X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Flifts.ma;h=5198b2d5810928cfb0cabd1c3c772f071b23adfa;hp=6f0cef9d0be252cf2e58847e28a0d538b1b36101;hb=222044da28742b24584549ba86b1805a87def070;hpb=2f00c2224c66757d00883602cfd0bbd2448eb3ca diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma index 6f0cef9d0..5198b2d58 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma @@ -25,10 +25,10 @@ include "basic_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 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 → @@ -41,25 +41,25 @@ interpretation "uniform relocation (term)" interpretation "generic relocation (term)" 'RLiftStar f T1 T2 = (lifts f T1 T2). -definition liftable2: predicate (relation term) ≝ - λR. ∀T1,T2. R T1 T2 → ∀f,U1. ⬆*[f] T1 ≡ U1 → - ∃∃U2. ⬆*[f] T2 ≡ U2 & R U1 U2. +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. + 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. (* 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 @@ -69,11 +69,11 @@ 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/ @@ -85,11 +85,11 @@ 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 @@ -98,12 +98,12 @@ 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 → +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 & + ∃∃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 @@ -116,14 +116,14 @@ 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 & +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 → +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 & + ∃∃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 @@ -136,12 +136,12 @@ 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 & +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 @@ -151,11 +151,11 @@ 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/ @@ -167,11 +167,11 @@ 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 @@ -180,12 +180,12 @@ 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 → +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 & + ∃∃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 @@ -198,14 +198,14 @@ 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 & +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 → +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 & + ∃∃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 @@ -218,16 +218,16 @@ 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 & +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) @@ -236,9 +236,9 @@ 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) @@ -248,7 +248,7 @@ 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 @@ -265,7 +265,7 @@ 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 @@ -282,22 +282,22 @@ 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-. @@ -305,14 +305,14 @@ 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/ @@ -320,8 +320,8 @@ 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/ @@ -330,49 +330,49 @@ qed-. (* Basic properties *********************************************************) -lemma lifts_eq_repl_back: ∀T1,T2. eq_repl_back … (λf. ⬆*[f] T1 ≡ T2). +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 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_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). +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: ∀f,T1,T2. ⬆*[f] T1 ≡ T2 → - ∀f1,f2. f2 ⊚ f1 ≡ f → - ∃∃T. ⬆*[f1] T1 ≡ T & ⬆*[f2] T ≡ T2. +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/ | #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/ | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f1 #f2 #Ht - elim (IHV … Ht) elim (IHT (↑f1) (↑f2)) -IHV -IHT + elim (IHV … 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 @@ -381,16 +381,16 @@ 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 → - ∀f2,f. f2 ⊚ f1 ≡ f → - ∃∃T. ⬆*[f2] T2 ≡ T & ⬆*[f] T1 ≡ T. +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/ | #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/ | #f1 #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #f #Ht - elim (IHV … Ht) elim (IHT (↑f2) (↑f)) -IHV -IHT + 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 elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht @@ -400,7 +400,7 @@ 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) // @@ -411,7 +411,7 @@ lemma is_lifts_dec: ∀T2,f. Decidable (∃T1. ⬆*[f] T1 ≡ T2). ] | * [ #p ] #I #V2 #T2 #IHV2 #IHT2 #f [ elim (IHV2 f) -IHV2 - [ * #V1 #HV12 elim (IHT2 (↑f)) -IHT2 + [ * #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/ @@ -434,7 +434,7 @@ 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: