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=6bdf97c003ff721ef0e1d857ab940999111d9168;hpb=bd53c4e895203eb049e75434f638f26b5a161a2b;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 6bdf97c00..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 *********************************************) @@ -22,7 +23,7 @@ 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_gref: ∀f,l. lifts f (§l) (§l) @@ -34,12 +35,12 @@ inductive lifts: rtmap → relation term ≝ 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. @@ -128,7 +129,7 @@ lemma lifts_inv_bind1: ∀f,p,I,V1,T1,Y. ⇧*[f] ⓑ[p,I]V1.T1 ≘ Y → 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,X,Y. ⇧*[f] X ≘ Y → ∀I,V1,T1. X = ⓕ[I]V1.T1 → ∃∃V2,T2. ⇧*[f] V1 ≘ V2 & ⇧*[f] T1 ≘ T2 & Y = ⓕ[I]V2.T2. @@ -143,7 +144,7 @@ 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 → +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-. @@ -210,7 +211,7 @@ lemma lifts_inv_bind2: ∀f,p,I,V2,T2,X. ⇧*[f] X ≘ ⓑ[p,I]V2.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,X,Y. ⇧*[f] X ≘ Y → ∀I,V2,T2. Y = ⓕ[I]V2.T2 → ∃∃V1,T1. ⇧*[f] V1 ≘ V2 & ⇧*[f] T1 ≘ T2 & X = ⓕ[I]V1.T1. @@ -225,7 +226,7 @@ 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 → +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-. @@ -291,7 +292,7 @@ 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 // +lapply (pr_pat_inv_unit_push … Hi ???) -Hi // qed-. lemma lifts_inv_push_succ_sn (f) (i1): @@ -299,30 +300,30 @@ lemma lifts_inv_push_succ_sn (f) (i1): ∃∃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 +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 *****************************************************) @@ -330,11 +331,11 @@ qed-. (* Basic_2A1: includes: lift_inv_O2 *) 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 → +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/ @@ -343,7 +344,7 @@ 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 → +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/ @@ -363,20 +364,20 @@ lemma deliftable2_sn_dx (R): symmetric … R → deliftable2_sn R → deliftable 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. #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 *) @@ -396,10 +397,10 @@ lemma lifts_push_zero (f): ⇧*[⫯f]#0 ≘ #0. 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. @@ -410,7 +411,7 @@ lemma lifts_split_trans: ∀f,T1,T2. ⇧*[f] T1 ≘ 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 @@ -428,7 +429,7 @@ lemma lifts_split_div: ∀f1,T1,T2. ⇧*[f1] T1 ≘ T2 → ∃∃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 @@ -445,7 +446,7 @@ qed-. 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 @@ -476,7 +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: