From: Ferruccio Guidi Date: Tue, 24 Jan 2017 17:21:08 +0000 (+0000) Subject: - updated equivalence on referred entries: it nust be degree-based X-Git-Tag: make_still_working~512 X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=commitdiff_plain;h=b4b5f03ffca4f250a1dc02f277b70e4f33ac8a9b - updated equivalence on referred entries: it nust be degree-based - refactoring --- diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/ceq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/ceq.etc new file mode 100644 index 000000000..57ecbaf68 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/ceq.etc @@ -0,0 +1,42 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/syntax/lenv.ma". + +(* CONTEXT-SENSITIVE EQUIVALENCES FOR TERMS *********************************) + +definition ceq: relation3 lenv term term ≝ λL,T1,T2. T1 = T2. + +definition cfull: relation3 lenv term term ≝ λL,T1,T2. ⊤. + +(* Basic properties *********************************************************) + +lemma ceq_refl (L): reflexive … (ceq L). +// qed. + +lemma cfull_refl (L): reflexive … (cfull L). +// qed. + +lemma ceq_sym (L): symmetric … (ceq L). +// qed-. + +lemma cfull_sym (L): symmetric … (cfull L). +// qed-. + +lemma cfull_top (R:relation3 lenv term term) (L) (T1) (T2): + R L T1 T2 → cfull L T1 T2. +// qed-. + +lemma ceq_cfull (L) (T1) (T2): ceq L T1 T2 → cfull L T1 T2. +// qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/ceq_ceq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/ceq_ceq.etc new file mode 100644 index 000000000..b17b186e2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/ceq_ceq.etc @@ -0,0 +1,37 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/syntax/ceq.ma". + +(* CONTEXT-SENSITIVE EQUIVALENCES FOR TERMS *********************************) + +(* Main properties **********************************************************) + +theorem ceq_trans (L): Transitive … (ceq L). +// qed-. + +lemma ceq_canc_sn (L): left_cancellable … (ceq L). +// qed-. + +lemma ceq_canc_dx (L): right_cancellable … (ceq L). +// qed-. + +theorem cfull_trans (L): Transitive … (cfull L). +// qed-. + +lemma cfull_canc_sn (L): left_cancellable … (cfull L). +// qed-. + +lemma cfull_canc_dx (L): right_cancellable … (cfull L). +// qed-. \ No newline at end of file diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cir/cir.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cir/cir.etc new file mode 100644 index 000000000..412297546 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cir/cir.etc @@ -0,0 +1,79 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/notation/relations/prednotreducible_3.ma". +include "basic_2/reduction/crr.ma". + +(* IRREDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION ************************) + +definition cir: relation3 genv lenv term ≝ λG,L,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⊥. + +interpretation "irreducibility for context-sensitive reduction (term)" + 'PRedNotReducible G L T = (cir G L T). + +(* Basic inversion lemmas ***************************************************) + +lemma cir_inv_delta: ∀G,L,K,V,i. ⬇[i] L ≡ K.ⓓV → ⦃G, L⦄ ⊢ ➡ 𝐈⦃#i⦄ → ⊥. +/3 width=3 by crr_delta/ qed-. + +lemma cir_inv_ri2: ∀I,G,L,V,T. ri2 I → ⦃G, L⦄ ⊢ ➡ 𝐈⦃②{I}V.T⦄ → ⊥. +/3 width=1 by crr_ri2/ qed-. + +lemma cir_inv_ib2: ∀a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓑ{a,I}V.T⦄ → + ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ➡ 𝐈⦃T⦄. +/4 width=1 by crr_ib2_sn, crr_ib2_dx, conj/ qed-. + +lemma cir_inv_bind: ∀a,I,G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓑ{a,I}V.T⦄ → + ∧∧ ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ & ⦃G, L.ⓑ{I}V⦄ ⊢ ➡ 𝐈⦃T⦄ & ib2 a I. +#a * [ elim a -a ] +#G #L #V #T #H [ elim H -H /3 width=1 by crr_ri2, or_introl/ ] +elim (cir_inv_ib2 … H) -H /3 width=1 by and3_intro, or_introl/ +qed-. + +lemma cir_inv_appl: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓐV.T⦄ → + ∧∧ ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄ & 𝐒⦃T⦄. +#G #L #V #T #HVT @and3_intro /3 width=1 by crr_appl_sn, crr_appl_dx/ +generalize in match HVT; -HVT elim T -T // +* // #a * #U #T #_ #_ #H elim H -H /2 width=1 by crr_beta, crr_theta/ +qed-. + +lemma cir_inv_flat: ∀I,G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓕ{I}V.T⦄ → + ∧∧ ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl. +* #G #L #V #T #H +[ elim (cir_inv_appl … H) -H /2 width=1 by and4_intro/ +| elim (cir_inv_ri2 … H) -H // +] +qed-. + +(* Basic properties *********************************************************) + +lemma cir_sort: ∀G,L,s. ⦃G, L⦄ ⊢ ➡ 𝐈⦃⋆s⦄. +/2 width=4 by crr_inv_sort/ qed. + +lemma cir_gref: ∀G,L,p. ⦃G, L⦄ ⊢ ➡ 𝐈⦃§p⦄. +/2 width=4 by crr_inv_gref/ qed. + +lemma tir_atom: ∀G,I. ⦃G, ⋆⦄ ⊢ ➡ 𝐈⦃⓪{I}⦄. +/2 width=3 by trr_inv_atom/ qed. + +lemma cir_ib2: ∀a,I,G,L,V,T. + ib2 a I → ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ ➡ 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓑ{a,I}V.T⦄. +#a #I #G #L #V #T #HI #HV #HT #H +elim (crr_inv_ib2 … HI H) -HI -H /2 width=1 by/ +qed. + +lemma cir_appl: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄ → 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓐV.T⦄. +#G #L #V #T #HV #HT #H1 #H2 +elim (crr_inv_appl … H2) -H2 /2 width=1 by/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cir/cir_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cir/cir_lift.etc new file mode 100644 index 000000000..147d4b43c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cir/cir_lift.etc @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/reduction/crr_lift.ma". +include "basic_2/reduction/cir.ma". + +(* IRREDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION ************************) + +(* Properties on relocation *************************************************) + +lemma cir_lift: ∀G,K,T. ⦃G, K⦄ ⊢ ➡ 𝐈⦃T⦄ → ∀L,c,l,k. ⬇[c, l, k] L ≡ K → + ∀U. ⬆[l, k] T ≡ U → ⦃G, L⦄ ⊢ ➡ 𝐈⦃U⦄. +/3 width=8 by crr_inv_lift/ qed. + +lemma cir_inv_lift: ∀G,L,U. ⦃G, L⦄ ⊢ ➡ 𝐈⦃U⦄ → ∀K,c,l,k. ⬇[c, l, k] L ≡ K → + ∀T. ⬆[l, k] T ≡ U → ⦃G, K⦄ ⊢ ➡ 𝐈⦃T⦄. +/3 width=8 by crr_lift/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cir/prednotreducible_3.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cir/prednotreducible_3.etc new file mode 100644 index 000000000..4ad7cbc72 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cir/prednotreducible_3.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ ➡ 𝐈 break ⦃ term 46 T ⦄ )" + non associative with precedence 45 + for @{ 'PRedNotReducible $G $L $T }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cix/cix.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cix/cix.etc new file mode 100644 index 000000000..5a7215ddf --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cix/cix.etc @@ -0,0 +1,93 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/notation/relations/prednotreducible_5.ma". +include "basic_2/reduction/cir.ma". +include "basic_2/reduction/crx.ma". + +(* IRREDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ***************) + +definition cix: ∀h. sd h → relation3 genv lenv term ≝ + λh,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → ⊥. + +interpretation "irreducibility for context-sensitive extended reduction (term)" + 'PRedNotReducible h o G L T = (cix h o G L T). + +(* Basic inversion lemmas ***************************************************) + +lemma cix_inv_sort: ∀h,o,G,L,s,d. deg h o s (d+1) → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃⋆s⦄ → ⊥. +/3 width=2 by crx_sort/ qed-. + +lemma cix_inv_delta: ∀h,o,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃#i⦄ → ⊥. +/3 width=4 by crx_delta/ qed-. + +lemma cix_inv_ri2: ∀h,o,I,G,L,V,T. ri2 I → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃②{I}V.T⦄ → ⊥. +/3 width=1 by crx_ri2/ qed-. + +lemma cix_inv_ib2: ∀h,o,a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓑ{a,I}V.T⦄ → + ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, o] 𝐈⦃T⦄. +/4 width=1 by crx_ib2_sn, crx_ib2_dx, conj/ qed-. + +lemma cix_inv_bind: ∀h,o,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓑ{a,I}V.T⦄ → + ∧∧ ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ & ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ & ib2 a I. +#h #o #a * [ elim a -a ] +#G #L #V #T #H [ elim H -H /3 width=1 by crx_ri2, or_introl/ ] +elim (cix_inv_ib2 … H) -H /3 width=1 by and3_intro, or_introl/ +qed-. + +lemma cix_inv_appl: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓐV.T⦄ → + ∧∧ ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ & 𝐒⦃T⦄. +#h #o #G #L #V #T #HVT @and3_intro /3 width=1 by crx_appl_sn, crx_appl_dx/ +generalize in match HVT; -HVT elim T -T // +* // #a * #U #T #_ #_ #H elim H -H /2 width=1 by crx_beta, crx_theta/ +qed-. + +lemma cix_inv_flat: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓕ{I}V.T⦄ → + ∧∧ ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl. +#h #o * #G #L #V #T #H +[ elim (cix_inv_appl … H) -H /2 width=1 by and4_intro/ +| elim (cix_inv_ri2 … H) -H // +] +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma cix_inv_cir: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄. +/3 width=1 by crr_crx/ qed-. + +(* Basic properties *********************************************************) + +lemma cix_sort: ∀h,o,G,L,s. deg h o s 0 → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃⋆s⦄. +#h #o #G #L #s #Hk #H elim (crx_inv_sort … H) -L #d #Hkd +lapply (deg_mono … Hk Hkd) -h -s (cpr_inv_sort1 … H) // +qed. + +lemma cnr_lref_free: ∀G,L,i. |L| ≤ i → ⦃G, L⦄ ⊢ ➡ 𝐍⦃#i⦄. +#G #L #i #Hi #X #H elim (cpr_inv_lref1 … H) -H // * +#K #V1 #V2 #HLK lapply (drop_fwd_length_lt2 … HLK) -HLK +#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/ +qed. + +(* Basic_1: was only: nf2_csort_lref *) +lemma cnr_lref_atom: ∀G,L,i. ⬇[i] L ≡ ⋆ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃#i⦄. +#G #L #i #HL @cnr_lref_free >(drop_fwd_length … HL) -HL // +qed. + +(* Basic_1: was: nf2_abst *) +lemma cnr_abst: ∀a,G,L,W,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ➡ 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓛ{a}W.T⦄. +#a #G #L #W #T #HW #HT #X #H +elim (cpr_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct +>(HW … HW0) -W0 >(HT … HT0) -T0 // +qed. + +(* Basic_1: was only: nf2_appl_lref *) +lemma cnr_appl_simple: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓐV.T⦄. +#G #L #V #T #HV #HT #HS #X #H +elim (cpr_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct +>(HV … HV0) -V0 >(HT … HT0) -T0 // +qed. + +(* Basic_1: was: nf2_dec *) +axiom cnr_dec: ∀G,L,T1. ⦃G, L⦄ ⊢ ➡ 𝐍⦃T1⦄ ∨ + ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡ T2 & (T1 = T2 → ⊥). + +(* Basic_1: removed theorems 1: nf2_abst_shift *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cnr/cnr_cir.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cnr/cnr_cir.etc new file mode 100644 index 000000000..480e359cb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cnr/cnr_cir.etc @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/reduction/cpr_cir.ma". +include "basic_2/reduction/cnr_crr.ma". + +(* NORMAL TERMS FOR CONTEXT-SENSITIVE REDUCTION *****************************) + +(* Main properties on irreducibility ****************************************) + +theorem cir_cnr: ∀G,L,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄. +/2 width=4 by cpr_fwd_cir/ qed. + +(* Main inversion lemmas on irreducibility **********************************) + +theorem cnr_inv_cir: ∀G,L,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄. +/2 width=5 by cnr_inv_crr/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cnr/cnr_crr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cnr/cnr_crr.etc new file mode 100644 index 000000000..31291bc3d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cnr/cnr_crr.etc @@ -0,0 +1,46 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/reduction/crr.ma". +include "basic_2/reduction/cnr.ma". + +(* NORMAL TERMS FOR CONTEXT-SENSITIVE REDUCTION *****************************) + +(* Advanced inversion lemmas on reducibility ********************************) + +(* Note: this property is unusual *) +lemma cnr_inv_crr: ∀G,L,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → ⊥. +#G #L #T #H elim H -L -T +[ #L #K #V #i #HLK #H + elim (cnr_inv_delta … HLK H) +| #L #V #T #_ #IHV #H + elim (cnr_inv_appl … H) -H /2 width=1 by/ +| #L #V #T #_ #IHT #H + elim (cnr_inv_appl … H) -H /2 width=1 by/ +| #I #L #V #T * #H1 #H2 destruct + [ elim (cnr_inv_zeta … H2) + | elim (cnr_inv_eps … H2) + ] +|5,6: #a * [ elim a ] #L #V #T * #H1 #_ #IH #H2 destruct + [1,3: elim (cnr_inv_abbr … H2) -H2 /2 width=1 by/ + |*: elim (cnr_inv_abst … H2) -H2 /2 width=1 by/ + ] +| #a #L #V #W #T #H + elim (cnr_inv_appl … H) -H #_ #_ #H + elim (simple_inv_bind … H) +| #a #L #V #W #T #H + elim (cnr_inv_appl … H) -H #_ #_ #H + elim (simple_inv_bind … H) +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cnr/cnr_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cnr/cnr_lift.etc new file mode 100644 index 000000000..eef9aefc8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cnr/cnr_lift.etc @@ -0,0 +1,49 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/reduction/cpr_lift.ma". +include "basic_2/reduction/cnr.ma". + +(* NORMAL TERMS FOR CONTEXT-SENSITIVE REDUCTION *****************************) + +(* Advanced properties ******************************************************) + +(* Basic_1: was: nf2_lref_abst *) +lemma cnr_lref_abst: ∀G,L,K,V,i. ⬇[i] L ≡ K. ⓛV → ⦃G, L⦄ ⊢ ➡ 𝐍⦃#i⦄. +#G #L #K #V #i #HLK #X #H +elim (cpr_inv_lref1 … H) -H // * +#K0 #V1 #V2 #HLK0 #_ #_ +lapply (drop_mono … HLK … HLK0) -L #H destruct +qed. + +(* Relocation properties ****************************************************) + +(* Basic_1: was: nf2_lift *) +lemma cnr_lift: ∀G,L0,L,T,T0,c,l,k. ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → + ⬇[c, l, k] L0 ≡ L → ⬆[l, k] T ≡ T0 → ⦃G, L0⦄ ⊢ ➡ 𝐍⦃T0⦄. +#G #L0 #L #T #T0 #c #l #k #HLT #HL0 #HT0 #X #H +elim (cpr_inv_lift1 … H … HL0 … HT0) -L0 #T1 #HT10 #HT1 +<(HLT … HT1) in HT0; -L #HT0 +>(lift_mono … HT10 … HT0) -T1 -X // +qed. + +(* Note: this was missing in basic_1 *) +lemma cnr_inv_lift: ∀G,L0,L,T,T0,c,l,k. ⦃G, L0⦄ ⊢ ➡ 𝐍⦃T0⦄ → + ⬇[c, l, k] L0 ≡ L → ⬆[l, k] T ≡ T0 → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄. +#G #L0 #L #T #T0 #c #l #k #HLT0 #HL0 #HT0 #X #H +elim (lift_total X l k) #X0 #HX0 +lapply (cpr_lift … H … HL0 … HT0 … HX0) -L #HTX0 +>(HLT0 … HTX0) in HX0; -L0 -X0 #H +>(lift_inj … H … HT0) -T0 -X -c -l -k // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cnr/prednormal_3.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cnr/prednormal_3.etc new file mode 100644 index 000000000..a8806a1c7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cnr/prednormal_3.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ ➡ 𝐍 break ⦃ term 46 T ⦄ )" + non associative with precedence 45 + for @{ 'PRedNormal $G $L $T }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cnx/cnx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cnx/cnx.etc new file mode 100644 index 000000000..0259a08aa --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cnx/cnx.etc @@ -0,0 +1,136 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/notation/relations/prednormal_5.ma". +include "basic_2/reduction/cnr.ma". +include "basic_2/reduction/cpx.ma". + +(* NORMAL TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ********************) + +definition cnx: ∀h. sd h → relation3 genv lenv term ≝ + λh,o,G,L. NF … (cpx h o G L) (eq …). + +interpretation + "normality for context-sensitive extended reduction (term)" + 'PRedNormal h o L T = (cnx h o L T). + +(* Basic inversion lemmas ***************************************************) + +lemma cnx_inv_sort: ∀h,o,G,L,s. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃⋆s⦄ → deg h o s 0. +#h #o #G #L #s #H elim (deg_total h o s) +#d @(nat_ind_plus … d) -d // #d #_ #Hkd +lapply (H (⋆(next h s)) ?) -H /2 width=2 by cpx_st/ -L -d #H +lapply (destruct_tatom_tatom_aux … H) -H #H (**) (* destruct lemma needed *) +lapply (destruct_sort_sort_aux … H) -H #H (**) (* destruct lemma needed *) +lapply (next_lt h s) >H -H #H elim (lt_refl_false … H) +qed-. + +lemma cnx_inv_delta: ∀h,o,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃#i⦄ → ⊥. +#h #o #I #G #L #K #V #i #HLK #H +elim (lift_total V 0 (i+1)) #W #HVW +lapply (H W ?) -H [ /3 width=7 by cpx_delta/ ] -HLK #H destruct +elim (lift_inv_lref2_be … HVW) -HVW /2 width=1 by ylt_inj/ +qed-. + +lemma cnx_inv_abst: ∀h,o,a,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓛ{a}V.T⦄ → + ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃V⦄ ∧ ⦃G, L.ⓛV⦄ ⊢ ➡[h, o] 𝐍⦃T⦄. +#h #o #a #G #L #V1 #T1 #HVT1 @conj +[ #V2 #HV2 lapply (HVT1 (ⓛ{a}V2.T1) ?) -HVT1 /2 width=2 by cpx_pair_sn/ -HV2 #H destruct // +| #T2 #HT2 lapply (HVT1 (ⓛ{a}V1.T2) ?) -HVT1 /2 width=2 by cpx_bind/ -HT2 #H destruct // +] +qed-. + +lemma cnx_inv_abbr: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃-ⓓV.T⦄ → + ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃V⦄ ∧ ⦃G, L.ⓓV⦄ ⊢ ➡[h, o] 𝐍⦃T⦄. +#h #o #G #L #V1 #T1 #HVT1 @conj +[ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2 by cpx_pair_sn/ -HV2 #H destruct // +| #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2 by cpx_bind/ -HT2 #H destruct // +] +qed-. + +lemma cnx_inv_zeta: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃+ⓓV.T⦄ → ⊥. +#h #o #G #L #V #T #H elim (is_lift_dec T 0 1) +[ * #U #HTU + lapply (H U ?) -H /2 width=3 by cpx_zeta/ #H destruct + elim (lift_inv_pair_xy_y … HTU) +| #HT + elim (cpr_delift G(⋆) V T (⋆.ⓓV) 0) // #T2 #T1 #HT2 #HT12 + lapply (H (+ⓓV.T2) ?) -H /5 width=1 by cpr_cpx, tpr_cpr, cpr_bind/ -HT2 + #H destruct /3 width=2 by ex_intro/ +] +qed-. + +lemma cnx_inv_appl: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓐV.T⦄ → + ∧∧ ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ & 𝐒⦃T⦄. +#h #o #G #L #V1 #T1 #HVT1 @and3_intro +[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1 by cpx_pair_sn/ -HV2 #H destruct // +| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1 by cpx_flat/ -HT2 #H destruct // +| generalize in match HVT1; -HVT1 elim T1 -T1 * // #a * #W1 #U1 #_ #_ #H + [ elim (lift_total V1 0 1) #V2 #HV12 + lapply (H (ⓓ{a}W1.ⓐV2.U1) ?) -H /3 width=3 by cpr_cpx, cpr_theta/ -HV12 #H destruct + | lapply (H (ⓓ{a}ⓝW1.V1.U1) ?) -H /3 width=1 by cpr_cpx, cpr_beta/ #H destruct + ] +] +qed-. + +lemma cnx_inv_eps: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓝV.T⦄ → ⊥. +#h #o #G #L #V #T #H lapply (H T ?) -H +/2 width=4 by cpx_eps, discr_tpair_xy_y/ +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma cnx_fwd_cnr: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄. +#h #o #G #L #T #H #U #HTU +@H /2 width=1 by cpr_cpx/ (**) (* auto fails because a δ-expansion gets in the way *) +qed-. + +(* Basic properties *********************************************************) + +lemma cnx_sort: ∀h,o,G,L,s. deg h o s 0 → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃⋆s⦄. +#h #o #G #L #s #Hk #X #H elim (cpx_inv_sort1 … H) -H // * #d #Hkd #_ +lapply (deg_mono … Hkd Hk) -h -L (drop_fwd_length … HL) -HL // +qed. + +lemma cnx_abst: ∀h,o,a,G,L,W,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → + ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓛ{a}W.T⦄. +#h #o #a #G #L #W #T #HW #HT #X #H +elim (cpx_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct +>(HW … HW0) -W0 >(HT … HT0) -T0 // +qed. + +lemma cnx_appl_simple: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → 𝐒⦃T⦄ → + ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓐV.T⦄. +#h #o #G #L #V #T #HV #HT #HS #X #H +elim (cpx_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct +>(HV … HV0) -V0 >(HT … HT0) -T0 // +qed. + +axiom cnx_dec: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T1⦄ ∨ + ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 & (T1 = T2 → ⊥). diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cnx/cnx_cix.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cnx/cnx_cix.etc new file mode 100644 index 000000000..32b9a76bf --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cnx/cnx_cix.etc @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/reduction/cpx_cix.ma". +include "basic_2/reduction/cnx_crx.ma". + +(* NORMAL TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ********************) + +(* Main properties on irreducibility ****************************************) + +theorem cix_cnx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄. +/2 width=6 by cpx_fwd_cix/ qed. + +(* Main inversion lemmas on irreducibility **********************************) + +theorem cnx_inv_cix: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄. +/2 width=7 by cnx_inv_crx/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cnx/cnx_crx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cnx/cnx_crx.etc new file mode 100644 index 000000000..bb251965e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cnx/cnx_crx.etc @@ -0,0 +1,49 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/reduction/crx.ma". +include "basic_2/reduction/cnx.ma". + +(* NORMAL TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ********************) + +(* Advanced inversion lemmas on reducibility ********************************) + +(* Note: this property is unusual *) +lemma cnx_inv_crx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ⊥. +#h #o #G #L #T #H elim H -L -T +[ #L #s #d #Hkd #H + lapply (cnx_inv_sort … H) -H #H + lapply (deg_mono … H Hkd) -h -L -s (lift_mono … HT10 … HT0) -T1 -X // +qed. + +lemma cnx_inv_lift: ∀h,o,G,L0,L,T,T0,c,l,k. ⦃G, L0⦄ ⊢ ➡[h, o] 𝐍⦃T0⦄ → ⬇[c, l, k] L0 ≡ L → + ⬆[l, k] T ≡ T0 → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄. +#h #o #G #L0 #L #T #T0 #c #l #k #HLT0 #HL0 #HT0 #X #H +elim (lift_total X l k) #X0 #HX0 +lapply (cpx_lift … H … HL0 … HT0 … HX0) -L #HTX0 +>(HLT0 … HTX0) in HX0; -L0 -X0 #H +>(lift_inj … H … HT0) -T0 -X -l -k // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cnx/prednormal_5.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cnx/prednormal_5.etc new file mode 100644 index 000000000..9df81e650 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cnx/prednormal_5.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ➡ break [ term 46 o , break term 46 h ] 𝐍 break ⦃ term 46 T ⦄ )" + non associative with precedence 45 + for @{ 'PRedNormal $h $o $G $L $T }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpg/cpg_drops.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpg/cpg_drops.etc new file mode 100644 index 000000000..c0447ffde --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpg/cpg_drops.etc @@ -0,0 +1,16 @@ +lemma cpg_delift: ∀c,h,I,G,K,V,T1,L,i. ⬇*[i] L ≡ (K.ⓑ{I}V) → + ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡[h, 𝟘𝟘] T2 & ⬆*[↑1] T ≡ T2. +#h #c #I #G #K #V #T1 elim T1 -T1 +[ * #i #L #l /2 width=4 by cpg_atom, lift_sort, lift_gref, ex2_2_intro/ + elim (lt_or_eq_or_gt i l) #Hil [1,3: /4 width=4 by cpg_atom, lift_lref_ge_minus, lift_lref_lt, ylt_inj, yle_inj, ex2_2_intro/ ] + destruct + elim (lift_total V 0 (i+1)) #W #HVW + elim (lift_split … HVW i i) /3 width=7 by cpg_delta, ex2_2_intro/ +| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK + elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2 + [ elim (IHU1 (L. ⓑ{I} W1) (l+1)) -IHU1 /3 width=9 by cpg_bind, drop_drop, lift_bind, ex2_2_intro/ + | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpg_flat, lift_flat, ex2_2_intro/ + ] +] +qed-. +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpg/cpg_length.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpg/cpg_length.etc new file mode 100644 index 000000000..142f0fd2c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpg/cpg_length.etc @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/relocation/drops.ma". +include "basic_2/rt_transition/cpg.ma". + +(* CONTEXT-SENSITIVE GENERIC PARALLEL RT-TRANSITION FOR TERMS ***************) + +(* Properties with length for local environments ****************************) + +lemma cpg_inv_lref1_ge: ∀h,c,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, c] T2 → |L| ≤ i → T2 = #i. +#h #c #G #L #T2 #i #H elim (cpg_inv_lref1 … H) -H // * +#I #K #V1 #V2 #HLK #_ #_ #HL -h -G -V2 lapply (drop_fwd_length_lt2 … HLK) -K -I -V1 +#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpr/cpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpr/cpr.etc new file mode 100644 index 000000000..3f30e621f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpr/cpr.etc @@ -0,0 +1,29 @@ +lemma cpr_delift: ∀G,K,V,T1,L,l. ⬇[l] L ≡ (K.ⓓV) → + ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡ T2 & ⬆[l, 1] T ≡ T2. +#G #K #V #T1 elim T1 -T1 +[ * /2 width=4 by cpr_atom, lift_sort, lift_gref, ex2_2_intro/ + #i #L #l #HLK elim (lt_or_eq_or_gt i l) + #Hil [1,3: /4 width=4 by lift_lref_ge_minus, lift_lref_lt, ylt_inj, yle_inj, ex2_2_intro/ ] + destruct + elim (lift_total V 0 (i+1)) #W #HVW + elim (lift_split … HVW i i) /3 width=6 by cpr_delta, ex2_2_intro/ +| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK + elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2 + [ elim (IHU1 (L. ⓑ{I}W1) (l+1)) -IHU1 /3 width=9 by drop_drop, cpr_bind, lift_bind, ex2_2_intro/ + | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpr_flat, lift_flat, ex2_2_intro/ + ] +] +qed-. + +fact lstas_cpr_aux: ∀h,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*[h, d] T2 → + d = 0 → ⦃G, L⦄ ⊢ T1 ➡ T2. +#h #G #L #T1 #T2 #d #H elim H -G -L -T1 -T2 -d +/3 width=1 by cpr_eps, cpr_flat, cpr_bind/ +[ #G #L #K #V1 #V2 #W2 #i #d #HLK #_ #HVW2 #IHV12 #H destruct + /3 width=6 by cpr_delta/ +| #G #L #K #V1 #V2 #W2 #i #d #_ #_ #_ #_ IHV1 -IHV1 // -HV1 >IHT1 -IHT1 // + | elim (cir_inv_ri2 … H) /2 width=1 by/ + ] +| #G #L #V1 #T1 #T #T2 #_ #_ #_ #H + elim (cir_inv_ri2 … H) /2 width=1 by or_introl/ +| #G #L #V1 #T1 #T2 #_ #_ #H + elim (cir_inv_ri2 … H) /2 width=1 by/ +| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H + elim (cir_inv_appl … H) -H #_ #_ #H + elim (simple_inv_bind … H) +| #a #G #L #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H + elim (cir_inv_appl … H) -H #_ #_ #H + elim (simple_inv_bind … H) +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpx/cpx_cix.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpx/cpx_cix.etc new file mode 100644 index 000000000..9bb51aebc --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpx/cpx_cix.etc @@ -0,0 +1,52 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/reduction/cix.ma". +include "basic_2/reduction/cpx.ma". + +(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************) + +(* Advanced forward lemmas on irreducibility ********************************) + +lemma cpx_fwd_cix: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T1⦄ → T2 = T1. +#h #o #G #L #T1 #T2 #H elim H -G -L -T1 -T2 +[ // +| #G #L #s #d #Hkd #H elim (cix_inv_sort … Hkd H) +| #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #H + elim (cix_inv_delta … HLK) // +| #a * #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #H + [ elim (cix_inv_bind … H) -H #HV1 #HT1 * #H destruct + lapply (IHV1 … HV1) -IHV1 -HV1 #H destruct + lapply (IHT1 … HT1) -IHT1 #H destruct // + | elim (cix_inv_ib2 … H) -H /3 width=2 by or_introl, eq_f2/ + ] +| * #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #H + [ elim (cix_inv_appl … H) -H #HV1 #HT1 #_ + >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 // + | elim (cix_inv_ri2 … H) /2 width=1 by/ + ] +| #G #L #V1 #T1 #T #T2 #_ #_ #_ #H + elim (cix_inv_ri2 … H) /2 width=1 by or_introl/ +| #G #L #V1 #T1 #T2 #_ #_ #H + elim (cix_inv_ri2 … H) /2 width=1 by/ +| #G #L #V1 #V2 #T #_ #_ #H + elim (cix_inv_ri2 … H) /2 width=1 by/ +| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H + elim (cix_inv_appl … H) -H #_ #_ #H + elim (simple_inv_bind … H) +| #a #G #L #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H + elim (cix_inv_appl … H) -H #_ #_ #H + elim (simple_inv_bind … H) +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpx/cpx_drops.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpx/cpx_drops.etc new file mode 100644 index 000000000..3dd5f3a48 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpx/cpx_drops.etc @@ -0,0 +1,18 @@ +lemma cpx_delift: ∀h,I,G,K,V,T1,L,l. ⬇[l] L ≡ (K.ⓑ{I}V) → + ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡[h] T2 & ⬆[l, 1] T ≡ T2. +#h #o #I #G #K #V #T1 elim T1 -T1 +[ * #i #L #l /2 width=4 by cpx_atom, lift_sort, lift_gref, ex2_2_intro/ + elim (lt_or_eq_or_gt i l) #Hil [1,3: /4 width=4 by cpx_atom, +lift_lref_ge_minus, lift_lref_lt, ylt_inj, yle_inj, ex2_2_intro/ ] + destruct + elim (lift_total V 0 (i+1)) #W #HVW + elim (lift_split … HVW i i) /3 width=7 by cpx_delta, ex2_2_intro/ +| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK + elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2 + [ elim (IHU1 (L. ⓑ{I} W1) (l+1)) -IHU1 /3 width=9 by cpx_bind, +drop_drop, lift_bind, ex2_2_intro/ + | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpx_flat, lift_flat, +ex2_2_intro/ + ] +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpx/cpx_length.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpx/cpx_length.etc new file mode 100644 index 000000000..c769990b2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpx/cpx_length.etc @@ -0,0 +1 @@ +lemma cpx_inv_lref1_ge: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h] T2 → |L| ≤ i → T2 = #i. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpx/cpx_sta.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpx/cpx_sta.etc new file mode 100644 index 000000000..3925f9d43 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpx/cpx_sta.etc @@ -0,0 +1,36 @@ +fact sta_cpx_aux: ∀h,o,G,L,T1,T2,d2,d1. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 → d2 = 1 → + ⦃G, L⦄ ⊢ T1 ▪[h, o] d1+1 → ⦃G, L⦄ ⊢ T1 ➡[h, o] T2. +#h #o #G #L #T1 #T2 #d2 #d1 #H elim H -G -L -T1 -T2 -d2 +[ #G #L #d2 #s #H0 destruct normalize + /3 width=4 by cpx_st, da_inv_sort/ +| #G #L #K #V1 #V2 #W2 #i #d2 #HLK #_ #HVW2 #IHV12 #H0 #H destruct + elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0 + lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct /3 width=7 by cpx_delta/ +| #G #L #K #V1 #V2 #i #_ #_ #_ #H destruct +| #G #L #K #V1 #V2 #W2 #i #d2 #HLK #HV12 #HVW2 #_ #H0 #H + lapply (discr_plus_xy_y … H0) -H0 #H0 destruct + elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0 + lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct + /4 width=7 by cpx_delta, cpr_cpx, lstas_cpr/ +| /4 width=2 by cpx_bind, da_inv_bind/ +| /4 width=3 by cpx_flat, da_inv_flat/ +| /4 width=3 by cpx_eps, da_inv_flat/ +] +qed-. + +lemma sta_cpx: ∀h,o,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 → + ⦃G, L⦄ ⊢ T1 ▪[h, o] d+1 → ⦃G, L⦄ ⊢ T1 ➡[h, o] T2. +/2 width=3 by sta_cpx_aux/ qed. + +lemma fqu_sta_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → + ∀U2. ⦃G2, L2⦄ ⊢ T2 •*[h, 1] U2 → + ∀d. ⦃G2, L2⦄ ⊢ T2 ▪[h, o] d+1 → + ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄. +/3 width=5 by fqu_cpx_trans, sta_cpx/ qed-. + +lemma fquq_sta_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → + ∀U2. ⦃G2, L2⦄ ⊢ T2 •*[h, 1] U2 → + ∀d. ⦃G2, L2⦄ ⊢ T2 ▪[h, o] d+1 → + ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄. +/3 width=5 by fquq_cpx_trans, sta_cpx/ qed-. + diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpy.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpy.etc new file mode 100644 index 000000000..4a27de0a4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpy.etc @@ -0,0 +1,286 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/notation/relations/psubst_6.ma". +include "basic_2/grammar/genv.ma". +include "basic_2/substitution/lsuby.ma". + +(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************) + +(* activate genv *) +inductive cpy: ynat → ynat → relation4 genv lenv term term ≝ +| cpy_atom : ∀I,G,L,l,m. cpy l m G L (⓪{I}) (⓪{I}) +| cpy_subst: ∀I,G,L,K,V,W,i,l,m. l ≤ i → i < l+m → + ⬇[i] L ≡ K.ⓑ{I}V → ⬆[0, ⫯i] V ≡ W → cpy l m G L (#i) W +| cpy_bind : ∀a,I,G,L,V1,V2,T1,T2,l,m. + cpy l m G L V1 V2 → cpy (⫯l) m G (L.ⓑ{I}V1) T1 T2 → + cpy l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) +| cpy_flat : ∀I,G,L,V1,V2,T1,T2,l,m. + cpy l m G L V1 V2 → cpy l m G L T1 T2 → + cpy l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) +. + +interpretation "context-sensitive extended ordinary substritution (term)" + 'PSubst G L T1 l m T2 = (cpy l m G L T1 T2). + +(* Basic properties *********************************************************) + +lemma lsuby_cpy_trans: ∀G,l,m. lsub_trans … (cpy l m G) (lsuby l m). +#G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m +[ // +| #I #G #L1 #K1 #V #W #i #l #m #Hli #Hilm #HLK1 #HVW #L2 #HL12 + elim (ylt_inv_plus_dx … Hilm) #m0 #H0 #_ + elim (lsuby_drop_trans_be … HL12 … HLK1 … H0) -HL12 -HLK1 -H0 /2 width=5 by cpy_subst/ +| /4 width=1 by lsuby_succ, cpy_bind/ +| /3 width=1 by cpy_flat/ +] +qed-. + +lemma cpy_refl: ∀G,T,L,l,m. ⦃G, L⦄ ⊢ T ▶[l, m] T. +#G #T elim T -T // * /2 width=1 by cpy_bind, cpy_flat/ +qed. + +(* Basic_1: was: subst1_ex *) +lemma cpy_full: ∀I,G,K,V,T1,L,l. ⬇[l] L ≡ K.ⓑ{I}V → + ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2 & ⬆[l, 1] T ≡ T2. +#I #G #K #V #T1 elim T1 -T1 +[ * #i #L #l #HLK + /2 width=4 by lift_sort, lift_gref, ex2_2_intro/ + elim (ylt_split_eq i l) /3 width=4 by lift_lref_pred, lift_lref_lt, ex2_2_intro/ + #H destruct lapply (drop_fwd_Y2 … HLK) #Hi + elim (lift_total (⫯i) … V 0) /2 width=1 by ylt_succ/ + #W #HVW elim (lift_split … HVW i i … 1) + /4 width=6 by cpy_subst, monotonic_ylt_plus_sn, ex2_2_intro/ +| * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #l #HLK + elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2 + [ elim (IHU1 (L.ⓑ{J}W1) (⫯l)) -IHU1 + /3 width=9 by cpy_bind, drop_drop, lift_bind, ex2_2_intro/ + | elim (IHU1 … HLK) -IHU1 -HLK + /3 width=8 by cpy_flat, lift_flat, ex2_2_intro/ + ] +] +qed-. + +lemma cpy_weak: ∀G,L,T1,T2,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T2 → + ∀l2,m2. l2 ≤ l1 → l1 + m1 ≤ l2 + m2 → + ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T2. +#G #L #T1 #T2 #l1 #m1 #H elim H -G -L -T1 -T2 -l1 -m1 // +[ /3 width=5 by cpy_subst, ylt_yle_trans, yle_trans/ +| /4 width=3 by cpy_bind, ylt_yle_trans, yle_succ/ +| /3 width=1 by cpy_flat/ +] +qed-. +(* +lemma cpy_weak_top: ∀G,L,T1,T2,l,m. + ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, ∞] T2. +/2 width=5 by cpy_weak/ qed-. + +lemma cpy_weak_full: ∀G,L,T1,T2,l,m. + ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[0, ∞] T2. +/2 width=5 by cpy_weak/ qed-. +*) +lemma cpy_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → + ∀i,m2. i + m2 = l + m → + ∀m1. i ≤ l + m1 → + ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l, m1] T & ⦃G, L⦄ ⊢ T ▶[i, m2] T2. +#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m +[ /2 width=3 by ex2_intro/ +| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #m2 #H2 #m1 #H1 + elim (ylt_split i j) [ -Hilm -H2 | -Hli ] + /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/ +| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #m2 #H2 #m1 #H1 + elim (IHV12 … H2 … H1) -IHV12 #V + elim (IHT12 (⫯i) … m2 … m1) -IHT12 /2 width=1 by yle_succ/ -H2 -H1 + #T #HT1 #HT2 lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 + /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/ +| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #m2 #H2 #m1 #H1 + elim (IHV12 … H2 … H1) -IHV12 elim (IHT12 … H2 … H1) -IHT12 -H2 -H1 + /3 width=5 by ex2_intro, cpy_flat/ +] +qed-. + +lemma cpy_split_down: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → + ∀m1,m2. m = m1 + m2 → + ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l+m2, m1] T & ⦃G, L⦄ ⊢ T ▶[l, m2] T2. +#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m +[ /2 width=3 by ex2_intro/ +| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #m1 #m2 #H destruct + elim (ylt_split i (l+m2)) [ -Hilm | -Hli ] + /3 width=9 by cpy_subst, ex2_intro/ +| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #m1 #m2 #H destruct + elim (IHV12 m1 m2) -IHV12 // #V + elim (IHT12 m1 m2) -IHT12 // + >yplus_succ1 #T #HT1 #HT2 + lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 + /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/ +| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #m1 #m2 #H destruct + elim (IHV12 m1 m2) -IHV12 // elim (IHT12 m1 m2) -IHT12 // + /3 width=5 by ex2_intro, cpy_flat/ +] +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma cpy_fwd_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 → + ∀T1,l,m. ⬆[l, m] T1 ≡ U1 → + l ≤ lt → l + m ≤ lt + mt → + ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶[l+m, lt+mt-(l+m)] U2 & ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt +[ * #i #G #L #lt #mt #T1 #l #m #H #_ + [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/ + | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/ + | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/ + ] +| #I #G #L #K #V #W #i #lt #mt #Hlti #Hilmt #HLK #HVW #T1 #l #m #H #Hllt #Hlmlmt + elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -V -Hilmt -Hlmlmt | -Hlti -Hllt ] + [ elim (ylt_yle_false … Hllt) -Hllt /3 width=3 by yle_ylt_trans, ylt_inj/ + | elim (yle_inv_plus_inj2 … Hil) #Hlim #Hmi + elim (lift_split … HVW l (⫯(i-m)) ? ? ?) [2,3,4: /2 width=1 by yle_succ_dx, le_S_S/ ] -Hlim + #T2 #_ >plus_minus /2 width=1 by yle_inv_inj/ ymax_pre_sn_comm // (**) (* explicit constructor *) + ] +| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt + elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct + elim (IHW12 … HVW1) -V1 -IHW12 // + elim (IHU12 … HTU1) -T1 -IHU12 /2 width=1 by yle_succ/ + yplus_SO2 >yplus_succ1 >yplus_succ1 + /3 width=2 by cpy_bind, lift_bind, ex2_intro/ +| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt + elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct + elim (IHW12 … HVW1) -V1 -IHW12 // elim (IHU12 … HTU1) -T1 -IHU12 + /3 width=2 by cpy_flat, lift_flat, ex2_intro/ +] +qed-. + +lemma cpy_fwd_tw: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ♯{T1} ≤ ♯{T2}. +#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m normalize +/3 width=1 by monotonic_le_plus_l, le_plus/ +qed-. + +(* Basic inversion lemmas ***************************************************) + +fact cpy_inv_atom1_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀J. T1 = ⓪{J} → + T2 = ⓪{J} ∨ + ∃∃I,K,V,i. l ≤ yinj i & i < l + m & + ⬇[i] L ≡ K.ⓑ{I}V & + ⬆[O, ⫯i] V ≡ T2 & + J = LRef i. +#G #L #T1 #T2 #l #m * -G -L -T1 -T2 -l -m +[ #I #G #L #l #m #J #H destruct /2 width=1 by or_introl/ +| #I #G #L #K #V #T2 #i #l #m #Hli #Hilm #HLK #HVT2 #J #H destruct /3 width=9 by ex5_4_intro, or_intror/ +| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct +| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct +] +qed-. + +lemma cpy_inv_atom1: ∀I,G,L,T2,l,m. ⦃G, L⦄ ⊢ ⓪{I} ▶[l, m] T2 → + T2 = ⓪{I} ∨ + ∃∃J,K,V,i. l ≤ yinj i & i < l + m & + ⬇[i] L ≡ K.ⓑ{J}V & + ⬆[O, ⫯i] V ≡ T2 & + I = LRef i. +/2 width=4 by cpy_inv_atom1_aux/ qed-. + +(* Basic_1: was: subst1_gen_sort *) +lemma cpy_inv_sort1: ∀G,L,T2,k,l,m. ⦃G, L⦄ ⊢ ⋆k ▶[l, m] T2 → T2 = ⋆k. +#G #L #T2 #k #l #m #H +elim (cpy_inv_atom1 … H) -H // +* #I #K #V #i #_ #_ #_ #_ #H destruct +qed-. + +(* Basic_1: was: subst1_gen_lref *) +lemma cpy_inv_lref1: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶[l, m] T2 → + T2 = #i ∨ + ∃∃I,K,V. l ≤ i & i < l + m & + ⬇[i] L ≡ K.ⓑ{I}V & + ⬆[O, ⫯i] V ≡ T2. +#G #L #T2 #i #l #m #H +elim (cpy_inv_atom1 … H) -H /2 width=1 by or_introl/ +* #I #K #V #j #Hlj #Hjlm #HLK #HVT2 #H destruct /3 width=5 by ex4_3_intro, or_intror/ +qed-. + +lemma cpy_inv_gref1: ∀G,L,T2,p,l,m. ⦃G, L⦄ ⊢ §p ▶[l, m] T2 → T2 = §p. +#G #L #T2 #p #l #m #H +elim (cpy_inv_atom1 … H) -H // +* #I #K #V #i #_ #_ #_ #_ #H destruct +qed-. + +fact cpy_inv_bind1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → + ∀a,I,V1,T1. U1 = ⓑ{a,I}V1.T1 → + ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 & + ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ▶[⫯l, m] T2 & + U2 = ⓑ{a,I}V2.T2. +#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m +[ #I #G #L #l #m #b #J #W1 #U1 #H destruct +| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #b #J #W1 #U1 #H destruct +| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/ +| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #b #J #W1 #U1 #H destruct +] +qed-. + +lemma cpy_inv_bind1: ∀a,I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓑ{a,I} V1. T1 ▶[l, m] U2 → + ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 & + ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶[⫯l, m] T2 & + U2 = ⓑ{a,I}V2.T2. +/2 width=3 by cpy_inv_bind1_aux/ qed-. + +fact cpy_inv_flat1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → + ∀I,V1,T1. U1 = ⓕ{I}V1.T1 → + ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 & + ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 & + U2 = ⓕ{I}V2.T2. +#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m +[ #I #G #L #l #m #J #W1 #U1 #H destruct +| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #J #W1 #U1 #H destruct +| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #W1 #U1 #H destruct +| #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/ +] +qed-. + +lemma cpy_inv_flat1: ∀I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ▶[l, m] U2 → + ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 & + ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 & + U2 = ⓕ{I}V2.T2. +/2 width=3 by cpy_inv_flat1_aux/ qed-. + + +fact cpy_inv_refl_O2_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → m = 0 → T1 = T2. +#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m +[ // +| #I #G #L #K #V #W #i #l #m #Hli #Hilm #_ #_ #H destruct + elim (ylt_yle_false … Hli) -Hli // +| /3 width=1 by eq_f2/ +| /3 width=1 by eq_f2/ +] +qed-. + +lemma cpy_inv_refl_O2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶[l, 0] T2 → T1 = T2. +/2 width=6 by cpy_inv_refl_O2_aux/ qed-. + +(* Basic_1: was: subst1_gen_lift_eq *) +lemma cpy_inv_lift1_eq: ∀G,T1,U1,l,m. ⬆[l, m] T1 ≡ U1 → + ∀L,U2. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → U1 = U2. +#G #T1 #U1 #l #m #HTU1 #L #U2 #HU12 elim (cpy_fwd_up … HU12 … HTU1) -HU12 -HTU1 +/2 width=4 by cpy_inv_refl_O2/ +qed-. + +(* Basic_1: removed theorems 25: + subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt + subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans + subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s + subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt + subst0_confluence_neq subst0_confluence_eq subst0_tlt_head + subst0_confluence_lift subst0_tlt + subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpy_cpy.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpy_cpy.etc new file mode 100644 index 000000000..7a65a2db5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpy_cpy.etc @@ -0,0 +1,123 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/substitution/cpy_lift.ma". + +(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************) + +(* Main properties **********************************************************) + +(* Basic_1: was: subst1_confluence_eq *) +theorem cpy_conf_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶[l1, m1] T1 → + ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → + ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶[l1, m1] T. +#G #L #T0 #T1 #l1 #m1 #H elim H -G -L -T0 -T1 -l1 -m1 +[ /2 width=3 by ex2_intro/ +| #I1 #G #L #K1 #V1 #T1 #i0 #l1 #m1 #Hl1 #Hlm1 #HLK1 #HVT1 #T2 #l2 #m2 #H + elim (cpy_inv_lref1 … H) -H + [ #HX destruct /3 width=7 by cpy_subst, ex2_intro/ + | -Hl1 -Hlm1 * #I2 #K2 #V2 #_ #_ #HLK2 #HVT2 + lapply (drop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct + >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3 by ex2_intro/ + ] +| #a #I #G #L #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #X #l2 #m2 #HX + elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV01 … HV02) -IHV01 -HV02 #V #HV1 #HV2 + elim (IHT01 … HT02) -T0 #T #HT1 #HT2 + lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/ + lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V2) ?) -HT2 + /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/ +| #I #G #L #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #X #l2 #m2 #HX + elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV01 … HV02) -V0 + elim (IHT01 … HT02) -T0 /3 width=5 by cpy_flat, ex2_intro/ +] +qed-. + +(* Basic_1: was: subst1_confluence_neq *) +theorem cpy_conf_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶[l1, m1] T1 → + ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶[l2, m2] T2 → + (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) → + ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶[l1, m1] T. +#G #L1 #T0 #T1 #l1 #m1 #H elim H -G -L1 -T0 -T1 -l1 -m1 +[ /2 width=3 by ex2_intro/ +| #I1 #G #L1 #K1 #V1 #T1 #i0 #l1 #m1 #Hl1 #Hlm1 #HLK1 #HVT1 #L2 #T2 #l2 #m2 #H1 #H2 + elim (cpy_inv_lref1 … H1) -H1 + [ #H destruct /3 width=7 by cpy_subst, ex2_intro/ + | -HLK1 -HVT1 * #I2 #K2 #V2 #Hl2 #Hlm2 #_ #_ elim H2 -H2 #Hlml [ -Hl1 -Hlm2 | -Hl2 -Hlm1 ] + [ elim (ylt_yle_false … Hlm1) -Hlm1 /2 width=3 by yle_trans/ + | elim (ylt_yle_false … Hlm2) -Hlm2 /2 width=3 by yle_trans/ + ] + ] +| #a #I #G #L1 #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #L2 #X #l2 #m2 #HX #H + elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV01 … HV02 H) -IHV01 -HV02 #V #HV1 #HV2 + elim (IHT01 … HT02) -T0 + [ -H #T #HT1 #HT2 + lapply (lsuby_cpy_trans … HT1 (L2.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/ + lapply (lsuby_cpy_trans … HT2 (L1.ⓑ{I}V2) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/ + | -HV1 -HV2 elim H -H /3 width=1 by yle_succ, or_introl, or_intror/ + ] +| #I #G #L1 #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #L2 #X #l2 #m2 #HX #H + elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV01 … HV02 H) -V0 + elim (IHT01 … HT02 H) -T0 -H /3 width=5 by cpy_flat, ex2_intro/ +] +qed-. + +(* Note: the constant 1 comes from cpy_subst *) +(* Basic_1: was: subst1_trans *) +theorem cpy_trans_ge: ∀G,L,T1,T0,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T0 → + ∀T2. ⦃G, L⦄ ⊢ T0 ▶[l, 1] T2 → 1 ≤ m → ⦃G, L⦄ ⊢ T1 ▶[l, m] T2. +#G #L #T1 #T0 #l #m #H elim H -G -L -T1 -T0 -l -m +[ #I #G #L #l #m #T2 #H #Hm + elim (cpy_inv_atom1 … H) -H + [ #H destruct // + | * #J #K #V #i #Hl2i #Hilm2 #HLK #HVT2 #H destruct + lapply (ylt_yle_trans … (l+m) … Hilm2) + /2 width=5 by cpy_subst, monotonic_yle_plus_sn/ + ] +| #I #G #L #K #V #V2 #i #l #m #Hli #Hilm #HLK #HVW #T2 #HVT2 #Hm + lapply (cpy_weak … HVT2 0 (i+1) ? ?) -HVT2 /3 width=1 by yle_plus_dx2_trans, yle_succ/ + >yplus_inj #HVT2 <(cpy_inv_lift1_eq … HVW … HVT2) -HVT2 /2 width=5 by cpy_subst/ +| #a #I #G #L #V1 #V0 #T1 #T0 #l #m #_ #_ #IHV10 #IHT10 #X #H #Hm + elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct + lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02 + lapply (IHT10 … HT02 Hm) -T0 /3 width=1 by cpy_bind/ +| #I #G #L #V1 #V0 #T1 #T0 #l #m #_ #_ #IHV10 #IHT10 #X #H #Hm + elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1 by cpy_flat/ +] +qed-. + +theorem cpy_trans_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T0 → + ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → l2 + m2 ≤ l1 → + ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T ▶[l1, m1] T2. +#G #L #T1 #T0 #l1 #m1 #H elim H -G -L -T1 -T0 -l1 -m1 +[ /2 width=3 by ex2_intro/ +| #I #G #L #K #V #W #i1 #l1 #m1 #Hli1 #Hilm1 #HLK #HVW #T2 #l2 #m2 #HWT2 #Hlm2l1 + lapply (yle_trans … Hlm2l1 … Hli1) -Hlm2l1 #Hlm2i1 + lapply (cpy_weak … HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hlm2i1 + >yplus_inj #HWT2 <(cpy_inv_lift1_eq … HVW … HWT2) -HWT2 /3 width=9 by cpy_subst, ex2_intro/ +| #a #I #G #L #V1 #V0 #T1 #T0 #l1 #m1 #_ #_ #IHV10 #IHT10 #X #l2 #m2 #HX #lm2l1 + elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02 + elim (IHV10 … HV02) -IHV10 -HV02 // #V + elim (IHT10 … HT02) -T0 /2 width=1 by yle_succ/ #T #HT1 #HT2 + lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /3 width=6 by cpy_bind, lsuby_succ, ex2_intro/ +| #I #G #L #V1 #V0 #T1 #T0 #l1 #m1 #_ #_ #IHV10 #IHT10 #X #l2 #m2 #HX #lm2l1 + elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct + elim (IHV10 … HV02) -V0 // + elim (IHT10 … HT02) -T0 /3 width=6 by cpy_flat, ex2_intro/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpy_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpy_lift.etc new file mode 100644 index 000000000..44a7f3d76 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpy_lift.etc @@ -0,0 +1,253 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/substitution/drop_drop.ma". +include "basic_2/substitution/cpy.ma". + +(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************) + +(* Properties on relocation *************************************************) + +(* Basic_1: was: subst1_lift_lt *) +lemma cpy_lift_le: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 → + ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K → + ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 → + lt + mt ≤ l → ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2. +#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt +[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_ + >(lift_mono … H1 … H2) -H1 -H2 // +| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hlmtl + lapply (ylt_yle_trans … Hlmtl … Hilmt) -Hlmtl #Hil + lapply (lift_inv_lref1_lt … H … Hil) -H #H destruct + elim (lift_trans_ge … HVW … HWU2) -W /2 width=1 by ylt_fwd_le_succ1/ + yplus_SO2 >yminus_succ2 #W #HVW #HWU2 + elim (drop_trans_le … HLK … HKV) -K /2 width=2 by ylt_fwd_le/ #X #HLK #H + elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K #Y #_ #HVY + >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=5 by cpy_subst/ +| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl + elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct + /4 width=7 by cpy_bind, drop_skip, yle_succ/ +| #G #I #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl + elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct + /3 width=7 by cpy_flat/ +] +qed-. + +lemma cpy_lift_be: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 → + ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K → + ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 → + lt ≤ l → l ≤ lt + mt → ⦃G, L⦄ ⊢ U1 ▶[lt, mt + m] U2. +#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt +[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_ #_ + >(lift_mono … H1 … H2) -H1 -H2 // +| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hltl #_ + elim (lift_inv_lref1 … H) -H * #Hil #H destruct + [ -Hltl + lapply (ylt_yle_trans … (lt+mt+m) … Hilmt) // -Hilmt #Hilmtm + elim (lift_trans_ge … HVW … HWU2) -W yplus_SO2 + [2: >yplus_O1 /2 width=1 by ylt_fwd_le_succ1/ ] >yminus_succ2 #W #HVW #HWU2 + elim (drop_trans_le … HLK … HKV) -K /2 width=1 by ylt_fwd_le/ #X #HLK #H + elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K #Y #_ #HVY + >(lift_mono … HVY … HVW) -V #H destruct /2 width=5 by cpy_subst/ + | -Hlti + lapply (yle_trans … Hltl … Hil) -Hltl #Hlti + lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by yle_succ_dx/ >plus_plus_comm_23 #HVU2 + lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hil + /3 width=5 by cpy_subst, drop_inv_gen, monotonic_ylt_plus_dx, yle_plus_dx1_trans/ + ] +| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hltl #Hllmt + elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct + /4 width=7 by cpy_bind, drop_skip, yle_succ/ +| #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl + elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct + /3 width=7 by cpy_flat/ +] +qed-. + +(* Basic_1: was: subst1_lift_ge *) +lemma cpy_lift_ge: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 → + ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K → + ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 → + l ≤ lt → ⦃G, L⦄ ⊢ U1 ▶[lt+m, mt] U2. +#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt +[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_ + >(lift_mono … H1 … H2) -H1 -H2 // +| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hllt + lapply (yle_trans … Hllt … Hlti) -Hllt #Hil + lapply (lift_inv_lref1_ge … H … Hil) -H #H destruct + lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by yle_succ_dx/ >plus_plus_comm_23 #HVU2 + lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hil + /3 width=5 by cpy_subst, drop_inv_gen, monotonic_ylt_plus_dx, monotonic_yle_plus_dx/ +| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hllt + elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct + /4 width=6 by cpy_bind, drop_skip, yle_succ/ +| #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hllt + elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct + /3 width=6 by cpy_flat/ +] +qed-. + +(* Inversion lemmas on relocation *******************************************) + +(* Basic_1: was: subst1_gen_lift_lt *) +lemma cpy_inv_lift1_le: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + lt + mt ≤ l → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 & ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt +[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_ + [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/ + | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/ + | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/ + ] +| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hlmtl + lapply (ylt_yle_trans … Hlmtl … Hilmt) -Hlmtl #Hil + lapply (lift_inv_lref2_lt … H … Hil) -H #H destruct + elim (drop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV + elim (lift_trans_le … HUV … HVW) -V // >minus_plus yplus_SO2 >ymax_pre_sn /2 width=1 by ylt_fwd_le_succ1/ -Hil + /3 width=5 by cpy_subst, ex2_intro/ +| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl + elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct + elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2 + elim (IHU12 … HTU1) -IHU12 -HTU1 + /3 width=6 by cpy_bind, yle_succ, drop_skip, lift_bind, ex2_intro/ +| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl + elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct + elim (IHW12 … HLK … HVW1) -W1 // + elim (IHU12 … HLK … HTU1) -U1 -HLK + /3 width=5 by cpy_flat, lift_flat, ex2_intro/ +] +qed-. + +lemma cpy_inv_lift1_be: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + lt ≤ l → l + m ≤ lt + mt → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, mt-m] T2 & ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt +[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_ #_ + [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/ + | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/ + | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/ + ] +| #I #G #L #KV #V #W #i #x #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hltl #Hlmlmt + elim (yle_inv_inj2 … Hlti) -Hlti #lt #Hlti #H destruct + lapply (yle_fwd_plus_yge … Hltl Hlmlmt) #Hmmt + elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -Hltl -Hilmt | -Hlti -Hlmlmt ] + [ lapply (ylt_yle_trans i l (lt+(mt-m)) ? ?) // + [ >yplus_minus_assoc_inj /2 width=1 by yle_plus1_to_minus_inj2/ ] -Hlmlmt #Hilmtm + elim (drop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV + elim (lift_trans_le … HUV … HVW) -V // + yplus_SO2 >ymax_pre_sn /2 width=1 by ylt_fwd_le_succ1/ -Hil + /4 width=5 by cpy_subst, ex2_intro, yle_inj/ + | elim (yle_inv_plus_inj2 … Hil) #Hlim #Hmi + lapply (yle_inv_inj … Hmi) -Hmi #Hmi + lapply (yle_trans … Hltl (i-m) ?) // -Hltl #Hltim + lapply (drop_conf_ge … HLK … HLKV ?) -L // #HKV + elim (lift_split … HVW l (i-m+1)) -HVW [2,3,4: /2 width=1 by yle_succ_dx, le_S_S/ ] -Hil -Hlim + #V1 #HV1 >plus_minus // yplus_minus_assoc_inj /3 width=1 by monotonic_ylt_minus_dx, yle_inj/ + ] +| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hltl #Hlmlmt + elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct + elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2 + elim (IHU12 … HTU1) -U1 + /3 width=6 by cpy_bind, drop_skip, lift_bind, yle_succ, ex2_intro/ +| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hltl #Hlmlmt + elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct + elim (IHW12 … HLK … HVW1) -W1 // + elim (IHU12 … HLK … HTU1) -U1 -HLK // + /3 width=5 by cpy_flat, lift_flat, ex2_intro/ +] +qed-. + +(* Basic_1: was: subst1_gen_lift_ge *) +lemma cpy_inv_lift1_ge: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + l + m ≤ lt → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt-m, mt] T2 & ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt +[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_ + [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/ + | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/ + | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/ + ] +| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hlmlt + lapply (yle_trans … Hlmlt … Hlti) #Hlmi + elim (yle_inv_plus_inj2 … Hlmlt) -Hlmlt #_ #Hmlt + elim (yle_inv_plus_inj2 … Hlmi) #Hlim #Hmi + lapply (yle_inv_inj … Hmi) -Hmi #Hmi + lapply (lift_inv_lref2_ge … H ?) -H // #H destruct + lapply (drop_conf_ge … HLK … HLKV ?) -L // #HKV + elim (lift_split … HVW l (i-m+1)) -HVW [2,3,4: /3 width=1 by yle_succ, yle_pred_sn, le_S_S/ ] -Hlmi -Hlim + #V0 #HV10 >plus_minus // yminus_succ1_inj /3 width=5 by cpy_bind, lift_bind, ex2_intro/ +| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl + elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct + elim (IHW12 … HLK … HVW1) -W1 // + elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpy_flat, lift_flat, ex2_intro/ +] +qed-. + +(* Advanced inversion lemmas on relocation ***********************************) + +lemma cpy_inv_lift1_ge_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + l ≤ lt → lt ≤ l + m → l + m ≤ lt + mt → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[l, lt + mt - (l + m)] T2 & ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt +elim (cpy_split_up … HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2 +lapply (cpy_weak … HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1 +lapply (cpy_inv_lift1_eq … HTU1 … HU1) -HU1 #HU1 destruct +elim (cpy_inv_lift1_ge … HU2 … HLK … HTU1) -U -L /2 width=3 by ex2_intro/ +qed-. + +lemma cpy_inv_lift1_be_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + lt ≤ l → lt + mt ≤ l + m → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, l-lt] T2 & ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm +lapply (cpy_weak … HU12 lt (l+m-lt) ? ?) -HU12 // +[ >ymax_pre_sn_comm /2 width=1 by yle_plus_dx1_trans/ ] -Hlmtlm #HU12 +elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) -U1 -L /2 width=3 by ex2_intro/ +qed-. + +lemma cpy_inv_lift1_le_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + lt ≤ l → l ≤ lt + mt → lt + mt ≤ l + m → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm +elim (cpy_split_up … HU12 l) -HU12 // #U #HU1 #HU2 +elim (cpy_inv_lift1_le … HU1 … HLK … HTU1) -U1 +[2: >ymax_pre_sn_comm // ] -Hltl #T #HT1 #HTU +lapply (cpy_weak … HU2 l m ? ?) -HU2 // +[ >ymax_pre_sn_comm // ] -Hllmt -Hlmtlm #HU2 +lapply (cpy_inv_lift1_eq … HTU … HU2) -L #H destruct /2 width=3 by ex2_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpy_nlift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpy_nlift.etc new file mode 100644 index 000000000..3a6450724 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpy_nlift.etc @@ -0,0 +1,65 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/substitution/lift_neg.ma". +include "basic_2/substitution/lift_lift.ma". +include "basic_2/substitution/cpy.ma". + +(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************) + +(* Inversion lemmas on negated relocation ***********************************) + +lemma cpy_fwd_nlift2_ge: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → + ∀i. l ≤ i → (∀T2. ⬆[i, 1] T2 ≡ U2 → ⊥) → + (∀T1. ⬆[i, 1] T1 ≡ U1 → ⊥) ∨ + ∃∃I,K,W,j. l ≤ yinj j & j < i & ⬇[j]L ≡ K.ⓑ{I}W & + (∀V. ⬆[⫰(i-j), 1] V ≡ W → ⊥) & (∀T1. ⬆[j, 1] T1 ≡ U1 → ⊥). +#G #L #U1 #U2 #l #m #H elim H -G -L -U1 -U2 -l -m +[ /3 width=2 by or_introl/ +| #I #G #L #K #V #W #j #l #m #Hlj #Hjlm #HLK #HVW #i #Hli #HnW + elim (ylt_split j i) #Hij + [ @or_intror @(ex5_4_intro … HLK) // -HLK + [ #X #HXV elim (lift_trans_le … HXV … HVW ?) -V // #Y #HXY + yplus_SO2 >ymax_pre_sn /2 width=2 by ylt_fwd_le_succ1/ + | -HnW /3 width=7 by lift_inv_lref2_be, ylt_inj/ + ] + | elim (lift_split … HVW i j) -HVW // + #X #_ #H elim HnW -HnW // + ] +| #a #I #G #L #W1 #W2 #U1 #U2 #l #m #_ #_ #IHW12 #IHU12 #i #Hli #H elim (nlift_inv_bind … H) -H + [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 // + [ /4 width=9 by nlift_bind_sn, or_introl/ + | * /5 width=9 by nlift_bind_sn, ex5_4_intro, or_intror/ + ] + | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 /2 width=1 by yle_succ/ + [ /4 width=9 by nlift_bind_dx, or_introl/ + | * #J #K #W #j #Hlj elim (yle_inv_succ1 … Hlj) -Hlj #Hlj #Hj + yminus_succ #Hji #HLK #HnW + lapply (drop_inv_drop1_lt … HLK ?) /2 width=1 by ylt_O/ -HLK #HLK + (cpy_inv_sort1 … HT2) -HT2 // +qed-. + +(* Note: this can be derived from cpys_inv_atom1 *) +lemma cpys_inv_gref1: ∀G,L,T2,p,l,m. ⦃G, L⦄ ⊢ §p ▶*[l, m] T2 → T2 = §p. +#G #L #T2 #p #l #m #H @(cpys_ind … H) -T2 // +#T #T2 #_ #HT2 #IHT1 destruct +>(cpy_inv_gref1 … HT2) -HT2 // +qed-. + +lemma cpys_inv_bind1: ∀a,I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[l, m] U2 → + ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 & + ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯l, m] T2 & + U2 = ⓑ{a,I}V2.T2. +#a #I #G #L #V1 #T1 #U2 #l #m #H @(cpys_ind … H) -U2 +[ /2 width=5 by ex3_2_intro/ +| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct + elim (cpy_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H + lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V1) ?) -HT2 + /3 width=5 by cpys_strap1, lsuby_succ, ex3_2_intro/ +] +qed-. + +lemma cpys_inv_flat1: ∀I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[l, m] U2 → + ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 & ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 & + U2 = ⓕ{I}V2.T2. +#I #G #L #V1 #T1 #U2 #l #m #H @(cpys_ind … H) -U2 +[ /2 width=5 by ex3_2_intro/ +| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct + elim (cpy_inv_flat1 … HU2) -HU2 + /3 width=5 by cpys_strap1, ex3_2_intro/ +] +qed-. + +lemma cpys_inv_refl_O2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶*[l, 0] T2 → T1 = T2. +#G #L #T1 #T2 #l #H @(cpys_ind … H) -T2 // +#T #T2 #_ #HT2 #IHT1 <(cpy_inv_refl_O2 … HT2) -HT2 // +qed-. + +lemma cpys_inv_lift1_eq: ∀G,L,U1,U2,l,m. + ⦃G, L⦄ ⊢ U1 ▶*[l, yinj m] U2 → ∀T1. ⬆[l, m] T1 ≡ U1 → U1 = U2. +#G #L #U1 #U2 #l #m #H #T1 #HTU1 @(cpys_ind … H) -U2 +/2 width=7 by cpy_inv_lift1_eq/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys_alt.etc new file mode 100644 index 000000000..d6b6c3096 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys_alt.etc @@ -0,0 +1,102 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/notation/relations/psubststaralt_6.ma". +include "basic_2/multiple/cpys_lift.ma". + +(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************) + +(* alternative definition of cpys *) +inductive cpysa: ynat → ynat → relation4 genv lenv term term ≝ +| cpysa_atom : ∀I,G,L,l,m. cpysa l m G L (⓪{I}) (⓪{I}) +| cpysa_subst: ∀I,G,L,K,V1,V2,W2,i,l,m. l ≤ yinj i → i < l+m → + ⬇[i] L ≡ K.ⓑ{I}V1 → cpysa 0 (⫰(l+m-i)) G K V1 V2 → + ⬆[0, i+1] V2 ≡ W2 → cpysa l m G L (#i) W2 +| cpysa_bind : ∀a,I,G,L,V1,V2,T1,T2,l,m. + cpysa l m G L V1 V2 → cpysa (⫯l) m G (L.ⓑ{I}V1) T1 T2 → + cpysa l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) +| cpysa_flat : ∀I,G,L,V1,V2,T1,T2,l,m. + cpysa l m G L V1 V2 → cpysa l m G L T1 T2 → + cpysa l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) +. + +interpretation + "context-sensitive extended multiple substritution (term) alternative" + 'PSubstStarAlt G L T1 l m T2 = (cpysa l m G L T1 T2). + +(* Basic properties *********************************************************) + +lemma lsuby_cpysa_trans: ∀G,l,m. lsub_trans … (cpysa l m G) (lsuby l m). +#G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m +[ // +| #I #G #L1 #K1 #V1 #V2 #W2 #i #l #m #Hli #Hilm #HLK1 #_ #HVW2 #IHV12 #L2 #HL12 + elim (lsuby_drop_trans_be … HL12 … HLK1) -HL12 -HLK1 /3 width=7 by cpysa_subst/ +| /4 width=1 by lsuby_succ, cpysa_bind/ +| /3 width=1 by cpysa_flat/ +] +qed-. + +lemma cpysa_refl: ∀G,T,L,l,m. ⦃G, L⦄ ⊢ T ▶▶*[l, m] T. +#G #T elim T -T // +#I elim I -I /2 width=1 by cpysa_bind, cpysa_flat/ +qed. + +lemma cpysa_cpy_trans: ∀G,L,T1,T,l,m. ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T → + ∀T2. ⦃G, L⦄ ⊢ T ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2. +#G #L #T1 #T #l #m #H elim H -G -L -T1 -T -l -m +[ #I #G #L #l #m #X #H + elim (cpy_inv_atom1 … H) -H // * /2 width=7 by cpysa_subst/ +| #I #G #L #K #V1 #V2 #W2 #i #l #m #Hli #Hilm #HLK #_ #HVW2 #IHV12 #T2 #H + lapply (drop_fwd_drop2 … HLK) #H0LK + lapply (cpy_weak … H 0 (l+m) ? ?) -H // #H + elim (cpy_inv_lift1_be … H … H0LK … HVW2) -H -H0LK -HVW2 + /3 width=7 by cpysa_subst, ylt_fwd_le_succ1/ +| #a #I #G #L #V1 #V #T1 #T #l #m #_ #_ #IHV1 #IHT1 #X #H + elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct + /5 width=5 by cpysa_bind, lsuby_cpy_trans, lsuby_succ/ +| #I #G #L #V1 #V #T1 #T #l #m #_ #_ #IHV1 #IHT1 #X #H + elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1 by cpysa_flat/ +] +qed-. + +lemma cpys_cpysa: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2. +/3 width=8 by cpysa_cpy_trans, cpys_ind/ qed. + +(* Basic inversion lemmas ***************************************************) + +lemma cpysa_inv_cpys: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2. +#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m +/2 width=7 by cpys_subst, cpys_flat, cpys_bind, cpy_cpys/ +qed-. + +(* Advanced eliminators *****************************************************) + +lemma cpys_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term. + (∀I,G,L,l,m. R l m G L (⓪{I}) (⓪{I})) → + (∀I,G,L,K,V1,V2,W2,i,l,m. l ≤ yinj i → i < l + m → + ⬇[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(l+m-i)] V2 → + ⬆[O, i+1] V2 ≡ W2 → R O (⫰(l+m-i)) G K V1 V2 → R l m G L (#i) W2 + ) → + (∀a,I,G,L,V1,V2,T1,T2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 → + ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯l, m] T2 → R l m G L V1 V2 → + R (⫯l) m G (L.ⓑ{I}V1) T1 T2 → R l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) + ) → + (∀I,G,L,V1,V2,T1,T2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 → + ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R l m G L V1 V2 → + R l m G L T1 T2 → R l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) + ) → + ∀l,m,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R l m G L T1 T2. +#R #H1 #H2 #H3 #H4 #l #m #G #L #T1 #T2 #H elim (cpys_cpysa … H) -G -L -T1 -T2 -l -m +/3 width=8 by cpysa_inv_cpys/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys_cpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys_cpys.etc new file mode 100644 index 000000000..7df95948e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys_cpys.etc @@ -0,0 +1,117 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/substitution/cpy_cpy.ma". +include "basic_2/multiple/cpys_alt.ma". + +(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************) + +(* Advanced inversion lemmas ************************************************) + +lemma cpys_inv_SO2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶*[l, 1] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2. +#G #L #T1 #T2 #l #H @(cpys_ind … H) -T2 /2 width=3 by cpy_trans_ge/ +qed-. + +(* Advanced properties ******************************************************) + +lemma cpys_strip_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶*[l1, m1] T1 → + ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → + ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶*[l1, m1] T. +normalize /3 width=3 by cpy_conf_eq, TC_strip1/ qed-. + +lemma cpys_strip_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶*[l1, m1] T1 → + ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶[l2, m2] T2 → + (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) → + ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶*[l1, m1] T. +normalize /3 width=3 by cpy_conf_neq, TC_strip1/ qed-. + +lemma cpys_strap1_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T0 → + ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → l2 + m2 ≤ l1 → + ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T ▶*[l1, m1] T2. +normalize /3 width=3 by cpy_trans_down, TC_strap1/ qed. + +lemma cpys_strap2_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T0 → + ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → l2 + m2 ≤ l1 → + ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T ▶[l1, m1] T2. +normalize /3 width=3 by cpy_trans_down, TC_strap2/ qed-. + +lemma cpys_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → + ∀i. l ≤ i → i ≤ l + m → + ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l, i - l] T & ⦃G, L⦄ ⊢ T ▶*[i, l + m - i] T2. +#G #L #T1 #T2 #l #m #H #i #Hli #Hilm @(cpys_ind … H) -T2 +[ /2 width=3 by ex2_intro/ +| #T #T2 #_ #HT12 * #T3 #HT13 #HT3 + elim (cpy_split_up … HT12 … Hilm) -HT12 -Hilm #T0 #HT0 #HT02 + elim (cpys_strap1_down … HT3 … HT0) -T /3 width=5 by cpys_strap1, ex2_intro/ + >ymax_pre_sn_comm // +] +qed-. + +lemma cpys_inv_lift1_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + l ≤ lt → lt ≤ l + m → l + m ≤ lt + mt → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[l, lt + mt - (l + m)] T2 & + ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt +elim (cpys_split_up … HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2 +lapply (cpys_weak … HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1 +lapply (cpys_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct +elim (cpys_inv_lift1_ge … HU2 … HLK … HTU1) -HU2 -HLK -HTU1 // +>yplus_minus_inj /2 width=3 by ex2_intro/ +qed-. + +(* Main properties **********************************************************) + +theorem cpys_conf_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶*[l1, m1] T1 → + ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → + ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶*[l1, m1] T. +normalize /3 width=3 by cpy_conf_eq, TC_confluent2/ qed-. + +theorem cpys_conf_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶*[l1, m1] T1 → + ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶*[l2, m2] T2 → + (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) → + ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶*[l1, m1] T. +normalize /3 width=3 by cpy_conf_neq, TC_confluent2/ qed-. + +theorem cpys_trans_eq: ∀G,L,T1,T,T2,l,m. + ⦃G, L⦄ ⊢ T1 ▶*[l, m] T → ⦃G, L⦄ ⊢ T ▶*[l, m] T2 → + ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2. +normalize /2 width=3 by trans_TC/ qed-. + +theorem cpys_trans_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T0 → + ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → l2 + m2 ≤ l1 → + ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T ▶*[l1, m1] T2. +normalize /3 width=3 by cpy_trans_down, TC_transitive2/ qed-. + +theorem cpys_antisym_eq: ∀G,L1,T1,T2,l,m. ⦃G, L1⦄ ⊢ T1 ▶*[l, m] T2 → + ∀L2. ⦃G, L2⦄ ⊢ T2 ▶*[l, m] T1 → T1 = T2. +#G #L1 #T1 #T2 #l #m #H @(cpys_ind_alt … H) -G -L1 -T1 -T2 // +[ #I1 #G #L1 #K1 #V1 #V2 #W2 #i #l #m #Hli #Hilm #_ #_ #HVW2 #_ #L2 #HW2 + elim (lt_or_ge (|L2|) (i+1)) #Hi [ -Hli -Hilm | ] + [ lapply (cpys_weak_full … HW2) -HW2 #HW2 + lapply (cpys_weak … HW2 0 (i+1) ? ?) -HW2 // + [ >yplus_O1 >yplus_O1 /3 width=1 by ylt_fwd_le, ylt_inj/ ] -Hi + #HW2 >(cpys_inv_lift1_eq … HW2) -HW2 // + | elim (drop_O1_le (Ⓕ) … Hi) -Hi #K2 #HLK2 + elim (cpys_inv_lift1_ge_up … HW2 … HLK2 … HVW2 ? ? ?) -HW2 -HLK2 -HVW2 + /2 width=1 by ylt_fwd_le_succ1, yle_succ_dx/ -Hli -Hilm + #X #_ #H elim (lift_inv_lref2_be … H) -H /2 width=1 by ylt_inj/ + ] +| #a #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_bind1 … H) -H + #V #T #HV2 #HT2 #H destruct + lapply (IHV12 … HV2) #H destruct -IHV12 -HV2 /3 width=2 by eq_f2/ +| #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_flat1 … H) -H + #V #T #HV2 #HT2 #H destruct /3 width=2 by eq_f2/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys_lift.etc new file mode 100644 index 000000000..59cddfbc9 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys_lift.etc @@ -0,0 +1,226 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/substitution/cpy_lift.ma". +include "basic_2/multiple/cpys.ma". + +(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************) + +(* Advanced properties ******************************************************) + +lemma cpys_subst: ∀I,G,L,K,V,U1,i,l,m. + l ≤ yinj i → i < l + m → + ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ⫰(l+m-i)] U1 → + ∀U2. ⬆[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[l, m] U2. +#I #G #L #K #V #U1 #i #l #m #Hli #Hilm #HLK #H @(cpys_ind … H) -U1 +[ /3 width=5 by cpy_cpys, cpy_subst/ +| #U #U1 #_ #HU1 #IHU #U2 #HU12 + elim (lift_total U 0 (i+1)) #U0 #HU0 + lapply (IHU … HU0) -IHU #H + lapply (drop_fwd_drop2 … HLK) -HLK #HLK + lapply (cpy_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // #HU02 + lapply (cpy_weak … HU02 l m ? ?) -HU02 + [2,3: /2 width=3 by cpys_strap1, yle_succ_dx/ ] + >yplus_O1 ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ1/ +] +qed. + +lemma cpys_subst_Y2: ∀I,G,L,K,V,U1,i,l. + l ≤ yinj i → + ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ∞] U1 → + ∀U2. ⬆[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[l, ∞] U2. +#I #G #L #K #V #U1 #i #l #Hli #HLK #HVU1 #U2 #HU12 +@(cpys_subst … HLK … HU12) >yminus_Y_inj // +qed. + +(* Advanced inversion lemmas *************************************************) + +lemma cpys_inv_atom1: ∀I,G,L,T2,l,m. ⦃G, L⦄ ⊢ ⓪{I} ▶*[l, m] T2 → + T2 = ⓪{I} ∨ + ∃∃J,K,V1,V2,i. l ≤ yinj i & i < l + m & + ⬇[i] L ≡ K.ⓑ{J}V1 & + ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(l+m-i)] V2 & + ⬆[O, i+1] V2 ≡ T2 & + I = LRef i. +#I #G #L #T2 #l #m #H @(cpys_ind … H) -T2 +[ /2 width=1 by or_introl/ +| #T #T2 #_ #HT2 * + [ #H destruct + elim (cpy_inv_atom1 … HT2) -HT2 [ /2 width=1 by or_introl/ | * /3 width=11 by ex6_5_intro, or_intror/ ] + | * #J #K #V1 #V #i #Hli #Hilm #HLK #HV1 #HVT #HI + lapply (drop_fwd_drop2 … HLK) #H + elim (cpy_inv_lift1_ge_up … HT2 … H … HVT) -HT2 -H -HVT + [2,3,4: /2 width=1 by ylt_fwd_le_succ1, yle_succ_dx/ ] + /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/ + ] +] +qed-. + +lemma cpys_inv_lref1: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶*[l, m] T2 → + T2 = #i ∨ + ∃∃I,K,V1,V2. l ≤ i & i < l + m & + ⬇[i] L ≡ K.ⓑ{I}V1 & + ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(l+m-i)] V2 & + ⬆[O, i+1] V2 ≡ T2. +#G #L #T2 #i #l #m #H elim (cpys_inv_atom1 … H) -H /2 width=1 by or_introl/ +* #I #K #V1 #V2 #j #Hlj #Hjlm #HLK #HV12 #HVT2 #H destruct /3 width=7 by ex5_4_intro, or_intror/ +qed-. + +lemma cpys_inv_lref1_Y2: ∀G,L,T2,i,l. ⦃G, L⦄ ⊢ #i ▶*[l, ∞] T2 → + T2 = #i ∨ + ∃∃I,K,V1,V2. l ≤ i & ⬇[i] L ≡ K.ⓑ{I}V1 & + ⦃G, K⦄ ⊢ V1 ▶*[0, ∞] V2 & ⬆[O, i+1] V2 ≡ T2. +#G #L #T2 #i #l #H elim (cpys_inv_lref1 … H) -H /2 width=1 by or_introl/ +* >yminus_Y_inj /3 width=7 by or_intror, ex4_4_intro/ +qed-. + +lemma cpys_inv_lref1_drop: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶*[l, m] T2 → + ∀I,K,V1. ⬇[i] L ≡ K.ⓑ{I}V1 → + ∀V2. ⬆[O, i+1] V2 ≡ T2 → + ∧∧ ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(l+m-i)] V2 + & l ≤ i + & i < l + m. +#G #L #T2 #i #l #m #H #I #K #V1 #HLK #V2 #HVT2 elim (cpys_inv_lref1 … H) -H +[ #H destruct elim (lift_inv_lref2_be … HVT2) -HVT2 -HLK /2 width=1 by ylt_inj/ +| * #Z #Y #X1 #X2 #Hli #Hilm #HLY #HX12 #HXT2 + lapply (lift_inj … HXT2 … HVT2) -T2 #H destruct + lapply (drop_mono … HLY … HLK) -L #H destruct + /2 width=1 by and3_intro/ +] +qed-. + +(* Properties on relocation *************************************************) + +lemma cpys_lift_le: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 → + ∀L,U1,s,l,m. lt + mt ≤ l → ⬇[s, l, m] L ≡ K → + ⬆[l, m] T1 ≡ U1 → ∀U2. ⬆[l, m] T2 ≡ U2 → + ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2. +#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hlmtl #HLK #HTU1 @(cpys_ind … H) -T2 +[ #U2 #H >(lift_mono … HTU1 … H) -H // +| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 + elim (lift_total T l m) #U #HTU + lapply (IHT … HTU) -IHT #HU1 + lapply (cpy_lift_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/ +] +qed-. + +lemma cpys_lift_be: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 → + ∀L,U1,s,l,m. lt ≤ l → l ≤ lt + mt → + ⬇[s, l, m] L ≡ K → ⬆[l, m] T1 ≡ U1 → + ∀U2. ⬆[l, m] T2 ≡ U2 → ⦃G, L⦄ ⊢ U1 ▶*[lt, mt + m] U2. +#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hltl #Hllmt #HLK #HTU1 @(cpys_ind … H) -T2 +[ #U2 #H >(lift_mono … HTU1 … H) -H // +| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 + elim (lift_total T l m) #U #HTU + lapply (IHT … HTU) -IHT #HU1 + lapply (cpy_lift_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/ +] +qed-. + +lemma cpys_lift_ge: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 → + ∀L,U1,s,l,m. l ≤ lt → ⬇[s, l, m] L ≡ K → + ⬆[l, m] T1 ≡ U1 → ∀U2. ⬆[l, m] T2 ≡ U2 → + ⦃G, L⦄ ⊢ U1 ▶*[lt+m, mt] U2. +#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hllt #HLK #HTU1 @(cpys_ind … H) -T2 +[ #U2 #H >(lift_mono … HTU1 … H) -H // +| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 + elim (lift_total T l m) #U #HTU + lapply (IHT … HTU) -IHT #HU1 + lapply (cpy_lift_ge … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/ +] +qed-. + +(* Inversion lemmas for relocation ******************************************) + +lemma cpys_inv_lift1_le: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + lt + mt ≤ l → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 & ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmtl @(cpys_ind … H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (cpy_inv_lift1_le … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +lemma cpys_inv_lift1_be: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + lt ≤ l → l + m ≤ lt + mt → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt - m] T2 & ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmlmt @(cpys_ind … H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (cpy_inv_lift1_be … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +lemma cpys_inv_lift1_ge: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + l + m ≤ lt → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt - m, mt] T2 & ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmlt @(cpys_ind … H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (cpy_inv_lift1_ge … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +(* Advanced inversion lemmas on relocation **********************************) + +lemma cpys_inv_lift1_ge_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + l ≤ lt → lt ≤ l + m → l + m ≤ lt + mt → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[l, lt + mt - (l + m)] T2 & + ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt @(cpys_ind … H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (cpy_inv_lift1_ge_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +lemma cpys_inv_lift1_be_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + lt ≤ l → lt + mt ≤ l + m → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm @(cpys_ind … H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (cpy_inv_lift1_be_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +lemma cpys_inv_lift1_le_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → + ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → + lt ≤ l → l ≤ lt + mt → lt + mt ≤ l + m → + ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2. +#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm @(cpys_ind … H) -U2 +[ /2 width=3 by ex2_intro/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (cpy_inv_lift1_le_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ +] +qed-. + +lemma cpys_inv_lift1_subst: ∀G,L,W1,W2,l,m. ⦃G, L⦄ ⊢ W1 ▶*[l, m] W2 → + ∀K,V1,i. ⬇[i+1] L ≡ K → ⬆[O, i+1] V1 ≡ W1 → + l ≤ yinj i → i < l + m → + ∃∃V2. ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(l+m-i)] V2 & ⬆[O, i+1] V2 ≡ W2. +#G #L #W1 #W2 #l #m #HW12 #K #V1 #i #HLK #HVW1 #Hli #Hilm +elim (cpys_inv_lift1_ge_up … HW12 … HLK … HVW1 ? ? ?) // +>yplus_O1 yplus_SO2 +[ >yminus_succ2 /2 width=3 by ex2_intro/ +| /2 width=1 by ylt_fwd_le_succ1/ +| /2 width=3 by yle_trans/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/crr/crr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/crr/crr.etc new file mode 100644 index 000000000..50107c4ab --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/crr/crr.etc @@ -0,0 +1,145 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/notation/relations/predreducible_3.ma". +include "basic_2/grammar/genv.ma". +include "basic_2/substitution/drop.ma". + +(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION **************************) + +(* reducible binary items *) +definition ri2: predicate item2 ≝ + λI. I = Bind2 true Abbr ∨ I = Flat2 Cast. + +(* irreducible binary binders *) +definition ib2: relation2 bool bind2 ≝ + λa,I. I = Abst ∨ Bind2 a I = Bind2 false Abbr. + +(* activate genv *) +(* reducible terms *) +inductive crr (G:genv): relation2 lenv term ≝ +| crr_delta : ∀L,K,V,i. ⬇[i] L ≡ K.ⓓV → crr G L (#i) +| crr_appl_sn: ∀L,V,T. crr G L V → crr G L (ⓐV.T) +| crr_appl_dx: ∀L,V,T. crr G L T → crr G L (ⓐV.T) +| crr_ri2 : ∀I,L,V,T. ri2 I → crr G L (②{I}V.T) +| crr_ib2_sn : ∀a,I,L,V,T. ib2 a I → crr G L V → crr G L (ⓑ{a,I}V.T) +| crr_ib2_dx : ∀a,I,L,V,T. ib2 a I → crr G (L.ⓑ{I}V) T → crr G L (ⓑ{a,I}V.T) +| crr_beta : ∀a,L,V,W,T. crr G L (ⓐV.ⓛ{a}W.T) +| crr_theta : ∀a,L,V,W,T. crr G L (ⓐV.ⓓ{a}W.T) +. + +interpretation + "reducibility for context-sensitive reduction (term)" + 'PRedReducible G L T = (crr G L T). + +(* Basic inversion lemmas ***************************************************) + +fact crr_inv_sort_aux: ∀G,L,T,s. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → T = ⋆s → ⊥. +#G #L #T #s0 * -L -T +[ #L #K #V #i #HLK #H destruct +| #L #V #T #_ #H destruct +| #L #V #T #_ #H destruct +| #I #L #V #T #_ #H destruct +| #a #I #L #V #T #_ #_ #H destruct +| #a #I #L #V #T #_ #_ #H destruct +| #a #L #V #W #T #H destruct +| #a #L #V #W #T #H destruct +] +qed-. + +lemma crr_inv_sort: ∀G,L,s. ⦃G, L⦄ ⊢ ➡ 𝐑⦃⋆s⦄ → ⊥. +/2 width=6 by crr_inv_sort_aux/ qed-. + +fact crr_inv_lref_aux: ∀G,L,T,i. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → T = #i → + ∃∃K,V. ⬇[i] L ≡ K.ⓓV. +#G #L #T #j * -L -T +[ #L #K #V #i #HLK #H destruct /2 width=3 by ex1_2_intro/ +| #L #V #T #_ #H destruct +| #L #V #T #_ #H destruct +| #I #L #V #T #_ #H destruct +| #a #I #L #V #T #_ #_ #H destruct +| #a #I #L #V #T #_ #_ #H destruct +| #a #L #V #W #T #H destruct +| #a #L #V #W #T #H destruct +] +qed-. + +lemma crr_inv_lref: ∀G,L,i. ⦃G, L⦄ ⊢ ➡ 𝐑⦃#i⦄ → ∃∃K,V. ⬇[i] L ≡ K.ⓓV. +/2 width=4 by crr_inv_lref_aux/ qed-. + +fact crr_inv_gref_aux: ∀G,L,T,p. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → T = §p → ⊥. +#G #L #T #q * -L -T +[ #L #K #V #i #HLK #H destruct +| #L #V #T #_ #H destruct +| #L #V #T #_ #H destruct +| #I #L #V #T #_ #H destruct +| #a #I #L #V #T #_ #_ #H destruct +| #a #I #L #V #T #_ #_ #H destruct +| #a #L #V #W #T #H destruct +| #a #L #V #W #T #H destruct +] +qed-. + +lemma crr_inv_gref: ∀G,L,p. ⦃G, L⦄ ⊢ ➡ 𝐑⦃§p⦄ → ⊥. +/2 width=6 by crr_inv_gref_aux/ qed-. + +lemma trr_inv_atom: ∀G,I. ⦃G, ⋆⦄ ⊢ ➡ 𝐑⦃⓪{I}⦄ → ⊥. +#G * #i #H +[ elim (crr_inv_sort … H) +| elim (crr_inv_lref … H) -H #L #V #H + elim (drop_inv_atom1 … H) -H #H destruct +| elim (crr_inv_gref … H) +] +qed-. + +fact crr_inv_ib2_aux: ∀a,I,G,L,W,U,T. ib2 a I → ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → T = ⓑ{a,I}W.U → + ⦃G, L⦄ ⊢ ➡ 𝐑⦃W⦄ ∨ ⦃G, L.ⓑ{I}W⦄ ⊢ ➡ 𝐑⦃U⦄. +#G #b #J #L #W0 #U #T #HI * -L -T +[ #L #K #V #i #_ #H destruct +| #L #V #T #_ #H destruct +| #L #V #T #_ #H destruct +| #I #L #V #T #H1 #H2 destruct + elim H1 -H1 #H destruct + elim HI -HI #H destruct +| #a #I #L #V #T #_ #HV #H destruct /2 width=1 by or_introl/ +| #a #I #L #V #T #_ #HT #H destruct /2 width=1 by or_intror/ +| #a #L #V #W #T #H destruct +| #a #L #V #W #T #H destruct +] +qed-. + +lemma crr_inv_ib2: ∀a,I,G,L,W,T. ib2 a I → ⦃G, L⦄ ⊢ ➡ 𝐑⦃ⓑ{a,I}W.T⦄ → + ⦃G, L⦄ ⊢ ➡ 𝐑⦃W⦄ ∨ ⦃G, L.ⓑ{I}W⦄ ⊢ ➡ 𝐑⦃T⦄. +/2 width=5 by crr_inv_ib2_aux/ qed-. + +fact crr_inv_appl_aux: ∀G,L,W,U,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → T = ⓐW.U → + ∨∨ ⦃G, L⦄ ⊢ ➡ 𝐑⦃W⦄ | ⦃G, L⦄ ⊢ ➡ 𝐑⦃U⦄ | (𝐒⦃U⦄ → ⊥). +#G #L #W0 #U #T * -L -T +[ #L #K #V #i #_ #H destruct +| #L #V #T #HV #H destruct /2 width=1 by or3_intro0/ +| #L #V #T #HT #H destruct /2 width=1 by or3_intro1/ +| #I #L #V #T #H1 #H2 destruct + elim H1 -H1 #H destruct +| #a #I #L #V #T #_ #_ #H destruct +| #a #I #L #V #T #_ #_ #H destruct +| #a #L #V #W #T #H destruct + @or3_intro2 #H elim (simple_inv_bind … H) +| #a #L #V #W #T #H destruct + @or3_intro2 #H elim (simple_inv_bind … H) +] +qed-. + +lemma crr_inv_appl: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃ⓐV.T⦄ → + ∨∨ ⦃G, L⦄ ⊢ ➡ 𝐑⦃V⦄ | ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ | (𝐒⦃T⦄ → ⊥). +/2 width=3 by crr_inv_appl_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/crr/crr_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/crr/crr_lift.etc new file mode 100644 index 000000000..9aab5dffd --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/crr/crr_lift.etc @@ -0,0 +1,74 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/substitution/drop_drop.ma". +include "basic_2/reduction/crr.ma". + +(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION **************************) + +(* Properties on relocation *************************************************) + +lemma crr_lift: ∀G,K,T. ⦃G, K⦄ ⊢ ➡ 𝐑⦃T⦄ → ∀L,c,l,k. ⬇[c, l, k] L ≡ K → + ∀U. ⬆[l, k] T ≡ U → ⦃G, L⦄ ⊢ ➡ 𝐑⦃U⦄. +#G #K #T #H elim H -K -T +[ #K #K0 #V #i #HK0 #L #c #l #k #HLK #X #H + elim (lift_inv_lref1 … H) -H * #Hil #H destruct + [ elim (drop_trans_lt … HLK … HK0) -K /2 width=4 by crr_delta/ + | lapply (drop_trans_ge … HLK … HK0 ?) -K /3 width=4 by crr_delta, drop_inv_gen/ + ] +| #K #V #T #_ #IHV #L #c #l #k #HLK #X #H + elim (lift_inv_flat1 … H) -H #W #U #HVW #_ #H destruct /3 width=5 by crr_appl_sn/ +| #K #V #T #_ #IHT #L #c #l #k #HLK #X #H + elim (lift_inv_flat1 … H) -H #W #U #_ #HTU #H destruct /3 width=5 by crr_appl_dx/ +| #I #K #V #T #HI #L #c #l #k #_ #X #H + elim (lift_fwd_pair1 … H) -H #W #U #_ #H destruct /2 width=1 by crr_ri2/ +| #a #I #K #V #T #HI #_ #IHV #L #c #l #k #HLK #X #H + elim (lift_inv_bind1 … H) -H #W #U #HVW #_ #H destruct /3 width=5 by crr_ib2_sn/ +| #a #I #K #V #T #HI #_ #IHT #L #c #l #k #HLK #X #H + elim (lift_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct /4 width=5 by crr_ib2_dx, drop_skip/ +| #a #K #V #V0 #T #L #c #l #k #_ #X #H + elim (lift_inv_flat1 … H) -H #W #X0 #_ #H0 #H destruct + elim (lift_inv_bind1 … H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crr_beta/ +| #a #K #V #V0 #T #L #c #l #k #_ #X #H + elim (lift_inv_flat1 … H) -H #W #X0 #_ #H0 #H destruct + elim (lift_inv_bind1 … H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crr_theta/ +] +qed. + +lemma crr_inv_lift: ∀G,L,U. ⦃G, L⦄ ⊢ ➡ 𝐑⦃U⦄ → ∀K,c,l,k. ⬇[c, l, k] L ≡ K → + ∀T. ⬆[l, k] T ≡ U → ⦃G, K⦄ ⊢ ➡ 𝐑⦃T⦄. +#G #L #U #H elim H -L -U +[ #L #L0 #W #i #HK0 #K #c #l #k #HLK #X #H + elim (lift_inv_lref2 … H) -H * #Hil #H destruct + [ elim (drop_conf_lt … HLK … HK0) -L /2 width=4 by crr_delta/ + | lapply (drop_conf_ge … HLK … HK0 ?) -L /2 width=4 by crr_delta/ + ] +| #L #W #U #_ #IHW #K #c #l #k #HLK #X #H + elim (lift_inv_flat2 … H) -H #V #T #HVW #_ #H destruct /3 width=5 by crr_appl_sn/ +| #L #W #U #_ #IHU #K #c #l #k #HLK #X #H + elim (lift_inv_flat2 … H) -H #V #T #_ #HTU #H destruct /3 width=5 by crr_appl_dx/ +| #I #L #W #U #HI #K #c #l #k #_ #X #H + elim (lift_fwd_pair2 … H) -H #V #T #_ #H destruct /2 width=1 by crr_ri2/ +| #a #I #L #W #U #HI #_ #IHW #K #c #l #k #HLK #X #H + elim (lift_inv_bind2 … H) -H #V #T #HVW #_ #H destruct /3 width=5 by crr_ib2_sn/ +| #a #I #L #W #U #HI #_ #IHU #K #c #l #k #HLK #X #H + elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct /4 width=5 by crr_ib2_dx, drop_skip/ +| #a #L #W #W0 #U #K #c #l #k #_ #X #H + elim (lift_inv_flat2 … H) -H #V #X0 #_ #H0 #H destruct + elim (lift_inv_bind2 … H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crr_beta/ +| #a #L #W #W0 #U #K #c #l #k #_ #X #H + elim (lift_inv_flat2 … H) -H #V #X0 #_ #H0 #H destruct + elim (lift_inv_bind2 … H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crr_theta/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/crr/predreducible_3.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/crr/predreducible_3.etc new file mode 100644 index 000000000..5d84a363d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/crr/predreducible_3.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ ➡ 𝐑 break ⦃ term 46 T ⦄ )" + non associative with precedence 45 + for @{ 'PRedReducible $G $L $T }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/crx/crx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/crx/crx.etc new file mode 100644 index 000000000..c6c962646 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/crx/crx.etc @@ -0,0 +1,152 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/notation/relations/predreducible_5.ma". +include "basic_2/static/sd.ma". +include "basic_2/reduction/crr.ma". + +(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION *****************) + +(* activate genv *) +(* extended reducible terms *) +inductive crx (h) (o) (G:genv): relation2 lenv term ≝ +| crx_sort : ∀L,s,d. deg h o s (d+1) → crx h o G L (⋆s) +| crx_delta : ∀I,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → crx h o G L (#i) +| crx_appl_sn: ∀L,V,T. crx h o G L V → crx h o G L (ⓐV.T) +| crx_appl_dx: ∀L,V,T. crx h o G L T → crx h o G L (ⓐV.T) +| crx_ri2 : ∀I,L,V,T. ri2 I → crx h o G L (②{I}V.T) +| crx_ib2_sn : ∀a,I,L,V,T. ib2 a I → crx h o G L V → crx h o G L (ⓑ{a,I}V.T) +| crx_ib2_dx : ∀a,I,L,V,T. ib2 a I → crx h o G (L.ⓑ{I}V) T → crx h o G L (ⓑ{a,I}V.T) +| crx_beta : ∀a,L,V,W,T. crx h o G L (ⓐV. ⓛ{a}W.T) +| crx_theta : ∀a,L,V,W,T. crx h o G L (ⓐV. ⓓ{a}W.T) +. + +interpretation + "reducibility for context-sensitive extended reduction (term)" + 'PRedReducible h o G L T = (crx h o G L T). + +(* Basic properties *********************************************************) + +lemma crr_crx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄. +#h #o #G #L #T #H elim H -L -T +/2 width=4 by crx_delta, crx_appl_sn, crx_appl_dx, crx_ri2, crx_ib2_sn, crx_ib2_dx, crx_beta, crx_theta/ +qed. + +(* Basic inversion lemmas ***************************************************) + +fact crx_inv_sort_aux: ∀h,o,G,L,T,s. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → T = ⋆s → + ∃d. deg h o s (d+1). +#h #o #G #L #T #s0 * -L -T +[ #L #s #d #Hkd #H destruct /2 width=2 by ex_intro/ +| #I #L #K #V #i #HLK #H destruct +| #L #V #T #_ #H destruct +| #L #V #T #_ #H destruct +| #I #L #V #T #_ #H destruct +| #a #I #L #V #T #_ #_ #H destruct +| #a #I #L #V #T #_ #_ #H destruct +| #a #L #V #W #T #H destruct +| #a #L #V #W #T #H destruct +] +qed-. + +lemma crx_inv_sort: ∀h,o,G,L,s. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃⋆s⦄ → ∃d. deg h o s (d+1). +/2 width=5 by crx_inv_sort_aux/ qed-. + +fact crx_inv_lref_aux: ∀h,o,G,L,T,i. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → T = #i → + ∃∃I,K,V. ⬇[i] L ≡ K.ⓑ{I}V. +#h #o #G #L #T #j * -L -T +[ #L #s #d #_ #H destruct +| #I #L #K #V #i #HLK #H destruct /2 width=4 by ex1_3_intro/ +| #L #V #T #_ #H destruct +| #L #V #T #_ #H destruct +| #I #L #V #T #_ #H destruct +| #a #I #L #V #T #_ #_ #H destruct +| #a #I #L #V #T #_ #_ #H destruct +| #a #L #V #W #T #H destruct +| #a #L #V #W #T #H destruct +] +qed-. + +lemma crx_inv_lref: ∀h,o,G,L,i. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃#i⦄ → ∃∃I,K,V. ⬇[i] L ≡ K.ⓑ{I}V. +/2 width=6 by crx_inv_lref_aux/ qed-. + +fact crx_inv_gref_aux: ∀h,o,G,L,T,p. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → T = §p → ⊥. +#h #o #G #L #T #q * -L -T +[ #L #s #d #_ #H destruct +| #I #L #K #V #i #HLK #H destruct +| #L #V #T #_ #H destruct +| #L #V #T #_ #H destruct +| #I #L #V #T #_ #H destruct +| #a #I #L #V #T #_ #_ #H destruct +| #a #I #L #V #T #_ #_ #H destruct +| #a #L #V #W #T #H destruct +| #a #L #V #W #T #H destruct +] +qed-. + +lemma crx_inv_gref: ∀h,o,G,L,p. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃§p⦄ → ⊥. +/2 width=8 by crx_inv_gref_aux/ qed-. + +lemma trx_inv_atom: ∀h,o,I,G. ⦃G, ⋆⦄ ⊢ ➡[h, o] 𝐑⦃⓪{I}⦄ → + ∃∃s,d. deg h o s (d+1) & I = Sort s. +#h #o * #i #G #H +[ elim (crx_inv_sort … H) -H /2 width=4 by ex2_2_intro/ +| elim (crx_inv_lref … H) -H #I #L #V #H + elim (drop_inv_atom1 … H) -H #H destruct +| elim (crx_inv_gref … H) +] +qed-. + +fact crx_inv_ib2_aux: ∀h,o,a,I,G,L,W,U,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → + T = ⓑ{a,I}W.U → ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃W⦄ ∨ ⦃G, L.ⓑ{I}W⦄ ⊢ ➡[h, o] 𝐑⦃U⦄. +#h #o #b #J #G #L #W0 #U #T #HI * -L -T +[ #L #s #d #_ #H destruct +| #I #L #K #V #i #_ #H destruct +| #L #V #T #_ #H destruct +| #L #V #T #_ #H destruct +| #I #L #V #T #H1 #H2 destruct + elim H1 -H1 #H destruct + elim HI -HI #H destruct +| #a #I #L #V #T #_ #HV #H destruct /2 width=1 by or_introl/ +| #a #I #L #V #T #_ #HT #H destruct /2 width=1 by or_intror/ +| #a #L #V #W #T #H destruct +| #a #L #V #W #T #H destruct +] +qed-. + +lemma crx_inv_ib2: ∀h,o,a,I,G,L,W,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃ⓑ{a,I}W.T⦄ → + ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃W⦄ ∨ ⦃G, L.ⓑ{I}W⦄ ⊢ ➡[h, o] 𝐑⦃T⦄. +/2 width=5 by crx_inv_ib2_aux/ qed-. + +fact crx_inv_appl_aux: ∀h,o,G,L,W,U,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → T = ⓐW.U → + ∨∨ ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃W⦄ | ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃U⦄ | (𝐒⦃U⦄ → ⊥). +#h #o #G #L #W0 #U #T * -L -T +[ #L #s #d #_ #H destruct +| #I #L #K #V #i #_ #H destruct +| #L #V #T #HV #H destruct /2 width=1 by or3_intro0/ +| #L #V #T #HT #H destruct /2 width=1 by or3_intro1/ +| #I #L #V #T #H1 #H2 destruct + elim H1 -H1 #H destruct +| #a #I #L #V #T #_ #_ #H destruct +| #a #I #L #V #T #_ #_ #H destruct +| #a #L #V #W #T #H destruct + @or3_intro2 #H elim (simple_inv_bind … H) +| #a #L #V #W #T #H destruct + @or3_intro2 #H elim (simple_inv_bind … H) +] +qed-. + +lemma crx_inv_appl: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃ⓐV.T⦄ → + ∨∨ ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃V⦄ | ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ | (𝐒⦃T⦄ → ⊥). +/2 width=3 by crx_inv_appl_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/crx/crx_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/crx/crx_lift.etc new file mode 100644 index 000000000..cbdc2718f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/crx/crx_lift.etc @@ -0,0 +1,78 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/substitution/drop_drop.ma". +include "basic_2/reduction/crx.ma". + +(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION *****************) + +(* Properties on relocation *************************************************) + +lemma crx_lift: ∀h,o,G,K,T. ⦃G, K⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → ∀L,c,l,k. ⬇[c, l, k] L ≡ K → + ∀U. ⬆[l, k] T ≡ U → ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃U⦄. +#h #o #G #K #T #H elim H -K -T +[ #K #s #d #Hkd #L #c #l #k #_ #X #H + >(lift_inv_sort1 … H) -X /2 width=2 by crx_sort/ +| #I #K #K0 #V #i #HK0 #L #c #l #k #HLK #X #H + elim (lift_inv_lref1 … H) -H * #Hil #H destruct + [ elim (drop_trans_lt … HLK … HK0) -K /2 width=4 by crx_delta/ + | lapply (drop_trans_ge … HLK … HK0 ?) -K /3 width=5 by crx_delta, drop_inv_gen/ + ] +| #K #V #T #_ #IHV #L #c #l #k #HLK #X #H + elim (lift_inv_flat1 … H) -H #W #U #HVW #_ #H destruct /3 width=5 by crx_appl_sn/ +| #K #V #T #_ #IHT #L #c #l #k #HLK #X #H + elim (lift_inv_flat1 … H) -H #W #U #_ #HTU #H destruct /3 width=5 by crx_appl_dx/ +| #I #K #V #T #HI #L #c #l #k #_ #X #H + elim (lift_fwd_pair1 … H) -H #W #U #_ #H destruct /2 width=1 by crx_ri2/ +| #a #I #K #V #T #HI #_ #IHV #L #c #l #k #HLK #X #H + elim (lift_inv_bind1 … H) -H #W #U #HVW #_ #H destruct /3 width=5 by crx_ib2_sn/ +| #a #I #K #V #T #HI #_ #IHT #L #c #l #k #HLK #X #H + elim (lift_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct /4 width=5 by crx_ib2_dx, drop_skip/ +| #a #K #V #V0 #T #L #c #l #k #_ #X #H + elim (lift_inv_flat1 … H) -H #W #X0 #_ #H0 #H destruct + elim (lift_inv_bind1 … H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crx_beta/ +| #a #K #V #V0 #T #L #c #l #k #_ #X #H + elim (lift_inv_flat1 … H) -H #W #X0 #_ #H0 #H destruct + elim (lift_inv_bind1 … H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crx_theta/ +] +qed. + +lemma crx_inv_lift: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃U⦄ → ∀K,c,l,k. ⬇[c, l, k] L ≡ K → + ∀T. ⬆[l, k] T ≡ U → ⦃G, K⦄ ⊢ ➡[h, o] 𝐑⦃T⦄. +#h #o #G #L #U #H elim H -L -U +[ #L #s #d #Hkd #K #c #l #k #_ #X #H + >(lift_inv_sort2 … H) -X /2 width=2 by crx_sort/ +| #I #L #L0 #W #i #HK0 #K #c #l #k #HLK #X #H + elim (lift_inv_lref2 … H) -H * #Hil #H destruct + [ elim (drop_conf_lt … HLK … HK0) -L /2 width=4 by crx_delta/ + | lapply (drop_conf_ge … HLK … HK0 ?) -L /2 width=4 by crx_delta/ + ] +| #L #W #U #_ #IHW #K #c #l #k #HLK #X #H + elim (lift_inv_flat2 … H) -H #V #T #HVW #_ #H destruct /3 width=5 by crx_appl_sn/ +| #L #W #U #_ #IHU #K #c #l #k #HLK #X #H + elim (lift_inv_flat2 … H) -H #V #T #_ #HTU #H destruct /3 width=5 by crx_appl_dx/ +| #I #L #W #U #HI #K #c #l #k #_ #X #H + elim (lift_fwd_pair2 … H) -H #V #T #_ #H destruct /2 width=1 by crx_ri2/ +| #a #I #L #W #U #HI #_ #IHW #K #c #l #k #HLK #X #H + elim (lift_inv_bind2 … H) -H #V #T #HVW #_ #H destruct /3 width=5 by crx_ib2_sn/ +| #a #I #L #W #U #HI #_ #IHU #K #c #l #k #HLK #X #H + elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct /4 width=5 by crx_ib2_dx, drop_skip/ +| #a #L #W #W0 #U #K #c #l #k #_ #X #H + elim (lift_inv_flat2 … H) -H #V #X0 #_ #H0 #H destruct + elim (lift_inv_bind2 … H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crx_beta/ +| #a #L #W #W0 #U #K #c #l #k #_ #X #H + elim (lift_inv_flat2 … H) -H #V #X0 #_ #H0 #H destruct + elim (lift_inv_bind2 … H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crx_theta/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/crx/predreducible_5.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/crx/predreducible_5.etc new file mode 100644 index 000000000..ca0d12b72 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/crx/predreducible_5.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ➡ break [ term 46 o , break term 46 h ] 𝐑 break ⦃ term 46 T ⦄ )" + non associative with precedence 45 + for @{ 'PRedReducible $h $o $G $L $T }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/da/da.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/da/da.etc new file mode 100644 index 000000000..b8d3d4315 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/da/da.etc @@ -0,0 +1,108 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/notation/relations/degree_6.ma". +include "basic_2/grammar/genv.ma". +include "basic_2/substitution/drop.ma". +include "basic_2/static/sd.ma". + +(* DEGREE ASSIGNMENT FOR TERMS **********************************************) + +(* activate genv *) +inductive da (h:sh) (o:sd h): relation4 genv lenv term nat ≝ +| da_sort: ∀G,L,s,d. deg h o s d → da h o G L (⋆s) d +| da_ldef: ∀G,L,K,V,i,d. ⬇[i] L ≡ K.ⓓV → da h o G K V d → da h o G L (#i) d +| da_ldec: ∀G,L,K,W,i,d. ⬇[i] L ≡ K.ⓛW → da h o G K W d → da h o G L (#i) (d+1) +| da_bind: ∀a,I,G,L,V,T,d. da h o G (L.ⓑ{I}V) T d → da h o G L (ⓑ{a,I}V.T) d +| da_flat: ∀I,G,L,V,T,d. da h o G L T d → da h o G L (ⓕ{I}V.T) d +. + +interpretation "degree assignment (term)" + 'Degree h o G L T d = (da h o G L T d). + +(* Basic inversion lemmas ***************************************************) + +fact da_inv_sort_aux: ∀h,o,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, o] d → + ∀s0. T = ⋆s0 → deg h o s0 d. +#h #o #G #L #T #d * -G -L -T -d +[ #G #L #s #d #Hkd #s0 #H destruct // +| #G #L #K #V #i #d #_ #_ #s0 #H destruct +| #G #L #K #W #i #d #_ #_ #s0 #H destruct +| #a #I #G #L #V #T #d #_ #s0 #H destruct +| #I #G #L #V #T #d #_ #s0 #H destruct +] +qed-. + +lemma da_inv_sort: ∀h,o,G,L,s,d. ⦃G, L⦄ ⊢ ⋆s ▪[h, o] d → deg h o s d. +/2 width=5 by da_inv_sort_aux/ qed-. + +fact da_inv_lref_aux: ∀h,o,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, o] d → ∀j. T = #j → + (∃∃K,V. ⬇[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ▪[h, o] d) ∨ + (∃∃K,W,d0. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, o] d0 & + d = d0 + 1 + ). +#h #o #G #L #T #d * -G -L -T -d +[ #G #L #s #d #_ #j #H destruct +| #G #L #K #V #i #d #HLK #HV #j #H destruct /3 width=4 by ex2_2_intro, or_introl/ +| #G #L #K #W #i #d #HLK #HW #j #H destruct /3 width=6 by ex3_3_intro, or_intror/ +| #a #I #G #L #V #T #d #_ #j #H destruct +| #I #G #L #V #T #d #_ #j #H destruct +] +qed-. + +lemma da_inv_lref: ∀h,o,G,L,j,d. ⦃G, L⦄ ⊢ #j ▪[h, o] d → + (∃∃K,V. ⬇[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ▪[h, o] d) ∨ + (∃∃K,W,d0. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, o] d0 & d = d0+1). +/2 width=3 by da_inv_lref_aux/ qed-. + +fact da_inv_gref_aux: ∀h,o,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, o] d → ∀p0. T = §p0 → ⊥. +#h #o #G #L #T #d * -G -L -T -d +[ #G #L #s #d #_ #p0 #H destruct +| #G #L #K #V #i #d #_ #_ #p0 #H destruct +| #G #L #K #W #i #d #_ #_ #p0 #H destruct +| #a #I #G #L #V #T #d #_ #p0 #H destruct +| #I #G #L #V #T #d #_ #p0 #H destruct +] +qed-. + +lemma da_inv_gref: ∀h,o,G,L,p,d. ⦃G, L⦄ ⊢ §p ▪[h, o] d → ⊥. +/2 width=9 by da_inv_gref_aux/ qed-. + +fact da_inv_bind_aux: ∀h,o,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, o] d → + ∀b,J,X,Y. T = ⓑ{b,J}Y.X → ⦃G, L.ⓑ{J}Y⦄ ⊢ X ▪[h, o] d. +#h #o #G #L #T #d * -G -L -T -d +[ #G #L #s #d #_ #b #J #X #Y #H destruct +| #G #L #K #V #i #d #_ #_ #b #J #X #Y #H destruct +| #G #L #K #W #i #d #_ #_ #b #J #X #Y #H destruct +| #a #I #G #L #V #T #d #HT #b #J #X #Y #H destruct // +| #I #G #L #V #T #d #_ #b #J #X #Y #H destruct +] +qed-. + +lemma da_inv_bind: ∀h,o,b,J,G,L,Y,X,d. ⦃G, L⦄ ⊢ ⓑ{b,J}Y.X ▪[h, o] d → ⦃G, L.ⓑ{J}Y⦄ ⊢ X ▪[h, o] d. +/2 width=4 by da_inv_bind_aux/ qed-. + +fact da_inv_flat_aux: ∀h,o,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, o] d → + ∀J,X,Y. T = ⓕ{J}Y.X → ⦃G, L⦄ ⊢ X ▪[h, o] d. +#h #o #G #L #T #d * -G -L -T -d +[ #G #L #s #d #_ #J #X #Y #H destruct +| #G #L #K #V #i #d #_ #_ #J #X #Y #H destruct +| #G #L #K #W #i #d #_ #_ #J #X #Y #H destruct +| #a #I #G #L #V #T #d #_ #J #X #Y #H destruct +| #I #G #L #V #T #d #HT #J #X #Y #H destruct // +] +qed-. + +lemma da_inv_flat: ∀h,o,J,G,L,Y,X,d. ⦃G, L⦄ ⊢ ⓕ{J}Y.X ▪[h, o] d → ⦃G, L⦄ ⊢ X ▪[h, o] d. +/2 width=5 by da_inv_flat_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/da/da_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/da/da_aaa.etc new file mode 100644 index 000000000..43d480c60 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/da/da_aaa.etc @@ -0,0 +1,31 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/static/aaa_lift.ma". +include "basic_2/static/da.ma". + +(* DEGREE ASSIGNMENT FOR TERMS **********************************************) + +(* Properties on atomic arity assignment for terms **************************) + +lemma aaa_da: ∀h,o,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∃d. ⦃G, L⦄ ⊢ T ▪[h, o] d. +#h #o #G #L #T #A #H elim H -G -L -T -A +[ #G #L #s elim (deg_total h o s) /3 width=2 by da_sort, ex_intro/ +| * #G #L #K #V #B #i #HLK #_ * /3 width=5 by da_ldef, da_ldec, ex_intro/ +| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by da_bind, ex_intro/ +| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by da_bind, ex_intro/ +| #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by da_flat, ex_intro/ +| #G #L #W #T #A #_ #_ #_ * /3 width=2 by da_flat, ex_intro/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/da/da_da.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/da/da_da.etc new file mode 100644 index 000000000..2bc863015 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/da/da_da.etc @@ -0,0 +1,38 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/static/da_lift.ma". + +(* DEGREE ASSIGNMENT FOR TERMS **********************************************) + +(* Main properties **********************************************************) + +theorem da_mono: ∀h,o,G,L,T,d1. ⦃G, L⦄ ⊢ T ▪[h, o] d1 → + ∀d2. ⦃G, L⦄ ⊢ T ▪[h, o] d2 → d1 = d2. +#h #o #G #L #T #d1 #H elim H -G -L -T -d1 +[ #G #L #s #d1 #Hkd1 #d2 #H + lapply (da_inv_sort … H) -G -L #Hkd2 + >(deg_mono … Hkd2 … Hkd1) -h -s -d2 // +| #G #L #K #V #i #d1 #HLK #_ #IHV #d2 #H + elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0 #HV0 [| #Hd0 ] + lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct /2 width=1 by/ +| #G #L #K #W #i #d1 #HLK #_ #IHW #d2 #H + elim (da_inv_lref … H) -H * #K0 #W0 [| #d0 ] #HLK0 #HW0 [| #Hd0 ] + lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct /3 width=1 by eq_f/ +| #a #I #G #L #V #T #d1 #_ #IHT #d2 #H + lapply (da_inv_bind … H) -H /2 width=1 by/ +| #I #G #L #V #T #d1 #_ #IHT #d2 #H + lapply (da_inv_flat … H) -H /2 width=1 by/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/da/da_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/da/da_lift.etc new file mode 100644 index 000000000..2f1a41554 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/da/da_lift.etc @@ -0,0 +1,78 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/substitution/drop_drop.ma". +include "basic_2/static/da.ma". + +(* DEGREE ASSIGNMENT FOR TERMS **********************************************) + +(* Properties on relocation *************************************************) + +lemma da_lift: ∀h,o,G,L1,T1,d. ⦃G, L1⦄ ⊢ T1 ▪[h, o] d → + ∀L2,c,l,k. ⬇[c, l, k] L2 ≡ L1 → ∀T2. ⬆[l, k] T1 ≡ T2 → + ⦃G, L2⦄ ⊢ T2 ▪[h, o] d. +#h #o #G #L1 #T1 #d #H elim H -G -L1 -T1 -d +[ #G #L1 #s #d #Hkd #L2 #c #l #k #_ #X #H + >(lift_inv_sort1 … H) -X /2 width=1 by da_sort/ +| #G #L1 #K1 #V1 #i #d #HLK1 #_ #IHV1 #L2 #c #l #k #HL21 #X #H + elim (lift_inv_lref1 … H) * #Hil #H destruct + [ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by ylt_fwd_le/ #X #HLK2 #H + elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K2 #V2 #HK21 #HV12 #H destruct + /3 width=9 by da_ldef/ + | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 + /3 width=8 by da_ldef, drop_inv_gen/ + ] +| #G #L1 #K1 #W1 #i #d #HLK1 #_ #IHW1 #L2 #c #l #k #HL21 #X #H + elim (lift_inv_lref1 … H) * #Hil #H destruct + [ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by ylt_fwd_le/ #X #HLK2 #H + elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K2 #W2 #HK21 #HW12 #H destruct + /3 width=8 by da_ldec/ + | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 + /3 width=8 by da_ldec, drop_inv_gen/ + ] +| #a #I #G #L1 #V1 #T1 #d #_ #IHT1 #L2 #c #l #k #HL21 #X #H + elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct + /4 width=5 by da_bind, drop_skip/ +| #I #G #L1 #V1 #T1 #d #_ #IHT1 #L2 #c #l #k #HL21 #X #H + elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct + /3 width=5 by da_flat/ +] +qed. + +(* Inversion lemmas on relocation *******************************************) + +lemma da_inv_lift: ∀h,o,G,L2,T2,d. ⦃G, L2⦄ ⊢ T2 ▪[h, o] d → + ∀L1,c,l,k. ⬇[c, l, k] L2 ≡ L1 → ∀T1. ⬆[l, k] T1 ≡ T2 → + ⦃G, L1⦄ ⊢ T1 ▪[h, o] d. +#h #o #G #L2 #T2 #d #H elim H -G -L2 -T2 -d +[ #G #L2 #s #d #Hkd #L1 #c #l #k #_ #X #H + >(lift_inv_sort2 … H) -X /2 width=1 by da_sort/ +| #G #L2 #K2 #V2 #i #d #HLK2 #HV2 #IHV2 #L1 #c #l #k #HL21 #X #H + elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HV2 | -IHV2 ] + [ elim (drop_conf_lt … HL21 … HLK2) -L2 /3 width=8 by da_ldef/ + | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 /2 width=4 by da_ldef/ + ] +| #G #L2 #K2 #W2 #i #d #HLK2 #HW2 #IHW2 #L1 #c #l #k #HL21 #X #H + elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HW2 | -IHW2 ] + [ elim (drop_conf_lt … HL21 … HLK2) -L2 /3 width=8 by da_ldec/ + | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 /2 width=4 by da_ldec/ + ] +| #a #I #G #L2 #V2 #T2 #d #_ #IHT2 #L1 #c #l #k #HL21 #X #H + elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct + /4 width=5 by da_bind, drop_skip/ +| #I #G #L2 #V2 #T2 #d #_ #IHT2 #L1 #c #l #k #HL21 #X #H + elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct + /3 width=5 by da_flat/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/da/degree_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/da/degree_6.etc new file mode 100644 index 000000000..bcc939ad1 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/da/degree_6.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T ▪ break [ term 46 o , break term 46 h ] break term 46 d )" + non associative with precedence 45 + for @{ 'Degree $h $o $G $L $T $d }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/droppreds_3.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/droppreds_3.etc new file mode 100644 index 000000000..b21fe5117 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/droppreds_3.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +notation "hvbox( ⫱ * [ term 46 L , break term 46 K ] term 46 f )" + non associative with precedence 46 + for @{ 'DropPreds $L $K $f }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/drops/drops.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/drops/drops.etc new file mode 100644 index 000000000..5d51b83c8 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/drops/drops.etc @@ -0,0 +1,2 @@ +lemma drop_inv_refl: ∀L,l,m. ⬇[Ⓕ, l, m] L ≡ L → m = 0. +/2 width=5 by drop_inv_length_eq/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/drops/drops_append.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/drops/drops_append.etc new file mode 100644 index 000000000..d134fe768 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/drops/drops_append.etc @@ -0,0 +1,65 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/grammar/lenv_append.ma". +include "basic_2/substitution/drop.ma". + +(* DROPPING *****************************************************************) + +(* Properties on append for local environments ******************************) + +fact drop_O1_append_sn_le_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 → + l = 0 → m ≤ |L1| → + ∀L. ⬇[s, 0, m] L @@ L1 ≡ L @@ L2. +#L1 #L2 #s #l #m #H elim H -L1 -L2 -l -m // +[ #l #m #_ #_ #H >(yle_inv_O2 … H) -m // +| /4 width=1 by drop_drop, yle_inv_succ/ +| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #_ #H elim (ysucc_inv_O_dx … H) +] +qed-. + +lemma drop_O1_append_sn_le: ∀L1,L2,s,m. ⬇[s, yinj 0, m] L1 ≡ L2 → m ≤ |L1| → + ∀L. ⬇[s, 0, m] L @@ L1 ≡ L @@ L2. +/2 width=3 by drop_O1_append_sn_le_aux/ qed. + +(* Inversion lemmas on append for local environments ************************) + +lemma drop_O1_inv_append1_ge: ∀K,L1,L2,s,m. ⬇[s, 0, m] L1 @@ L2 ≡ K → + ∀m0. |L2| + m0 = m → ⬇[s, 0, m0] L1 ≡ K. +#K #L1 #L2 elim L2 -L2 +[ #s #m #H #m0 >yplus_O1 #H0 destruct // +| #L2 #I #V #IHL2 #s #m #H #m0 >yplus_succ1 + #H0 elim (drop_inv_O1_pair1 … H) -H * #Hm #HL12 destruct + [ elim (ysucc_inv_O_dx … Hm) + | /2 width=3 by/ + ] +] +qed-. + +lemma drop_O1_inv_append1_le: ∀K,L1,L2,s,m. ⬇[s, 0, m] L1 @@ L2 ≡ K → m ≤ |L2| → + ∀K2. ⬇[s, 0, m] L2 ≡ K2 → K = L1 @@ K2. +#K #L1 #L2 elim L2 -L2 +[ #s #m #H1 #H2 #K2 #H3 lapply (yle_inv_O2 … H2) -H2 + #H2 elim (drop_inv_atom1 … H3) -H3 #H3 #_ destruct + >(drop_inv_O2 … H1) -H1 // +| #L2 #I #V #IHL2 #s #m @(ynat_ind … m) -m [ -IHL2 || -IHL2 ] + [ #H1 #_ #K2 #H2 + lapply (drop_inv_O2 … H1) -H1 #H1 + lapply (drop_inv_O2 … H2) -H2 #H2 destruct // + | /3 width=7 by drop_inv_drop1, yle_inv_succ/ + | #_ #H lapply (yle_inv_Y1 … H) -H + #H elim (ylt_yle_false (|L2.ⓑ{I}V|) (∞)) // + ] +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/drops/drops_drops.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/drops/drops_drops.etc new file mode 100644 index 000000000..906c85c73 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/drops/drops_drops.etc @@ -0,0 +1,13 @@ +(* Inversion lemmas on equivalence ******************************************) + +lemma drop_O1_inj: ∀i,L1,L2,K. ⬇[i] L1 ≡ K → ⬇[i] L2 ≡ K → L1 ⩬[i, ∞] L2. +#i @(ynat_ind … i) -i +[ #L1 #L2 #K #H <(drop_inv_O2 … H) -K #H <(drop_inv_O2 … H) -L1 // +| #i #IHi * [2: #L1 #I1 #V1 ] * [2,4: #L2 #I2 #V2 ] #K #HLK1 #HLK2 // + lapply (drop_fwd_length … HLK1) + <(drop_fwd_length … HLK2) [ /4 width=5 by drop_inv_drop1, lreq_succ/ ] + #H [ elim (ysucc_inv_O_sn … H) | elim (ysucc_inv_O_dx … H) ] +| #L1 #L2 #K #H1 lapply (drop_fwd_Y2 … H1) -H1 + #H elim (ylt_yle_false … H) // +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/drops/drops_length.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/drops/drops_length.etc new file mode 100644 index 000000000..667f9689d --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/drops/drops_length.etc @@ -0,0 +1,112 @@ +lemma drop_inv_O1_gt: ∀L,K,m,s. ⬇[s, 0, m] L ≡ K → |L| < m → + s = Ⓣ ∧ K = ⋆. +#L elim L -L [| #L #Z #X #IHL ] #K #m #s #H normalize in ⊢ (?%?→?); #H1m +[ elim (drop_inv_atom1 … H) -H elim s -s /2 width=1 by conj/ + #_ #Hs lapply (Hs ?) // -Hs #H destruct elim (ylt_yle_false … H1m) -H1m // +| elim (drop_inv_O1_pair1 … H) -H * #H2m #HLK destruct + [ elim (ylt_yle_false … H1m) -H1m // + | elim (IHL … HLK) -IHL -HLK /2 width=1 by ylt_pred, conj/ + ] +] +qed-. + +lemma drop_O1_le: ∀s,m,L. m ≤ |L| → ∃K. ⬇[s, 0, m] L ≡ K. +#s #m @(ynat_ind … m) -m /2 width=2 by ex_intro/ +[ #m #IHm * + [ #H elim (ylt_yle_false … H) -H // + | #L #I #V #H elim (IHm L) -IHm /3 width=2 by drop_drop, yle_inv_succ, ex_intro/ + ] +| #L #H elim (ylt_yle_false … H) -H // +] +qed-. + +lemma drop_O1_lt: ∀s,L,m. m < |L| → ∃∃I,K,V. ⬇[s, 0, m] L ≡ K.ⓑ{I}V. +#s #L elim L -L +[ #m #H elim (ylt_yle_false … H) -H // +| #L #I #V #IHL #m @(ynat_ind … m) -m /2 width=4 by drop_pair, ex1_3_intro/ + [ #m #_#H elim (IHL m) -IHL /3 width=4 by drop_drop, ylt_inv_succ, ex1_3_intro/ + | #H elim (ylt_yle_false … H) -H // + ] +] +qed-. + +lemma drop_O1_pair: ∀L,K,m,s. ⬇[s, 0, m] L ≡ K → m ≤ |L| → ∀I,V. + ∃∃J,W. ⬇[s, 0, m] L.ⓑ{I}V ≡ K.ⓑ{J}W. +#L elim L -L [| #L #Z #X #IHL ] #K #m #s #H #Hm #I #V +[ elim (drop_inv_atom1 … H) -H #H >(yle_inv_O2 … Hm) -m + #Hs destruct /2 width=3 by ex1_2_intro/ +| elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK destruct /2 width=3 by ex1_2_intro/ + elim (IHL … HLK … Z X) -IHL -HLK + /3 width=3 by yle_pred, drop_drop_lt, ex1_2_intro/ +] +qed-. + +lemma drop_O1_ge: ∀L,m. |L| ≤ m → ⬇[Ⓣ, 0, m] L ≡ ⋆. +#L elim L -L [ #m #_ @drop_atom #H destruct ] +#L #I #V #IHL #m @(ynat_ind … m) -m // +[ #H elim (ylt_yle_false … H) -H /2 width=1 by ylt_inj/ +| /4 width=1 by drop_drop, yle_inv_succ/ +] +qed. + +lemma drop_O1_eq: ∀L,s. ⬇[s, 0, |L|] L ≡ ⋆. +#L elim L -L /2 width=1 by drop_drop/ +qed. + +lemma drop_fwd_length_ge: ∀L1,L2,l,m,s. ⬇[s, l, m] L1 ≡ L2 → |L1| ≤ l → |L2| = |L1|. +#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m // +[ #I #L1 #L2 #V #m #_ #_ #H elim (ylt_yle_false … H) -H // +| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #IH #H + lapply (yle_inv_succ … H) -H #H + >length_pair >length_pair /3 width=1 by eq_f/ +] +qed-. + +lemma drop_fwd_length_le_le: ∀L1,L2,l,m,s. ⬇[s, l, m] L1 ≡ L2 → + ∀l0. l + m + l0 = |L1| → |L2| = l + l0. +#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m // +[ #l #m #Hm #l0 #H elim (yplus_inv_O … H) -H + #H #H0 elim (yplus_inv_O … H) -H + #H1 #_ destruct // +| #I #L1 #L2 #V #m #_ >yplus_O1 >yplus_O1 #IH #l0 + /3 width=1 by ysucc_inv_inj/ +| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #IHL12 #l0 >yplus_succ1 >yplus_succ1 #H + lapply (ysucc_inv_inj … H) -H #Hl1 + >yplus_succ1 /3 width=1 by eq_f/ +] +qed-. + +lemma drop_fwd_length_le_ge: ∀L1,L2,l,m,s. ⬇[s, l, m] L1 ≡ L2 → l ≤ |L1| → |L1| ≤ l + m → |L2| = l. +#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m +[ #l #m #_ #H #_ /2 width=1 by yle_inv_O2/ +| #I #L #V #_ #H elim (ylt_yle_false … H) -H // +| #I #L1 #L2 #V #m #_ >yplus_O1 >yplus_O1 + /3 width=1 by yle_inv_succ/ +| #I #L1 #L2 #V1 #v2 #l #m #_ #_ #IH + >yplus_SO2 >yplus_SO2 + /4 width=1 by yle_inv_succ, eq_f/ +] +qed-. + +lemma drop_fwd_be: ∀L,K,s,l,m,i. ⬇[s, l, m] L ≡ K → |K| ≤ i → i < l → |L| ≤ i. +#L #K #s #l #m #i #HLK #HK #Hl elim (ylt_split i (|L|)) // +#HL elim (drop_O1_lt (Ⓕ) … HL) #I #K0 #V #HLK0 -HL +elim (ylt_inv_plus_sn … Hl) -Hl #l0 #H0 +elim (drop_conf_lt … HLK … HLK0 … H0) -HLK -HLK0 -H0 +#K1 #V1 #HK1 #_ #_ lapply (drop_fwd_length_lt2 … HK1) -I -K1 -V1 +#H elim (ylt_yle_false … H) -H // +qed-. + +lemma drop_O1_ex: ∀K2,i,L1. |L1| = |K2| + i → + ∃∃L2. L1 ⩬[0, i] L2 & ⬇[i] L2 ≡ K2. +#K2 #i @(ynat_ind … i) -i +[ /3 width=3 by lreq_O2, ex2_intro/ +| #i #IHi #Y >yplus_succ2 #Hi + elim (drop_O1_lt (Ⓕ) Y 0) [2: >Hi // ] + #I #L1 #V #H lapply (drop_inv_O2 … H) -H #H destruct + >length_pair in Hi; #H lapply (ysucc_inv_inj … H) -H + #HL1K2 elim (IHi L1) -IHi // -HL1K2 + /3 width=5 by lreq_pair, drop_drop, ex2_intro/ +| #L1 >yplus_Y2 #H elim (ylt_yle_false (|L1|) (∞)) // +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/fqu/fqu.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/fqu/fqu.etc new file mode 100644 index 000000000..b5066d00f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/fqu/fqu.etc @@ -0,0 +1,16 @@ +(* Removed theorems *********************************************************) + +include "basic_2/substitution/drop.ma". + +| fqu_drop : ∀G,L,K,T,U,k. + ⬇[⫯k] L ≡ K → ⬆[0, ⫯k] T ≡ U → fqu G L U G K T + +lemma fqu_drop_lt: ∀G,L,K,T,U,k. 0 < k → + ⬇[k] L ≡ K → ⬆[0, k] T ≡ U → ⦃G, L, U⦄ ⊐ ⦃G, K, T⦄. +#G #L #K #T #U #k #Hm lapply (ylt_inv_O1 … Hm) -Hm +#Hm (drop_inv_O2 … HLK) -L >(lift_inv_O2 … HTU) -T // +qed. + +inductive fquq: tri_relation genv lenv term ≝ +| fquq_lref_O : ∀I,G,L,V. fquq G (L.ⓑ{I}V) (#0) G L V +| fquq_pair_sn: ∀I,G,L,V,T. fquq G L (②{I}V.T) G L V +| fquq_bind_dx: ∀a,I,G,L,V,T. fquq G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T +| fquq_flat_dx: ∀I,G, L,V,T. fquq G L (ⓕ{I}V.T) G L T +| fquq_drop : ∀G,L,K,T,U,k. + ⬇[k] L ≡ K → ⬆[0, k] T ≡ U → fquq G L U G K T +. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/frees/frees.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/frees/frees.etc new file mode 100644 index 000000000..a9349e870 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/frees/frees.etc @@ -0,0 +1,5 @@ +(* A Basic_A2 lemma we do not need so far *) +axiom frees_pair_flat: ∀L,T,f1,I1,V1. L.ⓑ{I1}V1 ⊢ 𝐅*⦃T⦄ ≡ f1 → + ∀f2,I2,V2. L.ⓑ{I2}V2 ⊢ 𝐅*⦃T⦄ ≡ f2 → + ∀f0. f1 ⋓ f2 ≡ f0 → + ∀I0,I. L.ⓑ{I0}ⓕ{I}V1.V2 ⊢ 𝐅*⦃T⦄ ≡ f0. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/frees/frees_append.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/frees/frees_append.etc new file mode 100644 index 000000000..ee8324cc9 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/frees/frees_append.etc @@ -0,0 +1,54 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/substitution/drop_append.ma". +include "basic_2/multiple/frees.ma". + +(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************) + +(* Properties on append for local environments ******************************) + +lemma frees_append: ∀L2,U,l,i. L2 ⊢ i ϵ 𝐅*[l]⦃U⦄ → i ≤ |L2| → + ∀L1. L1 @@ L2 ⊢ i ϵ 𝐅*[l]⦃U⦄. +#L2 #U #l #i #H elim H -L2 -U -l -i /3 width=2 by frees_eq/ +#I #L2 #K2 #U #W #l #i #j #Hlj #Hji #HnU #HLK2 #_ #IHW #Hi #L1 +lapply (drop_fwd_length_minus2 … HLK2) normalize #H0 +lapply (drop_O1_append_sn_le … HLK2 … L1) -HLK2 +[ -I -L1 -K2 -U -W -l /4 width=3 by ylt_yle_trans, ylt_inv_inj, lt_to_le/ +| #HLK2 @(frees_be … HnU HLK2) // -HnU -HLK2 @IHW -IHW + >(minus_plus_m_m (|K2|) 1) >H0 -H0 yminus_SO2 + /3 width=1 by monotonic_yle_minus_dx, yle_pred/ +] +qed. + +(* Inversion lemmas on append for local environments ************************) + +fact frees_inv_append_aux: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ → ∀L1,L2. L = L1 @@ L2 → + i ≤ |L2| → L2 ⊢ i ϵ 𝐅*[l]⦃U⦄. +#L #U #l #i #H elim H -L -U -l -i /3 width=2 by frees_eq/ +#Z #L #Y #U #X #l #i #j #Hlj #Hji #HnU #HLY #_ #IHW #L1 #L2 #H #Hi destruct +elim (drop_O1_lt (Ⓕ) L2 j) [2: -Z -Y -L1 -X -U -l /3 width=3 by ylt_yle_trans, ylt_inv_inj/ ] +#I #K2 #W #HLK2 lapply (drop_fwd_length_minus2 … HLK2) normalize #H0 +lapply (drop_O1_inv_append1_le … HLY … HLK2) -HLY +[ -Z -I -Y -K2 -L1 -X -U -W -l /4 width=3 by ylt_yle_trans, ylt_inv_inj, lt_to_le/ +| normalize #H destruct + @(frees_be … HnU HLK2) -HnU -HLK2 // @IHW -IHW // + >(minus_plus_m_m (|K2|) 1) >H0 -H0 yminus_SO2 + /3 width=1 by monotonic_yle_minus_dx, yle_pred/ +] +qed-. + +lemma frees_inv_append: ∀L1,L2,U,l,i. L1 @@ L2 ⊢ i ϵ 𝐅*[l]⦃U⦄ → + i ≤ |L2| → L2 ⊢ i ϵ 𝐅*[l]⦃U⦄. +/2 width=4 by frees_inv_append_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/frees/frees_fqus.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/frees/frees_fqus.etc new file mode 100644 index 000000000..618d858ff --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/frees/frees_fqus.etc @@ -0,0 +1,111 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/relocation/drops_weight.ma". +include "basic_2/s_computation/fqup_weight.ma". +include "basic_2/s_computation/fqus_weight.ma". +include "basic_2/static/frees.ma". + +(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************) + +(* Properties with star-iterated supclosure *********************************) + +(* Note: Restrict fqu_drop on atoms to prove frees_fqus_drops + Counterex: ⦃⋆, ⋆.ⓛ#0, ⓛ#1.#0⦄ ⊐ ⦃⋆, ⋆, ⓛ#0.#0⦄ ⊐ ⦃⋆, ⋆.ⓛ#0, #0⦄ +*) +lemma frees_fqus_drops: ∀G,L1,T1,f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 → + ∀L2,T2. ⦃G, L1, T1⦄ ⊐* ⦃G, L2, T2⦄ → + ∀I,n. ⬇*[n] L1 ≡ L2.ⓑ{I}T2 → + ∃∃f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ ⫱*[⫯n] f1. +#G #L1 #T1 @(fqup_wf_ind_eq … G L1 T1) -G -L1 -T1 +#G0 #L0 #T0 #IH #G #L1 * * +[ #s #HG #HL #HT #f1 #Hf1 #L2 #T2 #H12 #I #n #HL12 destruct + lapply (frees_inv_sort … Hf1) -Hf1 #Hf1 + elim (fqus_inv_sort1 … H12) -H12 * [| elim (nat_split n) ] + [ -IH #H1 #H2 #H3 destruct + lapply (drops_fwd_lw … HL12) -HL12 #HL12 + elim (lt_le_false … HL12) -HL12 // + | -IH #Hn #J #L #V #H12 #H destruct + lapply (drops_fwd_isid … HL12 ?) -HL12 // #H destruct + <(fqus_inv_refl_atom3 … H12) -H12 + /3 width=3 by frees_sort_gen, sle_isid_sn, ex2_intro/ + | * #m #Hn #J #L #V #H12 #H destruct + lapply (drops_inv_drop1 … HL12) -HL12 #HL12 + elim (IH … H12 … HL12) -IH -H12 -HL12 + /3 width=3 by frees_sort_gen, fqu_fqup, fqu_drop, lifts_sort, isid_tl, ex2_intro/ + ] +| #i #HG #HL #HT #f1 #Hf1 #L2 #T2 #H12 #I #n #HL12 destruct + elim (fqus_inv_lref1 … H12) -H12 * [|2,3: elim (nat_split n) ] + [ -IH #H1 #H2 #H3 destruct + lapply (drops_fwd_lw … HL12) -HL12 #HL12 + elim (lt_le_false … HL12) -HL12 // + | -IH #Hn #J #L #V #H12 #H1 #H2 destruct + elim (frees_inv_zero_pair … Hf1) -Hf1 #g1 #Hg1 #H destruct + lapply (drops_fwd_isid … HL12 ?) -HL12 -H12 // #H destruct + /2 width=3 by sle_refl, ex2_intro/ + | * #m #Hn #J #L #V #H12 #H1 #H2 destruct + elim (frees_inv_zero_pair … Hf1) -Hf1 #g1 #Hg1 #H destruct + lapply (drops_inv_drop1 … HL12) -HL12 #HL12 + elim (IH … H12 … HL12) -IH -H12 -HL12 + /2 width=3 by fqu_fqup, fqu_lref_O, ex2_intro/ + | -IH #Hn #J #L #V #j #H12 #H1 #H2 destruct + elim (frees_inv_lref_pair … Hf1) -Hf1 #g1 #Hg1 #H destruct + lapply (drops_fwd_isid … HL12 ?) -HL12 // #H destruct + <(fqus_inv_refl_atom3 … H12) -H12 /2 width=3 by sle_refl, ex2_intro/ + | * #m #Hn #J #L #V #j #H12 #H1 #H2 destruct + elim (frees_inv_lref_pair … Hf1) -Hf1 #g1 #Hg1 #H destruct + lapply (drops_inv_drop1 … HL12) -HL12 #HL12 + elim (IH … H12 … HL12) -IH -H12 -HL12 /2 width=3 by fqu_fqup, ex2_intro/ + ] +| #l #HG #HL #HT #f1 #Hf1 #L2 #T2 #H12 #I #n #HL12 destruct + lapply (frees_inv_gref … Hf1) -Hf1 #Hf1 + elim (fqus_inv_gref1 … H12) -H12 * [| elim (nat_split n) ] + [ -IH #H1 #H2 #H3 destruct + lapply (drops_fwd_lw … HL12) -HL12 #HL12 + elim (lt_le_false … HL12) -HL12 // + | -IH #Hn #J #L #V #H12 #H destruct + lapply (drops_fwd_isid … HL12 ?) -HL12 // #H destruct + <(fqus_inv_refl_atom3 … H12) -H12 + /3 width=3 by frees_gref_gen, sle_isid_sn, ex2_intro/ + | * #m #Hn #J #L #V #H12 #H destruct + lapply (drops_inv_drop1 … HL12) -HL12 #HL12 + elim (IH … H12 … HL12) -IH -H12 -HL12 + /3 width=3 by frees_gref_gen, fqu_fqup, fqu_drop, lifts_sort, isid_tl, ex2_intro/ + ] +| #p #I #V1 #T1 #HG #HL #HT #f1 #Hf1 #L2 #T2 #H12 #I #n #HL12 destruct + elim (frees_inv_bind … Hf1) -Hf1 #f1V #f1T #Hf1V #Hf1T #Hf1 + elim (fqus_inv_bind1 … H12) -H12 + [ -IH -f1V -f1T * #H1 #H2 #H3 destruct + lapply (drops_fwd_lw … HL12) -HL12 #HL12 + elim (lt_le_false … HL12) -HL12 // + | #H12 elim (IH … H12 … HL12) -IH -H12 -HL12 + /4 width=6 by sor_tls, sor_inv_sle_sn_trans, ex2_intro/ + | #H12 elim (IH … H12 I (⫯n)) -IH -H12 [|*: /2 width=3 by drops_drop, drops_refl/ ] -HL12 + H -H >commutative_plus #H (**) (* lemma needed here *) + lapply (le_plus_to_le_r … 0 H) -H #H + lapply (le_n_O_to_eq … H) -H #H destruct +| #I #G1 #G2 #V #H1 #_ #H2 + lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 normalize in ⊢ (? % ? → ?); >commutative_plus #H + lapply (lt_plus_to_lt_l … 0 H) -H #H + elim (lt_zero_false … H) +] +qed-. + +lemma gget_inv_eq: ∀G1,G2,m. ⬇[m] G1 ≡ G2 → |G1| = m + 1 → G1 = G2. +#G1 #G2 #m * -G1 -G2 // +[ #G #H1 #H2 >H2 in H1; -H2 >commutative_plus #H (**) (* lemma needed here *) + lapply (le_plus_to_le_r … 0 H) -H #H + lapply (le_n_O_to_eq … H) -H #H destruct +| #I #G1 #G2 #V #H1 #_ normalize #H2 + <(injective_plus_l … H2) in H1; -H2 #H + elim (lt_refl_false … H) +] +qed-. + +fact gget_inv_lt_aux: ∀I,G,G1,G2,V,m. ⬇[m] G ≡ G2 → G = G1. ⓑ{I} V → + m < |G1| → ⬇[m] G1 ≡ G2. +#I #G #G1 #G2 #V #m * -G -G2 +[ #G #H1 #H destruct #H2 + lapply (le_to_lt_to_lt … H1 H2) -H1 -H2 normalize in ⊢ (? % ? → ?); >commutative_plus #H + lapply (lt_plus_to_lt_l … 0 H) -H #H + elim (lt_zero_false … H) +| #G #H1 #H2 destruct >(injective_plus_l … H1) -H1 #H + elim (lt_refl_false … H) +| #J #G #G2 #W #_ #HG2 #H destruct // +] +qed-. + +lemma gget_inv_lt: ∀I,G1,G2,V,m. + ⬇[m] G1. ⓑ{I} V ≡ G2 → m < |G1| → ⬇[m] G1 ≡ G2. +/2 width=5 by gget_inv_lt_aux/ qed-. + +(* Basic properties *********************************************************) + +lemma gget_total: ∀m,G1. ∃G2. ⬇[m] G1 ≡ G2. +#m #G1 elim G1 -G1 /3 width=2 by gget_gt, ex_intro/ +#I #V #G1 * #G2 #HG12 +elim (lt_or_eq_or_gt m (|G1|)) #Hm +[ /3 width=2 by gget_lt, ex_intro/ +| destruct /3 width=2 by gget_eq, ex_intro/ +| @ex_intro [2: @gget_gt normalize /2 width=1 by/ | skip ] (**) (* explicit constructor *) +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/gget/gget_gget.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/gget/gget_gget.etc new file mode 100644 index 000000000..7bd39f09b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/gget/gget_gget.etc @@ -0,0 +1,40 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/substitution/gget.ma". + +(* GLOBAL ENVIRONMENT READING ***********************************************) + +(* Main properties **********************************************************) + +theorem gget_mono: ∀G,G1,m. ⬇[m] G ≡ G1 → ∀G2. ⬇[m] G ≡ G2 → G1 = G2. +#G #G1 #m #H elim H -G -G1 +[ #G #Hm #G2 #H + >(gget_inv_gt … H Hm) -H -Hm // +| #G #Hm #G2 #H + >(gget_inv_eq … H Hm) -H -Hm // +| #I #G #G1 #V #Hm #_ #IHG1 #G2 #H + lapply (gget_inv_lt … H Hm) -H -Hm /2 width=1 by/ +] +qed-. + +lemma gget_dec: ∀G1,G2,m. Decidable (⬇[m] G1 ≡ G2). +#G1 #G2 #m +elim (gget_total m G1) #G #HG1 +elim (eq_genv_dec G G2) #HG2 +[ destruct /2 width=1 by or_introl/ +| @or_intror #HG12 + lapply (gget_mono … HG1 … HG12) -HG1 -HG12 /2 width=1 by/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/gget/rdrop_3.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/gget/rdrop_3.etc new file mode 100644 index 000000000..a761d47c6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/gget/rdrop_3.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +notation "hvbox( ⬇ [ term 46 m ] break term 46 L1 ≡ break term 46 L2 )" + non associative with precedence 45 + for @{ 'RDrop $m $L1 $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lazyor_5.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lazyor_5.etc new file mode 100644 index 000000000..70d3a1c66 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lazyor_5.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +notation "hvbox( L1 ⋓ break [ term 46 T , break term 46 f ] break term 46 L2 ≡ break term 46 L )" + non associative with precedence 45 + for @{ 'LazyOr $L1 $T $f $L2 $L }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lfpr_main.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lfpr_main.etc new file mode 100644 index 000000000..bfe23e721 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lfpr_main.etc @@ -0,0 +1,80 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/rt_transition/lfpr_lfpr.ma". + +(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************) + +definition lfxs_confluent_R: relation2 … ≝ + λRP1,RP2. + ∀L0,T0,T1. RP1 L0 T0 T1 → ∀T2. RP2 L0 T0 T2 → + ∀L1. L0 ⦻*[RP1, T0] L1 → ∀L2. L0 ⦻*[RP2, T0] L2 → + ∃∃L. L1 ⦻*[RP2, T1] L & L2 ⦻*[RP1, T2] L. + +(* Main properties **********************************************************) + +fact lfpr_conf_cpr_atom_atom: + ∀h,I,G,L0. ( + ∀L,T. ⦃G, L0, ⓪{I}⦄ ⊐+ ⦃G, L, T⦄ → + ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → + ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 → + ∃∃K0. ⦃G, L1⦄ ⊢ ➡[h, T1] K0 & ⦃G, L2⦄ ⊢ ➡[h, T2] K0 + ) → + ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓪{I}] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓪{I}] L2 → + ∃∃L. ⦃G, L1⦄ ⊢ ➡[h, ⓪{I}] L & ⦃G, L2⦄ ⊢ ➡[h, ⓪{I}] L. +#h #I #G * [ | #K0 #J #V0 cases I -I [ | * | ] ] +[ #_ #L1 #HL01 #L2 #HL02 + lapply (lfpr_inv_atom_sn … HL01) -HL01 #H destruct + lapply (lfpr_inv_atom_sn … HL02) -HL02 #H destruct + /2 width=3 by ex2_intro/ +| #s #IH #L1 #HL01 #L2 #HL02 + elim (lfxs_inv_sort_pair_sn … HL01) -HL01 #K1 #V1 #HK01 #H destruct + elim (lfxs_inv_sort_pair_sn … HL02) -HL02 #K2 #V2 #HK02 #H destruct + elim (IH … HK01 … HK02) -IH -HK01 -HK02 + /3 width=5 by lfpr_sort, fqu_fqup, fqu_drop, ex2_intro/ +| #IH #L1 #HL01 #L2 #HL02 + elim (lfpr_inv_zero_pair_sn … HL01) -HL01 #K1 #V1 #HK01 #HV01 #H destruct + elim (lfpr_inv_zero_pair_sn … HL02) -HL02 #K2 #V2 #HK02 #HV02 #H destruct + elim (cpr_conf_lfpr … HV01 … HV02 … HK01 … HK02) #V #HV1 #HV2 + elim (IH … HV01 … HV02 … HK01 … HK02) -IH -HV01 -HV02 -HK01 -HK02 + /3 width=5 by lfpr_zero, fqu_fqup, fqu_drop, ex2_intro/ +| #i #IH #L1 #HL01 #L2 #HL02 + elim (lfxs_inv_lref_pair_sn … HL01) -HL01 #K1 #V1 #HK01 #H destruct + elim (lfxs_inv_lref_pair_sn … HL02) -HL02 #K2 #V2 #HK02 #H destruct + elim (IH … HK01 … HK02) -IH -HK01 -HK02 + /3 width=5 by lfpr_lref, fqu_fqup, fqu_drop, ex2_intro/ +| #l #IH #L1 #HL01 #L2 #HL02 + elim (lfxs_inv_gref_pair_sn … HL01) -HL01 #K1 #V1 #HK01 #H destruct + elim (lfxs_inv_gref_pair_sn … HL02) -HL02 #K2 #V2 #HK02 #H destruct + elim (IH … HK01 … HK02) -IH -HK01 -HK02 + /3 width=5 by lfpr_gref, fqu_fqup, fqu_drop, ex2_intro/ +] +qed-. + +theorem lfpr_conf_cpr: ∀h,G. lfxs_confluent_R (cpm 0 h G) (cpm 0 h G). +#h #G #L0 #T0 @(fqup_wf_ind_eq … G L0 T0) -G -L0 -T0 #G #L #T #IH #G0 #L0 * [| * ] +[ #I0 #HG #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct + elim (cpr_inv_atom1_drops … H1) -H1 + elim (cpr_inv_atom1_drops … H2) -H2 + [ #H2 #H1 destruct + /3 width=7 by lfpr_conf_cpr_atom_atom/ + | * #K0 #V0 #V2 * [2: #i2 ] #HLK0 #HV02 #HVT2 #H2 #H1 destruct + +(* + +theorem lpr_conf: ∀G. confluent … (lpr G). +/3 width=6 by lpx_sn_conf, cpr_conf_lpr/ +qed-. + +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs.etc new file mode 100644 index 000000000..f09a2cfc2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs.etc @@ -0,0 +1,264 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "ground_2/relocation/rtmap_id.ma". +include "basic_2/notation/relations/relationstar_4.ma". +include "basic_2/relocation/lexs.ma". +include "basic_2/static/frees.ma". + +(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) + +definition lfxs (R) (T): relation lenv ≝ + λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≡ f & L1 ⦻*[R, cfull, f] L2. + +interpretation "generic extension on referred entries (local environment)" + 'RelationStar R T L1 L2 = (lfxs R T L1 L2). + +definition R_frees_confluent: predicate (relation3 lenv term term) ≝ + λRN. + ∀f1,L,T1. L ⊢ 𝐅*⦃T1⦄ ≡ f1 → ∀T2. RN L T1 T2 → + ∃∃f2. L ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ f1. + +definition lexs_frees_confluent: relation (relation3 lenv term term) ≝ + λRN,RP. + ∀f1,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 → + ∀L2. L1 ⦻*[RN, RP, f1] L2 → + ∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 & f2 ⊆ f1. + +definition R_confluent2_lfxs: relation4 (relation3 lenv term term) + (relation3 lenv term term) … ≝ + λR1,R2,RP1,RP2. + ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 → + ∀L1. L0 ⦻*[RP1, T0] L1 → ∀L2. L0 ⦻*[RP2, T0] L2 → + ∃∃T. R2 L1 T1 T & R1 L2 T2 T. + +(* Basic properties ***********************************************************) + +lemma lfxs_atom: ∀R,I. ⋆ ⦻*[R, ⓪{I}] ⋆. +/3 width=3 by lexs_atom, frees_atom, ex2_intro/ +qed. + +lemma lfxs_sort: ∀R,I,L1,L2,V1,V2,s. + L1 ⦻*[R, ⋆s] L2 → L1.ⓑ{I}V1 ⦻*[R, ⋆s] L2.ⓑ{I}V2. +#R #I #L1 #L2 #V1 #V2 #s * /3 width=3 by lexs_push, frees_sort, ex2_intro/ +qed. + +lemma lfxs_zero: ∀R,I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 → + R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, #0] L2.ⓑ{I}V2. +#R #I #L1 #L2 #V1 #V2 * /3 width=3 by lexs_next, frees_zero, ex2_intro/ +qed. + +lemma lfxs_lref: ∀R,I,L1,L2,V1,V2,i. + L1 ⦻*[R, #i] L2 → L1.ⓑ{I}V1 ⦻*[R, #⫯i] L2.ⓑ{I}V2. +#R #I #L1 #L2 #V1 #V2 #i * /3 width=3 by lexs_push, frees_lref, ex2_intro/ +qed. + +lemma lfxs_gref: ∀R,I,L1,L2,V1,V2,l. + L1 ⦻*[R, §l] L2 → L1.ⓑ{I}V1 ⦻*[R, §l] L2.ⓑ{I}V2. +#R #I #L1 #L2 #V1 #V2 #l * /3 width=3 by lexs_push, frees_gref, ex2_intro/ +qed. + +lemma lfxs_pair_repl_dx: ∀R,I,L1,L2,T,V,V1. + L1.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V1 → + ∀V2. R L1 V V2 → + L1.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V2. +#R #I #L1 #L2 #T #V #V1 * #f #Hf #HL12 #V2 #HR +/3 width=5 by lexs_pair_repl, ex2_intro/ +qed-. + +lemma lfxs_sym: ∀R. lexs_frees_confluent R cfull → + (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) → + ∀T. symmetric … (lfxs R T). +#R #H1R #H2R #T #L1 #L2 * #f1 #Hf1 #HL12 elim (H1R … Hf1 … HL12) -Hf1 +/4 width=5 by sle_lexs_trans, lexs_sym, ex2_intro/ +qed-. + +lemma lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) → + ∀L1,L2,T. L1 ⦻*[R1, T] L2 → L1 ⦻*[R2, T] L2. +#R1 #R2 #HR #L1 #L2 #T * /4 width=7 by lexs_co, ex2_intro/ +qed-. + +(* Basic inversion lemmas ***************************************************) + +lemma lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⦻*[R, ⓪{I}] Y2 → Y2 = ⋆. +#R #I #Y2 * /2 width=4 by lexs_inv_atom1/ +qed-. + +lemma lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻*[R, ⓪{I}] ⋆ → Y1 = ⋆. +#R #I #Y1 * /2 width=4 by lexs_inv_atom2/ +qed-. + +lemma lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻*[R, ⋆s] Y2 → + (Y1 = ⋆ ∧ Y2 = ⋆) ∨ + ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, ⋆s] L2 & + Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. +#R * [ | #Y1 #I #V1 ] #Y2 #s * #f #H1 #H2 +[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/ +| lapply (frees_inv_sort … H1) -H1 #Hf + elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct + elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct + /5 width=8 by frees_sort_gen, ex3_5_intro, ex2_intro, or_intror/ +] +qed-. + +lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻*[R, #0] Y2 → + (Y1 = ⋆ ∧ Y2 = ⋆) ∨ + ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 & + Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. +#R #Y1 #Y2 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 * +[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/ +| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_next1_aux … H2 … HY1 Hg) -H2 -Hg + /4 width=9 by ex4_5_intro, ex2_intro, or_intror/ +] +qed-. + +lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 → + (Y1 = ⋆ ∧ Y2 = ⋆) ∨ + ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, #i] L2 & + Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. +#R #Y1 #Y2 #i * #f #H1 #H2 elim (frees_inv_lref … H1) -H1 * +[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/ +| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_push1_aux … H2 … HY1 Hg) -H2 -Hg + /4 width=8 by ex3_5_intro, ex2_intro, or_intror/ +] +qed-. + +lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 → + (Y1 = ⋆ ∧ Y2 = ⋆) ∨ + ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, §l] L2 & + Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. +#R * [ | #Y1 #I #V1 ] #Y2 #l * #f #H1 #H2 +[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/ +| lapply (frees_inv_gref … H1) -H1 #Hf + elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct + elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct + /5 width=8 by frees_gref_gen, ex3_5_intro, ex2_intro, or_intror/ +] +qed-. + +lemma lfxs_inv_bind: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 → + L1 ⦻*[R, V1] L2 ∧ L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2. +#R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf +/6 width=6 by sle_lexs_trans, lexs_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/ +qed-. + +lemma lfxs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → + L1 ⦻*[R, V] L2 ∧ L1 ⦻*[R, T] L2. +#R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf +/5 width=6 by sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/ +qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma lfxs_inv_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⦻*[R, ⋆s] Y2 → + ∃∃L2,V2. L1 ⦻*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2. +#R #I #Y2 #L1 #V1 #s #H elim (lfxs_inv_sort … H) -H * +[ #H destruct +| #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⦻*[R, ⋆s] L2.ⓑ{I}V2 → + ∃∃L1,V1. L1 ⦻*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1. +#R #I #Y1 #L2 #V2 #s #H elim (lfxs_inv_sort … H) -H * +[ #_ #H destruct +| #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma lfxs_inv_zero_pair_sn: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⦻*[R, #0] Y2 → + ∃∃L2,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 & + Y2 = L2.ⓑ{I}V2. +#R #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero … H) -H * +[ #H destruct +| #J #Y1 #L2 #X1 #V2 #HV1 #HV12 #H1 #H2 destruct + /2 width=5 by ex3_2_intro/ +] +qed-. + +lemma lfxs_inv_zero_pair_dx: ∀R,I,Y1,L2,V2. Y1 ⦻*[R, #0] L2.ⓑ{I}V2 → + ∃∃L1,V1. L1 ⦻*[R, V1] L2 & R L1 V1 V2 & + Y1 = L1.ⓑ{I}V1. +#R #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero … H) -H * +[ #_ #H destruct +| #J #L1 #Y2 #V1 #X2 #HV1 #HV12 #H1 #H2 destruct + /2 width=5 by ex3_2_intro/ +] +qed-. + +lemma lfxs_inv_lref_pair_sn: ∀R,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ⦻*[R, #⫯i] Y2 → + ∃∃L2,V2. L1 ⦻*[R, #i] L2 & Y2 = L2.ⓑ{I}V2. +#R #I #Y2 #L1 #V1 #i #H elim (lfxs_inv_lref … H) -H * +[ #H destruct +| #J #Y1 #L2 #X1 #V2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma lfxs_inv_lref_pair_dx: ∀R,I,Y1,L2,V2,i. Y1 ⦻*[R, #⫯i] L2.ⓑ{I}V2 → + ∃∃L1,V1. L1 ⦻*[R, #i] L2 & Y1 = L1.ⓑ{I}V1. +#R #I #Y1 #L2 #V2 #i #H elim (lfxs_inv_lref … H) -H * +[ #_ #H destruct +| #J #L1 #Y2 #V1 #X2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⦻*[R, §l] Y2 → + ∃∃L2,V2. L1 ⦻*[R, §l] L2 & Y2 = L2.ⓑ{I}V2. +#R #I #Y2 #L1 #V1 #l #H elim (lfxs_inv_gref … H) -H * +[ #H destruct +| #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⦻*[R, §l] L2.ⓑ{I}V2 → + ∃∃L1,V1. L1 ⦻*[R, §l] L2 & Y1 = L1.ⓑ{I}V1. +#R #I #Y1 #L2 #V2 #l #H elim (lfxs_inv_gref … H) -H * +[ #_ #H destruct +| #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lfxs_fwd_bind_sn: ∀R,p,I,L1,L2,V,T. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2. +#R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind … Hf) -Hf +/4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/ +qed-. + +lemma lfxs_fwd_bind_dx: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → + R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2. +#R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (lfxs_inv_bind … H HV) -H -HV // +qed-. + +lemma lfxs_fwd_flat_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, V] L2. +#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H // +qed-. + +lemma lfxs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, T] L2. +#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H // +qed-. + +lemma lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ②{I}V.T] L2 → L1 ⦻*[R, V] L2. +#R * /2 width=4 by lfxs_fwd_flat_sn, lfxs_fwd_bind_sn/ +qed-. + +(* Basic_2A1: removed theorems 24: + llpx_sn_sort llpx_sn_skip llpx_sn_lref llpx_sn_free llpx_sn_gref + llpx_sn_bind llpx_sn_flat + llpx_sn_inv_bind llpx_sn_inv_flat + llpx_sn_fwd_lref llpx_sn_fwd_pair_sn llpx_sn_fwd_length + llpx_sn_fwd_bind_sn llpx_sn_fwd_bind_dx llpx_sn_fwd_flat_sn llpx_sn_fwd_flat_dx + llpx_sn_refl llpx_sn_Y llpx_sn_bind_O llpx_sn_ge_up llpx_sn_ge llpx_sn_co + llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_drops.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_drops.etc new file mode 100644 index 000000000..0a3f287bb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_drops.etc @@ -0,0 +1,93 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/relocation/drops_ceq.ma". +include "basic_2/relocation/drops_lexs.ma". +include "basic_2/static/frees_drops.ma". +include "basic_2/static/lfxs.ma". + +(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) + +definition dedropable_sn: predicate (relation3 lenv term term) ≝ + λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 → + ∀K2,T. K1 ⦻*[R, T] K2 → ∀U. ⬆*[f] T ≡ U → + ∃∃L2. L1 ⦻*[R, U] L2 & ⬇*[b, f] L2 ≡ K2 & L1 ≡[f] L2. + +definition dropable_sn: predicate (relation3 lenv term term) ≝ + λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 → 𝐔⦃f⦄ → + ∀L2,U. L1 ⦻*[R, U] L2 → ∀T. ⬆*[f] T ≡ U → + ∃∃K2. K1 ⦻*[R, T] K2 & ⬇*[b, f] L2 ≡ K2. + +definition dropable_dx: predicate (relation3 lenv term term) ≝ + λR. ∀L1,L2,U. L1 ⦻*[R, U] L2 → + ∀b,f,K2. ⬇*[b, f] L2 ≡ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≡ U → + ∃∃K1. ⬇*[b, f] L1 ≡ K1 & K1 ⦻*[R, T] K2. + +(* Properties with generic slicing for local environments *******************) + +(* Basic_2A1: includes: llpx_sn_lift_le llpx_sn_lift_ge *) +lemma lfxs_liftable_dedropable: ∀R. (∀L. reflexive ? (R L)) → + d_liftable2 R → dedropable_sn R. +#R #H1R #H2R #b #f #L1 #K1 #HLK1 #K2 #T * #f1 #Hf1 #HK12 #U #HTU +elim (frees_total L1 U) #f2 #Hf2 +lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #Hf +elim (lexs_liftable_co_dedropable … H1R … H2R … HLK1 … HK12 … Hf) -f1 -K1 +/3 width=6 by cfull_lift, ex3_intro, ex2_intro/ +qed-. + +(* Inversion lemmas with generic slicing for local environments *************) + +(* Basic_2A1: restricts: llpx_sn_inv_lift_le llpx_sn_inv_lift_be llpx_sn_inv_lift_ge *) +(* Basic_2A1: was: llpx_sn_drop_conf_O *) +lemma lfxs_dropable_sn: ∀R. dropable_sn R. +#R #b #f #L1 #K1 #HLK1 #H1f #L2 #U * #f2 #Hf2 #HL12 #T #HTU +elim (frees_total K1 T) #f1 #Hf1 +lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #H2f +elim (lexs_co_dropable_sn … HLK1 … HL12 … H2f) -f2 -L1 +/3 width=3 by ex2_intro/ +qed-. + +(* Basic_2A1: was: llpx_sn_drop_trans_O *) +(* Note: the proof might be simplified *) +lemma lfxs_dropable_dx: ∀R. dropable_dx R. +#R #L1 #L2 #U * #f2 #Hf2 #HL12 #b #f #K2 #HLK2 #H1f #T #HTU +elim (drops_isuni_ex … H1f L1) #K1 #HLK1 +elim (frees_total K1 T) #f1 #Hf1 +lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -K1 #H2f +elim (lexs_co_dropable_dx … HL12 … HLK2 … H2f) -L2 +/4 width=9 by frees_inv_lifts, ex2_intro/ +qed-. + +(* Basic_2A1: was: llpx_sn_inv_lift_O *) +lemma lfxs_inv_lift_bi: ∀R,L1,L2,U. L1 ⦻*[R, U] L2 → + ∀K1,K2,i. ⬇*[i] L1 ≡ K1 → ⬇*[i] L2 ≡ K2 → + ∀T. ⬆*[i] T ≡ U → K1 ⦻*[R, T] K2. +#R #L1 #L2 #U #HL12 #K1 #K2 #i #HLK1 #HLK2 #T #HTU +elim (lfxs_dropable_sn … HLK1 … HL12 … HTU) -L1 -U // #Y #HK12 #HY +lapply (drops_mono … HY … HLK2) -L2 -i #H destruct // +qed-. + +lemma lfxs_inv_lref_sn: ∀R,L1,L2,i. L1 ⦻*[R, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 → + ∃∃K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 & K1 ⦻*[R, V1] K2 & R K1 V1 V2. +#R #L1 #L2 #i #HL12 #I #K1 #V1 #HLK1 elim (lfxs_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 // +#Y #HY #HLK2 elim (lfxs_inv_zero_pair_sn … HY) -HY +#K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/ +qed-. + +lemma lfxs_inv_lref_dx: ∀R,L1,L2,i. L1 ⦻*[R, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 → + ∃∃K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 & K1 ⦻*[R, V1] K2 & R K1 V1 V2. +#R #L1 #L2 #i #HL12 #I #K2 #V2 #HLK2 elim (lfxs_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 // +#Y #HLK1 #HY elim (lfxs_inv_zero_pair_dx … HY) -HY +#K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_fqup.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_fqup.etc new file mode 100644 index 000000000..62c5b0564 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_fqup.etc @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/static/frees_fqup.ma". +include "basic_2/static/lfxs.ma". + +(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) + +(* Advanced properties ******************************************************) + +lemma lfxs_refl: ∀R. (∀L. reflexive … (R L)) → ∀L,T. L ⦻*[R, T] L. +#R #HR #L #T elim (frees_total L T) /3 width=3 by lexs_refl, ex2_intro/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_length.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_length.etc new file mode 100644 index 000000000..01dd82cf4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_length.etc @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/relocation/lexs_length.ma". +include "basic_2/static/lfxs.ma". + +(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) + +(* Forward lemmas with length for local environments ************************) + +lemma lfxs_fwd_length: ∀R,L1,L2,T. L1 ⦻*[R, T] L2 → |L1| = |L2|. +#R #L1 #L2 #T * /2 width=4 by lexs_fwd_length/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_lfxs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_lfxs.etc new file mode 100644 index 000000000..307740bea --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_lfxs.etc @@ -0,0 +1,70 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/relocation/lexs_lexs.ma". +include "basic_2/static/frees_fqup.ma". +include "basic_2/static/frees_frees.ma". +include "basic_2/static/lfxs.ma". + +(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) + +(* Main properties **********************************************************) + +theorem lfxs_bind: ∀R,p,I,L1,L2,V1,V2,T. + L1 ⦻*[R, V1] L2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2 → + L1 ⦻*[R, ⓑ{p,I}V1.T] L2. +#R #p #I #L1 #L2 #V1 #V2 #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 +elim (lexs_fwd_pair … Hf2) -Hf2 #Hf2 #_ elim (sor_isfin_ex f1 (⫱f2)) +/3 width=7 by frees_fwd_isfin, frees_bind, lexs_join, isfin_tl, ex2_intro/ +qed. + +theorem lfxs_flat: ∀R,I,L1,L2,V,T. + L1 ⦻*[R, V] L2 → L1 ⦻*[R, T] L2 → + L1 ⦻*[R, ⓕ{I}V.T] L2. +#R #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 elim (sor_isfin_ex f1 f2) +/3 width=7 by frees_fwd_isfin, frees_flat, lexs_join, ex2_intro/ +qed. + +theorem lfxs_trans: ∀R. lexs_frees_confluent R cfull → + ∀T. Transitive … (lfxs R T). +#R #H1R #T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2 +elim (H1R … Hf1 … HL1) #f #H0 #H1 +lapply (frees_mono … Hf2 … H0) -Hf2 -H0 #Hf2 +lapply (lexs_eq_repl_back … HL2 … Hf2) -f2 #HL2 +lapply (sle_lexs_trans … HL1 … H1) -HL1 // #Hl1 +@(ex2_intro … f) + +/4 width=7 by lreq_trans, lexs_eq_repl_back, ex2_intro/ +qed-. + +theorem lfxs_conf: ∀R. lexs_frees_confluent R cfull → + R_confluent2_lfxs R R R R → + ∀T. confluent … (lfxs R T). +#R #H1R #H2R #T #L0 #L1 * #f1 #Hf1 #HL01 #L2 * #f #Hf #HL02 +lapply (frees_mono … Hf1 … Hf) -Hf1 #Hf12 +lapply (lexs_eq_repl_back … HL01 … Hf12) -f1 #HL01 +elim (lexs_conf … HL01 … HL02) /2 width=3 by ex2_intro/ [ | -HL01 -HL02 ] +[ #L #HL1 #HL2 + elim (H1R … Hf … HL01) -HL01 #f1 #Hf1 #H1 + elim (H1R … Hf … HL02) -HL02 #f2 #Hf2 #H2 + lapply (sle_lexs_trans … HL1 … H1) // -HL1 -H1 #HL1 + lapply (sle_lexs_trans … HL2 … H2) // -HL2 -H2 #HL2 + /3 width=5 by ex2_intro/ +| #g #I #K0 #V0 #n #HLK0 #Hgf #V1 #HV01 #V2 #HV02 #K1 #HK01 #K2 #HK02 + elim (frees_drops_next … Hf … HLK0 … Hgf) -Hf -HLK0 -Hgf #g0 #Hg0 #H0 + lapply (sle_lexs_trans … HK01 … H0) // -HK01 #HK01 + lapply (sle_lexs_trans … HK02 … H0) // -HK02 #HK02 + elim (H2R … HV01 … HV02 K1 … K2) /2 width=3 by ex2_intro/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lifts/lifts_neg.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lifts/lifts_neg.etc new file mode 100644 index 000000000..f4b8a70c1 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lifts/lifts_neg.etc @@ -0,0 +1,75 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/substitution/lift.ma". + +(* BASIC TERM RELOCATION ****************************************************) + +(* Properties on negated basic relocation ***********************************) + +lemma nlift_lref_be_SO: ∀X,j. j < ∞ → ⬆[j, 1] X ≡ #j → ⊥. +#X #j #Hj #H elim (lift_inv_lref2 … H) -H * +[ #H elim (ylt_yle_false … H) -H // +| #i #Hij #_ #H1 #H2 destruct + elim (ylt_inv_plus_Y … Hj) -Hj #Hi #_ + elim (ylt_yle_false … Hij) -Hij /2 width=1 by monotonic_ylt_plus_sn/ +] +qed-. + +lemma nlift_bind_sn: ∀W,l,m. (∀V. ⬆[l, m] V ≡ W → ⊥) → + ∀a,I,U. (∀X. ⬆[l, m] X ≡ ⓑ{a,I}W.U → ⊥). +#W #l #m #HW #a #I #U #X #H elim (lift_inv_bind2 … H) -H /2 width=2 by/ +qed-. + +lemma nlift_bind_dx: ∀U,l,m. (∀T. ⬆[⫯l, m] T ≡ U → ⊥) → + ∀a,I,W. (∀X. ⬆[l, m] X ≡ ⓑ{a,I}W.U → ⊥). +#U #l #m #HU #a #I #W #X #H elim (lift_inv_bind2 … H) -H /2 width=2 by/ +qed-. + +lemma nlift_flat_sn: ∀W,l,m. (∀V. ⬆[l, m] V ≡ W → ⊥) → + ∀I,U. (∀X. ⬆[l, m] X ≡ ⓕ{I}W.U → ⊥). +#W #l #m #HW #I #U #X #H elim (lift_inv_flat2 … H) -H /2 width=2 by/ +qed-. + +lemma nlift_flat_dx: ∀U,l,m. (∀T. ⬆[l, m] T ≡ U → ⊥) → + ∀I,W. (∀X. ⬆[l, m] X ≡ ⓕ{I}W.U → ⊥). +#U #l #m #HU #I #W #X #H elim (lift_inv_flat2 … H) -H /2 width=2 by/ +qed-. + +(* Inversion lemmas on negated basic relocation *****************************) + +lemma nlift_inv_lref_be_SO: ∀i,j. (∀X. ⬆[i, 1] X ≡ #j → ⊥) → j = i ∧ j < ∞. +#i #j elim (ylt_split_eq i j) #Hij #H destruct +[ elim (H (#⫰j)) -H /2 width=1 by lift_lref_pred/ +| elim (yle_split_eq i (∞)) /2 width=1 by conj/ #H0 destruct + elim (H (#∞)) -H /2 width=1 by lift_lref_plus, ylt_Y/ +| elim (H (#j)) -H /2 width=1 by lift_lref_lt/ +] +qed-. + +lemma nlift_inv_bind: ∀a,I,W,U,l,m. (∀X. ⬆[l, m] X ≡ ⓑ{a,I}W.U → ⊥) → + (∀V. ⬆[l, m] V ≡ W → ⊥) ∨ (∀T. ⬆[⫯l, m] T ≡ U → ⊥). +#a #I #W #U #l #m #H elim (is_lift_dec W l m) +[ * /4 width=2 by lift_bind, or_intror/ +| /4 width=2 by ex_intro, or_introl/ +] +qed-. + +lemma nlift_inv_flat: ∀I,W,U,l,m. (∀X. ⬆[l, m] X ≡ ⓕ{I}W.U → ⊥) → + (∀V. ⬆[l, m] V ≡ W → ⊥) ∨ (∀T. ⬆[l, m] T ≡ U → ⊥). +#I #W #U #l #m #H elim (is_lift_dec W l m) +[ * /4 width=2 by lift_flat, or_intror/ +| /4 width=2 by ex_intro, or_introl/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq.etc new file mode 100644 index 000000000..ac55a2c8a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq.etc @@ -0,0 +1,72 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/notation/relations/lazyeq_4.ma". +include "basic_2/multiple/llpx_sn.ma". + +(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) + +(* Basic inversion lemmas ***************************************************) + +lemma lleq_ind: ∀R:relation4 ynat term lenv lenv. ( + ∀L1,L2,l,s. |L1| = |L2| → R l (⋆s) L1 L2 + ) → ( + ∀L1,L2,l,i. |L1| = |L2| → yinj i < l → R l (#i) L1 L2 + ) → ( + ∀I,L1,L2,K1,K2,V,l,i. l ≤ yinj i → + ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V → + K1 ≡[V, yinj O] K2 → R (yinj O) V K1 K2 → R l (#i) L1 L2 + ) → ( + ∀L1,L2,l,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → R l (#i) L1 L2 + ) → ( + ∀L1,L2,l,p. |L1| = |L2| → R l (§p) L1 L2 + ) → ( + ∀a,I,L1,L2,V,T,l. + L1 ≡[V, l]L2 → L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V → + R l V L1 L2 → R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → R l (ⓑ{a,I}V.T) L1 L2 + ) → ( + ∀I,L1,L2,V,T,l. + L1 ≡[V, l]L2 → L1 ≡[T, l] L2 → + R l V L1 L2 → R l T L1 L2 → R l (ⓕ{I}V.T) L1 L2 + ) → + ∀l,T,L1,L2. L1 ≡[T, l] L2 → R l T L1 L2. +#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #l #T #L1 #L2 #H elim H -L1 -L2 -T -l /2 width=8 by/ +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lleq_fwd_lref: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → + ∨∨ |L1| ≤ i ∧ |L2| ≤ i + | yinj i < l + | ∃∃I,K1,K2,V. ⬇[i] L1 ≡ K1.ⓑ{I}V & + ⬇[i] L2 ≡ K2.ⓑ{I}V & + K1 ≡[V, yinj 0] K2 & l ≤ yinj i. +#L1 #L2 #l #i #H elim (llpx_sn_fwd_lref … H) /2 width=1 by or3_intro0, or3_intro1/ +* /3 width=7 by or3_intro2, ex4_4_intro/ +qed-. + +lemma lleq_fwd_drop_sn: ∀L1,L2,T,l. L1 ≡[l, T] L2 → ∀K1,i. ⬇[i] L1 ≡ K1 → + ∃K2. ⬇[i] L2 ≡ K2. +/2 width=7 by llpx_sn_fwd_drop_sn/ qed-. + +lemma lleq_fwd_drop_dx: ∀L1,L2,T,l. L1 ≡[l, T] L2 → ∀K2,i. ⬇[i] L2 ≡ K2 → + ∃K1. ⬇[i] L1 ≡ K1. +/2 width=7 by llpx_sn_fwd_drop_dx/ qed-. + +(* Basic properties *********************************************************) + +lemma lleq_lref: ∀I,L1,L2,K1,K2,V,l,i. l ≤ yinj i → + ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V → + K1 ≡[V, 0] K2 → L1 ≡[#i, l] L2. +/2 width=9 by llpx_sn_lref/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_alt.etc new file mode 100644 index 000000000..d2919f5b6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_alt.etc @@ -0,0 +1,41 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/multiple/llpx_sn_alt.ma". +include "basic_2/multiple/lleq.ma". + +(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) + +(* Alternative definition (not recursive) ***********************************) + +theorem lleq_intro_alt: ∀L1,L2,T,l. |L1| = |L2| → + (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + I1 = I2 ∧ V1 = V2 + ) → L1 ≡[T, l] L2. +#L1 #L2 #T #l #HL12 #IH @llpx_sn_alt_inv_llpx_sn @conj // -HL12 +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 +@(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 // +qed. + +theorem lleq_inv_alt: ∀L1,L2,T,l. L1 ≡[T, l] L2 → + |L1| = |L2| ∧ + ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + I1 = I2 ∧ V1 = V2. +#L1 #L2 #T #l #H elim (llpx_sn_llpx_sn_alt … H) -H +#HL12 #IH @conj // +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 +@(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_alt_rec.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_alt_rec.etc new file mode 100644 index 000000000..440e0f510 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_alt_rec.etc @@ -0,0 +1,54 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/multiple/llpx_sn_alt_rec.ma". +include "basic_2/multiple/lleq.ma". + +(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) + +(* Alternative definition (recursive) ***************************************) + +theorem lleq_intro_alt_r: ∀L1,L2,T,l. |L1| = |L2| → + (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2 + ) → L1 ≡[T, l] L2. +#L1 #L2 #T #l #HL12 #IH @llpx_sn_intro_alt_r // -HL12 +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 +elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/ +qed. + +theorem lleq_ind_alt_r: ∀S:relation4 ynat term lenv lenv. + (∀L1,L2,T,l. |L1| = |L2| → ( + ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2 & S 0 V1 K1 K2 + ) → S l T L1 L2) → + ∀L1,L2,T,l. L1 ≡[T, l] L2 → S l T L1 L2. +#S #IH1 #L1 #L2 #T #l #H @(llpx_sn_ind_alt_r … H) -L1 -L2 -T -l +#L1 #L2 #T #l #HL12 #IH2 @IH1 -IH1 // -HL12 +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 +elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /2 width=1 by and4_intro/ +qed-. + +theorem lleq_inv_alt_r: ∀L1,L2,T,l. L1 ≡[T, l] L2 → + |L1| = |L2| ∧ + ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2. +#L1 #L2 #T #l #H elim (llpx_sn_inv_alt_r … H) -H +#HL12 #IH @conj // +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 +elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_drop.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_drop.etc new file mode 100644 index 000000000..aa4fa9e8c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_drop.etc @@ -0,0 +1,150 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/multiple/llpx_sn_drop.ma". +include "basic_2/multiple/lleq.ma". + +(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) + +(* Advanced properties ******************************************************) + +lemma lleq_bind_repl_O: ∀I,L1,L2,V,T. L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V → + ∀J,W. L1 ≡[W, 0] L2 → L1.ⓑ{J}W ≡[T, 0] L2.ⓑ{J}W. +/2 width=7 by llpx_sn_bind_repl_O/ qed-. + +lemma lleq_dec: ∀T,L1,L2,l. Decidable (L1 ≡[T, l] L2). +/3 width=1 by llpx_sn_dec, eq_term_dec/ qed-. + +lemma lleq_llpx_sn_trans: ∀R. lleq_transitive R → + ∀L1,L2,T,l. L1 ≡[T, l] L2 → + ∀L. llpx_sn R l T L2 L → llpx_sn R l T L1 L. +#R #HR #L1 #L2 #T #l #H @(lleq_ind … H) -L1 -L2 -T -l +[1,2,5: /4 width=6 by llpx_sn_fwd_length, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, trans_eq/ +|4: /4 width=6 by llpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux, trans_eq/ +| #I #L1 #L2 #K1 #K2 #V #l #i #Hli #HLK1 #HLK2 #HK12 #IHK12 #L #H elim (llpx_sn_inv_lref_ge_sn … H … HLK2) -H -HLK2 + /3 width=11 by llpx_sn_lref/ +| #a #I #L1 #L2 #V #T #l #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_bind … H) -H + /3 width=1 by llpx_sn_bind/ +| #I #L1 #L2 #V #T #l #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_flat … H) -H + /3 width=1 by llpx_sn_flat/ +] +qed-. + +lemma lleq_llpx_sn_conf: ∀R. lleq_transitive R → + ∀L1,L2,T,l. L1 ≡[T, l] L2 → + ∀L. llpx_sn R l T L1 L → llpx_sn R l T L2 L. +/3 width=3 by lleq_llpx_sn_trans, lleq_sym/ qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma lleq_inv_lref_ge_dx: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i → + ∀I,K2,V. ⬇[i] L2 ≡ K2.ⓑ{I}V → + ∃∃K1. ⬇[i] L1 ≡ K1.ⓑ{I}V & K1 ≡[V, 0] K2. +#L1 #L2 #l #i #H #Hli #I #K2 #V #HLK2 elim (llpx_sn_inv_lref_ge_dx … H … HLK2) -L2 +/2 width=3 by ex2_intro/ +qed-. + +lemma lleq_inv_lref_ge_sn: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i → + ∀I,K1,V. ⬇[i] L1 ≡ K1.ⓑ{I}V → + ∃∃K2. ⬇[i] L2 ≡ K2.ⓑ{I}V & K1 ≡[V, 0] K2. +#L1 #L2 #l #i #H #Hli #I1 #K1 #V #HLK1 elim (llpx_sn_inv_lref_ge_sn … H … HLK1) -L1 +/2 width=3 by ex2_intro/ +qed-. + +lemma lleq_inv_lref_ge_bi: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i → + ∀I1,I2,K1,K2,V1,V2. + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + ∧∧ I1 = I2 & K1 ≡[V1, 0] K2 & V1 = V2. +/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-. + +lemma lleq_inv_lref_ge: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i → + ∀I,K1,K2,V. ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V → + K1 ≡[V, 0] K2. +#L1 #L2 #l #i #HL12 #Hli #I #K1 #K2 #V #HLK1 #HLK2 +elim (lleq_inv_lref_ge_bi … HL12 … HLK1 HLK2) // +qed-. + +lemma lleq_inv_S: ∀L1,L2,T,l. L1 ≡[T, l+1] L2 → + ∀I,K1,K2,V. ⬇[l] L1 ≡ K1.ⓑ{I}V → ⬇[l] L2 ≡ K2.ⓑ{I}V → + K1 ≡[V, 0] K2 → L1 ≡[T, l] L2. +/2 width=9 by llpx_sn_inv_S/ qed-. + +lemma lleq_inv_bind_O: ∀a,I,L1,L2,V,T. L1 ≡[ⓑ{a,I}V.T, 0] L2 → + L1 ≡[V, 0] L2 ∧ L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V. +/2 width=2 by llpx_sn_inv_bind_O/ qed-. + +(* Advanced forward lemmas **************************************************) + +lemma lleq_fwd_lref_dx: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → + ∀I,K2,V. ⬇[i] L2 ≡ K2.ⓑ{I}V → + i < l ∨ + ∃∃K1. ⬇[i] L1 ≡ K1.ⓑ{I}V & K1 ≡[V, 0] K2 & l ≤ i. +#L1 #L2 #l #i #H #I #K2 #V #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2 +[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/ +qed-. + +lemma lleq_fwd_lref_sn: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → + ∀I,K1,V. ⬇[i] L1 ≡ K1.ⓑ{I}V → + i < l ∨ + ∃∃K2. ⬇[i] L2 ≡ K2.ⓑ{I}V & K1 ≡[V, 0] K2 & l ≤ i. +#L1 #L2 #l #i #H #I #K1 #V #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1 +[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/ +qed-. + +lemma lleq_fwd_bind_O_dx: ∀a,I,L1,L2,V,T. L1 ≡[ⓑ{a,I}V.T, 0] L2 → + L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V. +/2 width=2 by llpx_sn_fwd_bind_O_dx/ qed-. + +(* Properties on relocation *************************************************) + +lemma lleq_lift_le: ∀K1,K2,T,lt. K1 ≡[T, lt] K2 → + ∀L1,L2,l,k. ⬇[Ⓕ, l, k] L1 ≡ K1 → ⬇[Ⓕ, l, k] L2 ≡ K2 → + ∀U. ⬆[l, k] T ≡ U → lt ≤ l → L1 ≡[U, lt] L2. +/3 width=10 by llpx_sn_lift_le, lift_mono/ qed-. + +lemma lleq_lift_ge: ∀K1,K2,T,lt. K1 ≡[T, lt] K2 → + ∀L1,L2,l,k. ⬇[Ⓕ, l, k] L1 ≡ K1 → ⬇[Ⓕ, l, k] L2 ≡ K2 → + ∀U. ⬆[l, k] T ≡ U → l ≤ lt → L1 ≡[U, lt+k] L2. +/2 width=9 by llpx_sn_lift_ge/ qed-. + +(* Inversion lemmas on relocation *******************************************) + +lemma lleq_inv_lift_le: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 → + ∀K1,K2,l,k. ⬇[Ⓕ, l, k] L1 ≡ K1 → ⬇[Ⓕ, l, k] L2 ≡ K2 → + ∀T. ⬆[l, k] T ≡ U → lt ≤ l → K1 ≡[T, lt] K2. +/3 width=10 by llpx_sn_inv_lift_le, ex2_intro/ qed-. + +lemma lleq_inv_lift_be: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 → + ∀K1,K2,l,k. ⬇[Ⓕ, l, k] L1 ≡ K1 → ⬇[Ⓕ, l, k] L2 ≡ K2 → + ∀T. ⬆[l, k] T ≡ U → l ≤ lt → lt ≤ l + k → K1 ≡[T, l] K2. +/2 width=11 by llpx_sn_inv_lift_be/ qed-. + +lemma lleq_inv_lift_ge: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 → + ∀K1,K2,l,k. ⬇[Ⓕ, l, k] L1 ≡ K1 → ⬇[Ⓕ, l, k] L2 ≡ K2 → + ∀T. ⬆[l, k] T ≡ U → l + k ≤ lt → K1 ≡[T, lt-k] K2. +/2 width=9 by llpx_sn_inv_lift_ge/ qed-. + +(* Inversion lemmas on negated lazy quivalence for local environments *******) + +lemma nlleq_inv_bind: ∀a,I,L1,L2,V,T,l. (L1 ≡[ⓑ{a,I}V.T, l] L2 → ⊥) → + (L1 ≡[V, l] L2 → ⊥) ∨ (L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V → ⊥). +/3 width=2 by nllpx_sn_inv_bind, eq_term_dec/ qed-. + +lemma nlleq_inv_flat: ∀I,L1,L2,V,T,l. (L1 ≡[ⓕ{I}V.T, l] L2 → ⊥) → + (L1 ≡[V, l] L2 → ⊥) ∨ (L1 ≡[T, l] L2 → ⊥). +/3 width=2 by nllpx_sn_inv_flat, eq_term_dec/ qed-. + +lemma nlleq_inv_bind_O: ∀a,I,L1,L2,V,T. (L1 ≡[ⓑ{a,I}V.T, 0] L2 → ⊥) → + (L1 ≡[V, 0] L2 → ⊥) ∨ (L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V → ⊥). +/3 width=2 by nllpx_sn_inv_bind_O, eq_term_dec/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_fqus.