+
(**************************************************************************)
(* ___ *)
(* ||M|| *)
(* *)
(**************************************************************************)
-include "ground_2/relocation/trace_isid.ma".
+include "ground_2/relocation/nstream_after.ma".
include "basic_2/notation/relations/rliftstar_3.ma".
-include "basic_2/grammar/term.ma".
+include "basic_2/syntax/term.ma".
-(* GENERIC TERM RELOCATION **************************************************)
+(* GENERIC RELOCATION FOR TERMS *********************************************)
(* Basic_1: includes:
lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
lifts_nil lifts_cons
*)
-inductive lifts: trace → relation term ≝
-| lifts_sort: ∀k,t. lifts t (⋆k) (⋆k)
-| lifts_lref: ∀i1,i2,t. @⦃i1, t⦄ ≡ i2 → lifts t (#i1) (#i2)
-| lifts_gref: ∀p,t. lifts t (§p) (§p)
-| lifts_bind: ∀a,I,V1,V2,T1,T2,t.
- lifts t V1 V2 → lifts (Ⓣ@t) T1 T2 →
- lifts t (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
-| lifts_flat: ∀I,V1,V2,T1,T2,t.
- lifts t V1 V2 → lifts t T1 T2 →
- lifts t (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+inductive lifts: rtmap → relation term ≝
+| lifts_sort: ∀f,s. lifts f (⋆s) (⋆s)
+| 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_flat: ∀f,I,V1,V2,T1,T2.
+ lifts f V1 V2 → lifts f T1 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 cs T1 T2 = (lifts cs T1 T2).
+ 'RLiftStar f T1 T2 = (lifts f T1 T2).
+
+definition liftable2_sn: predicate (relation term) ≝
+ λR. ∀T1,T2. R T1 T2 → ∀f,U1. ⬆*[f] T1 ≘ U1 →
+ ∃∃U2. ⬆*[f] T2 ≘ U2 & R U1 U2.
+
+definition deliftable2_sn: predicate (relation term) ≝
+ λR. ∀U1,U2. R U1 U2 → ∀f,T1. ⬆*[f] T1 ≘ U1 →
+ ∃∃T2. ⬆*[f] T2 ≘ U2 & R T1 T2.
+
+definition liftable2_bi: predicate (relation term) ≝
+ λR. ∀T1,T2. R T1 T2 → ∀f,U1. ⬆*[f] T1 ≘ U1 →
+ ∀U2. ⬆*[f] T2 ≘ U2 → R U1 U2.
+
+definition deliftable2_bi: predicate (relation term) ≝
+ λR. ∀U1,U2. R U1 U2 → ∀f,T1. ⬆*[f] T1 ≘ U1 →
+ ∀T2. ⬆*[f] T2 ≘ U2 → R T1 T2.
(* Basic inversion lemmas ***************************************************)
-fact lifts_inv_sort1_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → ∀k. X = ⋆k → Y = ⋆k.
-#X #Y #t * -X -Y -t //
-[ #i1 #i2 #t #_ #x #H destruct
-| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct
-| #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct
+fact lifts_inv_sort1_aux: ∀f,X,Y. ⬆*[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
+| #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
]
qed-.
(* Basic_1: was: lift1_sort *)
(* Basic_2A1: includes: lift_inv_sort1 *)
-lemma lifts_inv_sort1: ∀Y,k,t. ⬆*[t] ⋆k ≡ Y → Y = ⋆k.
+lemma lifts_inv_sort1: ∀f,Y,s. ⬆*[f] ⋆s ≘ Y → Y = ⋆s.
/2 width=4 by lifts_inv_sort1_aux/ qed-.
-fact lifts_inv_lref1_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → ∀i1. X = #i1 →
- ∃∃i2. @⦃i1, t⦄ ≡ i2 & Y = #i2.
-#X #Y #t * -X -Y -t
-[ #k #t #x #H destruct
-| #i1 #i2 #t #Hi12 #x #H destruct /2 width=3 by ex2_intro/
-| #p #t #x #H destruct
-| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct
-| #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct
+fact lifts_inv_lref1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀i1. X = #i1 →
+ ∃∃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/
+| #f #l #x #H destruct
+| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
+| #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
]
qed-.
(* Basic_1: was: lift1_lref *)
(* Basic_2A1: includes: lift_inv_lref1 lift_inv_lref1_lt lift_inv_lref1_ge *)
-lemma lifts_inv_lref1: ∀Y,i1,t. ⬆*[t] #i1 ≡ Y →
- ∃∃i2. @⦃i1, t⦄ ≡ i2 & Y = #i2.
+lemma lifts_inv_lref1: ∀f,Y,i1. ⬆*[f] #i1 ≘ Y →
+ ∃∃i2. @⦃i1, f⦄ ≘ i2 & Y = #i2.
/2 width=3 by lifts_inv_lref1_aux/ qed-.
-fact lifts_inv_gref1_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → ∀p. X = §p → Y = §p.
-#X #Y #t * -X -Y -t //
-[ #i1 #i2 #t #_ #x #H destruct
-| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct
-| #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct
+fact lifts_inv_gref1_aux: ∀f,X,Y. ⬆*[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
+| #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
]
qed-.
(* Basic_2A1: includes: lift_inv_gref1 *)
-lemma lifts_inv_gref1: ∀Y,p,t. ⬆*[t] §p ≡ Y → Y = §p.
+lemma lifts_inv_gref1: ∀f,Y,l. ⬆*[f] §l ≘ Y → Y = §l.
/2 width=4 by lifts_inv_gref1_aux/ qed-.
-fact lifts_inv_bind1_aux: ∀X,Y,t. ⬆*[t] X ≡ Y →
- ∀a,I,V1,T1. X = ⓑ{a,I}V1.T1 →
- ∃∃V2,T2. ⬆*[t] V1 ≡ V2 & ⬆*[Ⓣ@t] T1 ≡ T2 &
- Y = ⓑ{a,I}V2.T2.
-#X #Y #t * -X -Y -t
-[ #k #t #b #J #W1 #U1 #H destruct
-| #i1 #i2 #t #_ #b #J #W1 #U1 #H destruct
-| #p #t #b #J #W1 #U1 #H destruct
-| #a #I #V1 #V2 #T1 #T2 #t #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
-| #I #V1 #V2 #T1 #T2 #t #_ #_ #b #J #W1 #U1 #H destruct
+fact lifts_inv_bind1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y →
+ ∀p,I,V1,T1. X = ⓑ{p,I}V1.T1 →
+ ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ 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
+| #f #l #b #J #W1 #U1 #H destruct
+| #f #p #I #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+| #f #I #V1 #V2 #T1 #T2 #_ #_ #q #J #W1 #U1 #H destruct
]
qed-.
(* Basic_1: was: lift1_bind *)
(* Basic_2A1: includes: lift_inv_bind1 *)
-lemma lifts_inv_bind1: ∀a,I,V1,T1,Y,t. ⬆*[t] ⓑ{a,I}V1.T1 ≡ Y →
- ∃∃V2,T2. ⬆*[t] V1 ≡ V2 & ⬆*[Ⓣ@t] T1 ≡ T2 &
- Y = ⓑ{a,I}V2.T2.
+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.
/2 width=3 by lifts_inv_bind1_aux/ qed-.
-fact lifts_inv_flat1_aux: ∀X,Y,t. ⬆*[t] X ≡ Y →
+fact lifts_inv_flat1_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≘ Y →
∀I,V1,T1. X = ⓕ{I}V1.T1 →
- ∃∃V2,T2. ⬆*[t] V1 ≡ V2 & ⬆*[t] T1 ≡ T2 &
+ ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 &
Y = ⓕ{I}V2.T2.
