X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fstatic_2%2Frelocation%2Flifts.ma;h=6bdf97c003ff721ef0e1d857ab940999111d9168;hb=bd53c4e895203eb049e75434f638f26b5a161a2b;hp=54372cf8f4c0333fb8088c96a4b455309e71ec54;hpb=cc6fcb70ca4f3cf01205ed722d75a2fdb2aaf779;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/lifts.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/lifts.ma index 54372cf8f..6bdf97c00 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/lifts.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/lifts.ma @@ -24,14 +24,14 @@ include "static_2/syntax/term.ma". *) 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_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)" @@ -41,24 +41,32 @@ interpretation "generic relocation (term)" '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. + 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. + 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. + 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. + λR. ∀U1,U2. R U1 U2 → ∀f,T1. ⇧*[f] T1 ≘ U1 → + ∀T2. ⇧*[f] T2 ≘ U2 → R T1 T2. + +definition liftable2_dx: predicate (relation term) ≝ + λR. ∀T1,T2. R T1 T2 → ∀f,U2. ⇧*[f] T2 ≘ U2 → + ∃∃U1. ⇧*[f] T1 ≘ U1 & R U1 U2. + +definition deliftable2_dx: predicate (relation term) ≝ + λR. ∀U1,U2. R U1 U2 → ∀f,T2. ⇧*[f] T2 ≘ U2 → + ∃∃T1. ⇧*[f] T1 ≘ U1 & R T1 T2. (* Basic inversion lemmas ***************************************************) -fact lifts_inv_sort1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀s. X = ⋆s → Y = ⋆s. +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 @@ -68,11 +76,11 @@ qed-. (* Basic_1: was: lift1_sort *) (* Basic_2A1: includes: lift_inv_sort1 *) -lemma lifts_inv_sort1: ∀f,Y,s. ⬆*[f] ⋆s ≘ Y → Y = ⋆s. +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: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀i1. X = #i1 → - ∃∃i2. @⦃i1, f⦄ ≘ i2 & Y = #i2. +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/ @@ -84,11 +92,11 @@ qed-. (* 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. +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: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀l. X = §l → Y = §l. +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 @@ -97,13 +105,13 @@ fact lifts_inv_gref1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀l. X = §l → Y = § qed-. (* Basic_2A1: includes: lift_inv_gref1 *) -lemma lifts_inv_gref1: ∀f,Y,l. ⬆*[f] §l ≘ Y → Y = §l. +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: ∀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. +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 @@ -115,15 +123,15 @@ qed-. (* 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 → - ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 & - Y = ⓑ{p,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: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≘ Y → - ∀I,V1,T1. X = ⓕ{I}V1.T1 → - ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 & - Y = ⓕ{I}V2.T2. +fact lifts_inv_flat1_aux: ∀f:rtmap. ∀X,Y. ⇧*[f] X ≘ Y → + ∀I,V1,T1. X = ⓕ[I]V1.T1 → + ∃∃V2,T2. ⇧*[f] V1 ≘ V2 & ⇧*[f] T1 ≘ 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 @@ -135,12 +143,12 @@ qed-. (* 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 → - ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 & - Y = ⓕ{I}V2.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: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀s. Y = ⋆s → X = ⋆s. +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 @@ -150,11 +158,11 @@ qed-. (* Basic_1: includes: lift_gen_sort *) (* Basic_2A1: includes: lift_inv_sort2 *) -lemma lifts_inv_sort2: ∀f,X,s. ⬆*[f] X ≘ ⋆s → X = ⋆s. +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: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀i2. Y = #i2 → - ∃∃i1. @⦃i1, f⦄ ≘ i2 & X = #i1. +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/ @@ -166,11 +174,11 @@ 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: ∀f,X,i2. ⬆*[f] X ≘ #i2 → - ∃∃i1. @⦃i1, f⦄ ≘ 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: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀l. Y = §l → X = §l. +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 @@ -179,13 +187,13 @@ fact lifts_inv_gref2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀l. Y = §l → X = § qed-. (* Basic_2A1: includes: lift_inv_gref1 *) -lemma lifts_inv_gref2: ∀f,X,l. ⬆*[f] X ≘ §l → X = §l. +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: ∀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. +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 @@ -197,15 +205,15 @@ qed-. (* 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 → - ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 & - X = ⓑ{p,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: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≘ Y → - ∀I,V2,T2. Y = ⓕ{I}V2.T2 → - ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 & - X = ⓕ{I}V1.T1. +fact lifts_inv_flat2_aux: ∀f:rtmap. ∀X,Y. ⇧*[f] X ≘ Y → + ∀I,V2,T2. Y = ⓕ[I]V2.T2 → + ∃∃V1,T1. ⇧*[f] V1 ≘ V2 & ⇧*[f] T1 ≘ T2 & + 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 @@ -217,16 +225,16 @@ qed-. (* 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 → - ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 & - X = ⓕ{I}V1.T1. +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 → +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) @@ -235,9 +243,9 @@ lemma lifts_inv_atom1: ∀f,I,Y. ⬆*[f] ⓪{I} ≘ Y → ] -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) @@ -247,7 +255,7 @@ lemma lifts_inv_atom2: ∀f,I,X. ⬆*[f] X ≘ ⓪{I} → 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 @@ -264,7 +272,7 @@ qed-. (* 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 @@ -279,24 +287,40 @@ lemma lifts_inv_pair_xy_y: ∀I,T,V,f. ⬆*[f] ②{I}V.T ≘ T → ⊥. ] 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: ∀l,Y,i. ⬆*[l] #i ≘ Y → Y = #(l+i). +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 → +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. +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 → ⊥. +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-. @@ -304,14 +328,14 @@ qed-. (* Basic forward lemmas *****************************************************) (* Basic_2A1: includes: lift_inv_O2 *) -lemma lifts_fwd_isid: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → 𝐈⦃f⦄ → T1 = T2. +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: ∀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/ @@ -319,8 +343,8 @@ lemma lifts_fwd_pair1: ∀f:rtmap. ∀I,V1,T1,Y. ⬆*[f] ②{I}V1.T1 ≘ Y → 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/ @@ -329,24 +353,34 @@ qed-. (* Basic properties *********************************************************) -lemma lifts_eq_repl_back: ∀T1,T2. eq_repl_back … (λf. ⬆*[f] T1 ≘ T2). +lemma liftable2_sn_dx (R): symmetric … R → liftable2_sn R → liftable2_dx R. +#R #H2R #H1R #T1 #T2 #HT12 #f #U2 #HTU2 +elim (H1R … T1 … HTU2) -H1R /3 width=3 by ex2_intro/ +qed-. + +lemma deliftable2_sn_dx (R): symmetric … R → deliftable2_sn R → deliftable2_dx R. +#R #H2R #H1R #U1 #U2 #HU12 #f #T2 #HTU2 +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). #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). +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. +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. +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 @@ -356,15 +390,24 @@ elim (IHV1 f) -IHV1 #V2 #HV12 ] qed-. -lemma lifts_lref_uni: ∀l,i. ⬆*[l] #i ≘ #(l+i). +lemma lifts_push_zero (f): ⇧*[⫯f]#0 ≘ #0. +/2 width=1 by 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/ +qed. + +lemma lifts_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: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → +lemma lifts_split_trans: ∀f,T1,T2. ⇧*[f] T1 ≘ T2 → ∀f1,f2. f2 ⊚ f1 ≘ f → - ∃∃T. ⬆*[f1] T1 ≘ T & ⬆*[f2] T ≘ T2. + ∃∃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 @@ -380,9 +423,9 @@ lemma lifts_split_trans: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → qed-. (* Note: apparently, this was missing in Basic_2A1 *) -lemma lifts_split_div: ∀f1,T1,T2. ⬆*[f1] T1 ≘ T2 → +lemma lifts_split_div: ∀f1,T1,T2. ⇧*[f1] T1 ≘ T2 → ∀f2,f. f2 ⊚ f1 ≘ f → - ∃∃T. ⬆*[f2] T2 ≘ T & ⬆*[f] T1 ≘ T. + ∃∃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 @@ -399,7 +442,7 @@ qed-. (* 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. 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) // @@ -433,7 +476,7 @@ qed-. (* Properties with uniform relocation ***************************************) -lemma lifts_uni: ∀n1,n2,T,U. ⬆*[𝐔❴n1❵∘𝐔❴n2❵] T ≘ U → ⬆*[n1+n2] T ≘ U. +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: @@ -443,8 +486,8 @@ lemma lifts_uni: ∀n1,n2,T,U. ⬆*[𝐔❴n1❵∘𝐔❴n2❵] T ≘ U → ⬆ 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 *)