ⇧*[⫯f]#(𝟎) ≘ #(𝟎).
+/3 width=2 by pr_nat_refl, lifts_lref/ qed.
lemma lifts_push_lref (f) (i1) (i2): ⇧*[f]#i1 ≘ #i2 → ⇧*[⫯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/
+/3 width=7 by lifts_lref, pr_pat_push/
qed.
-lemma lifts_lref_uni: ∀l,i. ⇧*[l] #i ≘ #(l+i).
+lemma lifts_lref_uni: ∀l,i. ⇧[l] #i ≘ #(l+i).
#l elim l -l /2 width=1 by lifts_lref/
qed.
∃∃T. ⇧*[f1] T1 ≘ T & ⇧*[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
+| #f #i1 #i2 #Hi #f1 #f2 #Ht elim (pr_after_nat_des … 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
elim (IHV … Ht) elim (IHT (⫯f1) (⫯f2)) -IHV -IHT
- /3 width=5 by lifts_bind, after_O2, ex2_intro/
+ /3 width=7 by pr_after_refl, ex2_intro, lifts_bind/
| #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_flat, ex2_intro/
(* Note: apparently, this was missing in Basic_2A1 *)
lemma lifts_split_div: ∀f1,T1,T2. ⇧*[f1] T1 ≘ T2 →
- ∀f2,f. f2 ⊚ f1 ≘ f →
+ ∀f2. 𝐓❨f2❩ → ∀f. f2 ⊚ f1 ≘ f →
∃∃T. ⇧*[f2] T2 ≘ T & ⇧*[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
+| #f1 #i1 #i2 #Hi #f2 #Hf2 #f #Ht
+ elim (pr_after_des_ist_nat … 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
- elim (IHV … Ht) elim (IHT (⫯f2) (⫯f)) -IHV -IHT
- /3 width=5 by lifts_bind, after_O2, ex2_intro/
-| #f1 #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #f #Ht
+| #f1 #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #Hf2 #f #Ht
+ elim (IHV … Ht) elim (IHT (⫯f2) … (⫯f)) -IHV -IHT
+ /3 width=7 by pr_ist_push, pr_after_refl, ex2_intro, lifts_bind/
+| #f1 #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #Hf2 #f #Ht
elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht
/3 width=5 by lifts_flat, ex2_intro/
]
(* Basic_1: includes: dnf_dec2 dnf_dec *)
(* Basic_2A1: includes: is_lift_dec *)
-lemma is_lifts_dec: ∀T2,f. Decidable (∃T1. ⇧*[f] T1 ≘ T2).
+lemma is_lifts_dec: ∀T2,f. 𝐓❨f❩ → Decidable (∃T1. ⇧*[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) //
+ #i2 #f #Hf elim (is_pr_nat_dec f i2) //
[ * /4 width=3 by lifts_lref, ex_intro, or_introl/
| #H @or_intror *
#X #HX elim (lifts_inv_lref2 … HX) -HX
/3 width=2 by ex_intro/
]
-| * [ #p ] #I #V2 #T2 #IHV2 #IHT2 #f
- [ elim (IHV2 f) -IHV2
- [ * #V1 #HV12 elim (IHT2 (⫯f)) -IHT2
+| * [ #p ] #I #V2 #T2 #IHV2 #IHT2 #f #Hf
+ [ elim (IHV2 f) -IHV2 //
+ [ * #V1 #HV12 elim (IHT2 (⫯f)) -IHT2 /2 width=1 by pr_ist_push/
[ * #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/
| -IHT2 #HV2 @or_intror * #X #H
elim (lifts_inv_bind2 … H) -H /3 width=2 by ex_intro/
]
- | elim (IHV2 f) -IHV2
- [ * #V1 #HV12 elim (IHT2 f) -IHT2
+ | elim (IHV2 f) -IHV2 //
+ [ * #V1 #HV12 elim (IHT2 f) -IHT2 //
[ * #T1 #HT12 /4 width=2 by lifts_flat, ex_intro, or_introl/
| -V1 #HT2 @or_intror * #X #H
elim (lifts_inv_flat2 … H) -H /3 width=2 by ex_intro/
(* 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 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.
+lemma lifts_uni: â\88\80n1,n2,T,U. â\87§*[ð\9d\90®â\9d¨n2â\9d©â\88\98ð\9d\90®â\9d¨n1â\9d©] T â\89\98 U â\86\92 â\87§[n1+n2] T ≘ U.
+/3 width=4 by lifts_eq_repl_back, pr_after_inv_total/ qed.
(* Basic_2A1: removed theorems 14:
lifts_inv_nil lifts_inv_cons
lift_lref_ge_minus lift_lref_ge_minus_eq
*)
(* Basic_1: removed theorems 8:
- lift_lref_gt
- lift_head lift_gen_head
+ lift_lref_gt
+ lift_head lift_gen_head
lift_weight_map lift_weight lift_weight_add lift_weight_add_O
lift_tlt_dx
*)
projects / helm.git / blobdiff