-#X #Y #t * -X -Y -t
-[ #k #t #J #W1 #U1 #H destruct
-| #i1 #i2 #t #_ #J #W1 #U1 #H destruct
-| #p #t #J #W1 #U1 #H destruct
-| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #J #W1 #U1 #H destruct
-| #I #V1 #V2 #T1 #T2 #t #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+#f #X #Y * -f -X -Y
+[ #f #s #J #W1 #U1 #H destruct
+| #f #i1 #i2 #_ #J #W1 #U1 #H destruct
+| #f #l #J #W1 #U1 #H destruct
+| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
+| #f #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
]
qed-.
(* Basic_1: was: lift1_flat *)
(* Basic_2A1: includes: lift_inv_flat1 *)
-lemma lifts_inv_flat1: ∀I,V1,T1,Y,t. ⬆*[t] ⓕ{I}V1.T1 ≡ Y →
- ∃∃V2,T2. ⬆*[t] V1 ≡ V2 & ⬆*[t] T1 ≡ T2 &
+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.
/2 width=3 by lifts_inv_flat1_aux/ qed-.
-fact lifts_inv_sort2_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → ∀k. Y = ⋆k → X = ⋆k.
-#X #Y #t * -X -Y -t //
-[ #i1 #i2 #t #_ #x #H destruct
-| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct
-| #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct
+fact lifts_inv_sort2_aux: ∀f,X,Y. ⬆*[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
+| #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
]
qed-.
(* Basic_1: includes: lift_gen_sort *)
(* Basic_2A1: includes: lift_inv_sort2 *)
-lemma lifts_inv_sort2: ∀X,k,t. ⬆*[t] X ≡ ⋆k → X = ⋆k.
+lemma lifts_inv_sort2: ∀f,X,s. ⬆*[f] X ≘ ⋆s → X = ⋆s.
/2 width=4 by lifts_inv_sort2_aux/ qed-.
-fact lifts_inv_lref2_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → ∀i2. Y = #i2 →
- ∃∃i1. @⦃i1, t⦄ ≡ i2 & X = #i1.
-#X #Y #t * -X -Y -t
-[ #k #t #x #H destruct
-| #i1 #i2 #t #Hi12 #x #H destruct /2 width=3 by ex2_intro/
-| #p #t #x #H destruct
-| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct
-| #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct
+fact lifts_inv_lref2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀i2. Y = #i2 →
+ ∃∃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/
+| #f #l #x #H destruct
+| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
+| #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
]
qed-.
(* 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: ∀X,i2,t. ⬆*[t] X ≡ #i2 →
- ∃∃i1. @⦃i1, t⦄ ≡ i2 & X = #i1.
+lemma lifts_inv_lref2: ∀f,X,i2. ⬆*[f] X ≘ #i2 →
+ ∃∃i1. @⦃i1, f⦄ ≘ i2 & X = #i1.
/2 width=3 by lifts_inv_lref2_aux/ qed-.
-fact lifts_inv_gref2_aux: ∀X,Y,t. ⬆*[t] X ≡ Y → ∀p. Y = §p → X = §p.
-#X #Y #t * -X -Y -t //
-[ #i1 #i2 #t #_ #x #H destruct
-| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct
-| #I #V1 #V2 #T1 #T2 #t #_ #_ #x #H destruct
+fact lifts_inv_gref2_aux: ∀f,X,Y. ⬆*[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
+| #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
]
qed-.
(* Basic_2A1: includes: lift_inv_gref1 *)
-lemma lifts_inv_gref2: ∀X,p,t. ⬆*[t] X ≡ §p → X = §p.
+lemma lifts_inv_gref2: ∀f,X,l. ⬆*[f] X ≘ §l → X = §l.
/2 width=4 by lifts_inv_gref2_aux/ qed-.
-fact lifts_inv_bind2_aux: ∀X,Y,t. ⬆*[t] X ≡ Y →
- ∀a,I,V2,T2. Y = ⓑ{a,I}V2.T2 →
- ∃∃V1,T1. ⬆*[t] V1 ≡ V2 & ⬆*[Ⓣ@t] T1 ≡ T2 &
- X = ⓑ{a,I}V1.T1.
