X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fstatic_2%2Frelocation%2Flifts.ma;h=f019c66d6cb3fa61170a17457061e3f2d375a13f;hb=98e786e1a6bd7b621e37ba7cd4098d4a0a6f8278;hp=060f3b8057d843e509201ed69a526d62895c6f01;hpb=0d1dc967bc12041b9d23ee945db9dd91335e8c1d;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 060f3b805..f019c66d6 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/lifts.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/lifts.ma @@ -12,8 +12,9 @@ (* *) (**************************************************************************) -include "ground_2/relocation/nstream_after.ma". +include "ground/relocation/nstream_after.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 *********************************************) @@ -22,47 +23,51 @@ include "static_2/syntax/term.ma". 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 → - ∃∃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. + 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 @@ -72,11 +77,11 @@ 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/ @@ -88,11 +93,11 @@ 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 @@ -101,13 +106,13 @@ 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 @@ -119,15 +124,15 @@ 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,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 @@ -139,12 +144,12 @@ 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,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 @@ -154,11 +159,11 @@ 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/ @@ -170,11 +175,11 @@ 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 @@ -183,13 +188,13 @@ 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 @@ -201,15 +206,15 @@ 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,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 @@ -221,16 +226,16 @@ 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,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) @@ -239,9 +244,9 @@ 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) @@ -251,7 +256,7 @@ 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 @@ -268,7 +273,7 @@ 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 @@ -283,39 +288,55 @@ 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 (pr_pat_inv_unit_push … 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 (pr_pat_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 /4 width=4 by fr2_nat_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/ +/3 width=3 by pr_pat_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/ +#i1 #H1 #H2 destruct /4 width=2 by eq_inv_nplus_bi_sn, 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/ +#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: ∀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/ +/4 width=3 by pr_isi_pat_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/ @@ -323,8 +344,8 @@ 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,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/ @@ -333,29 +354,34 @@ qed-. (* Basic properties *********************************************************) +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). +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: ∀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/ +/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. ∃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 @@ -365,27 +391,27 @@ elim (IHV1 f) -IHV1 #V2 #HV12 ] qed-. -lemma lifts_push_zero (f): ⬆*[⫯f]#O ≘ #0. +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). +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. (* 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 +| #f #i1 #i2 #Hi #f1 #f2 #Ht elim (pr_after_pat_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 @@ -398,12 +424,12 @@ 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 +| #f1 #i1 #i2 #Hi #f2 #f #Ht elim (pr_after_des_ist_pat … 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 @@ -417,10 +443,10 @@ 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) // + #i2 #f elim (is_pr_pat_dec f i2) // [ * /4 width=3 by lifts_lref, ex_intro, or_introl/ | #H @or_intror * #X #HX elim (lifts_inv_lref2 … HX) -HX @@ -451,7 +477,7 @@ 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: @@ -461,8 +487,8 @@ 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 *)