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_fqus.etc new file mode 100644 index 000000000..bf0cec6d2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_fqus.etc @@ -0,0 +1,75 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/multiple/fqus_alt.ma". +include "basic_2/multiple/lleq_drop.ma". + +(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) + +(* Properties on supclosure *************************************************) + +lemma lleq_fqu_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐ ⦃G2, K2, U⦄ → + ∀L1. L1 ≡[T, 0] L2 → + ∃∃K1. ⦃G1, L1, T⦄ ⊐ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2. +#G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U +[ #I #G #L2 #V #L1 #H elim (lleq_inv_lref_ge_dx … H … I L2 V) -H // + #K1 #H1 #H2 lapply (drop_inv_O2 … H1) -H1 + #H destruct /2 width=3 by fqu_lref_O, ex2_intro/ +| * [ #a ] #I #G #L2 #V #T #L1 #H + [ elim (lleq_inv_bind … H) + | elim (lleq_inv_flat … H) + ] -H + /2 width=3 by fqu_pair_sn, ex2_intro/ +| #a #I #G #L2 #V #T #L1 #H elim (lleq_inv_bind_O … H) -H + #H3 #H4 /2 width=3 by fqu_bind_dx, ex2_intro/ +| #I #G #L2 #V #T #L1 #H elim (lleq_inv_flat … H) -H + /2 width=3 by fqu_flat_dx, ex2_intro/ +| #G #L2 #K2 #T #U #k #HLK2 #HTU #L1 #HL12 + elim (drop_O1_le (Ⓕ) (k+1) L1) + [ /3 width=12 by fqu_drop, lleq_inv_lift_le, ex2_intro/ + | lapply (drop_fwd_length_le2 … HLK2) -K2 + lapply (lleq_fwd_length … HL12) -T -U // + ] +] +qed-. + +lemma lleq_fquq_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐⸮ ⦃G2, K2, U⦄ → + ∀L1. L1 ≡[T, 0] L2 → + ∃∃K1. ⦃G1, L1, T⦄ ⊐⸮ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2. +#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fquq_inv_gen … H) -H +[ #H elim (lleq_fqu_trans … H … HL12) -L2 /3 width=3 by fqu_fquq, ex2_intro/ +| * #HG #HL #HT destruct /2 width=3 by ex2_intro/ +] +qed-. + +lemma lleq_fqup_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+ ⦃G2, K2, U⦄ → + ∀L1. L1 ≡[T, 0] L2 → + ∃∃K1. ⦃G1, L1, T⦄ ⊐+ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2. +#G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U +[ #G2 #K2 #U #HTU #L1 #HL12 elim (lleq_fqu_trans … HTU … HL12) -L2 + /3 width=3 by fqu_fqup, ex2_intro/ +| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12 elim (IHTU … HL12) -L2 + #K1 #HTU #HK1 elim (lleq_fqu_trans … HU2 … HK1) -K + /3 width=5 by fqup_strap1, ex2_intro/ +] +qed-. + +lemma lleq_fqus_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐* ⦃G2, K2, U⦄ → + ∀L1. L1 ≡[T, 0] L2 → + ∃∃K1. ⦃G1, L1, T⦄ ⊐* ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2. +#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_gen … H) -H +[ #H elim (lleq_fqup_trans … H … HL12) -L2 /3 width=3 by fqup_fqus, ex2_intro/ +| * #HG #HL #HT destruct /2 width=3 by ex2_intro/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_llor.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_llor.etc new file mode 100644 index 000000000..e702043ea --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_llor.etc @@ -0,0 +1,41 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/multiple/llor.ma". +include "basic_2/multiple/llpx_sn_frees.ma". +include "basic_2/multiple/lleq_alt.ma". + +(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) + +(* Properties on pointwise union for local environments **********************) + +lemma llpx_sn_llor_dx: ∀R. (c_r_confluent1 … R (llpx_sn R 0)) → (frees_trans R) → + ∀L1,L2,T,l. llpx_sn R l T L1 L2 → ∀L. L1 ⋓[T, l] L2 ≡ L → L2 ≡[T, l] L. +#R #H1R #H2R #L1 #L2 #T #l #H1 #L #H2 +lapply (llpx_sn_frees_trans … H1R H2R … H1) -H1R -H2R #HR +elim (llpx_sn_llpx_sn_alt … H1) -H1 #HL12 #IH1 +elim H2 -H2 #_ #HL1 #IH2 +@lleq_intro_alt // #I2 #I #K2 #K #V2 #V #i #Hi #HnT #HLK2 #HLK +lapply (drop_fwd_length_lt2 … HLK) #HiL +elim (drop_O1_lt (Ⓕ) L1 i) // -HiL #I1 #K1 #V1 #HLK1 +elim (IH1 … HLK1 HLK2) -IH1 /2 width=1 by/ #H #_ destruct +elim (IH2 … HLK1 HLK2 HLK) -IH2 -HLK1 -HLK2 -HLK * /2 width=1 by conj/ #H +[ elim (ylt_yle_false … H) -H // +| elim H -H /2 width=1 by/ +] +qed. + +lemma llpx_sn_llor_dx_sym: ∀R. (c_r_confluent1 … R (llpx_sn R 0)) → (frees_trans R) → + ∀L1,L2,T,l. llpx_sn R l T L1 L2 → ∀L. L1 ⋓[T, l] L2 ≡ L → L ≡[T, l] L2. +/3 width=6 by llpx_sn_llor_dx, lleq_sym/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_lreq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_lreq.etc new file mode 100644 index 000000000..8fa21a5da --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_lreq.etc @@ -0,0 +1,36 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/multiple/llpx_sn_lreq.ma". +include "basic_2/multiple/lleq.ma". + +(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) + +(* Properties on equivalence for local environments *************************) + +lemma lreq_lleq_trans: ∀L2,L,T,l. L2 ≡[T, l] L → + ∀L1. L1 ⩬[l, ∞] L2 → L1 ≡[T, l] L. +/2 width=3 by lreq_llpx_sn_trans/ qed-. + +lemma lleq_lreq_trans: ∀L,L1,T,l. L ≡[T, l] L1 → + ∀L2. L1 ⩬[l, ∞] L2 → L ≡[T, l] L2. +/2 width=3 by llpx_sn_lreq_trans/ qed-. + +lemma lleq_lreq_repl: ∀L1,L2,T,l. L1 ≡[T, l] L2 → ∀K1. K1 ⩬[l, ∞] L1 → + ∀K2. L2 ⩬[l, ∞] K2 → K1 ≡[T, l] K2. +/2 width=5 by llpx_sn_lreq_repl/ qed-. + +lemma lleq_bind_repl_SO: ∀I1,I2,L1,L2,V1,V2,T. L1.ⓑ{I1}V1 ≡[T, 0] L2.ⓑ{I2}V2 → + ∀J1,J2,W1,W2. L1.ⓑ{J1}W1 ≡[T, 1] L2.ⓑ{J2}W2. +/2 width=5 by llpx_sn_bind_repl_SO/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn.etc new file mode 100644 index 000000000..5a1124b5c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn.etc @@ -0,0 +1,45 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "ground_2/ynat/ynat_plus.ma". +include "basic_2/substitution/drop.ma". + +(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) + +| llpx_sn_lref: ∀I,L1,L2,K1,K2,V1,V2,l,i. l ≤ yinj i → + ⬇[i] L1 ≡ K1.ⓑ{I}V1 → ⬇[i] L2 ≡ K2.ⓑ{I}V2 → + llpx_sn R (yinj 0) V1 K1 K2 → R K1 V1 V2 → llpx_sn R l (#i) L1 L2 + +(* Basic forward lemmas *****************************************************) + +lemma llpx_sn_fwd_drop_sn: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → + ∀K1,i. ⬇[i] L1 ≡ K1 → ∃K2. ⬇[i] L2 ≡ K2. +#R #L1 #L2 #T #l #H #K1 #i #HLK1 lapply (llpx_sn_fwd_length … H) -H +#HL12 lapply (drop_fwd_length_le2 … HLK1) -HLK1 /2 width=1 by drop_O1_le/ +qed-. + +lemma llpx_sn_fwd_drop_dx: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → + ∀K2,i. ⬇[i] L2 ≡ K2 → ∃K1. ⬇[i] L1 ≡ K1. +#R #L1 #L2 #T #l #H #K2 #i #HLK2 lapply (llpx_sn_fwd_length … H) -H +#HL12 lapply (drop_fwd_length_le2 … HLK2) -HLK2 /2 width=1 by drop_O1_le/ +qed-. + +lemma llpx_sn_fwd_lref: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 → + ∨∨ |L1| ≤ i ∧ |L2| ≤ i + | yinj i < l + | ∃∃I,K1,K2,V1,V2. ⬇[i] L1 ≡ K1.ⓑ{I}V1 & + ⬇[i] L2 ≡ K2.ⓑ{I}V2 & + llpx_sn R (yinj 0) V1 K1 K2 & + R K1 V1 V2 & l ≤ yinj i. +/2 width=3 by llpx_sn_fwd_lref_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn_alt.etc new file mode 100644 index 000000000..ac0d12224 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn_alt.etc @@ -0,0 +1,63 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/multiple/frees.ma". +include "basic_2/multiple/llpx_sn_alt_rec.ma". + +(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) + +(* alternative definition of llpx_sn (not recursive) *) +definition llpx_sn_alt: relation3 lenv term term → relation4 ynat term lenv lenv ≝ + λR,l,T,L1,L2. |L1| = |L2| ∧ + (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + I1 = I2 ∧ R K1 V1 V2 + ). + +(* Main properties **********************************************************) + +theorem llpx_sn_llpx_sn_alt: ∀R,T,L1,L2,l. llpx_sn R l T L1 L2 → llpx_sn_alt R l T L1 L2. +#R #U #L1 @(f2_ind … rfw … L1 U) -L1 -U +#x #IHx #L1 #U #Hx #L2 #l #H elim (llpx_sn_inv_alt_r … H) -H +#HL12 #IHU @conj // +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2 elim (frees_inv … H) -H +[ -x #HnU elim (IHU … HnU HLK1 HLK2) -IHU -HnU -HLK1 -HLK2 /2 width=1 by conj/ +| * #J1 #K10 #W10 #j #Hlj #Hji #HnU #HLK10 yminus_inj >yminus_inj #HnW10 destruct + lapply (drop_fwd_drop2 … HLK10) #H + lapply (drop_conf_ge … H … HLK1 ?) -H /2 width=1 by ylt_fwd_le_succ1/ (minus_plus_k_k j (i+1)) in ⊢ (%→?); >commutative_plus yminus_SO2 +#HnV1 #HKY1 #HKY2 (**) (* full auto too slow *) +lapply (drop_trans_ge … H1 … HKY1 ?) -H1 -HKY1 // #HLY1 +lapply (drop_trans_ge … H2 … HKY2 ?) -H2 -HKY2 // #HLY2 +/4 width=9 by frees_be, yle_plus_dx2_trans, yle_succ_dx, ylt_inj/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn_alt_rec.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn_alt_rec.etc new file mode 100644 index 000000000..172ccf38a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn_alt_rec.etc @@ -0,0 +1,249 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/substitution/lift_neg.ma". +include "basic_2/substitution/drop_drop.ma". +include "basic_2/multiple/llpx_sn.ma". + +(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) + +(* alternative definition of llpx_sn (recursive) *) +inductive llpx_sn_alt_r (R:relation3 lenv term term): relation4 ynat term lenv lenv ≝ +| llpx_sn_alt_r_intro: ∀L1,L2,T,l. + (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R K1 V1 V2 + ) → + (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → llpx_sn_alt_r R 0 V1 K1 K2 + ) → |L1| = |L2| → llpx_sn_alt_r R l T L1 L2 +. + +(* Compact definition of llpx_sn_alt_r **************************************) + +lemma llpx_sn_alt_r_intro_alt: ∀R,L1,L2,T,l. |L1| = |L2| → + (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2 + ) → llpx_sn_alt_r R l T L1 L2. +#R #L1 #L2 #T #l #HL12 #IH @llpx_sn_alt_r_intro // -HL12 +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 +elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/ +qed. + +lemma llpx_sn_alt_r_ind_alt: ∀R. ∀S:relation4 ynat term lenv lenv. + (∀L1,L2,T,l. |L1| = |L2| → ( + ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2 & S 0 V1 K1 K2 + ) → S l T L1 L2) → + ∀L1,L2,T,l. llpx_sn_alt_r R l T L1 L2 → S l T L1 L2. +#R #S #IH #L1 #L2 #T #l #H elim H -L1 -L2 -T -l +#L1 #L2 #T #l #H1 #H2 #HL12 #IH2 @IH -IH // -HL12 +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 +elim (H1 … HnT HLK1 HLK2) -H1 /4 width=8 by and4_intro/ +qed-. + +lemma llpx_sn_alt_r_inv_alt: ∀R,L1,L2,T,l. llpx_sn_alt_r R l T L1 L2 → + |L1| = |L2| ∧ + ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2. +#R #L1 #L2 #T #l #H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -l +#L1 #L2 #T #l #HL12 #IH @conj // -HL12 +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 +elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/ +qed-. + +(* Basic inversion lemmas ***************************************************) + +lemma llpx_sn_alt_r_inv_flat: ∀R,I,L1,L2,V,T,l. llpx_sn_alt_r R l (ⓕ{I}V.T) L1 L2 → + llpx_sn_alt_r R l V L1 L2 ∧ llpx_sn_alt_r R l T L1 L2. +#R #I #L1 #L2 #V #T #l #H elim (llpx_sn_alt_r_inv_alt … H) -H +#HL12 #IH @conj @llpx_sn_alt_r_intro_alt // -HL12 +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2 +elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 // +/3 width=8 by nlift_flat_sn, nlift_flat_dx, and3_intro/ +qed-. + +lemma llpx_sn_alt_r_inv_bind: ∀R,a,I,L1,L2,V,T,l. llpx_sn_alt_r R l (ⓑ{a,I}V.T) L1 L2 → + llpx_sn_alt_r R l V L1 L2 ∧ llpx_sn_alt_r R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V). +#R #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_alt_r_inv_alt … H) -H +#HL12 #IH @conj @llpx_sn_alt_r_intro_alt [1,3: normalize // ] -HL12 +#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2 +[ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 + /3 width=9 by nlift_bind_sn, and3_intro/ +| lapply (yle_inv_succ1 … Hli) -Hli * #Hli #Hi yminus_Y_inj #K1 #HK12 #HLK1 + lapply (lreq_inv_O_Y … HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/ +| /4 width=5 by llpx_sn_free, lreq_fwd_length, le_repl_sn_trans_aux, trans_eq/ +| /4 width=1 by llpx_sn_bind, lreq_succ/ +] +qed-. + +lemma llpx_sn_lreq_trans: ∀R,L,L1,T,l. llpx_sn R l T L L1 → + ∀L2. L1 ⩬[l, ∞] L2 → llpx_sn R l T L L2. +#R #L #L1 #T #l #H elim H -L -L1 -T -l +/4 width=5 by llpx_sn_flat, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, lreq_fwd_length, trans_eq/ +[ #I #L #L1 #K #K1 #V #V1 #l #i #Hli #HLK #HLK1 #HK1 #HV1 #_ #L2 #HL12 + elim (lreq_drop_conf_be … HL12 … HLK1) -L1 // >yminus_Y_inj #K2 #HK12 #HLK2 + lapply (lreq_inv_O_Y … HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/ +| /4 width=5 by llpx_sn_free, lreq_fwd_length, le_repl_sn_conf_aux, trans_eq/ +| /4 width=1 by llpx_sn_bind, lreq_succ/ +] +qed-. + +lemma llpx_sn_lreq_repl: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → ∀K1. K1 ⩬[l, ∞] L1 → + ∀K2. L2 ⩬[l, ∞] K2 → llpx_sn R l T K1 K2. +/3 width=4 by llpx_sn_lreq_trans, lreq_llpx_sn_trans/ qed-. + +lemma llpx_sn_bind_repl_SO: ∀R,I1,I2,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) → + ∀J1,J2,W1,W2. llpx_sn R 1 T (L1.ⓑ{J1}W1) (L2.ⓑ{J2}W2). +#R #I1 #I2 #L1 #L2 #V1 #V2 #T #HT #J1 #J2 #W1 #W2 lapply (llpx_sn_ge R … 1 … HT) -HT +/3 width=7 by llpx_sn_lreq_repl, lreq_succ/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn_tc.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn_tc.etc new file mode 100644 index 000000000..13fc14e82 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn_tc.etc @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/multiple/llpx_sn_drop.ma". + +(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) + +(* Properties about transitive closure **************************************) + +lemma llpx_sn_TC_pair_dx: ∀R. (∀L. reflexive … (R L)) → + ∀I,L,V1,V2,T. LTC … R L V1 V2 → + LTC … (llpx_sn R 0) T (L.ⓑ{I}V1) (L.ⓑ{I}V2). +#R #HR #I #L #V1 #V2 #T #H @(TC_star_ind … V2 H) -V2 +/4 width=9 by llpx_sn_bind_repl_O, llpx_sn_refl, step, inj/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lpx/lpx_fquq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lpx/lpx_fquq.etc new file mode 100644 index 000000000..799808fc6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lpx/lpx_fquq.etc @@ -0,0 +1,47 @@ +(* Properties on supclosure *************************************************) + +lemma fqu_lpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → + ∀K2. ⦃G2, L2⦄ ⊢ ➡[h, o] K2 → + ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h, o] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h, o] T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄. +#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 +/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_flat_dx, lpx_pair, ex3_2_intro/ +[ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpx_inv_pair1 … H) -H + #K2 #W2 #HLK2 #HVW2 #H destruct + /3 width=5 by cpx_pair_sn, fqu_bind_dx, ex3_2_intro/ +| #G #L1 #K1 #T1 #U1 #k #HLK1 #HTU1 #K2 #HK12 + elim (drop_lpx_trans … HLK1 … HK12) -HK12 + /3 width=7 by fqu_drop, ex3_2_intro/ +] +qed-. + +lemma fquq_lpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → + ∀K2. ⦃G2, L2⦄ ⊢ ➡[h, o] K2 → + ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h, o] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h, o] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄. +#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 elim (fquq_inv_gen … H) -H +[ #HT12 elim (fqu_lpx_trans … HT12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/ +| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/ +] +qed-. + +lemma lpx_fqu_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → + ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → + ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, o] T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2. +#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 +/3 width=7 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpx_pair, ex3_2_intro/ +[ #I #G1 #L1 #V1 #X #H elim (lpx_inv_pair2 … H) -H + #K1 #W1 #HKL1 #HWV1 #H destruct elim (lift_total V1 0 1) + /4 width=7 by cpx_delta, fqu_drop, drop_drop, ex3_2_intro/ +| #G #L1 #K1 #T1 #U1 #k #HLK1 #HTU1 #L0 #HL01 + elim (lpx_drop_trans_O1 … HL01 … HLK1) -L1 + /3 width=5 by fqu_drop, ex3_2_intro/ +] +qed-. + +lemma lpx_fquq_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → + ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → + ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, o] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2. +#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fquq_inv_gen … H) -H +[ #HT12 elim (lpx_fqu_trans … HT12 … HKL1) /3 width=5 by fqu_fquq, ex3_2_intro/ +| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lpx_sn/lpx_sn_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lpx_sn/lpx_sn_alt.etc new file mode 100644 index 000000000..5099ffa1a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lpx_sn/lpx_sn_alt.etc @@ -0,0 +1,127 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/substitution/drop.ma". +include "basic_2/substitution/lpx_sn.ma". + +(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********) + +(* alternative definition of lpx_sn *) +definition lpx_sn_alt: relation3 lenv term term → relation lenv ≝ + λR,L1,L2. |L1| = |L2| ∧ + (∀I1,I2,K1,K2,V1,V2,i. + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + I1 = I2 ∧ R K1 V1 V2 + ). + +(* Basic forward lemmas ******************************************************) + +lemma lpx_sn_alt_fwd_length: ∀R,L1,L2. lpx_sn_alt R L1 L2 → |L1| = |L2|. +#R #L1 #L2 #H elim H // +qed-. + +(* Basic inversion lemmas ***************************************************) + +lemma lpx_sn_alt_inv_atom1: ∀R,L2. lpx_sn_alt R (⋆) L2 → L2 = ⋆. +#R #L2 #H lapply (lpx_sn_alt_fwd_length … H) -H +normalize /2 width=1 by length_inv_zero_sn/ +qed-. + +lemma lpx_sn_alt_inv_pair1: ∀R,I,L2,K1,V1. lpx_sn_alt R (K1.ⓑ{I}V1) L2 → + ∃∃K2,V2. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2. +#R #I1 #L2 #K1 #V1 #H elim H -H +#H #IH elim (length_inv_pos_sn … H) -H +#I2 #K2 #V2 #HK12 #H destruct +elim (IH I1 I2 K1 K2 V1 V2 0) // +#H #HV12 destruct @(ex3_2_intro … K2 V2) // -HV12 +@conj // -HK12 +#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (⫯i)) -IH +/2 width=1 by drop_drop, conj/ +qed-. + +lemma lpx_sn_alt_inv_atom2: ∀R,L1. lpx_sn_alt R L1 (⋆) → L1 = ⋆. +#R #L1 #H lapply (lpx_sn_alt_fwd_length … H) -H +normalize /2 width=1 by length_inv_zero_dx/ +qed-. + +lemma lpx_sn_alt_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn_alt R L1 (K2.ⓑ{I}V2) → + ∃∃K1,V1. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1. +#R #I2 #L1 #K2 #V2 #H elim H -H +#H #IH elim (length_inv_pos_dx … H) -H +#I1 #K1 #V1 #HK12 #H destruct +elim (IH I1 I2 K1 K2 V1 V2 0) // +#H #HV12 destruct @(ex3_2_intro … K1 V1) // -HV12 +@conj // -HK12 +#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (⫯i)) -IH +/2 width=1 by drop_drop, conj/ +qed-. + +(* Basic properties *********************************************************) + +lemma lpx_sn_alt_atom: ∀R. lpx_sn_alt R (⋆) (⋆). +#R @conj // +#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 elim (drop_inv_atom1 … HLK1) -HLK1 +#H destruct +qed. + +lemma lpx_sn_alt_pair: ∀R,I,L1,L2,V1,V2. + lpx_sn_alt R L1 L2 → R L1 V1 V2 → + lpx_sn_alt R (L1.ⓑ{I}V1) (L2.ⓑ{I}V2). +#R #I #L1 #L2 #V1 #V2 #H #HV12 elim H -H +#HL12 #IH @conj // +#I1 #I2 #K1 #K2 #W1 #W2 #i @(ynat_ind … i) -i +[ #HLK1 #HLK2 + lapply (drop_inv_O2 … HLK1) -HLK1 #H destruct + lapply (drop_inv_O2 … HLK2) -HLK2 #H destruct + /2 width=1 by conj/ +| -HL12 -HV12 /3 width=6 by drop_inv_drop1/ +| #H lapply (drop_fwd_Y2 … H) -H + #H elim (ylt_yle_false … H) -H // +] +qed. + +(* Main properties **********************************************************) + +theorem lpx_sn_lpx_sn_alt: ∀R,L1,L2. lpx_sn R L1 L2 → lpx_sn_alt R L1 L2. +#R #L1 #L2 #H elim H -L1 -L2 +/2 width=1 by lpx_sn_alt_atom, lpx_sn_alt_pair/ +qed. + +(* Main inversion lemmas ****************************************************) + +theorem lpx_sn_alt_inv_lpx_sn: ∀R,L1,L2. lpx_sn_alt R L1 L2 → lpx_sn R L1 L2. +#R #L1 elim L1 -L1 +[ #L2 #H lapply (lpx_sn_alt_inv_atom1 … H) -H // +| #L1 #I #V1 #IH #X #H elim (lpx_sn_alt_inv_pair1 … H) -H + #L2 #V2 #HL12 #HV12 #H destruct /3 width=1 by lpx_sn_pair/ +] +qed-. + +(* alternative definition of lpx_sn *****************************************) + +lemma lpx_sn_intro_alt: ∀R,L1,L2. |L1| = |L2| → + (∀I1,I2,K1,K2,V1,V2,i. + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + I1 = I2 ∧ R K1 V1 V2 + ) → lpx_sn R L1 L2. +/4 width=4 by lpx_sn_alt_inv_lpx_sn, conj/ qed. + +lemma lpx_sn_inv_alt: ∀R,L1,L2. lpx_sn R L1 L2 → + |L1| = |L2| ∧ + ∀I1,I2,K1,K2,V1,V2,i. + ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → + I1 = I2 ∧ R K1 V1 V2. +#R #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H +#H elim H -H /3 width=4 by conj/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lpx_sn/lpx_sn_tc.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lpx_sn/lpx_sn_tc.etc new file mode 100644 index 000000000..23eaab6ab --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lpx_sn/lpx_sn_tc.etc @@ -0,0 +1,119 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/substitution/lpx_sn.ma". + +(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********) + +(* Properties on transitive_closure *****************************************) + +lemma TC_lpx_sn_pair_refl: ∀R. (∀L. reflexive … (R L)) → + ∀L1,L2. TC … (lpx_sn R) L1 L2 → + ∀I,V. TC … (lpx_sn R) (L1. ⓑ{I} V) (L2. ⓑ{I} V). +#R #HR #L1 #L2 #H @(TC_star_ind … L2 H) -L2 +[ /2 width=1 by lpx_sn_refl/ +| /3 width=1 by TC_reflexive, lpx_sn_refl/ +| /3 width=5 by lpx_sn_pair, step/ +] +qed-. + +lemma TC_lpx_sn_pair: ∀R. (∀L. reflexive … (R L)) → + ∀I,L1,L2. TC … (lpx_sn R) L1 L2 → + ∀V1,V2. LTC … R L1 V1 V2 → + TC … (lpx_sn R) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2). +#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx … V1 H) -V1 // +[ /2 width=1 by TC_lpx_sn_pair_refl/ +| /4 width=3 by TC_strap, lpx_sn_pair, lpx_sn_refl/ +] +qed-. + +lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀L. reflexive … (R L)) → + ∀L1,L2. lpx_sn (LTC … R) L1 L2 → + TC … (lpx_sn R) L1 L2. +#R #HR #L1 #L2 #H elim H -L1 -L2 +/2 width=1 by TC_lpx_sn_pair, lpx_sn_atom, inj/ +qed-. + +(* Inversion lemmas on transitive closure ***********************************) + +lemma TC_lpx_sn_inv_atom2: ∀R,L1. TC … (lpx_sn R) L1 (⋆) → L1 = ⋆. +#R #L1 #H @(TC_ind_dx … L1 H) -L1 +[ /2 width=2 by lpx_sn_inv_atom2/ +| #L1 #L #HL1 #_ #IHL2 destruct /2 width=2 by lpx_sn_inv_atom2/ +] +qed-. + +lemma TC_lpx_sn_inv_pair2: ∀R. c_rs_transitive … R (λ_. lpx_sn R) → + ∀I,L1,K2,V2. TC … (lpx_sn R) L1 (K2.ⓑ{I}V2) → + ∃∃K1,V1. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L1 = K1. ⓑ{I} V1. +#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx … L1 H) -L1 +[ #L1 #H elim (lpx_sn_inv_pair2 … H) -H /3 width=5 by inj, ex3_2_intro/ +| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct + elim (lpx_sn_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct + lapply (HR … HV2 … HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/ +] +qed-. + +lemma TC_lpx_sn_ind: ∀R. c_rs_transitive … R (λ_. lpx_sn R) → + ∀S:relation lenv. + S (⋆) (⋆) → ( + ∀I,K1,K2,V1,V2. + TC … (lpx_sn R) K1 K2 → LTC … R K1 V1 V2 → + S K1 K2 → S (K1.ⓑ{I}V1) (K2.ⓑ{I}V2) + ) → + ∀L2,L1. TC … (lpx_sn R) L1 L2 → S L1 L2. +#R #HR #S #IH1 #IH2 #L2 elim L2 -L2 +[ #X #H >(TC_lpx_sn_inv_atom2 … H) -X // +| #L2 #I #V2 #IHL2 #X #H + elim (TC_lpx_sn_inv_pair2 … H) // -H -HR + #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/ +] +qed-. + +lemma TC_lpx_sn_inv_atom1: ∀R,L2. TC … (lpx_sn R) (⋆) L2 → L2 = ⋆. +#R #L2 #H elim H -L2 +[ /2 width=2 by lpx_sn_inv_atom1/ +| #L #L2 #_ #HL2 #IHL1 destruct /2 width=2 by lpx_sn_inv_atom1/ +] +qed-. + +fact TC_lpx_sn_inv_pair1_aux: ∀R. c_rs_transitive … R (λ_. lpx_sn R) → + ∀L1,L2. TC … (lpx_sn R) L1 L2 → + ∀I,K1,V1. L1 = K1.ⓑ{I}V1 → + ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2. +#R #HR #L1 #L2 #H @(TC_lpx_sn_ind … H) // -HR -L1 -L2 +[ #J #K #W #H destruct +| #I #L1 #L2 #V1 #V2 #HL12 #HV12 #_ #J #K #W #H destruct /2 width=5 by ex3_2_intro/ +] +qed-. + +lemma TC_lpx_sn_inv_pair1: ∀R. c_rs_transitive … R (λ_. lpx_sn R) → + ∀I,K1,L2,V1. TC … (lpx_sn R) (K1.ⓑ{I}V1) L2 → + ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2. +/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-. + +lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. c_rs_transitive … R (λ_. lpx_sn R) → + ∀L1,L2. TC … (lpx_sn R) L1 L2 → + lpx_sn (LTC … R) L1 L2. +/3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-. + +(* Forward lemmas on transitive closure *************************************) + +lemma TC_lpx_sn_fwd_length: ∀R,L1,L2. TC … (lpx_sn R) L1 L2 → |L1| = |L2|. +#R #L1 #L2 #H elim H -L2 +[ #L2 #HL12 >(lpx_sn_fwd_length … HL12) -HL12 // +| #L #L2 #_ #HL2 #IHL1 + >IHL1 -L1 >(lpx_sn_fwd_length … HL2) -HL2 // +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lrsubeq_4.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lrsubeq_4.etc new file mode 100644 index 000000000..6135be4d4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lrsubeq_4.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +notation "hvbox( L1 break ⊆ [ term 46 l , break term 46 m ] break term 46 L2 )" + non associative with precedence 45 + for @{ 'LRSubEq $L1 $l $k $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lstas/lstas.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lstas/lstas.etc new file mode 100644 index 000000000..e9f034116 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lstas/lstas.etc @@ -0,0 +1,190 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/notation/relations/statictypestar_6.ma". +include "basic_2/grammar/genv.ma". +include "basic_2/substitution/drop.ma". +include "basic_2/static/sh.ma". + +(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************) + +(* activate genv *) +inductive lstas (h): nat → relation4 genv lenv term term ≝ +| lstas_sort: ∀G,L,d,s. lstas h d G L (⋆s) (⋆((next h)^d s)) +| lstas_ldef: ∀G,L,K,V,W,U,i,d. ⬇[i] L ≡ K.ⓓV → lstas h d G K V W → + ⬆[0, i+1] W ≡ U → lstas h d G L (#i) U +| lstas_zero: ∀G,L,K,W,V,i. ⬇[i] L ≡ K.ⓛW → lstas h 0 G K W V → + lstas h 0 G L (#i) (#i) +| lstas_succ: ∀G,L,K,W,V,U,i,d. ⬇[i] L ≡ K.ⓛW → lstas h d G K W V → + ⬆[0, i+1] V ≡ U → lstas h (d+1) G L (#i) U +| lstas_bind: ∀a,I,G,L,V,T,U,d. lstas h d G (L.ⓑ{I}V) T U → + lstas h d G L (ⓑ{a,I}V.T) (ⓑ{a,I}V.U) +| lstas_appl: ∀G,L,V,T,U,d. lstas h d G L T U → lstas h d G L (ⓐV.T) (ⓐV.U) +| lstas_cast: ∀G,L,W,T,U,d. lstas h d G L T U → lstas h d G L (ⓝW.T) U +. + +interpretation "nat-iterated static type assignment (term)" + 'StaticTypeStar h G L d T U = (lstas h d G L T U). + +(* Basic inversion lemmas ***************************************************) + +fact lstas_inv_sort1_aux: ∀h,G,L,T,U,d. ⦃G, L⦄ ⊢ T •*[h, d] U → ∀s0. T = ⋆s0 → + U = ⋆((next h)^d s0). +#h #G #L #T #U #d * -G -L -T -U -d +[ #G #L #d #s #s0 #H destruct // +| #G #L #K #V #W #U #i #d #_ #_ #_ #s0 #H destruct +| #G #L #K #W #V #i #_ #_ #s0 #H destruct +| #G #L #K #W #V #U #i #d #_ #_ #_ #s0 #H destruct +| #a #I #G #L #V #T #U #d #_ #s0 #H destruct +| #G #L #V #T #U #d #_ #s0 #H destruct +| #G #L #W #T #U #d #_ #s0 #H destruct +qed-. + +(* Basic_1: was just: sty0_gen_sort *) +lemma lstas_inv_sort1: ∀h,G,L,X,s,d. ⦃G, L⦄ ⊢ ⋆s •*[h, d] X → X = ⋆((next h)^d s). +/2 width=5 by lstas_inv_sort1_aux/ +qed-. + +fact lstas_inv_lref1_aux: ∀h,G,L,T,U,d. ⦃G, L⦄ ⊢ T •*[h, d] U → ∀j. T = #j → ∨∨ + (∃∃K,V,W. ⬇[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, d] W & + ⬆[0, j+1] W ≡ U + ) | + (∃∃K,W,V. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, 0] V & + U = #j & d = 0 + ) | + (∃∃K,W,V,d0. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, d0] V & + ⬆[0, j+1] V ≡ U & d = d0+1 + ). +#h #G #L #T #U #d * -G -L -T -U -d +[ #G #L #d #s #j #H destruct +| #G #L #K #V #W #U #i #d #HLK #HVW #HWU #j #H destruct /3 width=6 by or3_intro0, ex3_3_intro/ +| #G #L #K #W #V #i #HLK #HWV #j #H destruct /3 width=5 by or3_intro1, ex4_3_intro/ +| #G #L #K #W #V #U #i #d #HLK #HWV #HWU #j #H destruct /3 width=8 by or3_intro2, ex4_4_intro/ +| #a #I #G #L #V #T #U #d #_ #j #H destruct +| #G #L #V #T #U #d #_ #j #H destruct +| #G #L #W #T #U #d #_ #j #H destruct +] +qed-. + +lemma lstas_inv_lref1: ∀h,G,L,X,i,d. ⦃G, L⦄ ⊢ #i •*[h, d] X → ∨∨ + (∃∃K,V,W. ⬇[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, d] W & + ⬆[0, i+1] W ≡ X + ) | + (∃∃K,W,V. ⬇[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, 0] V & + X = #i & d = 0 + ) | + (∃∃K,W,V,d0. ⬇[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, d0] V & + ⬆[0, i+1] V ≡ X & d = d0+1 + ). +/2 width=3 by lstas_inv_lref1_aux/ +qed-. + +lemma lstas_inv_lref1_O: ∀h,G,L,X,i. ⦃G, L⦄ ⊢ #i •*[h, 0] X → + (∃∃K,V,W. ⬇[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, 0] W & + ⬆[0, i+1] W ≡ X + ) ∨ + (∃∃K,W,V. ⬇[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, 0] V & + X = #i + ). +#h #G #L #X #i #H elim (lstas_inv_lref1 … H) -H * /3 width=6 by ex3_3_intro, or_introl, or_intror/ +#K #W #V #d #_ #_ #_ (lift_inv_sort1 … H1) -X1 + >(lift_inv_sort1 … H2) -X2 // +| #G #L1 #K1 #V1 #W1 #W #i #d #HLK1 #_ #HW1 #IHVW1 #L2 #c #l #k #HL21 #X #H #U2 #HWU2 + elim (lift_inv_lref1 … H) * #Hil #H destruct + [ elim (lift_trans_ge … HW1 … HWU2) -W /2 width=1 by ylt_fwd_le_succ1/ #W2 #HW12 #HWU2 + elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by ylt_fwd_le/ #X #HLK2 #H + elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K2 #V2 #HK21 #HV12 #H destruct + /3 width=9 by lstas_ldef/ + | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W /2 width=1 by yle_succ_dx/ #HW1U2 + lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 /3 width=9 by lstas_ldef, drop_inv_gen/ + ] +| #G #L1 #K1 #V1 #W1 #i #HLK1 #_ #IHVW1 #L2 #c #l #k #HL21 #X #H #U2 #HWU2 + >(lift_mono … HWU2 … H) -U2 + elim (lift_inv_lref1 … H) * #Hil #H destruct + [ elim (lift_total W1 (l-i-1) k) #W2 #HW12 + elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by ylt_fwd_le/ #X #HLK2 #H + elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K2 #V2 #HK21 #HV12 #H destruct + /3 width=10 by lstas_zero/ + | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 + /3 width=10 by lstas_zero, drop_inv_gen/ + ] +| #G #L1 #K1 #W1 #V1 #W #i #d #HLK1 #_ #HW1 #IHWV1 #L2 #c #l #k #HL21 #X #H #U2 #HWU2 + elim (lift_inv_lref1 … H) * #Hil #H destruct + [ elim (lift_trans_ge … HW1 … HWU2) -W /2 width=1 by ylt_fwd_le_succ1/ #W #HW1 #HWU2 + elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by ylt_fwd_le/ #X #HLK2 #H + elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K2 #W2 #HK21 #HW12 #H destruct + /3 width=9 by lstas_succ/ + | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W /2 width=1 by yle_succ_dx/ #HW1U2 + lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 /3 width=9 by lstas_succ, drop_inv_gen/ + ] +| #a #I #G #L1 #V1 #T1 #U1 #d #_ #IHTU1 #L2 #c #l #k #HL21 #X1 #H1 #X2 #H2 + elim (lift_inv_bind1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct + elim (lift_inv_bind1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct + lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=6 by lstas_bind, drop_skip/ +| #G #L1 #V1 #T1 #U1 #d #_ #IHTU1 #L2 #c #l #k #HL21 #X1 #H1 #X2 #H2 + elim (lift_inv_flat1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct + elim (lift_inv_flat1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct + lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=6 by lstas_appl/ +| #G #L1 #W1 #T1 #U1 #d #_ #IHTU1 #L2 #c #l #k #HL21 #X #H #U2 #HU12 + elim (lift_inv_flat1 … H) -H #W2 #T2 #_ #HT12 #H destruct /3 width=6 by lstas_cast/ +] +qed. + +(* Inversion lemmas on relocation *******************************************) + +(* Note: apparently this was missing in basic_1 *) +lemma lstas_inv_lift1: ∀h,G,d. d_deliftable_sn (lstas h G d). +#h #G #d #L2 #T2 #U2 #H elim H -G -L2 -T2 -U2 -d +[ #G #L2 #d #s #L1 #c #l #k #_ #X #H + >(lift_inv_sort2 … H) -X /2 width=3 by lstas_sort, lift_sort, ex2_intro/ +| #G #L2 #K2 #V2 #W2 #W #i #d #HLK2 #HVW2 #HW2 #IHVW2 #L1 #c #l #k #HL21 #X #H + elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HVW2 | -IHVW2 ] + [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #V1 #HLK1 #HK21 #HV12 + elim (IHVW2 … HK21 … HV12) -K2 -V2 #W1 #HW12 #HVW1 + elim (lift_trans_le … HW12 … HW2) -W2 // yplus_SO2 >ymax_pre_sn /3 width=8 by lstas_ldef, ylt_fwd_le_succ1, ex2_intro/ + | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2 + elim (yle_inv_plus_inj2 … Hil) -Hil #Hlim #mi + elim (lift_split … HW2 l (i-k+1)) -HW2 /2 width=1 by yle_succ_dx, le_S_S/ + #W0 #HW20 minus_minus_k_k /3 width=8 by lstas_ldef, yle_inv_inj, le_S, ex2_intro/ + ] +| #G #L2 #K2 #W2 #V2 #i #HLK2 #HWV2 #IHWV2 #L1 #c #l #k #HL21 #X #H + elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HWV2 | -IHWV2 ] + [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12 + elim (IHWV2 … HK21 … HW12) -K2 + /3 width=5 by lstas_zero, lift_lref_lt, ex2_intro/ + | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 + /3 width=5 by lstas_zero, lift_lref_ge_minus, ex2_intro/ + ] +| #G #L2 #K2 #W2 #V2 #W #i #d #HLK2 #HWV2 #HW2 #IHWV2 #L1 #c #l #k #HL21 #X #H + elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HWV2 | -IHWV2 ] + [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12 + elim (IHWV2 … HK21 … HW12) -K2 #V1 #HV12 #HWV1 + elim (lift_trans_le … HV12 … HW2) -W2 // yplus_SO2 >ymax_pre_sn /3 width=8 by lstas_succ, ylt_fwd_le_succ1, ex2_intro/ + | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2 + elim (yle_inv_plus_inj2 … Hil) -Hil #Hlim #mi + elim (lift_split … HW2 l (i-k+1)) -HW2 /2 width=1 by yle_succ_dx, le_S_S/ + #W0 #HW20 minus_minus_k_k /3 width=8 by lstas_succ, yle_inv_inj, le_S, ex2_intro/ + ] +| #a #I #G #L2 #V2 #T2 #U2 #d #_ #IHTU2 #L1 #c #l #k #HL21 #X #H + elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct + elim (IHTU2 (L1.ⓑ{I}V1) … HT12) -IHTU2 -HT12 /3 width=5 by lstas_bind, drop_skip, lift_bind, ex2_intro/ +| #G #L2 #V2 #T2 #U2 #d #_ #IHTU2 #L1 #c #l #k #HL21 #X #H + elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct + elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=5 by lstas_appl, lift_flat, ex2_intro/ +| #G #L2 #W2 #T2 #U2 #d #_ #IHTU2 #L1 #c #l #k #HL21 #X #H + elim (lift_inv_flat2 … H) -H #W1 #T1 #_ #HT12 #H destruct + elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=3 by lstas_cast, ex2_intro/ +] +qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma lstas_split_aux: ∀h,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*[h, d] T2 → ∀d1,d2. d = d1 + d2 → + ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, d1] T & ⦃G, L⦄ ⊢ T •*[h, d2] T2. +#h #G #L #T1 #T2 #d #H elim H -G -L -T1 -T2 -d +[ #G #L #d #s #d1 #d2 #H destruct + >commutative_plus >iter_plus /2 width=3 by lstas_sort, ex2_intro/ +| #G #L #K #V1 #V2 #U2 #i #d #HLK #_ #VU2 #IHV12 #d1 #d2 #H destruct + elim (IHV12 d1 d2) -IHV12 // #V + elim (lift_total V 0 (i+1)) + lapply (drop_fwd_drop2 … HLK) + /3 width=12 by lstas_lift, lstas_ldef, ex2_intro/ +| #G #L #K #W1 #W2 #i #HLK #HW12 #_ #d1 #d2 #H + elim (zero_eq_plus … H) -H #H1 #H2 destruct + /3 width=5 by lstas_zero, ex2_intro/ +| #G #L #K #W1 #W2 #U2 #i #d #HLK #HW12 #HWU2 #IHW12 #d1 @(nat_ind_plus … d1) -d1 + [ #d2 normalize #H destruct + elim (IHW12 0 d) -IHW12 // + lapply (drop_fwd_drop2 … HLK) + /3 width=8 by lstas_succ, lstas_zero, ex2_intro/ + | #d1 #_ #d2 (lstas_inv_sort1 … H) -X + (lstas_inv_sort1 … H) -X // +| #G #L #K #V #V1 #U1 #i #d #HLK #_ #HVU1 #IHV1 #X #H + elim (lstas_inv_lref1 … H) -H * + #K0 #V0 #W0 [3: #d0 ] #HLK0 + lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct + #HVW0 #HX lapply (IHV1 … HVW0) -IHV1 -HVW0 #H destruct + /2 width=5 by lift_mono/ +| #G #L #K #W #W1 #i #HLK #_ #_ #X #H + elim (lstas_inv_lref1_O … H) -H * + #K0 #V0 #W0 #HLK0 + lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct // +| #G #L #K #W #W1 #U1 #i #d #HLK #_ #HWU1 #IHWV #X #H + elim (lstas_inv_lref1_S … H) -H * #K0 #W0 #V0 #HLK0 + lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct + #HW0 #HX lapply (IHWV … HW0) -IHWV -HW0 #H destruct + /2 width=5 by lift_mono/ +| #a #I #G #L #V #T #U1 #d #_ #IHTU1 #X #H + elim (lstas_inv_bind1 … H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/ +| #G #L #V #T #U1 #d #_ #IHTU1 #X #H + elim (lstas_inv_appl1 … H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/ +| #G #L #W #T #U1 #d #_ #IHTU1 #U2 #H + lapply (lstas_inv_cast1 … H) -H /2 width=1 by/ +] +qed-. + +(* Advanced inversion lemmas ************************************************) + +(* Basic_1: was just: sty0_correct *) +lemma lstas_correct: ∀h,G,L,T1,T,d1. ⦃G, L⦄ ⊢ T1 •*[h, d1] T → + ∀d2. ∃T2. ⦃G, L⦄ ⊢ T •*[h, d2] T2. +#h #G #L #T1 #T #d1 #H elim H -G -L -T1 -T -d1 +[ /2 width=2 by lstas_sort, ex_intro/ +| #G #L #K #V1 #V #U #i #d #HLK #_ #HVU #IHV1 #d2 + elim (IHV1 d2) -IHV1 #V2 + elim (lift_total V2 0 (i+1)) + lapply (drop_fwd_drop2 … HLK) -HLK + /3 width=11 by ex_intro, lstas_lift/ +| #G #L #K #W1 #W #i #HLK #HW1 #IHW1 #d2 + @(nat_ind_plus … d2) -d2 /3 width=5 by lstas_zero, ex_intro/ + #d2 #_ elim (IHW1 d2) -IHW1 #W2 #HW2 + lapply (lstas_trans … HW1 … HW2) -W + elim (lift_total W2 0 (i+1)) + /3 width=7 by lstas_succ, ex_intro/ +| #G #L #K #W1 #W #U #i #d #HLK #_ #HWU #IHW1 #d2 + elim (IHW1 d2) -IHW1 #W2 + elim (lift_total W2 0 (i+1)) + lapply (drop_fwd_drop2 … HLK) -HLK + /3 width=11 by ex_intro, lstas_lift/ +| #a #I #G #L #V #T #U #d #_ #IHTU #d2 + elim (IHTU d2) -IHTU /3 width=2 by lstas_bind, ex_intro/ +| #G #L #V #T #U #d #_ #IHTU #d2 + elim (IHTU d2) -IHTU /3 width=2 by lstas_appl, ex_intro/ +| #G #L #W #T #U #d #_ #IHTU #d2 + elim (IHTU d2) -IHTU /2 width=2 by ex_intro/ +] +qed-. + +(* more main properties *****************************************************) + +theorem lstas_conf_le: ∀h,G,L,T,U1,d1. ⦃G, L⦄ ⊢ T •*[h, d1] U1 → + ∀U2,d2. d1 ≤ d2 → ⦃G, L⦄ ⊢ T •*[h, d2] U2 → + ⦃G, L⦄ ⊢ U1 •*[h, d2-d1] U2. +#h #G #L #T #U1 #d1 #HTU1 #U2 #d2 #Hd12 +>(plus_minus_k_k … Hd12) in ⊢ (%→?); -Hd12 >commutative_plus #H +elim (lstas_split … H) -H #U #HTU +>(lstas_mono … HTU … HTU1) -T // +qed-. + +theorem lstas_conf: ∀h,G,L,T0,T1,d1. ⦃G, L⦄ ⊢ T0 •*[h, d1] T1 → + ∀T2,d2. ⦃G, L⦄ ⊢ T0 •*[h, d2] T2 → + ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, d2] T & ⦃G, L⦄ ⊢ T2 •*[h, d1] T. +#h #G #L #T0 #T1 #d1 #HT01 #T2 #d2 #HT02 +elim (lstas_lstas … HT01 (d1+d2)) #T #HT0 +lapply (lstas_conf_le … HT01 … HT0) // -HT01 (lsubd_inv_atom1 … H) -H // +| #I #L1 #L #Y #HL1 #IHL1 #X #H + elim (lsubd_inv_pair1 … H) -H * #L2 + [ #HL2 #H destruct /3 width=1 by lsubd_pair/ + | #W #V #d #HV #HW #HL2 #H1 #H2 #H3 destruct + /3 width=3 by lsubd_beta, lsubd_da_trans/ + ] +| #L1 #L #W #V #d #HV #HW #HL1 #IHL1 #X #H + elim (lsubd_inv_pair1 … H) -H * #L2 + [ #HL2 #H destruct /3 width=5 by lsubd_beta, lsubd_da_conf/ + | #W0 #V0 #d0 #_ #_ #_ #H destruct + ] +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lsuby.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lsuby.etc new file mode 100644 index 000000000..e2ee7c7c2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lsuby.etc @@ -0,0 +1,236 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/notation/relations/lrsubeq_4.ma". +include "basic_2/substitution/drop.ma". + +(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************) + +inductive lsuby: relation4 ynat ynat lenv lenv ≝ +| lsuby_atom: ∀L,l,m. lsuby l m L (⋆) +| lsuby_zero: ∀I1,I2,L1,L2,V1,V2. + lsuby 0 0 L1 L2 → lsuby 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) +| lsuby_pair: ∀I1,I2,L1,L2,V,m. lsuby 0 m L1 L2 → + lsuby 0 (⫯m) (L1.ⓑ{I1}V) (L2.ⓑ{I2}V) +| lsuby_succ: ∀I1,I2,L1,L2,V1,V2,l,m. + lsuby l m L1 L2 → lsuby (⫯l) m (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) +. + +interpretation + "local environment refinement (extended substitution)" + 'LRSubEq L1 l m L2 = (lsuby l m L1 L2). + +(* Basic properties *********************************************************) + +lemma lsuby_pair_lt: ∀I1,I2,L1,L2,V,m. L1 ⊆[0, ⫰m] L2 → 0 < m → + L1.ⓑ{I1}V ⊆[0, m] L2.ⓑ{I2}V. +#I1 #I2 #L1 #L2 #V #m #HL12 #Hm <(ylt_inv_O1 … Hm) /2 width=1 by lsuby_pair/ +qed. + +lemma lsuby_succ_lt: ∀I1,I2,L1,L2,V1,V2,l,m. L1 ⊆[⫰l, m] L2 → 0 < l → + L1.ⓑ{I1}V1 ⊆[l, m] L2. ⓑ{I2}V2. +#I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #Hl <(ylt_inv_O1 … Hl) /2 width=1 by lsuby_succ/ +qed. + +lemma lsuby_pair_O_Y: ∀L1,L2. L1 ⊆[0, ∞] L2 → + ∀I1,I2,V. L1.ⓑ{I1}V ⊆[0,∞] L2.ⓑ{I2}V. +#L1 #L2 #HL12 #I1 #I2 #V lapply (lsuby_pair I1 I2 … V … HL12) -HL12 // +qed. + +lemma lsuby_refl: ∀L,l,m. L ⊆[l, m] L. +#L elim L -L // +#L #I #V #IHL #l elim (ynat_cases … l) [| * #x ] +#Hl destruct /2 width=1 by lsuby_succ/ +#m elim (ynat_cases … m) [| * #x ] +#Hm destruct /2 width=1 by lsuby_zero, lsuby_pair/ +qed. + +lemma lsuby_O2: ∀L2,L1,l. |L2| ≤ |L1| → L1 ⊆[l, 0] L2. +#L2 elim L2 -L2 // #L2 #I2 #V2 #IHL2 * +[ #l #H elim (ylt_yle_false … H) -H // +| #L1 #I1 #V1 #l + #H lapply (yle_inv_succ … H) -H #HL12 + elim (ynat_cases l) /3 width=1 by lsuby_zero/ + * /3 width=1 by lsuby_succ/ +] +qed. + +lemma lsuby_sym: ∀l,m,L1,L2. L1 ⊆[l, m] L2 → |L1| = |L2| → L2 ⊆[l, m] L1. +#l #m #L1 #L2 #H elim H -l -m -L1 -L2 +[ #L1 #l #m #H >(length_inv_zero_dx … H) -L1 // +| /2 width=1 by lsuby_O2/ +| #I1 #I2 #L1 #L2 #V #m #_ #IHL12 #H lapply (ysucc_inv_inj … H) -H + /3 width=1 by lsuby_pair/ +| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #H lapply (ysucc_inv_inj … H) -H + /3 width=1 by lsuby_succ/ +] +qed-. + +(* Basic inversion lemmas ***************************************************) + +fact lsuby_inv_atom1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → L1 = ⋆ → L2 = ⋆. +#L1 #L2 #l #m * -L1 -L2 -l -m // +[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct +| #I1 #I2 #L1 #L2 #V #m #_ #H destruct +| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #H destruct +] +qed-. + +lemma lsuby_inv_atom1: ∀L2,l,m. ⋆ ⊆[l, m] L2 → L2 = ⋆. +/2 width=5 by lsuby_inv_atom1_aux/ qed-. + +fact lsuby_inv_zero1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → + ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → l = 0 → m = 0 → + L2 = ⋆ ∨ + ∃∃J2,K2,W2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{J2}W2. +#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/ +[ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct + /3 width=5 by ex2_3_intro, or_intror/ +| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #_ #H + elim (ysucc_inv_O_dx … H) +| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W1 #_ #H + elim (ysucc_inv_O_dx … H) +] +qed-. + +lemma lsuby_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⊆[0, 0] L2 → + L2 = ⋆ ∨ + ∃∃I2,K2,V2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{I2}V2. +/2 width=9 by lsuby_inv_zero1_aux/ qed-. + +fact lsuby_inv_pair1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → + ∀J1,K1,W. L1 = K1.ⓑ{J1}W → l = 0 → 0 < m → + L2 = ⋆ ∨ + ∃∃J2,K2. K1 ⊆[0, ⫰m] K2 & L2 = K2.ⓑ{J2}W. +#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/ +[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W #_ #_ #H + elim (ylt_yle_false … H) // +| #I1 #I2 #L1 #L2 #V #m #HL12 #J1 #K1 #W #H #_ #_ destruct + /3 width=4 by ex2_2_intro, or_intror/ +| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W #_ #H + elim (ysucc_inv_O_dx … H) +] +qed-. + +lemma lsuby_inv_pair1: ∀I1,K1,L2,V,m. K1.ⓑ{I1}V ⊆[0, m] L2 → 0 < m → + L2 = ⋆ ∨ + ∃∃I2,K2. K1 ⊆[0, ⫰m] K2 & L2 = K2.ⓑ{I2}V. +/2 width=6 by lsuby_inv_pair1_aux/ qed-. + +fact lsuby_inv_succ1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → + ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < l → + L2 = ⋆ ∨ + ∃∃J2,K2,W2. K1 ⊆[⫰l, m] K2 & L2 = K2.ⓑ{J2}W2. +#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/ +[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H + elim (ylt_yle_false … H) // +| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #H + elim (ylt_yle_false … H) // +| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J1 #K1 #W1 #H #_ destruct + /3 width=5 by ex2_3_intro, or_intror/ +] +qed-. + +lemma lsuby_inv_succ1: ∀I1,K1,L2,V1,l,m. K1.ⓑ{I1}V1 ⊆[l, m] L2 → 0 < l → + L2 = ⋆ ∨ + ∃∃I2,K2,V2. K1 ⊆[⫰l, m] K2 & L2 = K2.ⓑ{I2}V2. +/2 width=5 by lsuby_inv_succ1_aux/ qed-. + +fact lsuby_inv_zero2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → + ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → l = 0 → m = 0 → + ∃∃J1,K1,W1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{J1}W1. +#L1 #L2 #l #m * -L1 -L2 -l -m +[ #L1 #l #m #J2 #K2 #W1 #H destruct +| #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J2 #K2 #W2 #H #_ #_ destruct + /2 width=5 by ex2_3_intro/ +| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K2 #W2 #_ #_ #H + elim (ysucc_inv_O_dx … H) +| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W2 #_ #H + elim (ysucc_inv_O_dx … H) +] +qed-. + +lemma lsuby_inv_zero2: ∀I2,K2,L1,V2. L1 ⊆[0, 0] K2.ⓑ{I2}V2 → + ∃∃I1,K1,V1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{I1}V1. +/2 width=9 by lsuby_inv_zero2_aux/ qed-. + +fact lsuby_inv_pair2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → + ∀J2,K2,W. L2 = K2.ⓑ{J2}W → l = 0 → 0 < m → + ∃∃J1,K1. K1 ⊆[0, ⫰m] K2 & L1 = K1.ⓑ{J1}W. +#L1 #L2 #l #m * -L1 -L2 -l -m +[ #L1 #l #m #J2 #K2 #W #H destruct +| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W #_ #_ #H + elim (ylt_yle_false … H) // +| #I1 #I2 #L1 #L2 #V #m #HL12 #J2 #K2 #W #H #_ #_ destruct + /2 width=4 by ex2_2_intro/ +| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W #_ #H + elim (ysucc_inv_O_dx … H) +] +qed-. + +lemma lsuby_inv_pair2: ∀I2,K2,L1,V,m. L1 ⊆[0, m] K2.ⓑ{I2}V → 0 < m → + ∃∃I1,K1. K1 ⊆[0, ⫰m] K2 & L1 = K1.ⓑ{I1}V. +/2 width=6 by lsuby_inv_pair2_aux/ qed-. + +fact lsuby_inv_succ2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → + ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → 0 < l → + ∃∃J1,K1,W1. K1 ⊆[⫰l, m] K2 & L1 = K1.ⓑ{J1}W1. +#L1 #L2 #l #m * -L1 -L2 -l -m +[ #L1 #l #m #J2 #K2 #W2 #H destruct +| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W2 #_ #H + elim (ylt_yle_false … H) // +| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K1 #W2 #_ #H + elim (ylt_yle_false … H) // +| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J2 #K2 #W2 #H #_ destruct + /2 width=5 by ex2_3_intro/ +] +qed-. + +lemma lsuby_inv_succ2: ∀I2,K2,L1,V2,l,m. L1 ⊆[l, m] K2.ⓑ{I2}V2 → 0 < l → + ∃∃I1,K1,V1. K1 ⊆[⫰l, m] K2 & L1 = K1.ⓑ{I1}V1. +/2 width=5 by lsuby_inv_succ2_aux/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lsuby_fwd_length: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → |L2| ≤ |L1|. +#L1 #L2 #l #m #H elim H -L1 -L2 -l -m /2 width=1 by yle_succ/ +qed-. + +(* Properties on basic slicing **********************************************) + +lemma lsuby_drop_trans_be: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → + ∀I2,K2,W,s,i. ⬇[s, 0, i] L2 ≡ K2.ⓑ{I2}W → + l ≤ i → ∀m0. i + ⫯m0 = l + m → + ∃∃I1,K1. K1 ⊆[0, m0] K2 & ⬇[s, 0, i] L1 ≡ K1.ⓑ{I1}W. +#L1 #L2 #l #m #H elim H -L1 -L2 -l -m +[ #L1 #l #m #J2 #K2 #W #s #i #H + elim (drop_inv_atom1 … H) -H #H destruct +| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #s #i #_ #_ #m0 + >yplus_O2 >yplus_succ2 #H elim (ysucc_inv_O_dx … H) +| #I1 #I2 #L1 #L2 #V #m #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ #m0 + >yplus_succ2 >yplus_succ2 #H0 lapply (ysucc_inv_inj … H0) -H0 + elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ] + [ destruct -I2 /2 width=4 by drop_pair, ex2_2_intro/ + | lapply (ylt_inv_O1 … Hi) + #H yplus_succ1 >yplus_succ2 #H lapply (ysucc_inv_inj … H) -H + tls_S >S_pred // +qed-. + +definition ltls (f): lenv → lenv → rtmap ≝ λL,K. ⫱*[|L|-|K|] f. + +interpretation "ltls (rtmap)" 'DropPreds L K f = (ltls f L K). + +lemma ltls_refl: ∀f,L1,L2. |L1| ≤ |L2| → ⫱*[L1, L2] f = f. +#f #L1 #L2 #HL12 whd in ⊢ (??%?); >(eq_minus_O … HL12) -HL12 // +qed. + +lemma ltls_pair2: ∀f,I,L1,L2,V. |L2| < |L1| → ⫱⫱*[L1, L2.ⓑ{I}V] f = ⫱*[L1, L2] f. +#f #I #L1 #L2 #V #HL12 whd in ⊢ (??(?%)%); minus_Sn_m // +qed. + +lemma ltls_pair1_next: ∀f,I,L1,L2,V. |L2| ≤ |L1| → ⫱*[L1.ⓑ{I}V, L2] ⫯f = ⫱*[L1, L2] f. +#f #I #L1 #L2 #V #HL12 whd in ⊢ (??%%); >minus_Sn_m // +qed. + +lemma ltls_sle_pair: ∀f1,f2,L1,L2. ⫱*[L2, L1] f2 ⊆ ⫱*[L1, L2] f1 → + ∀I,V1. ⫱*[L2, L1.ⓑ{I}V1] f2 ⊆ ⫱*[L1.ⓑ{I}V1, L2] ⫯f1. +#f1 #f2 #L1 #L2 elim (lt_or_ge (|L1|) (|L2|)) +[ #HL12 >ltls_refl in ⊢ (??%→?); /2 width=1 by lt_to_le/ + #Hf21 #I #V1 >ltls_refl in ⊢ (??%); // + <(ltls_pair2 … I … V1 HL12) in Hf21; -HL12 /2 width=1 by sle_inv_tl1/ +| #HL21 >ltls_refl // #Hf21 #I #V1 >ltls_refl /2 width=1 by le_S/ + >ltls_pair1_next // +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/psubst_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/psubst_6.etc new file mode 100644 index 000000000..56cb72ec4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/psubst_6.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ [ term 46 l , break term 46 m ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'PSubst $G $L $T1 $l $k $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/psubststar_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/psubststar_6.etc new file mode 100644 index 000000000..90362edb6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/psubststar_6.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ * [ term 46 l , break term 46 m ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'PSubstStar $G $L $T1 $l $k $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/psubststaralt_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/psubststaralt_6.etc new file mode 100644 index 000000000..80a29a841 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/psubststaralt_6.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ ▶ * [ term 46 l , break term 46 m ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'PSubstStarAlt $G $L $T1 $l $k $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/unfold/unfold.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/unfold/unfold.etc new file mode 100644 index 000000000..445421935 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/unfold/unfold.etc @@ -0,0 +1,35 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/notation/relations/unfold_4.ma". +include "basic_2/grammar/lenv_append.ma". +include "basic_2/grammar/genv.ma". +include "basic_2/substitution/drop.ma". + +(* CONTEXT-SENSITIVE UNFOLD FOR TERMS ***************************************) + +(* activate genv *) +inductive unfold: relation4 genv lenv term lenv ≝ +| unfold_sort: ∀G,L,s. unfold G L (⋆s) L +| unfold_lref: ∀I,G,L1,L2,K1,K2,V,i. ⬇[i] L1 ≡ K1. ⓑ{I}V → + unfold G K1 V K2 → ⬇[Ⓣ, |L2|, i] L2 ≡ K2 → + unfold G L1 (#i) (L1@@L2) +| unfold_bind: ∀a,I,G,L1,L2,V,T. + unfold G (L1.ⓑ{I}V) T L2 → unfold G L1 (ⓑ{a,I}V.T) L2 +| unfold_flat: ∀I,G,L1,L2,V,T. + unfold G L1 T L2 → unfold G L1 (ⓕ{I}V.T) L2 +. + +interpretation "context-sensitive unfold (term)" + 'Unfold G L1 T L2 = (unfold G L1 T L2). diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/unfold/unfold_4.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/unfold/unfold_4.etc new file mode 100644 index 000000000..3381c5632 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/unfold/unfold_4.etc @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +notation "hvbox( ⦃ term 46 G, break term 46 L1 ⦄ ⊢ ⧫ * break term 46 T ≡ break term 46 L2 )" + non associative with precedence 45 + for @{ 'Unfold $G $L1 $T $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/ceq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/ceq.etc deleted file mode 100644 index 57ecbaf68..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/ceq.etc +++ /dev/null @@ -1,42 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/syntax/lenv.ma". - -(* CONTEXT-SENSITIVE EQUIVALENCES FOR TERMS *********************************) - -definition ceq: relation3 lenv term term ≝ λL,T1,T2. T1 = T2. - -definition cfull: relation3 lenv term term ≝ λL,T1,T2. ⊤. - -(* Basic properties *********************************************************) - -lemma ceq_refl (L): reflexive … (ceq L). -// qed. - -lemma cfull_refl (L): reflexive … (cfull L). -// qed. - -lemma ceq_sym (L): symmetric … (ceq L). -// qed-. - -lemma cfull_sym (L): symmetric … (cfull L). -// qed-. - -lemma cfull_top (R:relation3 lenv term term) (L) (T1) (T2): - R L T1 T2 → cfull L T1 T2. -// qed-. - -lemma ceq_cfull (L) (T1) (T2): ceq L T1 T2 → cfull L T1 T2. -// qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/ceq_ceq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/ceq_ceq.etc deleted file mode 100644 index b17b186e2..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/ceq_ceq.etc +++ /dev/null @@ -1,37 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/syntax/ceq.ma". - -(* CONTEXT-SENSITIVE EQUIVALENCES FOR TERMS *********************************) - -(* Main properties **********************************************************) - -theorem ceq_trans (L): Transitive … (ceq L). -// qed-. - -lemma ceq_canc_sn (L): left_cancellable … (ceq L). -// qed-. - -lemma ceq_canc_dx (L): right_cancellable … (ceq L). -// qed-. - -theorem cfull_trans (L): Transitive … (cfull L). -// qed-. - -lemma cfull_canc_sn (L): left_cancellable … (cfull L). -// qed-. - -lemma cfull_canc_dx (L): right_cancellable … (cfull L). -// qed-. \ No newline at end of file diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cir/cir.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cir/cir.etc deleted file mode 100644 index 412297546..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cir/cir.etc +++ /dev/null @@ -1,79 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/notation/relations/prednotreducible_3.ma". -include "basic_2/reduction/crr.ma". - -(* IRREDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION ************************) - -definition cir: relation3 genv lenv term ≝ λG,L,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⊥. - -interpretation "irreducibility for context-sensitive reduction (term)" - 'PRedNotReducible G L T = (cir G L T). - -(* Basic inversion lemmas ***************************************************) - -lemma cir_inv_delta: ∀G,L,K,V,i. ⬇[i] L ≡ K.ⓓV → ⦃G, L⦄ ⊢ ➡ 𝐈⦃#i⦄ → ⊥. -/3 width=3 by crr_delta/ qed-. - -lemma cir_inv_ri2: ∀I,G,L,V,T. ri2 I → ⦃G, L⦄ ⊢ ➡ 𝐈⦃②{I}V.T⦄ → ⊥. -/3 width=1 by crr_ri2/ qed-. - -lemma cir_inv_ib2: ∀a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓑ{a,I}V.T⦄ → - ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ➡ 𝐈⦃T⦄. -/4 width=1 by crr_ib2_sn, crr_ib2_dx, conj/ qed-. - -lemma cir_inv_bind: ∀a,I,G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓑ{a,I}V.T⦄ → - ∧∧ ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ & ⦃G, L.ⓑ{I}V⦄ ⊢ ➡ 𝐈⦃T⦄ & ib2 a I. -#a * [ elim a -a ] -#G #L #V #T #H [ elim H -H /3 width=1 by crr_ri2, or_introl/ ] -elim (cir_inv_ib2 … H) -H /3 width=1 by and3_intro, or_introl/ -qed-. - -lemma cir_inv_appl: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓐV.T⦄ → - ∧∧ ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄ & 𝐒⦃T⦄. -#G #L #V #T #HVT @and3_intro /3 width=1 by crr_appl_sn, crr_appl_dx/ -generalize in match HVT; -HVT elim T -T // -* // #a * #U #T #_ #_ #H elim H -H /2 width=1 by crr_beta, crr_theta/ -qed-. - -lemma cir_inv_flat: ∀I,G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓕ{I}V.T⦄ → - ∧∧ ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl. -* #G #L #V #T #H -[ elim (cir_inv_appl … H) -H /2 width=1 by and4_intro/ -| elim (cir_inv_ri2 … H) -H // -] -qed-. - -(* Basic properties *********************************************************) - -lemma cir_sort: ∀G,L,s. ⦃G, L⦄ ⊢ ➡ 𝐈⦃⋆s⦄. -/2 width=4 by crr_inv_sort/ qed. - -lemma cir_gref: ∀G,L,p. ⦃G, L⦄ ⊢ ➡ 𝐈⦃§p⦄. -/2 width=4 by crr_inv_gref/ qed. - -lemma tir_atom: ∀G,I. ⦃G, ⋆⦄ ⊢ ➡ 𝐈⦃⓪{I}⦄. -/2 width=3 by trr_inv_atom/ qed. - -lemma cir_ib2: ∀a,I,G,L,V,T. - ib2 a I → ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ ➡ 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓑ{a,I}V.T⦄. -#a #I #G #L #V #T #HI #HV #HT #H -elim (crr_inv_ib2 … HI H) -HI -H /2 width=1 by/ -qed. - -lemma cir_appl: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄ → 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓐV.T⦄. -#G #L #V #T #HV #HT #H1 #H2 -elim (crr_inv_appl … H2) -H2 /2 width=1 by/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cir/cir_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cir/cir_lift.etc deleted file mode 100644 index 147d4b43c..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cir/cir_lift.etc +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/reduction/crr_lift.ma". -include "basic_2/reduction/cir.ma". - -(* IRREDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION ************************) - -(* Properties on relocation *************************************************) - -lemma cir_lift: ∀G,K,T. ⦃G, K⦄ ⊢ ➡ 𝐈⦃T⦄ → ∀L,c,l,k. ⬇[c, l, k] L ≡ K → - ∀U. ⬆[l, k] T ≡ U → ⦃G, L⦄ ⊢ ➡ 𝐈⦃U⦄. -/3 width=8 by crr_inv_lift/ qed. - -lemma cir_inv_lift: ∀G,L,U. ⦃G, L⦄ ⊢ ➡ 𝐈⦃U⦄ → ∀K,c,l,k. ⬇[c, l, k] L ≡ K → - ∀T. ⬆[l, k] T ≡ U → ⦃G, K⦄ ⊢ ➡ 𝐈⦃T⦄. -/3 width=8 by crr_lift/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cir/prednotreducible_3.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cir/prednotreducible_3.etc deleted file mode 100644 index 4ad7cbc72..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cir/prednotreducible_3.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ ➡ 𝐈 break ⦃ term 46 T ⦄ )" - non associative with precedence 45 - for @{ 'PRedNotReducible $G $L $T }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cix/cix.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cix/cix.etc deleted file mode 100644 index 5a7215ddf..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cix/cix.etc +++ /dev/null @@ -1,93 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/notation/relations/prednotreducible_5.ma". -include "basic_2/reduction/cir.ma". -include "basic_2/reduction/crx.ma". - -(* IRREDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ***************) - -definition cix: ∀h. sd h → relation3 genv lenv term ≝ - λh,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → ⊥. - -interpretation "irreducibility for context-sensitive extended reduction (term)" - 'PRedNotReducible h o G L T = (cix h o G L T). - -(* Basic inversion lemmas ***************************************************) - -lemma cix_inv_sort: ∀h,o,G,L,s,d. deg h o s (d+1) → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃⋆s⦄ → ⊥. -/3 width=2 by crx_sort/ qed-. - -lemma cix_inv_delta: ∀h,o,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃#i⦄ → ⊥. -/3 width=4 by crx_delta/ qed-. - -lemma cix_inv_ri2: ∀h,o,I,G,L,V,T. ri2 I → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃②{I}V.T⦄ → ⊥. -/3 width=1 by crx_ri2/ qed-. - -lemma cix_inv_ib2: ∀h,o,a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓑ{a,I}V.T⦄ → - ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, o] 𝐈⦃T⦄. -/4 width=1 by crx_ib2_sn, crx_ib2_dx, conj/ qed-. - -lemma cix_inv_bind: ∀h,o,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓑ{a,I}V.T⦄ → - ∧∧ ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ & ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ & ib2 a I. -#h #o #a * [ elim a -a ] -#G #L #V #T #H [ elim H -H /3 width=1 by crx_ri2, or_introl/ ] -elim (cix_inv_ib2 … H) -H /3 width=1 by and3_intro, or_introl/ -qed-. - -lemma cix_inv_appl: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓐV.T⦄ → - ∧∧ ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ & 𝐒⦃T⦄. -#h #o #G #L #V #T #HVT @and3_intro /3 width=1 by crx_appl_sn, crx_appl_dx/ -generalize in match HVT; -HVT elim T -T // -* // #a * #U #T #_ #_ #H elim H -H /2 width=1 by crx_beta, crx_theta/ -qed-. - -lemma cix_inv_flat: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓕ{I}V.T⦄ → - ∧∧ ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl. -#h #o * #G #L #V #T #H -[ elim (cix_inv_appl … H) -H /2 width=1 by and4_intro/ -| elim (cix_inv_ri2 … H) -H // -] -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma cix_inv_cir: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄. -/3 width=1 by crr_crx/ qed-. - -(* Basic properties *********************************************************) - -lemma cix_sort: ∀h,o,G,L,s. deg h o s 0 → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃⋆s⦄. -#h #o #G #L #s #Hk #H elim (crx_inv_sort … H) -L #d #Hkd -lapply (deg_mono … Hk Hkd) -h -s (cpr_inv_sort1 … H) // -qed. - -lemma cnr_lref_free: ∀G,L,i. |L| ≤ i → ⦃G, L⦄ ⊢ ➡ 𝐍⦃#i⦄. -#G #L #i #Hi #X #H elim (cpr_inv_lref1 … H) -H // * -#K #V1 #V2 #HLK lapply (drop_fwd_length_lt2 … HLK) -HLK -#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/ -qed. - -(* Basic_1: was only: nf2_csort_lref *) -lemma cnr_lref_atom: ∀G,L,i. ⬇[i] L ≡ ⋆ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃#i⦄. -#G #L #i #HL @cnr_lref_free >(drop_fwd_length … HL) -HL // -qed. - -(* Basic_1: was: nf2_abst *) -lemma cnr_abst: ∀a,G,L,W,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ➡ 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓛ{a}W.T⦄. -#a #G #L #W #T #HW #HT #X #H -elim (cpr_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct ->(HW … HW0) -W0 >(HT … HT0) -T0 // -qed. - -(* Basic_1: was only: nf2_appl_lref *) -lemma cnr_appl_simple: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓐV.T⦄. -#G #L #V #T #HV #HT #HS #X #H -elim (cpr_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct ->(HV … HV0) -V0 >(HT … HT0) -T0 // -qed. - -(* Basic_1: was: nf2_dec *) -axiom cnr_dec: ∀G,L,T1. ⦃G, L⦄ ⊢ ➡ 𝐍⦃T1⦄ ∨ - ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡ T2 & (T1 = T2 → ⊥). - -(* Basic_1: removed theorems 1: nf2_abst_shift *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnr/cnr_cir.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnr/cnr_cir.etc deleted file mode 100644 index 480e359cb..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnr/cnr_cir.etc +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/reduction/cpr_cir.ma". -include "basic_2/reduction/cnr_crr.ma". - -(* NORMAL TERMS FOR CONTEXT-SENSITIVE REDUCTION *****************************) - -(* Main properties on irreducibility ****************************************) - -theorem cir_cnr: ∀G,L,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄. -/2 width=4 by cpr_fwd_cir/ qed. - -(* Main inversion lemmas on irreducibility **********************************) - -theorem cnr_inv_cir: ∀G,L,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄. -/2 width=5 by cnr_inv_crr/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnr/cnr_crr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnr/cnr_crr.etc deleted file mode 100644 index 31291bc3d..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnr/cnr_crr.etc +++ /dev/null @@ -1,46 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/reduction/crr.ma". -include "basic_2/reduction/cnr.ma". - -(* NORMAL TERMS FOR CONTEXT-SENSITIVE REDUCTION *****************************) - -(* Advanced inversion lemmas on reducibility ********************************) - -(* Note: this property is unusual *) -lemma cnr_inv_crr: ∀G,L,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → ⊥. -#G #L #T #H elim H -L -T -[ #L #K #V #i #HLK #H - elim (cnr_inv_delta … HLK H) -| #L #V #T #_ #IHV #H - elim (cnr_inv_appl … H) -H /2 width=1 by/ -| #L #V #T #_ #IHT #H - elim (cnr_inv_appl … H) -H /2 width=1 by/ -| #I #L #V #T * #H1 #H2 destruct - [ elim (cnr_inv_zeta … H2) - | elim (cnr_inv_eps … H2) - ] -|5,6: #a * [ elim a ] #L #V #T * #H1 #_ #IH #H2 destruct - [1,3: elim (cnr_inv_abbr … H2) -H2 /2 width=1 by/ - |*: elim (cnr_inv_abst … H2) -H2 /2 width=1 by/ - ] -| #a #L #V #W #T #H - elim (cnr_inv_appl … H) -H #_ #_ #H - elim (simple_inv_bind … H) -| #a #L #V #W #T #H - elim (cnr_inv_appl … H) -H #_ #_ #H - elim (simple_inv_bind … H) -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnr/cnr_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnr/cnr_lift.etc deleted file mode 100644 index eef9aefc8..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnr/cnr_lift.etc +++ /dev/null @@ -1,49 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/reduction/cpr_lift.ma". -include "basic_2/reduction/cnr.ma". - -(* NORMAL TERMS FOR CONTEXT-SENSITIVE REDUCTION *****************************) - -(* Advanced properties ******************************************************) - -(* Basic_1: was: nf2_lref_abst *) -lemma cnr_lref_abst: ∀G,L,K,V,i. ⬇[i] L ≡ K. ⓛV → ⦃G, L⦄ ⊢ ➡ 𝐍⦃#i⦄. -#G #L #K #V #i #HLK #X #H -elim (cpr_inv_lref1 … H) -H // * -#K0 #V1 #V2 #HLK0 #_ #_ -lapply (drop_mono … HLK … HLK0) -L #H destruct -qed. - -(* Relocation properties ****************************************************) - -(* Basic_1: was: nf2_lift *) -lemma cnr_lift: ∀G,L0,L,T,T0,c,l,k. ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → - ⬇[c, l, k] L0 ≡ L → ⬆[l, k] T ≡ T0 → ⦃G, L0⦄ ⊢ ➡ 𝐍⦃T0⦄. -#G #L0 #L #T #T0 #c #l #k #HLT #HL0 #HT0 #X #H -elim (cpr_inv_lift1 … H … HL0 … HT0) -L0 #T1 #HT10 #HT1 -<(HLT … HT1) in HT0; -L #HT0 ->(lift_mono … HT10 … HT0) -T1 -X // -qed. - -(* Note: this was missing in basic_1 *) -lemma cnr_inv_lift: ∀G,L0,L,T,T0,c,l,k. ⦃G, L0⦄ ⊢ ➡ 𝐍⦃T0⦄ → - ⬇[c, l, k] L0 ≡ L → ⬆[l, k] T ≡ T0 → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄. -#G #L0 #L #T #T0 #c #l #k #HLT0 #HL0 #HT0 #X #H -elim (lift_total X l k) #X0 #HX0 -lapply (cpr_lift … H … HL0 … HT0 … HX0) -L #HTX0 ->(HLT0 … HTX0) in HX0; -L0 -X0 #H ->(lift_inj … H … HT0) -T0 -X -c -l -k // -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnr/prednormal_3.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnr/prednormal_3.etc deleted file mode 100644 index a8806a1c7..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnr/prednormal_3.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ ➡ 𝐍 break ⦃ term 46 T ⦄ )" - non associative with precedence 45 - for @{ 'PRedNormal $G $L $T }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnx/cnx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnx/cnx.etc deleted file mode 100644 index 0259a08aa..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnx/cnx.etc +++ /dev/null @@ -1,136 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/notation/relations/prednormal_5.ma". -include "basic_2/reduction/cnr.ma". -include "basic_2/reduction/cpx.ma". - -(* NORMAL TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ********************) - -definition cnx: ∀h. sd h → relation3 genv lenv term ≝ - λh,o,G,L. NF … (cpx h o G L) (eq …). - -interpretation - "normality for context-sensitive extended reduction (term)" - 'PRedNormal h o L T = (cnx h o L T). - -(* Basic inversion lemmas ***************************************************) - -lemma cnx_inv_sort: ∀h,o,G,L,s. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃⋆s⦄ → deg h o s 0. -#h #o #G #L #s #H elim (deg_total h o s) -#d @(nat_ind_plus … d) -d // #d #_ #Hkd -lapply (H (⋆(next h s)) ?) -H /2 width=2 by cpx_st/ -L -d #H -lapply (destruct_tatom_tatom_aux … H) -H #H (**) (* destruct lemma needed *) -lapply (destruct_sort_sort_aux … H) -H #H (**) (* destruct lemma needed *) -lapply (next_lt h s) >H -H #H elim (lt_refl_false … H) -qed-. - -lemma cnx_inv_delta: ∀h,o,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃#i⦄ → ⊥. -#h #o #I #G #L #K #V #i #HLK #H -elim (lift_total V 0 (i+1)) #W #HVW -lapply (H W ?) -H [ /3 width=7 by cpx_delta/ ] -HLK #H destruct -elim (lift_inv_lref2_be … HVW) -HVW /2 width=1 by ylt_inj/ -qed-. - -lemma cnx_inv_abst: ∀h,o,a,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓛ{a}V.T⦄ → - ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃V⦄ ∧ ⦃G, L.ⓛV⦄ ⊢ ➡[h, o] 𝐍⦃T⦄. -#h #o #a #G #L #V1 #T1 #HVT1 @conj -[ #V2 #HV2 lapply (HVT1 (ⓛ{a}V2.T1) ?) -HVT1 /2 width=2 by cpx_pair_sn/ -HV2 #H destruct // -| #T2 #HT2 lapply (HVT1 (ⓛ{a}V1.T2) ?) -HVT1 /2 width=2 by cpx_bind/ -HT2 #H destruct // -] -qed-. - -lemma cnx_inv_abbr: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃-ⓓV.T⦄ → - ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃V⦄ ∧ ⦃G, L.ⓓV⦄ ⊢ ➡[h, o] 𝐍⦃T⦄. -#h #o #G #L #V1 #T1 #HVT1 @conj -[ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2 by cpx_pair_sn/ -HV2 #H destruct // -| #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2 by cpx_bind/ -HT2 #H destruct // -] -qed-. - -lemma cnx_inv_zeta: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃+ⓓV.T⦄ → ⊥. -#h #o #G #L #V #T #H elim (is_lift_dec T 0 1) -[ * #U #HTU - lapply (H U ?) -H /2 width=3 by cpx_zeta/ #H destruct - elim (lift_inv_pair_xy_y … HTU) -| #HT - elim (cpr_delift G(⋆) V T (⋆.ⓓV) 0) // #T2 #T1 #HT2 #HT12 - lapply (H (+ⓓV.T2) ?) -H /5 width=1 by cpr_cpx, tpr_cpr, cpr_bind/ -HT2 - #H destruct /3 width=2 by ex_intro/ -] -qed-. - -lemma cnx_inv_appl: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓐV.T⦄ → - ∧∧ ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ & 𝐒⦃T⦄. -#h #o #G #L #V1 #T1 #HVT1 @and3_intro -[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1 by cpx_pair_sn/ -HV2 #H destruct // -| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1 by cpx_flat/ -HT2 #H destruct // -| generalize in match HVT1; -HVT1 elim T1 -T1 * // #a * #W1 #U1 #_ #_ #H - [ elim (lift_total V1 0 1) #V2 #HV12 - lapply (H (ⓓ{a}W1.ⓐV2.U1) ?) -H /3 width=3 by cpr_cpx, cpr_theta/ -HV12 #H destruct - | lapply (H (ⓓ{a}ⓝW1.V1.U1) ?) -H /3 width=1 by cpr_cpx, cpr_beta/ #H destruct - ] -] -qed-. - -lemma cnx_inv_eps: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓝV.T⦄ → ⊥. -#h #o #G #L #V #T #H lapply (H T ?) -H -/2 width=4 by cpx_eps, discr_tpair_xy_y/ -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma cnx_fwd_cnr: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄. -#h #o #G #L #T #H #U #HTU -@H /2 width=1 by cpr_cpx/ (**) (* auto fails because a δ-expansion gets in the way *) -qed-. - -(* Basic properties *********************************************************) - -lemma cnx_sort: ∀h,o,G,L,s. deg h o s 0 → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃⋆s⦄. -#h #o #G #L #s #Hk #X #H elim (cpx_inv_sort1 … H) -H // * #d #Hkd #_ -lapply (deg_mono … Hkd Hk) -h -L (drop_fwd_length … HL) -HL // -qed. - -lemma cnx_abst: ∀h,o,a,G,L,W,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → - ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓛ{a}W.T⦄. -#h #o #a #G #L #W #T #HW #HT #X #H -elim (cpx_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct ->(HW … HW0) -W0 >(HT … HT0) -T0 // -qed. - -lemma cnx_appl_simple: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → 𝐒⦃T⦄ → - ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓐV.T⦄. -#h #o #G #L #V #T #HV #HT #HS #X #H -elim (cpx_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct ->(HV … HV0) -V0 >(HT … HT0) -T0 // -qed. - -axiom cnx_dec: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T1⦄ ∨ - ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 & (T1 = T2 → ⊥). diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnx/cnx_cix.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnx/cnx_cix.etc deleted file mode 100644 index 32b9a76bf..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnx/cnx_cix.etc +++ /dev/null @@ -1,28 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/reduction/cpx_cix.ma". -include "basic_2/reduction/cnx_crx.ma". - -(* NORMAL TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ********************) - -(* Main properties on irreducibility ****************************************) - -theorem cix_cnx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄. -/2 width=6 by cpx_fwd_cix/ qed. - -(* Main inversion lemmas on irreducibility **********************************) - -theorem cnx_inv_cix: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄. -/2 width=7 by cnx_inv_crx/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnx/cnx_crx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnx/cnx_crx.etc deleted file mode 100644 index bb251965e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnx/cnx_crx.etc +++ /dev/null @@ -1,49 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/reduction/crx.ma". -include "basic_2/reduction/cnx.ma". - -(* NORMAL TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ********************) - -(* Advanced inversion lemmas on reducibility ********************************) - -(* Note: this property is unusual *) -lemma cnx_inv_crx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ⊥. -#h #o #G #L #T #H elim H -L -T -[ #L #s #d #Hkd #H - lapply (cnx_inv_sort … H) -H #H - lapply (deg_mono … H Hkd) -h -L -s (lift_mono … HT10 … HT0) -T1 -X // -qed. - -lemma cnx_inv_lift: ∀h,o,G,L0,L,T,T0,c,l,k. ⦃G, L0⦄ ⊢ ➡[h, o] 𝐍⦃T0⦄ → ⬇[c, l, k] L0 ≡ L → - ⬆[l, k] T ≡ T0 → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄. -#h #o #G #L0 #L #T #T0 #c #l #k #HLT0 #HL0 #HT0 #X #H -elim (lift_total X l k) #X0 #HX0 -lapply (cpx_lift … H … HL0 … HT0 … HX0) -L #HTX0 ->(HLT0 … HTX0) in HX0; -L0 -X0 #H ->(lift_inj … H … HT0) -T0 -X -l -k // -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnx/prednormal_5.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnx/prednormal_5.etc deleted file mode 100644 index 9df81e650..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cnx/prednormal_5.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ➡ break [ term 46 o , break term 46 h ] 𝐍 break ⦃ term 46 T ⦄ )" - non associative with precedence 45 - for @{ 'PRedNormal $h $o $G $L $T }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpg/cpg_drops.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpg/cpg_drops.etc deleted file mode 100644 index c0447ffde..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpg/cpg_drops.etc +++ /dev/null @@ -1,16 +0,0 @@ -lemma cpg_delift: ∀c,h,I,G,K,V,T1,L,i. ⬇*[i] L ≡ (K.ⓑ{I}V) → - ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡[h, 𝟘𝟘] T2 & ⬆*[↑1] T ≡ T2. -#h #c #I #G #K #V #T1 elim T1 -T1 -[ * #i #L #l /2 width=4 by cpg_atom, lift_sort, lift_gref, ex2_2_intro/ - elim (lt_or_eq_or_gt i l) #Hil [1,3: /4 width=4 by cpg_atom, lift_lref_ge_minus, lift_lref_lt, ylt_inj, yle_inj, ex2_2_intro/ ] - destruct - elim (lift_total V 0 (i+1)) #W #HVW - elim (lift_split … HVW i i) /3 width=7 by cpg_delta, ex2_2_intro/ -| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK - elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2 - [ elim (IHU1 (L. ⓑ{I} W1) (l+1)) -IHU1 /3 width=9 by cpg_bind, drop_drop, lift_bind, ex2_2_intro/ - | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpg_flat, lift_flat, ex2_2_intro/ - ] -] -qed-. -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpg/cpg_length.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpg/cpg_length.etc deleted file mode 100644 index 142f0fd2c..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpg/cpg_length.etc +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/relocation/drops.ma". -include "basic_2/rt_transition/cpg.ma". - -(* CONTEXT-SENSITIVE GENERIC PARALLEL RT-TRANSITION FOR TERMS ***************) - -(* Properties with length for local environments ****************************) - -lemma cpg_inv_lref1_ge: ∀h,c,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, c] T2 → |L| ≤ i → T2 = #i. -#h #c #G #L #T2 #i #H elim (cpg_inv_lref1 … H) -H // * -#I #K #V1 #V2 #HLK #_ #_ #HL -h -G -V2 lapply (drop_fwd_length_lt2 … HLK) -K -I -V1 -#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpr/cpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpr/cpr.etc deleted file mode 100644 index 3f30e621f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpr/cpr.etc +++ /dev/null @@ -1,29 +0,0 @@ -lemma cpr_delift: ∀G,K,V,T1,L,l. ⬇[l] L ≡ (K.ⓓV) → - ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡ T2 & ⬆[l, 1] T ≡ T2. -#G #K #V #T1 elim T1 -T1 -[ * /2 width=4 by cpr_atom, lift_sort, lift_gref, ex2_2_intro/ - #i #L #l #HLK elim (lt_or_eq_or_gt i l) - #Hil [1,3: /4 width=4 by lift_lref_ge_minus, lift_lref_lt, ylt_inj, yle_inj, ex2_2_intro/ ] - destruct - elim (lift_total V 0 (i+1)) #W #HVW - elim (lift_split … HVW i i) /3 width=6 by cpr_delta, ex2_2_intro/ -| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK - elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2 - [ elim (IHU1 (L. ⓑ{I}W1) (l+1)) -IHU1 /3 width=9 by drop_drop, cpr_bind, lift_bind, ex2_2_intro/ - | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpr_flat, lift_flat, ex2_2_intro/ - ] -] -qed-. - -fact lstas_cpr_aux: ∀h,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*[h, d] T2 → - d = 0 → ⦃G, L⦄ ⊢ T1 ➡ T2. -#h #G #L #T1 #T2 #d #H elim H -G -L -T1 -T2 -d -/3 width=1 by cpr_eps, cpr_flat, cpr_bind/ -[ #G #L #K #V1 #V2 #W2 #i #d #HLK #_ #HVW2 #IHV12 #H destruct - /3 width=6 by cpr_delta/ -| #G #L #K #V1 #V2 #W2 #i #d #_ #_ #_ #_ IHV1 -IHV1 // -HV1 >IHT1 -IHT1 // - | elim (cir_inv_ri2 … H) /2 width=1 by/ - ] -| #G #L #V1 #T1 #T #T2 #_ #_ #_ #H - elim (cir_inv_ri2 … H) /2 width=1 by or_introl/ -| #G #L #V1 #T1 #T2 #_ #_ #H - elim (cir_inv_ri2 … H) /2 width=1 by/ -| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H - elim (cir_inv_appl … H) -H #_ #_ #H - elim (simple_inv_bind … H) -| #a #G #L #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H - elim (cir_inv_appl … H) -H #_ #_ #H - elim (simple_inv_bind … H) -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpx/cpx_cix.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpx/cpx_cix.etc deleted file mode 100644 index 9bb51aebc..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpx/cpx_cix.etc +++ /dev/null @@ -1,52 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/reduction/cix.ma". -include "basic_2/reduction/cpx.ma". - -(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************) - -(* Advanced forward lemmas on irreducibility ********************************) - -lemma cpx_fwd_cix: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T1⦄ → T2 = T1. -#h #o #G #L #T1 #T2 #H elim H -G -L -T1 -T2 -[ // -| #G #L #s #d #Hkd #H elim (cix_inv_sort … Hkd H) -| #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #H - elim (cix_inv_delta … HLK) // -| #a * #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #H - [ elim (cix_inv_bind … H) -H #HV1 #HT1 * #H destruct - lapply (IHV1 … HV1) -IHV1 -HV1 #H destruct - lapply (IHT1 … HT1) -IHT1 #H destruct // - | elim (cix_inv_ib2 … H) -H /3 width=2 by or_introl, eq_f2/ - ] -| * #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #H - [ elim (cix_inv_appl … H) -H #HV1 #HT1 #_ - >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 // - | elim (cix_inv_ri2 … H) /2 width=1 by/ - ] -| #G #L #V1 #T1 #T #T2 #_ #_ #_ #H - elim (cix_inv_ri2 … H) /2 width=1 by or_introl/ -| #G #L #V1 #T1 #T2 #_ #_ #H - elim (cix_inv_ri2 … H) /2 width=1 by/ -| #G #L #V1 #V2 #T #_ #_ #H - elim (cix_inv_ri2 … H) /2 width=1 by/ -| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H - elim (cix_inv_appl … H) -H #_ #_ #H - elim (simple_inv_bind … H) -| #a #G #L #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H - elim (cix_inv_appl … H) -H #_ #_ #H - elim (simple_inv_bind … H) -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpx/cpx_drops.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpx/cpx_drops.etc deleted file mode 100644 index 3dd5f3a48..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpx/cpx_drops.etc +++ /dev/null @@ -1,18 +0,0 @@ -lemma cpx_delift: ∀h,I,G,K,V,T1,L,l. ⬇[l] L ≡ (K.ⓑ{I}V) → - ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡[h] T2 & ⬆[l, 1] T ≡ T2. -#h #o #I #G #K #V #T1 elim T1 -T1 -[ * #i #L #l /2 width=4 by cpx_atom, lift_sort, lift_gref, ex2_2_intro/ - elim (lt_or_eq_or_gt i l) #Hil [1,3: /4 width=4 by cpx_atom, -lift_lref_ge_minus, lift_lref_lt, ylt_inj, yle_inj, ex2_2_intro/ ] - destruct - elim (lift_total V 0 (i+1)) #W #HVW - elim (lift_split … HVW i i) /3 width=7 by cpx_delta, ex2_2_intro/ -| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK - elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2 - [ elim (IHU1 (L. ⓑ{I} W1) (l+1)) -IHU1 /3 width=9 by cpx_bind, -drop_drop, lift_bind, ex2_2_intro/ - | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpx_flat, lift_flat, -ex2_2_intro/ - ] -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpx/cpx_length.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpx/cpx_length.etc deleted file mode 100644 index c769990b2..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpx/cpx_length.etc +++ /dev/null @@ -1 +0,0 @@ -lemma cpx_inv_lref1_ge: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h] T2 → |L| ≤ i → T2 = #i. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpx/cpx_sta.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpx/cpx_sta.etc deleted file mode 100644 index 3925f9d43..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpx/cpx_sta.etc +++ /dev/null @@ -1,36 +0,0 @@ -fact sta_cpx_aux: ∀h,o,G,L,T1,T2,d2,d1. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 → d2 = 1 → - ⦃G, L⦄ ⊢ T1 ▪[h, o] d1+1 → ⦃G, L⦄ ⊢ T1 ➡[h, o] T2. -#h #o #G #L #T1 #T2 #d2 #d1 #H elim H -G -L -T1 -T2 -d2 -[ #G #L #d2 #s #H0 destruct normalize - /3 width=4 by cpx_st, da_inv_sort/ -| #G #L #K #V1 #V2 #W2 #i #d2 #HLK #_ #HVW2 #IHV12 #H0 #H destruct - elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0 - lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct /3 width=7 by cpx_delta/ -| #G #L #K #V1 #V2 #i #_ #_ #_ #H destruct -| #G #L #K #V1 #V2 #W2 #i #d2 #HLK #HV12 #HVW2 #_ #H0 #H - lapply (discr_plus_xy_y … H0) -H0 #H0 destruct - elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0 - lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct - /4 width=7 by cpx_delta, cpr_cpx, lstas_cpr/ -| /4 width=2 by cpx_bind, da_inv_bind/ -| /4 width=3 by cpx_flat, da_inv_flat/ -| /4 width=3 by cpx_eps, da_inv_flat/ -] -qed-. - -lemma sta_cpx: ∀h,o,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 → - ⦃G, L⦄ ⊢ T1 ▪[h, o] d+1 → ⦃G, L⦄ ⊢ T1 ➡[h, o] T2. -/2 width=3 by sta_cpx_aux/ qed. - -lemma fqu_sta_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 •*[h, 1] U2 → - ∀d. ⦃G2, L2⦄ ⊢ T2 ▪[h, o] d+1 → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄. -/3 width=5 by fqu_cpx_trans, sta_cpx/ qed-. - -lemma fquq_sta_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 •*[h, 1] U2 → - ∀d. ⦃G2, L2⦄ ⊢ T2 ▪[h, o] d+1 → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄. -/3 width=5 by fquq_cpx_trans, sta_cpx/ qed-. - diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpy/cpy.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpy/cpy.etc deleted file mode 100644 index 4a27de0a4..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpy/cpy.etc +++ /dev/null @@ -1,286 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/notation/relations/psubst_6.ma". -include "basic_2/grammar/genv.ma". -include "basic_2/substitution/lsuby.ma". - -(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************) - -(* activate genv *) -inductive cpy: ynat → ynat → relation4 genv lenv term term ≝ -| cpy_atom : ∀I,G,L,l,m. cpy l m G L (⓪{I}) (⓪{I}) -| cpy_subst: ∀I,G,L,K,V,W,i,l,m. l ≤ i → i < l+m → - ⬇[i] L ≡ K.ⓑ{I}V → ⬆[0, ⫯i] V ≡ W → cpy l m G L (#i) W -| cpy_bind : ∀a,I,G,L,V1,V2,T1,T2,l,m. - cpy l m G L V1 V2 → cpy (⫯l) m G (L.ⓑ{I}V1) T1 T2 → - cpy l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) -| cpy_flat : ∀I,G,L,V1,V2,T1,T2,l,m. - cpy l m G L V1 V2 → cpy l m G L T1 T2 → - cpy l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) -. - -interpretation "context-sensitive extended ordinary substritution (term)" - 'PSubst G L T1 l m T2 = (cpy l m G L T1 T2). - -(* Basic properties *********************************************************) - -lemma lsuby_cpy_trans: ∀G,l,m. lsub_trans … (cpy l m G) (lsuby l m). -#G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m -[ // -| #I #G #L1 #K1 #V #W #i #l #m #Hli #Hilm #HLK1 #HVW #L2 #HL12 - elim (ylt_inv_plus_dx … Hilm) #m0 #H0 #_ - elim (lsuby_drop_trans_be … HL12 … HLK1 … H0) -HL12 -HLK1 -H0 /2 width=5 by cpy_subst/ -| /4 width=1 by lsuby_succ, cpy_bind/ -| /3 width=1 by cpy_flat/ -] -qed-. - -lemma cpy_refl: ∀G,T,L,l,m. ⦃G, L⦄ ⊢ T ▶[l, m] T. -#G #T elim T -T // * /2 width=1 by cpy_bind, cpy_flat/ -qed. - -(* Basic_1: was: subst1_ex *) -lemma cpy_full: ∀I,G,K,V,T1,L,l. ⬇[l] L ≡ K.ⓑ{I}V → - ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2 & ⬆[l, 1] T ≡ T2. -#I #G #K #V #T1 elim T1 -T1 -[ * #i #L #l #HLK - /2 width=4 by lift_sort, lift_gref, ex2_2_intro/ - elim (ylt_split_eq i l) /3 width=4 by lift_lref_pred, lift_lref_lt, ex2_2_intro/ - #H destruct lapply (drop_fwd_Y2 … HLK) #Hi - elim (lift_total (⫯i) … V 0) /2 width=1 by ylt_succ/ - #W #HVW elim (lift_split … HVW i i … 1) - /4 width=6 by cpy_subst, monotonic_ylt_plus_sn, ex2_2_intro/ -| * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #l #HLK - elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2 - [ elim (IHU1 (L.ⓑ{J}W1) (⫯l)) -IHU1 - /3 width=9 by cpy_bind, drop_drop, lift_bind, ex2_2_intro/ - | elim (IHU1 … HLK) -IHU1 -HLK - /3 width=8 by cpy_flat, lift_flat, ex2_2_intro/ - ] -] -qed-. - -lemma cpy_weak: ∀G,L,T1,T2,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T2 → - ∀l2,m2. l2 ≤ l1 → l1 + m1 ≤ l2 + m2 → - ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T2. -#G #L #T1 #T2 #l1 #m1 #H elim H -G -L -T1 -T2 -l1 -m1 // -[ /3 width=5 by cpy_subst, ylt_yle_trans, yle_trans/ -| /4 width=3 by cpy_bind, ylt_yle_trans, yle_succ/ -| /3 width=1 by cpy_flat/ -] -qed-. -(* -lemma cpy_weak_top: ∀G,L,T1,T2,l,m. - ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, ∞] T2. -/2 width=5 by cpy_weak/ qed-. - -lemma cpy_weak_full: ∀G,L,T1,T2,l,m. - ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[0, ∞] T2. -/2 width=5 by cpy_weak/ qed-. -*) -lemma cpy_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → - ∀i,m2. i + m2 = l + m → - ∀m1. i ≤ l + m1 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l, m1] T & ⦃G, L⦄ ⊢ T ▶[i, m2] T2. -#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m -[ /2 width=3 by ex2_intro/ -| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #m2 #H2 #m1 #H1 - elim (ylt_split i j) [ -Hilm -H2 | -Hli ] - /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/ -| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #m2 #H2 #m1 #H1 - elim (IHV12 … H2 … H1) -IHV12 #V - elim (IHT12 (⫯i) … m2 … m1) -IHT12 /2 width=1 by yle_succ/ -H2 -H1 - #T #HT1 #HT2 lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 - /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/ -| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #m2 #H2 #m1 #H1 - elim (IHV12 … H2 … H1) -IHV12 elim (IHT12 … H2 … H1) -IHT12 -H2 -H1 - /3 width=5 by ex2_intro, cpy_flat/ -] -qed-. - -lemma cpy_split_down: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → - ∀m1,m2. m = m1 + m2 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l+m2, m1] T & ⦃G, L⦄ ⊢ T ▶[l, m2] T2. -#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m -[ /2 width=3 by ex2_intro/ -| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #m1 #m2 #H destruct - elim (ylt_split i (l+m2)) [ -Hilm | -Hli ] - /3 width=9 by cpy_subst, ex2_intro/ -| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #m1 #m2 #H destruct - elim (IHV12 m1 m2) -IHV12 // #V - elim (IHT12 m1 m2) -IHT12 // - >yplus_succ1 #T #HT1 #HT2 - lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 - /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/ -| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #m1 #m2 #H destruct - elim (IHV12 m1 m2) -IHV12 // elim (IHT12 m1 m2) -IHT12 // - /3 width=5 by ex2_intro, cpy_flat/ -] -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma cpy_fwd_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 → - ∀T1,l,m. ⬆[l, m] T1 ≡ U1 → - l ≤ lt → l + m ≤ lt + mt → - ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶[l+m, lt+mt-(l+m)] U2 & ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt -[ * #i #G #L #lt #mt #T1 #l #m #H #_ - [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/ - | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/ - | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/ - ] -| #I #G #L #K #V #W #i #lt #mt #Hlti #Hilmt #HLK #HVW #T1 #l #m #H #Hllt #Hlmlmt - elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -V -Hilmt -Hlmlmt | -Hlti -Hllt ] - [ elim (ylt_yle_false … Hllt) -Hllt /3 width=3 by yle_ylt_trans, ylt_inj/ - | elim (yle_inv_plus_inj2 … Hil) #Hlim #Hmi - elim (lift_split … HVW l (⫯(i-m)) ? ? ?) [2,3,4: /2 width=1 by yle_succ_dx, le_S_S/ ] -Hlim - #T2 #_ >plus_minus /2 width=1 by yle_inv_inj/ ymax_pre_sn_comm // (**) (* explicit constructor *) - ] -| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt - elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct - elim (IHW12 … HVW1) -V1 -IHW12 // - elim (IHU12 … HTU1) -T1 -IHU12 /2 width=1 by yle_succ/ - yplus_SO2 >yplus_succ1 >yplus_succ1 - /3 width=2 by cpy_bind, lift_bind, ex2_intro/ -| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt - elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct - elim (IHW12 … HVW1) -V1 -IHW12 // elim (IHU12 … HTU1) -T1 -IHU12 - /3 width=2 by cpy_flat, lift_flat, ex2_intro/ -] -qed-. - -lemma cpy_fwd_tw: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ♯{T1} ≤ ♯{T2}. -#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m normalize -/3 width=1 by monotonic_le_plus_l, le_plus/ -qed-. - -(* Basic inversion lemmas ***************************************************) - -fact cpy_inv_atom1_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀J. T1 = ⓪{J} → - T2 = ⓪{J} ∨ - ∃∃I,K,V,i. l ≤ yinj i & i < l + m & - ⬇[i] L ≡ K.ⓑ{I}V & - ⬆[O, ⫯i] V ≡ T2 & - J = LRef i. -#G #L #T1 #T2 #l #m * -G -L -T1 -T2 -l -m -[ #I #G #L #l #m #J #H destruct /2 width=1 by or_introl/ -| #I #G #L #K #V #T2 #i #l #m #Hli #Hilm #HLK #HVT2 #J #H destruct /3 width=9 by ex5_4_intro, or_intror/ -| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct -| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct -] -qed-. - -lemma cpy_inv_atom1: ∀I,G,L,T2,l,m. ⦃G, L⦄ ⊢ ⓪{I} ▶[l, m] T2 → - T2 = ⓪{I} ∨ - ∃∃J,K,V,i. l ≤ yinj i & i < l + m & - ⬇[i] L ≡ K.ⓑ{J}V & - ⬆[O, ⫯i] V ≡ T2 & - I = LRef i. -/2 width=4 by cpy_inv_atom1_aux/ qed-. - -(* Basic_1: was: subst1_gen_sort *) -lemma cpy_inv_sort1: ∀G,L,T2,k,l,m. ⦃G, L⦄ ⊢ ⋆k ▶[l, m] T2 → T2 = ⋆k. -#G #L #T2 #k #l #m #H -elim (cpy_inv_atom1 … H) -H // -* #I #K #V #i #_ #_ #_ #_ #H destruct -qed-. - -(* Basic_1: was: subst1_gen_lref *) -lemma cpy_inv_lref1: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶[l, m] T2 → - T2 = #i ∨ - ∃∃I,K,V. l ≤ i & i < l + m & - ⬇[i] L ≡ K.ⓑ{I}V & - ⬆[O, ⫯i] V ≡ T2. -#G #L #T2 #i #l #m #H -elim (cpy_inv_atom1 … H) -H /2 width=1 by or_introl/ -* #I #K #V #j #Hlj #Hjlm #HLK #HVT2 #H destruct /3 width=5 by ex4_3_intro, or_intror/ -qed-. - -lemma cpy_inv_gref1: ∀G,L,T2,p,l,m. ⦃G, L⦄ ⊢ §p ▶[l, m] T2 → T2 = §p. -#G #L #T2 #p #l #m #H -elim (cpy_inv_atom1 … H) -H // -* #I #K #V #i #_ #_ #_ #_ #H destruct -qed-. - -fact cpy_inv_bind1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → - ∀a,I,V1,T1. U1 = ⓑ{a,I}V1.T1 → - ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 & - ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ▶[⫯l, m] T2 & - U2 = ⓑ{a,I}V2.T2. -#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m -[ #I #G #L #l #m #b #J #W1 #U1 #H destruct -| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #b #J #W1 #U1 #H destruct -| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/ -| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #b #J #W1 #U1 #H destruct -] -qed-. - -lemma cpy_inv_bind1: ∀a,I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓑ{a,I} V1. T1 ▶[l, m] U2 → - ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 & - ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶[⫯l, m] T2 & - U2 = ⓑ{a,I}V2.T2. -/2 width=3 by cpy_inv_bind1_aux/ qed-. - -fact cpy_inv_flat1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → - ∀I,V1,T1. U1 = ⓕ{I}V1.T1 → - ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 & - ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 & - U2 = ⓕ{I}V2.T2. -#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m -[ #I #G #L #l #m #J #W1 #U1 #H destruct -| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #J #W1 #U1 #H destruct -| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #W1 #U1 #H destruct -| #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/ -] -qed-. - -lemma cpy_inv_flat1: ∀I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ▶[l, m] U2 → - ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 & - ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 & - U2 = ⓕ{I}V2.T2. -/2 width=3 by cpy_inv_flat1_aux/ qed-. - - -fact cpy_inv_refl_O2_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → m = 0 → T1 = T2. -#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m -[ // -| #I #G #L #K #V #W #i #l #m #Hli #Hilm #_ #_ #H destruct - elim (ylt_yle_false … Hli) -Hli // -| /3 width=1 by eq_f2/ -| /3 width=1 by eq_f2/ -] -qed-. - -lemma cpy_inv_refl_O2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶[l, 0] T2 → T1 = T2. -/2 width=6 by cpy_inv_refl_O2_aux/ qed-. - -(* Basic_1: was: subst1_gen_lift_eq *) -lemma cpy_inv_lift1_eq: ∀G,T1,U1,l,m. ⬆[l, m] T1 ≡ U1 → - ∀L,U2. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → U1 = U2. -#G #T1 #U1 #l #m #HTU1 #L #U2 #HU12 elim (cpy_fwd_up … HU12 … HTU1) -HU12 -HTU1 -/2 width=4 by cpy_inv_refl_O2/ -qed-. - -(* Basic_1: removed theorems 25: - subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt - subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans - subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s - subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt - subst0_confluence_neq subst0_confluence_eq subst0_tlt_head - subst0_confluence_lift subst0_tlt - subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpy/cpy_cpy.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpy/cpy_cpy.etc deleted file mode 100644 index 7a65a2db5..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpy/cpy_cpy.etc +++ /dev/null @@ -1,123 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/substitution/cpy_lift.ma". - -(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************) - -(* Main properties **********************************************************) - -(* Basic_1: was: subst1_confluence_eq *) -theorem cpy_conf_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶[l1, m1] T1 → - ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶[l1, m1] T. -#G #L #T0 #T1 #l1 #m1 #H elim H -G -L -T0 -T1 -l1 -m1 -[ /2 width=3 by ex2_intro/ -| #I1 #G #L #K1 #V1 #T1 #i0 #l1 #m1 #Hl1 #Hlm1 #HLK1 #HVT1 #T2 #l2 #m2 #H - elim (cpy_inv_lref1 … H) -H - [ #HX destruct /3 width=7 by cpy_subst, ex2_intro/ - | -Hl1 -Hlm1 * #I2 #K2 #V2 #_ #_ #HLK2 #HVT2 - lapply (drop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct - >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3 by ex2_intro/ - ] -| #a #I #G #L #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #X #l2 #m2 #HX - elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - elim (IHV01 … HV02) -IHV01 -HV02 #V #HV1 #HV2 - elim (IHT01 … HT02) -T0 #T #HT1 #HT2 - lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/ - lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V2) ?) -HT2 - /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/ -| #I #G #L #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #X #l2 #m2 #HX - elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - elim (IHV01 … HV02) -V0 - elim (IHT01 … HT02) -T0 /3 width=5 by cpy_flat, ex2_intro/ -] -qed-. - -(* Basic_1: was: subst1_confluence_neq *) -theorem cpy_conf_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶[l1, m1] T1 → - ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶[l2, m2] T2 → - (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) → - ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶[l1, m1] T. -#G #L1 #T0 #T1 #l1 #m1 #H elim H -G -L1 -T0 -T1 -l1 -m1 -[ /2 width=3 by ex2_intro/ -| #I1 #G #L1 #K1 #V1 #T1 #i0 #l1 #m1 #Hl1 #Hlm1 #HLK1 #HVT1 #L2 #T2 #l2 #m2 #H1 #H2 - elim (cpy_inv_lref1 … H1) -H1 - [ #H destruct /3 width=7 by cpy_subst, ex2_intro/ - | -HLK1 -HVT1 * #I2 #K2 #V2 #Hl2 #Hlm2 #_ #_ elim H2 -H2 #Hlml [ -Hl1 -Hlm2 | -Hl2 -Hlm1 ] - [ elim (ylt_yle_false … Hlm1) -Hlm1 /2 width=3 by yle_trans/ - | elim (ylt_yle_false … Hlm2) -Hlm2 /2 width=3 by yle_trans/ - ] - ] -| #a #I #G #L1 #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #L2 #X #l2 #m2 #HX #H - elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - elim (IHV01 … HV02 H) -IHV01 -HV02 #V #HV1 #HV2 - elim (IHT01 … HT02) -T0 - [ -H #T #HT1 #HT2 - lapply (lsuby_cpy_trans … HT1 (L2.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/ - lapply (lsuby_cpy_trans … HT2 (L1.ⓑ{I}V2) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/ - | -HV1 -HV2 elim H -H /3 width=1 by yle_succ, or_introl, or_intror/ - ] -| #I #G #L1 #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #L2 #X #l2 #m2 #HX #H - elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - elim (IHV01 … HV02 H) -V0 - elim (IHT01 … HT02 H) -T0 -H /3 width=5 by cpy_flat, ex2_intro/ -] -qed-. - -(* Note: the constant 1 comes from cpy_subst *) -(* Basic_1: was: subst1_trans *) -theorem cpy_trans_ge: ∀G,L,T1,T0,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T0 → - ∀T2. ⦃G, L⦄ ⊢ T0 ▶[l, 1] T2 → 1 ≤ m → ⦃G, L⦄ ⊢ T1 ▶[l, m] T2. -#G #L #T1 #T0 #l #m #H elim H -G -L -T1 -T0 -l -m -[ #I #G #L #l #m #T2 #H #Hm - elim (cpy_inv_atom1 … H) -H - [ #H destruct // - | * #J #K #V #i #Hl2i #Hilm2 #HLK #HVT2 #H destruct - lapply (ylt_yle_trans … (l+m) … Hilm2) - /2 width=5 by cpy_subst, monotonic_yle_plus_sn/ - ] -| #I #G #L #K #V #V2 #i #l #m #Hli #Hilm #HLK #HVW #T2 #HVT2 #Hm - lapply (cpy_weak … HVT2 0 (i+1) ? ?) -HVT2 /3 width=1 by yle_plus_dx2_trans, yle_succ/ - >yplus_inj #HVT2 <(cpy_inv_lift1_eq … HVW … HVT2) -HVT2 /2 width=5 by cpy_subst/ -| #a #I #G #L #V1 #V0 #T1 #T0 #l #m #_ #_ #IHV10 #IHT10 #X #H #Hm - elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct - lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02 - lapply (IHT10 … HT02 Hm) -T0 /3 width=1 by cpy_bind/ -| #I #G #L #V1 #V0 #T1 #T0 #l #m #_ #_ #IHV10 #IHT10 #X #H #Hm - elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1 by cpy_flat/ -] -qed-. - -theorem cpy_trans_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T0 → - ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → l2 + m2 ≤ l1 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T ▶[l1, m1] T2. -#G #L #T1 #T0 #l1 #m1 #H elim H -G -L -T1 -T0 -l1 -m1 -[ /2 width=3 by ex2_intro/ -| #I #G #L #K #V #W #i1 #l1 #m1 #Hli1 #Hilm1 #HLK #HVW #T2 #l2 #m2 #HWT2 #Hlm2l1 - lapply (yle_trans … Hlm2l1 … Hli1) -Hlm2l1 #Hlm2i1 - lapply (cpy_weak … HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hlm2i1 - >yplus_inj #HWT2 <(cpy_inv_lift1_eq … HVW … HWT2) -HWT2 /3 width=9 by cpy_subst, ex2_intro/ -| #a #I #G #L #V1 #V0 #T1 #T0 #l1 #m1 #_ #_ #IHV10 #IHT10 #X #l2 #m2 #HX #lm2l1 - elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02 - elim (IHV10 … HV02) -IHV10 -HV02 // #V - elim (IHT10 … HT02) -T0 /2 width=1 by yle_succ/ #T #HT1 #HT2 - lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /3 width=6 by cpy_bind, lsuby_succ, ex2_intro/ -| #I #G #L #V1 #V0 #T1 #T0 #l1 #m1 #_ #_ #IHV10 #IHT10 #X #l2 #m2 #HX #lm2l1 - elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct - elim (IHV10 … HV02) -V0 // - elim (IHT10 … HT02) -T0 /3 width=6 by cpy_flat, ex2_intro/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpy/cpy_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpy/cpy_lift.etc deleted file mode 100644 index 44a7f3d76..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpy/cpy_lift.etc +++ /dev/null @@ -1,253 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/substitution/drop_drop.ma". -include "basic_2/substitution/cpy.ma". - -(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************) - -(* Properties on relocation *************************************************) - -(* Basic_1: was: subst1_lift_lt *) -lemma cpy_lift_le: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 → - ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K → - ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 → - lt + mt ≤ l → ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2. -#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt -[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_ - >(lift_mono … H1 … H2) -H1 -H2 // -| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hlmtl - lapply (ylt_yle_trans … Hlmtl … Hilmt) -Hlmtl #Hil - lapply (lift_inv_lref1_lt … H … Hil) -H #H destruct - elim (lift_trans_ge … HVW … HWU2) -W /2 width=1 by ylt_fwd_le_succ1/ - yplus_SO2 >yminus_succ2 #W #HVW #HWU2 - elim (drop_trans_le … HLK … HKV) -K /2 width=2 by ylt_fwd_le/ #X #HLK #H - elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K #Y #_ #HVY - >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=5 by cpy_subst/ -| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl - elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct - /4 width=7 by cpy_bind, drop_skip, yle_succ/ -| #G #I #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl - elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct - /3 width=7 by cpy_flat/ -] -qed-. - -lemma cpy_lift_be: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 → - ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K → - ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 → - lt ≤ l → l ≤ lt + mt → ⦃G, L⦄ ⊢ U1 ▶[lt, mt + m] U2. -#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt -[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_ #_ - >(lift_mono … H1 … H2) -H1 -H2 // -| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hltl #_ - elim (lift_inv_lref1 … H) -H * #Hil #H destruct - [ -Hltl - lapply (ylt_yle_trans … (lt+mt+m) … Hilmt) // -Hilmt #Hilmtm - elim (lift_trans_ge … HVW … HWU2) -W yplus_SO2 - [2: >yplus_O1 /2 width=1 by ylt_fwd_le_succ1/ ] >yminus_succ2 #W #HVW #HWU2 - elim (drop_trans_le … HLK … HKV) -K /2 width=1 by ylt_fwd_le/ #X #HLK #H - elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K #Y #_ #HVY - >(lift_mono … HVY … HVW) -V #H destruct /2 width=5 by cpy_subst/ - | -Hlti - lapply (yle_trans … Hltl … Hil) -Hltl #Hlti - lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by yle_succ_dx/ >plus_plus_comm_23 #HVU2 - lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hil - /3 width=5 by cpy_subst, drop_inv_gen, monotonic_ylt_plus_dx, yle_plus_dx1_trans/ - ] -| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hltl #Hllmt - elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct - /4 width=7 by cpy_bind, drop_skip, yle_succ/ -| #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl - elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct - /3 width=7 by cpy_flat/ -] -qed-. - -(* Basic_1: was: subst1_lift_ge *) -lemma cpy_lift_ge: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 → - ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K → - ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 → - l ≤ lt → ⦃G, L⦄ ⊢ U1 ▶[lt+m, mt] U2. -#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt -[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_ - >(lift_mono … H1 … H2) -H1 -H2 // -| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hllt - lapply (yle_trans … Hllt … Hlti) -Hllt #Hil - lapply (lift_inv_lref1_ge … H … Hil) -H #H destruct - lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by yle_succ_dx/ >plus_plus_comm_23 #HVU2 - lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hil - /3 width=5 by cpy_subst, drop_inv_gen, monotonic_ylt_plus_dx, monotonic_yle_plus_dx/ -| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hllt - elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct - /4 width=6 by cpy_bind, drop_skip, yle_succ/ -| #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hllt - elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 - elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct - /3 width=6 by cpy_flat/ -] -qed-. - -(* Inversion lemmas on relocation *******************************************) - -(* Basic_1: was: subst1_gen_lift_lt *) -lemma cpy_inv_lift1_le: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - lt + mt ≤ l → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 & ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt -[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_ - [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/ - | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/ - | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/ - ] -| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hlmtl - lapply (ylt_yle_trans … Hlmtl … Hilmt) -Hlmtl #Hil - lapply (lift_inv_lref2_lt … H … Hil) -H #H destruct - elim (drop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV - elim (lift_trans_le … HUV … HVW) -V // >minus_plus yplus_SO2 >ymax_pre_sn /2 width=1 by ylt_fwd_le_succ1/ -Hil - /3 width=5 by cpy_subst, ex2_intro/ -| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl - elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct - elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2 - elim (IHU12 … HTU1) -IHU12 -HTU1 - /3 width=6 by cpy_bind, yle_succ, drop_skip, lift_bind, ex2_intro/ -| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl - elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct - elim (IHW12 … HLK … HVW1) -W1 // - elim (IHU12 … HLK … HTU1) -U1 -HLK - /3 width=5 by cpy_flat, lift_flat, ex2_intro/ -] -qed-. - -lemma cpy_inv_lift1_be: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - lt ≤ l → l + m ≤ lt + mt → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, mt-m] T2 & ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt -[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_ #_ - [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/ - | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/ - | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/ - ] -| #I #G #L #KV #V #W #i #x #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hltl #Hlmlmt - elim (yle_inv_inj2 … Hlti) -Hlti #lt #Hlti #H destruct - lapply (yle_fwd_plus_yge … Hltl Hlmlmt) #Hmmt - elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -Hltl -Hilmt | -Hlti -Hlmlmt ] - [ lapply (ylt_yle_trans i l (lt+(mt-m)) ? ?) // - [ >yplus_minus_assoc_inj /2 width=1 by yle_plus1_to_minus_inj2/ ] -Hlmlmt #Hilmtm - elim (drop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV - elim (lift_trans_le … HUV … HVW) -V // - yplus_SO2 >ymax_pre_sn /2 width=1 by ylt_fwd_le_succ1/ -Hil - /4 width=5 by cpy_subst, ex2_intro, yle_inj/ - | elim (yle_inv_plus_inj2 … Hil) #Hlim #Hmi - lapply (yle_inv_inj … Hmi) -Hmi #Hmi - lapply (yle_trans … Hltl (i-m) ?) // -Hltl #Hltim - lapply (drop_conf_ge … HLK … HLKV ?) -L // #HKV - elim (lift_split … HVW l (i-m+1)) -HVW [2,3,4: /2 width=1 by yle_succ_dx, le_S_S/ ] -Hil -Hlim - #V1 #HV1 >plus_minus // yplus_minus_assoc_inj /3 width=1 by monotonic_ylt_minus_dx, yle_inj/ - ] -| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hltl #Hlmlmt - elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct - elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2 - elim (IHU12 … HTU1) -U1 - /3 width=6 by cpy_bind, drop_skip, lift_bind, yle_succ, ex2_intro/ -| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hltl #Hlmlmt - elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct - elim (IHW12 … HLK … HVW1) -W1 // - elim (IHU12 … HLK … HTU1) -U1 -HLK // - /3 width=5 by cpy_flat, lift_flat, ex2_intro/ -] -qed-. - -(* Basic_1: was: subst1_gen_lift_ge *) -lemma cpy_inv_lift1_ge: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - l + m ≤ lt → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt-m, mt] T2 & ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt -[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_ - [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/ - | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/ - | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/ - ] -| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hlmlt - lapply (yle_trans … Hlmlt … Hlti) #Hlmi - elim (yle_inv_plus_inj2 … Hlmlt) -Hlmlt #_ #Hmlt - elim (yle_inv_plus_inj2 … Hlmi) #Hlim #Hmi - lapply (yle_inv_inj … Hmi) -Hmi #Hmi - lapply (lift_inv_lref2_ge … H ?) -H // #H destruct - lapply (drop_conf_ge … HLK … HLKV ?) -L // #HKV - elim (lift_split … HVW l (i-m+1)) -HVW [2,3,4: /3 width=1 by yle_succ, yle_pred_sn, le_S_S/ ] -Hlmi -Hlim - #V0 #HV10 >plus_minus // yminus_succ1_inj /3 width=5 by cpy_bind, lift_bind, ex2_intro/ -| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl - elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct - elim (IHW12 … HLK … HVW1) -W1 // - elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpy_flat, lift_flat, ex2_intro/ -] -qed-. - -(* Advanced inversion lemmas on relocation ***********************************) - -lemma cpy_inv_lift1_ge_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - l ≤ lt → lt ≤ l + m → l + m ≤ lt + mt → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[l, lt + mt - (l + m)] T2 & ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt -elim (cpy_split_up … HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2 -lapply (cpy_weak … HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1 -lapply (cpy_inv_lift1_eq … HTU1 … HU1) -HU1 #HU1 destruct -elim (cpy_inv_lift1_ge … HU2 … HLK … HTU1) -U -L /2 width=3 by ex2_intro/ -qed-. - -lemma cpy_inv_lift1_be_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - lt ≤ l → lt + mt ≤ l + m → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, l-lt] T2 & ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm -lapply (cpy_weak … HU12 lt (l+m-lt) ? ?) -HU12 // -[ >ymax_pre_sn_comm /2 width=1 by yle_plus_dx1_trans/ ] -Hlmtlm #HU12 -elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) -U1 -L /2 width=3 by ex2_intro/ -qed-. - -lemma cpy_inv_lift1_le_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - lt ≤ l → l ≤ lt + mt → lt + mt ≤ l + m → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm -elim (cpy_split_up … HU12 l) -HU12 // #U #HU1 #HU2 -elim (cpy_inv_lift1_le … HU1 … HLK … HTU1) -U1 -[2: >ymax_pre_sn_comm // ] -Hltl #T #HT1 #HTU -lapply (cpy_weak … HU2 l m ? ?) -HU2 // -[ >ymax_pre_sn_comm // ] -Hllmt -Hlmtlm #HU2 -lapply (cpy_inv_lift1_eq … HTU … HU2) -L #H destruct /2 width=3 by ex2_intro/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpy/cpy_nlift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpy/cpy_nlift.etc deleted file mode 100644 index 3a6450724..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpy/cpy_nlift.etc +++ /dev/null @@ -1,65 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/substitution/lift_neg.ma". -include "basic_2/substitution/lift_lift.ma". -include "basic_2/substitution/cpy.ma". - -(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************) - -(* Inversion lemmas on negated relocation ***********************************) - -lemma cpy_fwd_nlift2_ge: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → - ∀i. l ≤ i → (∀T2. ⬆[i, 1] T2 ≡ U2 → ⊥) → - (∀T1. ⬆[i, 1] T1 ≡ U1 → ⊥) ∨ - ∃∃I,K,W,j. l ≤ yinj j & j < i & ⬇[j]L ≡ K.ⓑ{I}W & - (∀V. ⬆[⫰(i-j), 1] V ≡ W → ⊥) & (∀T1. ⬆[j, 1] T1 ≡ U1 → ⊥). -#G #L #U1 #U2 #l #m #H elim H -G -L -U1 -U2 -l -m -[ /3 width=2 by or_introl/ -| #I #G #L #K #V #W #j #l #m #Hlj #Hjlm #HLK #HVW #i #Hli #HnW - elim (ylt_split j i) #Hij - [ @or_intror @(ex5_4_intro … HLK) // -HLK - [ #X #HXV elim (lift_trans_le … HXV … HVW ?) -V // #Y #HXY - yplus_SO2 >ymax_pre_sn /2 width=2 by ylt_fwd_le_succ1/ - | -HnW /3 width=7 by lift_inv_lref2_be, ylt_inj/ - ] - | elim (lift_split … HVW i j) -HVW // - #X #_ #H elim HnW -HnW // - ] -| #a #I #G #L #W1 #W2 #U1 #U2 #l #m #_ #_ #IHW12 #IHU12 #i #Hli #H elim (nlift_inv_bind … H) -H - [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 // - [ /4 width=9 by nlift_bind_sn, or_introl/ - | * /5 width=9 by nlift_bind_sn, ex5_4_intro, or_intror/ - ] - | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 /2 width=1 by yle_succ/ - [ /4 width=9 by nlift_bind_dx, or_introl/ - | * #J #K #W #j #Hlj elim (yle_inv_succ1 … Hlj) -Hlj #Hlj #Hj - yminus_succ #Hji #HLK #HnW - lapply (drop_inv_drop1_lt … HLK ?) /2 width=1 by ylt_O/ -HLK #HLK - (cpy_inv_sort1 … HT2) -HT2 // -qed-. - -(* Note: this can be derived from cpys_inv_atom1 *) -lemma cpys_inv_gref1: ∀G,L,T2,p,l,m. ⦃G, L⦄ ⊢ §p ▶*[l, m] T2 → T2 = §p. -#G #L #T2 #p #l #m #H @(cpys_ind … H) -T2 // -#T #T2 #_ #HT2 #IHT1 destruct ->(cpy_inv_gref1 … HT2) -HT2 // -qed-. - -lemma cpys_inv_bind1: ∀a,I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[l, m] U2 → - ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 & - ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯l, m] T2 & - U2 = ⓑ{a,I}V2.T2. -#a #I #G #L #V1 #T1 #U2 #l #m #H @(cpys_ind … H) -U2 -[ /2 width=5 by ex3_2_intro/ -| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct - elim (cpy_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H - lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V1) ?) -HT2 - /3 width=5 by cpys_strap1, lsuby_succ, ex3_2_intro/ -] -qed-. - -lemma cpys_inv_flat1: ∀I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[l, m] U2 → - ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 & ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 & - U2 = ⓕ{I}V2.T2. -#I #G #L #V1 #T1 #U2 #l #m #H @(cpys_ind … H) -U2 -[ /2 width=5 by ex3_2_intro/ -| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct - elim (cpy_inv_flat1 … HU2) -HU2 - /3 width=5 by cpys_strap1, ex3_2_intro/ -] -qed-. - -lemma cpys_inv_refl_O2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶*[l, 0] T2 → T1 = T2. -#G #L #T1 #T2 #l #H @(cpys_ind … H) -T2 // -#T #T2 #_ #HT2 #IHT1 <(cpy_inv_refl_O2 … HT2) -HT2 // -qed-. - -lemma cpys_inv_lift1_eq: ∀G,L,U1,U2,l,m. - ⦃G, L⦄ ⊢ U1 ▶*[l, yinj m] U2 → ∀T1. ⬆[l, m] T1 ≡ U1 → U1 = U2. -#G #L #U1 #U2 #l #m #H #T1 #HTU1 @(cpys_ind … H) -U2 -/2 width=7 by cpy_inv_lift1_eq/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys_alt.etc deleted file mode 100644 index d6b6c3096..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys_alt.etc +++ /dev/null @@ -1,102 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/notation/relations/psubststaralt_6.ma". -include "basic_2/multiple/cpys_lift.ma". - -(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************) - -(* alternative definition of cpys *) -inductive cpysa: ynat → ynat → relation4 genv lenv term term ≝ -| cpysa_atom : ∀I,G,L,l,m. cpysa l m G L (⓪{I}) (⓪{I}) -| cpysa_subst: ∀I,G,L,K,V1,V2,W2,i,l,m. l ≤ yinj i → i < l+m → - ⬇[i] L ≡ K.ⓑ{I}V1 → cpysa 0 (⫰(l+m-i)) G K V1 V2 → - ⬆[0, i+1] V2 ≡ W2 → cpysa l m G L (#i) W2 -| cpysa_bind : ∀a,I,G,L,V1,V2,T1,T2,l,m. - cpysa l m G L V1 V2 → cpysa (⫯l) m G (L.ⓑ{I}V1) T1 T2 → - cpysa l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) -| cpysa_flat : ∀I,G,L,V1,V2,T1,T2,l,m. - cpysa l m G L V1 V2 → cpysa l m G L T1 T2 → - cpysa l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) -. - -interpretation - "context-sensitive extended multiple substritution (term) alternative" - 'PSubstStarAlt G L T1 l m T2 = (cpysa l m G L T1 T2). - -(* Basic properties *********************************************************) - -lemma lsuby_cpysa_trans: ∀G,l,m. lsub_trans … (cpysa l m G) (lsuby l m). -#G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m -[ // -| #I #G #L1 #K1 #V1 #V2 #W2 #i #l #m #Hli #Hilm #HLK1 #_ #HVW2 #IHV12 #L2 #HL12 - elim (lsuby_drop_trans_be … HL12 … HLK1) -HL12 -HLK1 /3 width=7 by cpysa_subst/ -| /4 width=1 by lsuby_succ, cpysa_bind/ -| /3 width=1 by cpysa_flat/ -] -qed-. - -lemma cpysa_refl: ∀G,T,L,l,m. ⦃G, L⦄ ⊢ T ▶▶*[l, m] T. -#G #T elim T -T // -#I elim I -I /2 width=1 by cpysa_bind, cpysa_flat/ -qed. - -lemma cpysa_cpy_trans: ∀G,L,T1,T,l,m. ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T → - ∀T2. ⦃G, L⦄ ⊢ T ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2. -#G #L #T1 #T #l #m #H elim H -G -L -T1 -T -l -m -[ #I #G #L #l #m #X #H - elim (cpy_inv_atom1 … H) -H // * /2 width=7 by cpysa_subst/ -| #I #G #L #K #V1 #V2 #W2 #i #l #m #Hli #Hilm #HLK #_ #HVW2 #IHV12 #T2 #H - lapply (drop_fwd_drop2 … HLK) #H0LK - lapply (cpy_weak … H 0 (l+m) ? ?) -H // #H - elim (cpy_inv_lift1_be … H … H0LK … HVW2) -H -H0LK -HVW2 - /3 width=7 by cpysa_subst, ylt_fwd_le_succ1/ -| #a #I #G #L #V1 #V #T1 #T #l #m #_ #_ #IHV1 #IHT1 #X #H - elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct - /5 width=5 by cpysa_bind, lsuby_cpy_trans, lsuby_succ/ -| #I #G #L #V1 #V #T1 #T #l #m #_ #_ #IHV1 #IHT1 #X #H - elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1 by cpysa_flat/ -] -qed-. - -lemma cpys_cpysa: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2. -/3 width=8 by cpysa_cpy_trans, cpys_ind/ qed. - -(* Basic inversion lemmas ***************************************************) - -lemma cpysa_inv_cpys: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2. -#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m -/2 width=7 by cpys_subst, cpys_flat, cpys_bind, cpy_cpys/ -qed-. - -(* Advanced eliminators *****************************************************) - -lemma cpys_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term. - (∀I,G,L,l,m. R l m G L (⓪{I}) (⓪{I})) → - (∀I,G,L,K,V1,V2,W2,i,l,m. l ≤ yinj i → i < l + m → - ⬇[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(l+m-i)] V2 → - ⬆[O, i+1] V2 ≡ W2 → R O (⫰(l+m-i)) G K V1 V2 → R l m G L (#i) W2 - ) → - (∀a,I,G,L,V1,V2,T1,T2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 → - ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯l, m] T2 → R l m G L V1 V2 → - R (⫯l) m G (L.ⓑ{I}V1) T1 T2 → R l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) - ) → - (∀I,G,L,V1,V2,T1,T2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 → - ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R l m G L V1 V2 → - R l m G L T1 T2 → R l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) - ) → - ∀l,m,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R l m G L T1 T2. -#R #H1 #H2 #H3 #H4 #l #m #G #L #T1 #T2 #H elim (cpys_cpysa … H) -G -L -T1 -T2 -l -m -/3 width=8 by cpysa_inv_cpys/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys_cpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys_cpys.etc deleted file mode 100644 index 7df95948e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys_cpys.etc +++ /dev/null @@ -1,117 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/substitution/cpy_cpy.ma". -include "basic_2/multiple/cpys_alt.ma". - -(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************) - -(* Advanced inversion lemmas ************************************************) - -lemma cpys_inv_SO2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶*[l, 1] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2. -#G #L #T1 #T2 #l #H @(cpys_ind … H) -T2 /2 width=3 by cpy_trans_ge/ -qed-. - -(* Advanced properties ******************************************************) - -lemma cpys_strip_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶*[l1, m1] T1 → - ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶*[l1, m1] T. -normalize /3 width=3 by cpy_conf_eq, TC_strip1/ qed-. - -lemma cpys_strip_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶*[l1, m1] T1 → - ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶[l2, m2] T2 → - (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) → - ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶*[l1, m1] T. -normalize /3 width=3 by cpy_conf_neq, TC_strip1/ qed-. - -lemma cpys_strap1_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T0 → - ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → l2 + m2 ≤ l1 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T ▶*[l1, m1] T2. -normalize /3 width=3 by cpy_trans_down, TC_strap1/ qed. - -lemma cpys_strap2_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T0 → - ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → l2 + m2 ≤ l1 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T ▶[l1, m1] T2. -normalize /3 width=3 by cpy_trans_down, TC_strap2/ qed-. - -lemma cpys_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → - ∀i. l ≤ i → i ≤ l + m → - ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l, i - l] T & ⦃G, L⦄ ⊢ T ▶*[i, l + m - i] T2. -#G #L #T1 #T2 #l #m #H #i #Hli #Hilm @(cpys_ind … H) -T2 -[ /2 width=3 by ex2_intro/ -| #T #T2 #_ #HT12 * #T3 #HT13 #HT3 - elim (cpy_split_up … HT12 … Hilm) -HT12 -Hilm #T0 #HT0 #HT02 - elim (cpys_strap1_down … HT3 … HT0) -T /3 width=5 by cpys_strap1, ex2_intro/ - >ymax_pre_sn_comm // -] -qed-. - -lemma cpys_inv_lift1_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - l ≤ lt → lt ≤ l + m → l + m ≤ lt + mt → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[l, lt + mt - (l + m)] T2 & - ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt -elim (cpys_split_up … HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2 -lapply (cpys_weak … HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1 -lapply (cpys_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct -elim (cpys_inv_lift1_ge … HU2 … HLK … HTU1) -HU2 -HLK -HTU1 // ->yplus_minus_inj /2 width=3 by ex2_intro/ -qed-. - -(* Main properties **********************************************************) - -theorem cpys_conf_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶*[l1, m1] T1 → - ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶*[l1, m1] T. -normalize /3 width=3 by cpy_conf_eq, TC_confluent2/ qed-. - -theorem cpys_conf_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶*[l1, m1] T1 → - ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶*[l2, m2] T2 → - (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) → - ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶*[l1, m1] T. -normalize /3 width=3 by cpy_conf_neq, TC_confluent2/ qed-. - -theorem cpys_trans_eq: ∀G,L,T1,T,T2,l,m. - ⦃G, L⦄ ⊢ T1 ▶*[l, m] T → ⦃G, L⦄ ⊢ T ▶*[l, m] T2 → - ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2. -normalize /2 width=3 by trans_TC/ qed-. - -theorem cpys_trans_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T0 → - ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → l2 + m2 ≤ l1 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T ▶*[l1, m1] T2. -normalize /3 width=3 by cpy_trans_down, TC_transitive2/ qed-. - -theorem cpys_antisym_eq: ∀G,L1,T1,T2,l,m. ⦃G, L1⦄ ⊢ T1 ▶*[l, m] T2 → - ∀L2. ⦃G, L2⦄ ⊢ T2 ▶*[l, m] T1 → T1 = T2. -#G #L1 #T1 #T2 #l #m #H @(cpys_ind_alt … H) -G -L1 -T1 -T2 // -[ #I1 #G #L1 #K1 #V1 #V2 #W2 #i #l #m #Hli #Hilm #_ #_ #HVW2 #_ #L2 #HW2 - elim (lt_or_ge (|L2|) (i+1)) #Hi [ -Hli -Hilm | ] - [ lapply (cpys_weak_full … HW2) -HW2 #HW2 - lapply (cpys_weak … HW2 0 (i+1) ? ?) -HW2 // - [ >yplus_O1 >yplus_O1 /3 width=1 by ylt_fwd_le, ylt_inj/ ] -Hi - #HW2 >(cpys_inv_lift1_eq … HW2) -HW2 // - | elim (drop_O1_le (Ⓕ) … Hi) -Hi #K2 #HLK2 - elim (cpys_inv_lift1_ge_up … HW2 … HLK2 … HVW2 ? ? ?) -HW2 -HLK2 -HVW2 - /2 width=1 by ylt_fwd_le_succ1, yle_succ_dx/ -Hli -Hilm - #X #_ #H elim (lift_inv_lref2_be … H) -H /2 width=1 by ylt_inj/ - ] -| #a #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_bind1 … H) -H - #V #T #HV2 #HT2 #H destruct - lapply (IHV12 … HV2) #H destruct -IHV12 -HV2 /3 width=2 by eq_f2/ -| #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_flat1 … H) -H - #V #T #HV2 #HT2 #H destruct /3 width=2 by eq_f2/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys_lift.etc deleted file mode 100644 index 59cddfbc9..