-#X #Y #t * -X -Y -t
-[ #k #t #b #J #W2 #U2 #H destruct
-| #i1 #i2 #t #_ #b #J #W2 #U2 #H destruct
-| #p #t #b #J #W2 #U2 #H destruct
-| #a #I #V1 #V2 #T1 #T2 #t #HV12 #HT12 #b #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/
-| #I #V1 #V2 #T1 #T2 #t #_ #_ #b #J #W2 #U2 #H destruct
+fact lifts_inv_bind2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y →
+ ∀p,I,V2,T2. Y = ⓑ{p,I}V2.T2 →
+ ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 &
+ 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
+| #f #l #q #J #W2 #U2 #H destruct
+| #f #p #I #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/
+| #f #I #V1 #V2 #T1 #T2 #_ #_ #q #J #W2 #U2 #H destruct
]
qed-.
(* Basic_1: includes: lift_gen_bind *)
(* Basic_2A1: includes: lift_inv_bind2 *)
-lemma lifts_inv_bind2: ∀a,I,V2,T2,X,t. ⬆*[t] X ≡ ⓑ{a,I}V2.T2 →
- ∃∃V1,T1. ⬆*[t] V1 ≡ V2 & ⬆*[Ⓣ@t] T1 ≡ T2 &
- X = ⓑ{a,I}V1.T1.
+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.
/2 width=3 by lifts_inv_bind2_aux/ qed-.
-fact lifts_inv_flat2_aux: ∀X,Y,t. ⬆*[t] X ≡ Y →
+fact lifts_inv_flat2_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≘ Y →
∀I,V2,T2. Y = ⓕ{I}V2.T2 →
- ∃∃V1,T1. ⬆*[t] V1 ≡ V2 & ⬆*[t] T1 ≡ T2 &
+ ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 &
X = ⓕ{I}V1.T1.
-#X #Y #t * -X -Y -t
-[ #k #t #J #W2 #U2 #H destruct
-| #i1 #i2 #t #_ #J #W2 #U2 #H destruct
-| #p #t #J #W2 #U2 #H destruct
-| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #J #W2 #U2 #H destruct
-| #I #V1 #V2 #T1 #T2 #t #HV12 #HT12 #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/
+#f #X #Y * -f -X -Y
+[ #f #s #J #W2 #U2 #H destruct
+| #f #i1 #i2 #_ #J #W2 #U2 #H destruct
+| #f #l #J #W2 #U2 #H destruct
+| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #J #W2 #U2 #H destruct
+| #f #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/
]
qed-.
(* Basic_1: includes: lift_gen_flat *)
(* Basic_2A1: includes: lift_inv_flat2 *)
-lemma lifts_inv_flat2: ∀I,V2,T2,X,t. ⬆*[t] X ≡ ⓕ{I}V2.T2 →
- ∃∃V1,T1. ⬆*[t] V1 ≡ V2 & ⬆*[t] T1 ≡ 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.
/2 width=3 by lifts_inv_flat2_aux/ qed-.
+(* Advanced inversion lemmas ************************************************)
+
+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
+ | ∃∃l. I = GRef l & Y = §l.
+#f * #n #Y #H
+[ lapply (lifts_inv_sort1 … H)
+| elim (lifts_inv_lref1 … H)
+| lapply (lifts_inv_gref1 … 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} →
+ ∨∨ ∃∃s. X = ⋆s & I = Sort s
+ | ∃∃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)
+| elim (lifts_inv_lref2 … H)
+| lapply (lifts_inv_gref2 … H)
+] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
+qed-.
+
(* Basic_2A1: includes: lift_inv_pair_xy_x *)
-lemma lifts_inv_pair_xy_x: ∀I,V,T,t. ⬆*[t] ②{I}V.T ≡ V → ⊥.
-#J #V elim V -V
-[ * #i #U #t #H
+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
| elim (lifts_inv_lref2 … H) -H
#x #_ #H destruct
| lapply (lifts_inv_gref2 … H) -H #H destruct
]
-| * [ #a ] #I #V2 #T2 #IHV2 #_ #U #t #H
+| * [ #p ] #I #V2 #T2 #IHV2 #_ #U #H
[ elim (lifts_inv_bind2 … H) -H #V1 #T1 #HV12 #_ #H destruct /2 width=3 by/
| elim (lifts_inv_flat2 … H) -H #V1 #T1 #HV12 #_ #H destruct /2 width=3 by/
]
(* 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,t. ⬆*[t] 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 #t #H
+[ * #i #W #f #H
[ lapply (lifts_inv_sort2 … H) -H #H destruct
| elim (lifts_inv_lref2 … H) -H
#x #_ #H destruct
| lapply (lifts_inv_gref2 … H) -H #H destruct
]
-| * [ #a ] #I #V2 #T2 #_ #IHT2 #W #t #H
+| * [ #p ] #I #V2 #T2 #_ #IHT2 #W #f #H
[ elim (lifts_inv_bind2 … H) -H #V1 #T1 #_ #HT12 #H destruct /2 width=4 by/
| elim (lifts_inv_flat2 … H) -H #V1 #T1 #_ #HT12 #H destruct /2 width=4 by/
]
]
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/
+qed-.
+
+lemma lifts_inv_lref2_uni: ∀l,X,i2. ⬆*[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: ∀l,X,i. ⬆*[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: ∀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/
+qed-.
+
(* Basic forward lemmas *****************************************************)
(* Basic_2A1: includes: lift_inv_O2 *)
-lemma lifts_fwd_isid: ∀T1,T2,t. ⬆*[t] T1 ≡ T2 → 𝐈⦃t⦄ → T1 = T2.
-#T1 #T2 #t #H elim H -T1 -T2 -t /4 width=3 by isid_inv_at, eq_f2, eq_f/
+lemma lifts_fwd_isid: ∀f,T1,T2. ⬆*[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: ∀I,V1,T1,Y,t. ⬆*[t] ②{I}V1.T1 ≡ Y →
- ∃∃V2,T2. ⬆*[t] V1 ≡ V2 & Y = ②{I}V2.T2.
-* [ #a ] #I #V1 #T1 #Y #t #H
+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: ∀I,V2,T2,X,t. ⬆*[t] X ≡ ②{I}V2.T2 →
- ∃∃V1,T1. ⬆*[t] V1 ≡ V2 & X = ②{I}V1.T1.
-* [ #a ] #I #V2 #T2 #X #t #H
+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 properties *********************************************************)
+lemma lifts_eq_repl_back: ∀T1,T2. 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/
+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 *)
+qed-.
+
+(* Basic_1: includes: lift_r *)
+(* Basic_2A1: includes: lift_refl *)
+lemma lifts_refl: ∀T,f. 𝐈⦃f⦄ → ⬆*[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: ∀T1,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/
+| elim (IHT1 f) -IHT1 /3 width=2 by lifts_flat, ex_intro/
+]
+qed-.
+
+lemma lift_lref_uni: ∀l,i. ⬆*[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: ∀T1,T2,t. ⬆*[t] T1 ≡ T2 →
- ∀t1,t2. t2 ⊚ t1 ≡ t →
- ∃∃T. ⬆*[t1] T1 ≡ T & ⬆*[t2] T ≡ T2.
-#T1 #T2 #t #H elim H -T1 -T2 -t
+lemma lifts_split_trans: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 →
+ ∀f1,f2. f2 ⊚ f1 ≘ f →
+ ∃∃T. ⬆*[f1] T1 ≘ T & ⬆*[f2] T ≘ T2.
