X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Flifts.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Flifts.ma;h=45cd021ebbb0e8d580f590c35570340114e08060;hb=802e118337ebd0f8b732d4939973aae6415b5bec;hp=6a66289bd6f28f6e56ad0222650fc5bb080d59f7;hpb=a961a1237063702ed9c32a9a4b7994671cb40818;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma index 6a66289bd..45cd021eb 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma @@ -12,7 +12,7 @@ (* *) (**************************************************************************) -include "ground_2/relocation/trace_isid.ma". +include "ground_2/relocation/nstream_id.ma". include "basic_2/notation/relations/rliftstar_3.ma". include "basic_2/grammar/term.ma". @@ -22,16 +22,16 @@ include "basic_2/grammar/term.ma". 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: ∀s,f. lifts f (⋆s) (⋆s) +| lifts_lref: ∀i1,i2,f. @⦃i1, f⦄ ≡ i2 → lifts f (#i1) (#i2) +| lifts_gref: ∀l,f. lifts f (§l) (§l) +| lifts_bind: ∀p,I,V1,V2,T1,T2,f. + lifts f V1 V2 → lifts (↑f) T1 T2 → + lifts f (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2) +| lifts_flat: ∀I,V1,V2,T1,T2,f. + lifts f V1 V2 → lifts f T1 T2 → + lifts f (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) . interpretation "generic relocation (term)" @@ -39,180 +39,180 @@ interpretation "generic relocation (term)" (* 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: ∀X,Y,f. ⬆*[f] X ≡ Y → ∀s. X = ⋆s → Y = ⋆s. +#X #Y #f * -X -Y -f // +[ #i1 #i2 #f #_ #x #H destruct +| #p #I #V1 #V2 #T1 #T2 #f #_ #_ #x #H destruct +| #I #V1 #V2 #T1 #T2 #f #_ #_ #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: ∀Y,s,f. ⬆*[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: ∀X,Y,f. ⬆*[f] X ≡ Y → ∀i1. X = #i1 → + ∃∃i2. @⦃i1, f⦄ ≡ i2 & Y = #i2. +#X #Y #f * -X -Y -f +[ #s #f #x #H destruct +| #i1 #i2 #f #Hi12 #x #H destruct /2 width=3 by ex2_intro/ +| #l #f #x #H destruct +| #p #I #V1 #V2 #T1 #T2 #f #_ #_ #x #H destruct +| #I #V1 #V2 #T1 #T2 #f #_ #_ #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: ∀Y,i1,f. ⬆*[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: ∀X,Y,f. ⬆*[f] X ≡ Y → ∀l. X = §l → Y = §l. +#X #Y #f * -X -Y -f // +[ #i1 #i2 #f #_ #x #H destruct +| #p #I #V1 #V2 #T1 #T2 #f #_ #_ #x #H destruct +| #I #V1 #V2 #T1 #T2 #f #_ #_ #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: ∀Y,l,f. ⬆*[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: ∀X,Y,f. ⬆*[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. +#X #Y #f * -X -Y -f +[ #s #f #q #J #W1 #U1 #H destruct +| #i1 #i2 #f #_ #q #J #W1 #U1 #H destruct +| #l #f #b #J #W1 #U1 #H destruct +| #p #I #V1 #V2 #T1 #T2 #f #HV12 #HT12 #q #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/ +| #I #V1 #V2 #T1 #T2 #f #_ #_ #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: ∀p,I,V1,T1,Y,f. ⬆*[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: ∀X,Y,f. ⬆*[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/ +#X #Y #f * -X -Y -f +[ #s #f #J #W1 #U1 #H destruct +| #i1 #i2 #f #_ #J #W1 #U1 #H destruct +| #l #f #J #W1 #U1 #H destruct +| #p #I #V1 #V2 #T1 #T2 #f #_ #_ #J #W1 #U1 #H destruct +| #I #V1 #V2 #T1 #T2 #f #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: ∀I,V1,T1,Y,f. ⬆*[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: ∀X,Y,f. ⬆*[f] X ≡ Y → ∀s. Y = ⋆s → X = ⋆s. +#X #Y #f * -X -Y -f // +[ #i1 #i2 #f #_ #x #H destruct +| #p #I #V1 #V2 #T1 #T2 #f #_ #_ #x #H destruct +| #I #V1 #V2 #T1 #T2 #f #_ #_ #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: ∀X,s,f. ⬆*[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: ∀X,Y,f. ⬆*[f] X ≡ Y → ∀i2. Y = #i2 → + ∃∃i1. @⦃i1, f⦄ ≡ i2 & X = #i1. +#X #Y #f * -X -Y -f +[ #s #f #x #H destruct +| #i1 #i2 #f #Hi12 #x #H destruct /2 width=3 by ex2_intro/ +| #l #f #x #H destruct +| #p #I #V1 #V2 #T1 #T2 #f #_ #_ #x #H destruct +| #I #V1 #V2 #T1 #T2 #f #_ #_ #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: ∀X,i2,f. ⬆*[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: ∀X,Y,f. ⬆*[f] X ≡ Y → ∀l. Y = §l → X = §l. +#X #Y #f * -X -Y -f // +[ #i1 #i2 #f #_ #x #H destruct +| #p #I #V1 #V2 #T1 #T2 #f #_ #_ #x #H destruct +| #I #V1 #V2 #T1 #T2 #f #_ #_ #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: ∀X,l,f. ⬆*[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: ∀X,Y,f. ⬆*[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. +#X #Y #f * -X -Y -f +[ #s #f #q #J #W2 #U2 #H destruct +| #i1 #i2 #f #_ #q #J #W2 #U2 #H destruct +| #l #f #q #J #W2 #U2 #H destruct +| #p #I #V1 #V2 #T1 #T2 #f #HV12 #HT12 #q #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/ +| #I #V1 #V2 #T1 #T2 #f #_ #_ #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: ∀p,I,V2,T2,X,f. ⬆*[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: ∀X,Y,f. ⬆*[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/ +#X #Y #f * -X -Y -f +[ #s #f #J #W2 #U2 #H destruct +| #i1 #i2 #f #_ #J #W2 #U2 #H destruct +| #l #f #J #W2 #U2 #H destruct +| #p #I #V1 #V2 #T1 #T2 #f #_ #_ #J #W2 #U2 #H destruct +| #I #V1 #V2 #T1 #T2 #f #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: ∀I,V2,T2,X,f. ⬆*[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-. (* Basic_2A1: includes: lift_inv_pair_xy_x *) -lemma lifts_inv_pair_xy_x: ∀I,V,T,t. ⬆*[t] ②{I}V.T ≡ V → ⊥. +lemma lifts_inv_pair_xy_x: ∀I,V,T,f. ⬆*[f] ②{I}V.T ≡ V → ⊥. #J #V elim V -V -[ * #i #U #t #H +[ * #i #U #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 #IHV2 #_ #U #t #H +| * [ #p ] #I #V2 #T2 #IHV2 #_ #U #f #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/ ] @@ -221,15 +221,15 @@ 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,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/ ] @@ -239,23 +239,24 @@ 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: ∀T1,T2,f. ⬆*[f] T1 ≡ T2 → 𝐈⦃f⦄ → T1 = T2. +#T1 #T2 #f #H elim H -T1 -T2 -f +/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: ∀I,V1,T1,Y,f. ⬆*[f] ②{I}V1.T1 ≡ Y → + ∃∃V2,T2. ⬆*[f] V1 ≡ V2 & Y = ②{I}V2.T2. +* [ #p ] #I #V1 #T1 #Y #f #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: ∀I,V2,T2,X,f. ⬆*[f] X ≡ ②{I}V2.T2 → + ∃∃V1,T1. ⬆*[f] V1 ≡ V2 & X = ②{I}V1.T1. +* [ #p ] #I #V2 #T2 #X #f #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/ ] @@ -263,38 +264,65 @@ qed-. (* Basic properties *********************************************************) +lemma lifts_eq_repl_back: ∀T1,T2. eq_stream_repl_back … (λf. ⬆*[f] T1 ≡ T2). +#T1 #T2 #f1 #H elim H -T1 -T2 -f1 +/4 width=3 by lifts_flat, lifts_bind, lifts_lref, at_eq_repl_back, push_eq_repl/ +qed-. + +lemma lifts_eq_repl_fwd: ∀T1,T2. eq_stream_repl_fwd … (λf. ⬆*[f] T1 ≡ T2). +#T1 #T2 @eq_stream_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=1 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-. + (* 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: ∀T1,T2,f. ⬆*[f] T1 ≡ T2 → + ∀f1,f2. f2 ⊚ f1 ≡ f → + ∃∃T. ⬆*[f1] T1 ≡ T & ⬆*[f2] T ≡ T2. +#T1 #T2 #f #H elim H -T1 -T2 -f [ /3 width=3 by lifts_sort, ex2_intro/ -| #i1 #i2 #t #Hi #t1 #t2 #Ht elim (after_at_fwd … Ht … Hi) -Ht -Hi +| #i1 #i2 #f #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 … Ht) elim (IHT (Ⓣ@t1) (Ⓣ@t2)) -IHV -IHT - /3 width=5 by lifts_bind, after_true, ex2_intro/ -| #I #V1 #V2 #T1 #T2 #t #_ #_ #IHV #IHT #t1 #t2 #Ht +| #p #I #V1 #V2 #T1 #T2 #f #_ #_ #IHV #IHT #f1 #f2 #Ht + elim (IHV … Ht) elim (IHT (↑f1) (↑f2)) -IHV -IHT + /3 width=5 by lifts_bind, after_O2, ex2_intro/ +| #I #V1 #V2 #T1 #T2 #f #_ #_ #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: ∀T1,T2,f1. ⬆*[f1] T1 ≡ T2 → + ∀f2,f. f2 ⊚ f1 ≡ f → + ∃∃T. ⬆*[f2] T2 ≡ T & ⬆*[f] T1 ≡ T. +#T1 #T2 #f1 #H elim H -T1 -T2 -f1 [ /3 width=3 by lifts_sort, ex2_intro/ -| #i1 #i2 #t1 #Hi #t2 #t #Ht elim (after_at1_fwd … Ht … Hi) -Ht -Hi +| #i1 #i2 #f1 #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 … Ht) elim (IHT (Ⓣ@t2) (Ⓣ@t)) -IHV -IHT - /3 width=5 by lifts_bind, after_true, ex2_intro/ -| #I #V1 #V2 #T1 #T2 #t1 #_ #_ #IHV #IHT #t2 #t #Ht +| #p #I #V1 #V2 #T1 #T2 #f1 #_ #_ #IHV #IHT #f2 #f #Ht + elim (IHV … Ht) elim (IHT (↑f2) (↑f)) -IHV -IHT + /3 width=5 by lifts_bind, after_O2, ex2_intro/ +| #I #V1 #V2 #T1 #T2 #f1 #_ #_ #IHV #IHT #f2 #f #Ht elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht /3 width=5 by lifts_flat, ex2_intro/ ] @@ -302,18 +330,18 @@ qed-. (* 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 (Ⓣ@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/ @@ -321,8 +349,8 @@ lemma is_lifts_dec: ∀T2,t. Decidable (∃T1. ⬆*[t] T1 ≡ T2). | -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/