+#f #T1 #T2 #H elim H -f -T1 -T2
[ /3 width=3 by lifts_sort, ex2_intro/
-| #i1 #i2 #t #Hi #t1 #t2 #Ht elim (after_at_fwd … Ht … Hi) -Ht -Hi
+| #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/
-| #a #I #V1 #V2 #T1 #T2 #t #_ #_ #IHV #IHT #t1 #t2 #Ht
- elim (IHV â\80¦ Ht) elim (IHT (â\93\89@t1) (â\93\89@t2)) -IHV -IHT
- /3 width=5 by lifts_bind, after_true, ex2_intro/
-| #I #V1 #V2 #T1 #T2 #t #_ #_ #IHV #IHT #t1 #t2 #Ht
+| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f1 #f2 #Ht
+ elim (IHV â\80¦ Ht) elim (IHT (⫯f1) (⫯f2)) -IHV -IHT
+ /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_flat, ex2_intro/
]
qed-.
(* Note: apparently, this was missing in Basic_2A1 *)
-lemma lifts_split_div: ∀T1,T2,t1. ⬆*[t1] T1 ≡ T2 →
- ∀t2,t. t2 ⊚ t1 ≡ t →
- ∃∃T. ⬆*[t2] T2 ≡ T & ⬆*[t] T1 ≡ T.
-#T1 #T2 #t1 #H elim H -T1 -T2 -t1
+lemma lifts_split_div: ∀f1,T1,T2. ⬆*[f1] T1 ≘ T2 →
+ ∀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/
-| #i1 #i2 #t1 #Hi #t2 #t #Ht elim (after_at1_fwd … Ht … Hi) -Ht -Hi
+| #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/
-| #a #I #V1 #V2 #T1 #T2 #t1 #_ #_ #IHV #IHT #t2 #t #Ht
- elim (IHV â\80¦ Ht) elim (IHT (â\93\89@t2) (â\93\89@t)) -IHV -IHT
- /3 width=5 by lifts_bind, after_true, ex2_intro/
-| #I #V1 #V2 #T1 #T2 #t1 #_ #_ #IHV #IHT #t2 #t #Ht
+| #f1 #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #f #Ht
+ elim (IHV â\80¦ 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
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,t. Decidable (∃T1. ⬆*[t] T1 ≡ T2).
+lemma is_lifts_dec: ∀T2,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 #t elim (is_at_dec t i2)
+ #i2 #f elim (is_at_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/
]
-| * [ #a ] #I #V2 #T2 #IHV2 #IHT2 #t
- [ elim (IHV2 t) -IHV2
- [ * #V1 #HV12 elim (IHT2 (â\93\89@t)) -IHT2
+| * [ #p ] #I #V2 #T2 #IHV2 #IHT2 #f
+ [ elim (IHV2 f) -IHV2
+ [ * #V1 #HV12 elim (IHT2 (⫯f)) -IHT2
[ * #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 t) -IHV2
- [ * #V1 #HV12 elim (IHT2 t) -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/
]
qed-.
-(* Basic_2A1: removed theorems 17:
- lifts_inv_nil lifts_inv_cons lifts_total
+(* Properties with uniform relocation ***************************************)
+
+lemma lifts_uni: ∀n1,n2,T,U. ⬆*[𝐔❴n1❵∘𝐔❴n2❵] T ≘ U → ⬆*[n1+n2] T ≘ U.
+/3 width=4 by lifts_eq_repl_back, after_inv_total/ qed.
+
+(* Basic_2A1: removed theorems 14:
+ lifts_inv_nil lifts_inv_cons
lift_inv_Y1 lift_inv_Y2 lift_inv_lref_Y1 lift_inv_lref_Y2 lift_lref_Y lift_Y1
lift_lref_lt_eq lift_lref_ge_eq lift_lref_plus lift_lref_pred
- lift_lref_ge_minus lift_lref_ge_minus_eq lift_total lift_refl
+ lift_lref_ge_minus lift_lref_ge_minus_eq
*)
(* Basic_1: removed theorems 8:
lift_lref_gt
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