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/cpys/cpys_lift.etc +++ /dev/null @@ -1,226 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/substitution/cpy_lift.ma". -include "basic_2/multiple/cpys.ma". - -(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************) - -(* Advanced properties ******************************************************) - -lemma cpys_subst: ∀I,G,L,K,V,U1,i,l,m. - l ≤ yinj i → i < l + m → - ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ⫰(l+m-i)] U1 → - ∀U2. ⬆[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[l, m] U2. -#I #G #L #K #V #U1 #i #l #m #Hli #Hilm #HLK #H @(cpys_ind … H) -U1 -[ /3 width=5 by cpy_cpys, cpy_subst/ -| #U #U1 #_ #HU1 #IHU #U2 #HU12 - elim (lift_total U 0 (i+1)) #U0 #HU0 - lapply (IHU … HU0) -IHU #H - lapply (drop_fwd_drop2 … HLK) -HLK #HLK - lapply (cpy_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // #HU02 - lapply (cpy_weak … HU02 l m ? ?) -HU02 - [2,3: /2 width=3 by cpys_strap1, yle_succ_dx/ ] - >yplus_O1 ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ1/ -] -qed. - -lemma cpys_subst_Y2: ∀I,G,L,K,V,U1,i,l. - l ≤ yinj i → - ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ∞] U1 → - ∀U2. ⬆[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[l, ∞] U2. -#I #G #L #K #V #U1 #i #l #Hli #HLK #HVU1 #U2 #HU12 -@(cpys_subst … HLK … HU12) >yminus_Y_inj // -qed. - -(* Advanced inversion lemmas *************************************************) - -lemma cpys_inv_atom1: ∀I,G,L,T2,l,m. ⦃G, L⦄ ⊢ ⓪{I} ▶*[l, m] T2 → - T2 = ⓪{I} ∨ - ∃∃J,K,V1,V2,i. l ≤ yinj i & i < l + m & - ⬇[i] L ≡ K.ⓑ{J}V1 & - ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(l+m-i)] V2 & - ⬆[O, i+1] V2 ≡ T2 & - I = LRef i. -#I #G #L #T2 #l #m #H @(cpys_ind … H) -T2 -[ /2 width=1 by or_introl/ -| #T #T2 #_ #HT2 * - [ #H destruct - elim (cpy_inv_atom1 … HT2) -HT2 [ /2 width=1 by or_introl/ | * /3 width=11 by ex6_5_intro, or_intror/ ] - | * #J #K #V1 #V #i #Hli #Hilm #HLK #HV1 #HVT #HI - lapply (drop_fwd_drop2 … HLK) #H - elim (cpy_inv_lift1_ge_up … HT2 … H … HVT) -HT2 -H -HVT - [2,3,4: /2 width=1 by ylt_fwd_le_succ1, yle_succ_dx/ ] - /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/ - ] -] -qed-. - -lemma cpys_inv_lref1: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶*[l, m] T2 → - T2 = #i ∨ - ∃∃I,K,V1,V2. l ≤ i & i < l + m & - ⬇[i] L ≡ K.ⓑ{I}V1 & - ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(l+m-i)] V2 & - ⬆[O, i+1] V2 ≡ T2. -#G #L #T2 #i #l #m #H elim (cpys_inv_atom1 … H) -H /2 width=1 by or_introl/ -* #I #K #V1 #V2 #j #Hlj #Hjlm #HLK #HV12 #HVT2 #H destruct /3 width=7 by ex5_4_intro, or_intror/ -qed-. - -lemma cpys_inv_lref1_Y2: ∀G,L,T2,i,l. ⦃G, L⦄ ⊢ #i ▶*[l, ∞] T2 → - T2 = #i ∨ - ∃∃I,K,V1,V2. l ≤ i & ⬇[i] L ≡ K.ⓑ{I}V1 & - ⦃G, K⦄ ⊢ V1 ▶*[0, ∞] V2 & ⬆[O, i+1] V2 ≡ T2. -#G #L #T2 #i #l #H elim (cpys_inv_lref1 … H) -H /2 width=1 by or_introl/ -* >yminus_Y_inj /3 width=7 by or_intror, ex4_4_intro/ -qed-. - -lemma cpys_inv_lref1_drop: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶*[l, m] T2 → - ∀I,K,V1. ⬇[i] L ≡ K.ⓑ{I}V1 → - ∀V2. ⬆[O, i+1] V2 ≡ T2 → - ∧∧ ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(l+m-i)] V2 - & l ≤ i - & i < l + m. -#G #L #T2 #i #l #m #H #I #K #V1 #HLK #V2 #HVT2 elim (cpys_inv_lref1 … H) -H -[ #H destruct elim (lift_inv_lref2_be … HVT2) -HVT2 -HLK /2 width=1 by ylt_inj/ -| * #Z #Y #X1 #X2 #Hli #Hilm #HLY #HX12 #HXT2 - lapply (lift_inj … HXT2 … HVT2) -T2 #H destruct - lapply (drop_mono … HLY … HLK) -L #H destruct - /2 width=1 by and3_intro/ -] -qed-. - -(* Properties on relocation *************************************************) - -lemma cpys_lift_le: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 → - ∀L,U1,s,l,m. lt + mt ≤ l → ⬇[s, l, m] L ≡ K → - ⬆[l, m] T1 ≡ U1 → ∀U2. ⬆[l, m] T2 ≡ U2 → - ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2. -#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hlmtl #HLK #HTU1 @(cpys_ind … H) -T2 -[ #U2 #H >(lift_mono … HTU1 … H) -H // -| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 - elim (lift_total T l m) #U #HTU - lapply (IHT … HTU) -IHT #HU1 - lapply (cpy_lift_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/ -] -qed-. - -lemma cpys_lift_be: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 → - ∀L,U1,s,l,m. lt ≤ l → l ≤ lt + mt → - ⬇[s, l, m] L ≡ K → ⬆[l, m] T1 ≡ U1 → - ∀U2. ⬆[l, m] T2 ≡ U2 → ⦃G, L⦄ ⊢ U1 ▶*[lt, mt + m] U2. -#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hltl #Hllmt #HLK #HTU1 @(cpys_ind … H) -T2 -[ #U2 #H >(lift_mono … HTU1 … H) -H // -| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 - elim (lift_total T l m) #U #HTU - lapply (IHT … HTU) -IHT #HU1 - lapply (cpy_lift_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/ -] -qed-. - -lemma cpys_lift_ge: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 → - ∀L,U1,s,l,m. l ≤ lt → ⬇[s, l, m] L ≡ K → - ⬆[l, m] T1 ≡ U1 → ∀U2. ⬆[l, m] T2 ≡ U2 → - ⦃G, L⦄ ⊢ U1 ▶*[lt+m, mt] U2. -#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hllt #HLK #HTU1 @(cpys_ind … H) -T2 -[ #U2 #H >(lift_mono … HTU1 … H) -H // -| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 - elim (lift_total T l m) #U #HTU - lapply (IHT … HTU) -IHT #HU1 - lapply (cpy_lift_ge … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/ -] -qed-. - -(* Inversion lemmas for relocation ******************************************) - -lemma cpys_inv_lift1_le: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - lt + mt ≤ l → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 & ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmtl @(cpys_ind … H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (cpy_inv_lift1_le … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -lemma cpys_inv_lift1_be: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - lt ≤ l → l + m ≤ lt + mt → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt - m] T2 & ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmlmt @(cpys_ind … H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (cpy_inv_lift1_be … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -lemma cpys_inv_lift1_ge: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - l + m ≤ lt → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt - m, mt] T2 & ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmlt @(cpys_ind … H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (cpy_inv_lift1_ge … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -(* Advanced inversion lemmas on relocation **********************************) - -lemma cpys_inv_lift1_ge_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - l ≤ lt → lt ≤ l + m → l + m ≤ lt + mt → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[l, lt + mt - (l + m)] T2 & - ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt @(cpys_ind … H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (cpy_inv_lift1_ge_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -lemma cpys_inv_lift1_be_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - lt ≤ l → lt + mt ≤ l + m → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm @(cpys_ind … H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (cpy_inv_lift1_be_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -lemma cpys_inv_lift1_le_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 → - ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 → - lt ≤ l → l ≤ lt + mt → lt + mt ≤ l + m → - ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2. -#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm @(cpys_ind … H) -U2 -[ /2 width=3 by ex2_intro/ -| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU - elim (cpy_inv_lift1_le_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/ -] -qed-. - -lemma cpys_inv_lift1_subst: ∀G,L,W1,W2,l,m. ⦃G, L⦄ ⊢ W1 ▶*[l, m] W2 → - ∀K,V1,i. ⬇[i+1] L ≡ K → ⬆[O, i+1] V1 ≡ W1 → - l ≤ yinj i → i < l + m → - ∃∃V2. ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(l+m-i)] V2 & ⬆[O, i+1] V2 ≡ W2. -#G #L #W1 #W2 #l #m #HW12 #K #V1 #i #HLK #HVW1 #Hli #Hilm -elim (cpys_inv_lift1_ge_up … HW12 … HLK … HVW1 ? ? ?) // ->yplus_O1 yplus_SO2 -[ >yminus_succ2 /2 width=3 by ex2_intro/ -| /2 width=1 by ylt_fwd_le_succ1/ -| /2 width=3 by yle_trans/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/crr/crr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/crr/crr.etc deleted file mode 100644 index 50107c4ab..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/crr/crr.etc +++ /dev/null @@ -1,145 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/notation/relations/predreducible_3.ma". -include "basic_2/grammar/genv.ma". -include "basic_2/substitution/drop.ma". - -(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION **************************) - -(* reducible binary items *) -definition ri2: predicate item2 ≝ - λI. I = Bind2 true Abbr ∨ I = Flat2 Cast. - -(* irreducible binary binders *) -definition ib2: relation2 bool bind2 ≝ - λa,I. I = Abst ∨ Bind2 a I = Bind2 false Abbr. - -(* activate genv *) -(* reducible terms *) -inductive crr (G:genv): relation2 lenv term ≝ -| crr_delta : ∀L,K,V,i. ⬇[i] L ≡ K.ⓓV → crr G L (#i) -| crr_appl_sn: ∀L,V,T. crr G L V → crr G L (ⓐV.T) -| crr_appl_dx: ∀L,V,T. crr G L T → crr G L (ⓐV.T) -| crr_ri2 : ∀I,L,V,T. ri2 I → crr G L (②{I}V.T) -| crr_ib2_sn : ∀a,I,L,V,T. ib2 a I → crr G L V → crr G L (ⓑ{a,I}V.T) -| crr_ib2_dx : ∀a,I,L,V,T. ib2 a I → crr G (L.ⓑ{I}V) T → crr G L (ⓑ{a,I}V.T) -| crr_beta : ∀a,L,V,W,T. crr G L (ⓐV.ⓛ{a}W.T) -| crr_theta : ∀a,L,V,W,T. crr G L (ⓐV.ⓓ{a}W.T) -. - -interpretation - "reducibility for context-sensitive reduction (term)" - 'PRedReducible G L T = (crr G L T). - -(* Basic inversion lemmas ***************************************************) - -fact crr_inv_sort_aux: ∀G,L,T,s. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → T = ⋆s → ⊥. -#G #L #T #s0 * -L -T -[ #L #K #V #i #HLK #H destruct -| #L #V #T #_ #H destruct -| #L #V #T #_ #H destruct -| #I #L #V #T #_ #H destruct -| #a #I #L #V #T #_ #_ #H destruct -| #a #I #L #V #T #_ #_ #H destruct -| #a #L #V #W #T #H destruct -| #a #L #V #W #T #H destruct -] -qed-. - -lemma crr_inv_sort: ∀G,L,s. ⦃G, L⦄ ⊢ ➡ 𝐑⦃⋆s⦄ → ⊥. -/2 width=6 by crr_inv_sort_aux/ qed-. - -fact crr_inv_lref_aux: ∀G,L,T,i. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → T = #i → - ∃∃K,V. ⬇[i] L ≡ K.ⓓV. -#G #L #T #j * -L -T -[ #L #K #V #i #HLK #H destruct /2 width=3 by ex1_2_intro/ -| #L #V #T #_ #H destruct -| #L #V #T #_ #H destruct -| #I #L #V #T #_ #H destruct -| #a #I #L #V #T #_ #_ #H destruct -| #a #I #L #V #T #_ #_ #H destruct -| #a #L #V #W #T #H destruct -| #a #L #V #W #T #H destruct -] -qed-. - -lemma crr_inv_lref: ∀G,L,i. ⦃G, L⦄ ⊢ ➡ 𝐑⦃#i⦄ → ∃∃K,V. ⬇[i] L ≡ K.ⓓV. -/2 width=4 by crr_inv_lref_aux/ qed-. - -fact crr_inv_gref_aux: ∀G,L,T,p. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → T = §p → ⊥. -#G #L #T #q * -L -T -[ #L #K #V #i #HLK #H destruct -| #L #V #T #_ #H destruct -| #L #V #T #_ #H destruct -| #I #L #V #T #_ #H destruct -| #a #I #L #V #T #_ #_ #H destruct -| #a #I #L #V #T #_ #_ #H destruct -| #a #L #V #W #T #H destruct -| #a #L #V #W #T #H destruct -] -qed-. - -lemma crr_inv_gref: ∀G,L,p. ⦃G, L⦄ ⊢ ➡ 𝐑⦃§p⦄ → ⊥. -/2 width=6 by crr_inv_gref_aux/ qed-. - -lemma trr_inv_atom: ∀G,I. ⦃G, ⋆⦄ ⊢ ➡ 𝐑⦃⓪{I}⦄ → ⊥. -#G * #i #H -[ elim (crr_inv_sort … H) -| elim (crr_inv_lref … H) -H #L #V #H - elim (drop_inv_atom1 … H) -H #H destruct -| elim (crr_inv_gref … H) -] -qed-. - -fact crr_inv_ib2_aux: ∀a,I,G,L,W,U,T. ib2 a I → ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → T = ⓑ{a,I}W.U → - ⦃G, L⦄ ⊢ ➡ 𝐑⦃W⦄ ∨ ⦃G, L.ⓑ{I}W⦄ ⊢ ➡ 𝐑⦃U⦄. -#G #b #J #L #W0 #U #T #HI * -L -T -[ #L #K #V #i #_ #H destruct -| #L #V #T #_ #H destruct -| #L #V #T #_ #H destruct -| #I #L #V #T #H1 #H2 destruct - elim H1 -H1 #H destruct - elim HI -HI #H destruct -| #a #I #L #V #T #_ #HV #H destruct /2 width=1 by or_introl/ -| #a #I #L #V #T #_ #HT #H destruct /2 width=1 by or_intror/ -| #a #L #V #W #T #H destruct -| #a #L #V #W #T #H destruct -] -qed-. - -lemma crr_inv_ib2: ∀a,I,G,L,W,T. ib2 a I → ⦃G, L⦄ ⊢ ➡ 𝐑⦃ⓑ{a,I}W.T⦄ → - ⦃G, L⦄ ⊢ ➡ 𝐑⦃W⦄ ∨ ⦃G, L.ⓑ{I}W⦄ ⊢ ➡ 𝐑⦃T⦄. -/2 width=5 by crr_inv_ib2_aux/ qed-. - -fact crr_inv_appl_aux: ∀G,L,W,U,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → T = ⓐW.U → - ∨∨ ⦃G, L⦄ ⊢ ➡ 𝐑⦃W⦄ | ⦃G, L⦄ ⊢ ➡ 𝐑⦃U⦄ | (𝐒⦃U⦄ → ⊥). -#G #L #W0 #U #T * -L -T -[ #L #K #V #i #_ #H destruct -| #L #V #T #HV #H destruct /2 width=1 by or3_intro0/ -| #L #V #T #HT #H destruct /2 width=1 by or3_intro1/ -| #I #L #V #T #H1 #H2 destruct - elim H1 -H1 #H destruct -| #a #I #L #V #T #_ #_ #H destruct -| #a #I #L #V #T #_ #_ #H destruct -| #a #L #V #W #T #H destruct - @or3_intro2 #H elim (simple_inv_bind … H) -| #a #L #V #W #T #H destruct - @or3_intro2 #H elim (simple_inv_bind … H) -] -qed-. - -lemma crr_inv_appl: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃ⓐV.T⦄ → - ∨∨ ⦃G, L⦄ ⊢ ➡ 𝐑⦃V⦄ | ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ | (𝐒⦃T⦄ → ⊥). -/2 width=3 by crr_inv_appl_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/crr/crr_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/crr/crr_lift.etc deleted file mode 100644 index 9aab5dffd..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/crr/crr_lift.etc +++ /dev/null @@ -1,74 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/substitution/drop_drop.ma". -include "basic_2/reduction/crr.ma". - -(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION **************************) - -(* Properties on relocation *************************************************) - -lemma crr_lift: ∀G,K,T. ⦃G, K⦄ ⊢ ➡ 𝐑⦃T⦄ → ∀L,c,l,k. ⬇[c, l, k] L ≡ K → - ∀U. ⬆[l, k] T ≡ U → ⦃G, L⦄ ⊢ ➡ 𝐑⦃U⦄. -#G #K #T #H elim H -K -T -[ #K #K0 #V #i #HK0 #L #c #l #k #HLK #X #H - elim (lift_inv_lref1 … H) -H * #Hil #H destruct - [ elim (drop_trans_lt … HLK … HK0) -K /2 width=4 by crr_delta/ - | lapply (drop_trans_ge … HLK … HK0 ?) -K /3 width=4 by crr_delta, drop_inv_gen/ - ] -| #K #V #T #_ #IHV #L #c #l #k #HLK #X #H - elim (lift_inv_flat1 … H) -H #W #U #HVW #_ #H destruct /3 width=5 by crr_appl_sn/ -| #K #V #T #_ #IHT #L #c #l #k #HLK #X #H - elim (lift_inv_flat1 … H) -H #W #U #_ #HTU #H destruct /3 width=5 by crr_appl_dx/ -| #I #K #V #T #HI #L #c #l #k #_ #X #H - elim (lift_fwd_pair1 … H) -H #W #U #_ #H destruct /2 width=1 by crr_ri2/ -| #a #I #K #V #T #HI #_ #IHV #L #c #l #k #HLK #X #H - elim (lift_inv_bind1 … H) -H #W #U #HVW #_ #H destruct /3 width=5 by crr_ib2_sn/ -| #a #I #K #V #T #HI #_ #IHT #L #c #l #k #HLK #X #H - elim (lift_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct /4 width=5 by crr_ib2_dx, drop_skip/ -| #a #K #V #V0 #T #L #c #l #k #_ #X #H - elim (lift_inv_flat1 … H) -H #W #X0 #_ #H0 #H destruct - elim (lift_inv_bind1 … H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crr_beta/ -| #a #K #V #V0 #T #L #c #l #k #_ #X #H - elim (lift_inv_flat1 … H) -H #W #X0 #_ #H0 #H destruct - elim (lift_inv_bind1 … H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crr_theta/ -] -qed. - -lemma crr_inv_lift: ∀G,L,U. ⦃G, L⦄ ⊢ ➡ 𝐑⦃U⦄ → ∀K,c,l,k. ⬇[c, l, k] L ≡ K → - ∀T. ⬆[l, k] T ≡ U → ⦃G, K⦄ ⊢ ➡ 𝐑⦃T⦄. -#G #L #U #H elim H -L -U -[ #L #L0 #W #i #HK0 #K #c #l #k #HLK #X #H - elim (lift_inv_lref2 … H) -H * #Hil #H destruct - [ elim (drop_conf_lt … HLK … HK0) -L /2 width=4 by crr_delta/ - | lapply (drop_conf_ge … HLK … HK0 ?) -L /2 width=4 by crr_delta/ - ] -| #L #W #U #_ #IHW #K #c #l #k #HLK #X #H - elim (lift_inv_flat2 … H) -H #V #T #HVW #_ #H destruct /3 width=5 by crr_appl_sn/ -| #L #W #U #_ #IHU #K #c #l #k #HLK #X #H - elim (lift_inv_flat2 … H) -H #V #T #_ #HTU #H destruct /3 width=5 by crr_appl_dx/ -| #I #L #W #U #HI #K #c #l #k #_ #X #H - elim (lift_fwd_pair2 … H) -H #V #T #_ #H destruct /2 width=1 by crr_ri2/ -| #a #I #L #W #U #HI #_ #IHW #K #c #l #k #HLK #X #H - elim (lift_inv_bind2 … H) -H #V #T #HVW #_ #H destruct /3 width=5 by crr_ib2_sn/ -| #a #I #L #W #U #HI #_ #IHU #K #c #l #k #HLK #X #H - elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct /4 width=5 by crr_ib2_dx, drop_skip/ -| #a #L #W #W0 #U #K #c #l #k #_ #X #H - elim (lift_inv_flat2 … H) -H #V #X0 #_ #H0 #H destruct - elim (lift_inv_bind2 … H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crr_beta/ -| #a #L #W #W0 #U #K #c #l #k #_ #X #H - elim (lift_inv_flat2 … H) -H #V #X0 #_ #H0 #H destruct - elim (lift_inv_bind2 … H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crr_theta/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/crr/predreducible_3.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/crr/predreducible_3.etc deleted file mode 100644 index 5d84a363d..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/crr/predreducible_3.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ ➡ 𝐑 break ⦃ term 46 T ⦄ )" - non associative with precedence 45 - for @{ 'PRedReducible $G $L $T }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/crx/crx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/crx/crx.etc deleted file mode 100644 index c6c962646..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/crx/crx.etc +++ /dev/null @@ -1,152 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/notation/relations/predreducible_5.ma". -include "basic_2/static/sd.ma". -include "basic_2/reduction/crr.ma". - -(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION *****************) - -(* activate genv *) -(* extended reducible terms *) -inductive crx (h) (o) (G:genv): relation2 lenv term ≝ -| crx_sort : ∀L,s,d. deg h o s (d+1) → crx h o G L (⋆s) -| crx_delta : ∀I,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → crx h o G L (#i) -| crx_appl_sn: ∀L,V,T. crx h o G L V → crx h o G L (ⓐV.T) -| crx_appl_dx: ∀L,V,T. crx h o G L T → crx h o G L (ⓐV.T) -| crx_ri2 : ∀I,L,V,T. ri2 I → crx h o G L (②{I}V.T) -| crx_ib2_sn : ∀a,I,L,V,T. ib2 a I → crx h o G L V → crx h o G L (ⓑ{a,I}V.T) -| crx_ib2_dx : ∀a,I,L,V,T. ib2 a I → crx h o G (L.ⓑ{I}V) T → crx h o G L (ⓑ{a,I}V.T) -| crx_beta : ∀a,L,V,W,T. crx h o G L (ⓐV. ⓛ{a}W.T) -| crx_theta : ∀a,L,V,W,T. crx h o G L (ⓐV. ⓓ{a}W.T) -. - -interpretation - "reducibility for context-sensitive extended reduction (term)" - 'PRedReducible h o G L T = (crx h o G L T). - -(* Basic properties *********************************************************) - -lemma crr_crx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄. -#h #o #G #L #T #H elim H -L -T -/2 width=4 by crx_delta, crx_appl_sn, crx_appl_dx, crx_ri2, crx_ib2_sn, crx_ib2_dx, crx_beta, crx_theta/ -qed. - -(* Basic inversion lemmas ***************************************************) - -fact crx_inv_sort_aux: ∀h,o,G,L,T,s. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → T = ⋆s → - ∃d. deg h o s (d+1). -#h #o #G #L #T #s0 * -L -T -[ #L #s #d #Hkd #H destruct /2 width=2 by ex_intro/ -| #I #L #K #V #i #HLK #H destruct -| #L #V #T #_ #H destruct -| #L #V #T #_ #H destruct -| #I #L #V #T #_ #H destruct -| #a #I #L #V #T #_ #_ #H destruct -| #a #I #L #V #T #_ #_ #H destruct -| #a #L #V #W #T #H destruct -| #a #L #V #W #T #H destruct -] -qed-. - -lemma crx_inv_sort: ∀h,o,G,L,s. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃⋆s⦄ → ∃d. deg h o s (d+1). -/2 width=5 by crx_inv_sort_aux/ qed-. - -fact crx_inv_lref_aux: ∀h,o,G,L,T,i. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → T = #i → - ∃∃I,K,V. ⬇[i] L ≡ K.ⓑ{I}V. -#h #o #G #L #T #j * -L -T -[ #L #s #d #_ #H destruct -| #I #L #K #V #i #HLK #H destruct /2 width=4 by ex1_3_intro/ -| #L #V #T #_ #H destruct -| #L #V #T #_ #H destruct -| #I #L #V #T #_ #H destruct -| #a #I #L #V #T #_ #_ #H destruct -| #a #I #L #V #T #_ #_ #H destruct -| #a #L #V #W #T #H destruct -| #a #L #V #W #T #H destruct -] -qed-. - -lemma crx_inv_lref: ∀h,o,G,L,i. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃#i⦄ → ∃∃I,K,V. ⬇[i] L ≡ K.ⓑ{I}V. -/2 width=6 by crx_inv_lref_aux/ qed-. - -fact crx_inv_gref_aux: ∀h,o,G,L,T,p. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → T = §p → ⊥. -#h #o #G #L #T #q * -L -T -[ #L #s #d #_ #H destruct -| #I #L #K #V #i #HLK #H destruct -| #L #V #T #_ #H destruct -| #L #V #T #_ #H destruct -| #I #L #V #T #_ #H destruct -| #a #I #L #V #T #_ #_ #H destruct -| #a #I #L #V #T #_ #_ #H destruct -| #a #L #V #W #T #H destruct -| #a #L #V #W #T #H destruct -] -qed-. - -lemma crx_inv_gref: ∀h,o,G,L,p. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃§p⦄ → ⊥. -/2 width=8 by crx_inv_gref_aux/ qed-. - -lemma trx_inv_atom: ∀h,o,I,G. ⦃G, ⋆⦄ ⊢ ➡[h, o] 𝐑⦃⓪{I}⦄ → - ∃∃s,d. deg h o s (d+1) & I = Sort s. -#h #o * #i #G #H -[ elim (crx_inv_sort … H) -H /2 width=4 by ex2_2_intro/ -| elim (crx_inv_lref … H) -H #I #L #V #H - elim (drop_inv_atom1 … H) -H #H destruct -| elim (crx_inv_gref … H) -] -qed-. - -fact crx_inv_ib2_aux: ∀h,o,a,I,G,L,W,U,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → - T = ⓑ{a,I}W.U → ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃W⦄ ∨ ⦃G, L.ⓑ{I}W⦄ ⊢ ➡[h, o] 𝐑⦃U⦄. -#h #o #b #J #G #L #W0 #U #T #HI * -L -T -[ #L #s #d #_ #H destruct -| #I #L #K #V #i #_ #H destruct -| #L #V #T #_ #H destruct -| #L #V #T #_ #H destruct -| #I #L #V #T #H1 #H2 destruct - elim H1 -H1 #H destruct - elim HI -HI #H destruct -| #a #I #L #V #T #_ #HV #H destruct /2 width=1 by or_introl/ -| #a #I #L #V #T #_ #HT #H destruct /2 width=1 by or_intror/ -| #a #L #V #W #T #H destruct -| #a #L #V #W #T #H destruct -] -qed-. - -lemma crx_inv_ib2: ∀h,o,a,I,G,L,W,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃ⓑ{a,I}W.T⦄ → - ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃W⦄ ∨ ⦃G, L.ⓑ{I}W⦄ ⊢ ➡[h, o] 𝐑⦃T⦄. -/2 width=5 by crx_inv_ib2_aux/ qed-. - -fact crx_inv_appl_aux: ∀h,o,G,L,W,U,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → T = ⓐW.U → - ∨∨ ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃W⦄ | ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃U⦄ | (𝐒⦃U⦄ → ⊥). -#h #o #G #L #W0 #U #T * -L -T -[ #L #s #d #_ #H destruct -| #I #L #K #V #i #_ #H destruct -| #L #V #T #HV #H destruct /2 width=1 by or3_intro0/ -| #L #V #T #HT #H destruct /2 width=1 by or3_intro1/ -| #I #L #V #T #H1 #H2 destruct - elim H1 -H1 #H destruct -| #a #I #L #V #T #_ #_ #H destruct -| #a #I #L #V #T #_ #_ #H destruct -| #a #L #V #W #T #H destruct - @or3_intro2 #H elim (simple_inv_bind … H) -| #a #L #V #W #T #H destruct - @or3_intro2 #H elim (simple_inv_bind … H) -] -qed-. - -lemma crx_inv_appl: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃ⓐV.T⦄ → - ∨∨ ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃V⦄ | ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ | (𝐒⦃T⦄ → ⊥). -/2 width=3 by crx_inv_appl_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/crx/crx_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/crx/crx_lift.etc deleted file mode 100644 index cbdc2718f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/crx/crx_lift.etc +++ /dev/null @@ -1,78 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/substitution/drop_drop.ma". -include "basic_2/reduction/crx.ma". - -(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION *****************) - -(* Properties on relocation *************************************************) - -lemma crx_lift: ∀h,o,G,K,T. ⦃G, K⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → ∀L,c,l,k. ⬇[c, l, k] L ≡ K → - ∀U. ⬆[l, k] T ≡ U → ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃U⦄. -#h #o #G #K #T #H elim H -K -T -[ #K #s #d #Hkd #L #c #l #k #_ #X #H - >(lift_inv_sort1 … H) -X /2 width=2 by crx_sort/ -| #I #K #K0 #V #i #HK0 #L #c #l #k #HLK #X #H - elim (lift_inv_lref1 … H) -H * #Hil #H destruct - [ elim (drop_trans_lt … HLK … HK0) -K /2 width=4 by crx_delta/ - | lapply (drop_trans_ge … HLK … HK0 ?) -K /3 width=5 by crx_delta, drop_inv_gen/ - ] -| #K #V #T #_ #IHV #L #c #l #k #HLK #X #H - elim (lift_inv_flat1 … H) -H #W #U #HVW #_ #H destruct /3 width=5 by crx_appl_sn/ -| #K #V #T #_ #IHT #L #c #l #k #HLK #X #H - elim (lift_inv_flat1 … H) -H #W #U #_ #HTU #H destruct /3 width=5 by crx_appl_dx/ -| #I #K #V #T #HI #L #c #l #k #_ #X #H - elim (lift_fwd_pair1 … H) -H #W #U #_ #H destruct /2 width=1 by crx_ri2/ -| #a #I #K #V #T #HI #_ #IHV #L #c #l #k #HLK #X #H - elim (lift_inv_bind1 … H) -H #W #U #HVW #_ #H destruct /3 width=5 by crx_ib2_sn/ -| #a #I #K #V #T #HI #_ #IHT #L #c #l #k #HLK #X #H - elim (lift_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct /4 width=5 by crx_ib2_dx, drop_skip/ -| #a #K #V #V0 #T #L #c #l #k #_ #X #H - elim (lift_inv_flat1 … H) -H #W #X0 #_ #H0 #H destruct - elim (lift_inv_bind1 … H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crx_beta/ -| #a #K #V #V0 #T #L #c #l #k #_ #X #H - elim (lift_inv_flat1 … H) -H #W #X0 #_ #H0 #H destruct - elim (lift_inv_bind1 … H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crx_theta/ -] -qed. - -lemma crx_inv_lift: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃U⦄ → ∀K,c,l,k. ⬇[c, l, k] L ≡ K → - ∀T. ⬆[l, k] T ≡ U → ⦃G, K⦄ ⊢ ➡[h, o] 𝐑⦃T⦄. -#h #o #G #L #U #H elim H -L -U -[ #L #s #d #Hkd #K #c #l #k #_ #X #H - >(lift_inv_sort2 … H) -X /2 width=2 by crx_sort/ -| #I #L #L0 #W #i #HK0 #K #c #l #k #HLK #X #H - elim (lift_inv_lref2 … H) -H * #Hil #H destruct - [ elim (drop_conf_lt … HLK … HK0) -L /2 width=4 by crx_delta/ - | lapply (drop_conf_ge … HLK … HK0 ?) -L /2 width=4 by crx_delta/ - ] -| #L #W #U #_ #IHW #K #c #l #k #HLK #X #H - elim (lift_inv_flat2 … H) -H #V #T #HVW #_ #H destruct /3 width=5 by crx_appl_sn/ -| #L #W #U #_ #IHU #K #c #l #k #HLK #X #H - elim (lift_inv_flat2 … H) -H #V #T #_ #HTU #H destruct /3 width=5 by crx_appl_dx/ -| #I #L #W #U #HI #K #c #l #k #_ #X #H - elim (lift_fwd_pair2 … H) -H #V #T #_ #H destruct /2 width=1 by crx_ri2/ -| #a #I #L #W #U #HI #_ #IHW #K #c #l #k #HLK #X #H - elim (lift_inv_bind2 … H) -H #V #T #HVW #_ #H destruct /3 width=5 by crx_ib2_sn/ -| #a #I #L #W #U #HI #_ #IHU #K #c #l #k #HLK #X #H - elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct /4 width=5 by crx_ib2_dx, drop_skip/ -| #a #L #W #W0 #U #K #c #l #k #_ #X #H - elim (lift_inv_flat2 … H) -H #V #X0 #_ #H0 #H destruct - elim (lift_inv_bind2 … H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crx_beta/ -| #a #L #W #W0 #U #K #c #l #k #_ #X #H - elim (lift_inv_flat2 … H) -H #V #X0 #_ #H0 #H destruct - elim (lift_inv_bind2 … H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crx_theta/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/crx/predreducible_5.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/crx/predreducible_5.etc deleted file mode 100644 index ca0d12b72..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/crx/predreducible_5.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ➡ break [ term 46 o , break term 46 h ] 𝐑 break ⦃ term 46 T ⦄ )" - non associative with precedence 45 - for @{ 'PRedReducible $h $o $G $L $T }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/da/da.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/da/da.etc deleted file mode 100644 index b8d3d4315..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/da/da.etc +++ /dev/null @@ -1,108 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/notation/relations/degree_6.ma". -include "basic_2/grammar/genv.ma". -include "basic_2/substitution/drop.ma". -include "basic_2/static/sd.ma". - -(* DEGREE ASSIGNMENT FOR TERMS **********************************************) - -(* activate genv *) -inductive da (h:sh) (o:sd h): relation4 genv lenv term nat ≝ -| da_sort: ∀G,L,s,d. deg h o s d → da h o G L (⋆s) d -| da_ldef: ∀G,L,K,V,i,d. ⬇[i] L ≡ K.ⓓV → da h o G K V d → da h o G L (#i) d -| da_ldec: ∀G,L,K,W,i,d. ⬇[i] L ≡ K.ⓛW → da h o G K W d → da h o G L (#i) (d+1) -| da_bind: ∀a,I,G,L,V,T,d. da h o G (L.ⓑ{I}V) T d → da h o G L (ⓑ{a,I}V.T) d -| da_flat: ∀I,G,L,V,T,d. da h o G L T d → da h o G L (ⓕ{I}V.T) d -. - -interpretation "degree assignment (term)" - 'Degree h o G L T d = (da h o G L T d). - -(* Basic inversion lemmas ***************************************************) - -fact da_inv_sort_aux: ∀h,o,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, o] d → - ∀s0. T = ⋆s0 → deg h o s0 d. -#h #o #G #L #T #d * -G -L -T -d -[ #G #L #s #d #Hkd #s0 #H destruct // -| #G #L #K #V #i #d #_ #_ #s0 #H destruct -| #G #L #K #W #i #d #_ #_ #s0 #H destruct -| #a #I #G #L #V #T #d #_ #s0 #H destruct -| #I #G #L #V #T #d #_ #s0 #H destruct -] -qed-. - -lemma da_inv_sort: ∀h,o,G,L,s,d. ⦃G, L⦄ ⊢ ⋆s ▪[h, o] d → deg h o s d. -/2 width=5 by da_inv_sort_aux/ qed-. - -fact da_inv_lref_aux: ∀h,o,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, o] d → ∀j. T = #j → - (∃∃K,V. ⬇[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ▪[h, o] d) ∨ - (∃∃K,W,d0. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, o] d0 & - d = d0 + 1 - ). -#h #o #G #L #T #d * -G -L -T -d -[ #G #L #s #d #_ #j #H destruct -| #G #L #K #V #i #d #HLK #HV #j #H destruct /3 width=4 by ex2_2_intro, or_introl/ -| #G #L #K #W #i #d #HLK #HW #j #H destruct /3 width=6 by ex3_3_intro, or_intror/ -| #a #I #G #L #V #T #d #_ #j #H destruct -| #I #G #L #V #T #d #_ #j #H destruct -] -qed-. - -lemma da_inv_lref: ∀h,o,G,L,j,d. ⦃G, L⦄ ⊢ #j ▪[h, o] d → - (∃∃K,V. ⬇[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ▪[h, o] d) ∨ - (∃∃K,W,d0. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, o] d0 & d = d0+1). -/2 width=3 by da_inv_lref_aux/ qed-. - -fact da_inv_gref_aux: ∀h,o,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, o] d → ∀p0. T = §p0 → ⊥. -#h #o #G #L #T #d * -G -L -T -d -[ #G #L #s #d #_ #p0 #H destruct -| #G #L #K #V #i #d #_ #_ #p0 #H destruct -| #G #L #K #W #i #d #_ #_ #p0 #H destruct -| #a #I #G #L #V #T #d #_ #p0 #H destruct -| #I #G #L #V #T #d #_ #p0 #H destruct -] -qed-. - -lemma da_inv_gref: ∀h,o,G,L,p,d. ⦃G, L⦄ ⊢ §p ▪[h, o] d → ⊥. -/2 width=9 by da_inv_gref_aux/ qed-. - -fact da_inv_bind_aux: ∀h,o,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, o] d → - ∀b,J,X,Y. T = ⓑ{b,J}Y.X → ⦃G, L.ⓑ{J}Y⦄ ⊢ X ▪[h, o] d. -#h #o #G #L #T #d * -G -L -T -d -[ #G #L #s #d #_ #b #J #X #Y #H destruct -| #G #L #K #V #i #d #_ #_ #b #J #X #Y #H destruct -| #G #L #K #W #i #d #_ #_ #b #J #X #Y #H destruct -| #a #I #G #L #V #T #d #HT #b #J #X #Y #H destruct // -| #I #G #L #V #T #d #_ #b #J #X #Y #H destruct -] -qed-. - -lemma da_inv_bind: ∀h,o,b,J,G,L,Y,X,d. ⦃G, L⦄ ⊢ ⓑ{b,J}Y.X ▪[h, o] d → ⦃G, L.ⓑ{J}Y⦄ ⊢ X ▪[h, o] d. -/2 width=4 by da_inv_bind_aux/ qed-. - -fact da_inv_flat_aux: ∀h,o,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, o] d → - ∀J,X,Y. T = ⓕ{J}Y.X → ⦃G, L⦄ ⊢ X ▪[h, o] d. -#h #o #G #L #T #d * -G -L -T -d -[ #G #L #s #d #_ #J #X #Y #H destruct -| #G #L #K #V #i #d #_ #_ #J #X #Y #H destruct -| #G #L #K #W #i #d #_ #_ #J #X #Y #H destruct -| #a #I #G #L #V #T #d #_ #J #X #Y #H destruct -| #I #G #L #V #T #d #HT #J #X #Y #H destruct // -] -qed-. - -lemma da_inv_flat: ∀h,o,J,G,L,Y,X,d. ⦃G, L⦄ ⊢ ⓕ{J}Y.X ▪[h, o] d → ⦃G, L⦄ ⊢ X ▪[h, o] d. -/2 width=5 by da_inv_flat_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/da/da_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/da/da_aaa.etc deleted file mode 100644 index 43d480c60..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/da/da_aaa.etc +++ /dev/null @@ -1,31 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/static/aaa_lift.ma". -include "basic_2/static/da.ma". - -(* DEGREE ASSIGNMENT FOR TERMS **********************************************) - -(* Properties on atomic arity assignment for terms **************************) - -lemma aaa_da: ∀h,o,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∃d. ⦃G, L⦄ ⊢ T ▪[h, o] d. -#h #o #G #L #T #A #H elim H -G -L -T -A -[ #G #L #s elim (deg_total h o s) /3 width=2 by da_sort, ex_intro/ -| * #G #L #K #V #B #i #HLK #_ * /3 width=5 by da_ldef, da_ldec, ex_intro/ -| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by da_bind, ex_intro/ -| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by da_bind, ex_intro/ -| #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by da_flat, ex_intro/ -| #G #L #W #T #A #_ #_ #_ * /3 width=2 by da_flat, ex_intro/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/da/da_da.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/da/da_da.etc deleted file mode 100644 index 2bc863015..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/da/da_da.etc +++ /dev/null @@ -1,38 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/static/da_lift.ma". - -(* DEGREE ASSIGNMENT FOR TERMS **********************************************) - -(* Main properties **********************************************************) - -theorem da_mono: ∀h,o,G,L,T,d1. ⦃G, L⦄ ⊢ T ▪[h, o] d1 → - ∀d2. ⦃G, L⦄ ⊢ T ▪[h, o] d2 → d1 = d2. -#h #o #G #L #T #d1 #H elim H -G -L -T -d1 -[ #G #L #s #d1 #Hkd1 #d2 #H - lapply (da_inv_sort … H) -G -L #Hkd2 - >(deg_mono … Hkd2 … Hkd1) -h -s -d2 // -| #G #L #K #V #i #d1 #HLK #_ #IHV #d2 #H - elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0 #HV0 [| #Hd0 ] - lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct /2 width=1 by/ -| #G #L #K #W #i #d1 #HLK #_ #IHW #d2 #H - elim (da_inv_lref … H) -H * #K0 #W0 [| #d0 ] #HLK0 #HW0 [| #Hd0 ] - lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct /3 width=1 by eq_f/ -| #a #I #G #L #V #T #d1 #_ #IHT #d2 #H - lapply (da_inv_bind … H) -H /2 width=1 by/ -| #I #G #L #V #T #d1 #_ #IHT #d2 #H - lapply (da_inv_flat … H) -H /2 width=1 by/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/da/da_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/da/da_lift.etc deleted file mode 100644 index 2f1a41554..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/da/da_lift.etc +++ /dev/null @@ -1,78 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/substitution/drop_drop.ma". -include "basic_2/static/da.ma". - -(* DEGREE ASSIGNMENT FOR TERMS **********************************************) - -(* Properties on relocation *************************************************) - -lemma da_lift: ∀h,o,G,L1,T1,d. ⦃G, L1⦄ ⊢ T1 ▪[h, o] d → - ∀L2,c,l,k. ⬇[c, l, k] L2 ≡ L1 → ∀T2. ⬆[l, k] T1 ≡ T2 → - ⦃G, L2⦄ ⊢ T2 ▪[h, o] d. -#h #o #G #L1 #T1 #d #H elim H -G -L1 -T1 -d -[ #G #L1 #s #d #Hkd #L2 #c #l #k #_ #X #H - >(lift_inv_sort1 … H) -X /2 width=1 by da_sort/ -| #G #L1 #K1 #V1 #i #d #HLK1 #_ #IHV1 #L2 #c #l #k #HL21 #X #H - elim (lift_inv_lref1 … H) * #Hil #H destruct - [ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by ylt_fwd_le/ #X #HLK2 #H - elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K2 #V2 #HK21 #HV12 #H destruct - /3 width=9 by da_ldef/ - | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 - /3 width=8 by da_ldef, drop_inv_gen/ - ] -| #G #L1 #K1 #W1 #i #d #HLK1 #_ #IHW1 #L2 #c #l #k #HL21 #X #H - elim (lift_inv_lref1 … H) * #Hil #H destruct - [ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by ylt_fwd_le/ #X #HLK2 #H - elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K2 #W2 #HK21 #HW12 #H destruct - /3 width=8 by da_ldec/ - | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 - /3 width=8 by da_ldec, drop_inv_gen/ - ] -| #a #I #G #L1 #V1 #T1 #d #_ #IHT1 #L2 #c #l #k #HL21 #X #H - elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct - /4 width=5 by da_bind, drop_skip/ -| #I #G #L1 #V1 #T1 #d #_ #IHT1 #L2 #c #l #k #HL21 #X #H - elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct - /3 width=5 by da_flat/ -] -qed. - -(* Inversion lemmas on relocation *******************************************) - -lemma da_inv_lift: ∀h,o,G,L2,T2,d. ⦃G, L2⦄ ⊢ T2 ▪[h, o] d → - ∀L1,c,l,k. ⬇[c, l, k] L2 ≡ L1 → ∀T1. ⬆[l, k] T1 ≡ T2 → - ⦃G, L1⦄ ⊢ T1 ▪[h, o] d. -#h #o #G #L2 #T2 #d #H elim H -G -L2 -T2 -d -[ #G #L2 #s #d #Hkd #L1 #c #l #k #_ #X #H - >(lift_inv_sort2 … H) -X /2 width=1 by da_sort/ -| #G #L2 #K2 #V2 #i #d #HLK2 #HV2 #IHV2 #L1 #c #l #k #HL21 #X #H - elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HV2 | -IHV2 ] - [ elim (drop_conf_lt … HL21 … HLK2) -L2 /3 width=8 by da_ldef/ - | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 /2 width=4 by da_ldef/ - ] -| #G #L2 #K2 #W2 #i #d #HLK2 #HW2 #IHW2 #L1 #c #l #k #HL21 #X #H - elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HW2 | -IHW2 ] - [ elim (drop_conf_lt … HL21 … HLK2) -L2 /3 width=8 by da_ldec/ - | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 /2 width=4 by da_ldec/ - ] -| #a #I #G #L2 #V2 #T2 #d #_ #IHT2 #L1 #c #l #k #HL21 #X #H - elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct - /4 width=5 by da_bind, drop_skip/ -| #I #G #L2 #V2 #T2 #d #_ #IHT2 #L1 #c #l #k #HL21 #X #H - elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct - /3 width=5 by da_flat/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/da/degree_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/da/degree_6.etc deleted file mode 100644 index bcc939ad1..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/da/degree_6.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T ▪ break [ term 46 o , break term 46 h ] break term 46 d )" - non associative with precedence 45 - for @{ 'Degree $h $o $G $L $T $d }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/droppreds_3.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/droppreds_3.etc deleted file mode 100644 index b21fe5117..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/droppreds_3.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( ⫱ * [ term 46 L , break term 46 K ] term 46 f )" - non associative with precedence 46 - for @{ 'DropPreds $L $K $f }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/drops/drops.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/drops/drops.etc deleted file mode 100644 index 5d51b83c8..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/drops/drops.etc +++ /dev/null @@ -1,2 +0,0 @@ -lemma drop_inv_refl: ∀L,l,m. ⬇[Ⓕ, l, m] L ≡ L → m = 0. -/2 width=5 by drop_inv_length_eq/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/drops/drops_append.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/drops/drops_append.etc deleted file mode 100644 index d134fe768..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/drops/drops_append.etc +++ /dev/null @@ -1,65 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/grammar/lenv_append.ma". -include "basic_2/substitution/drop.ma". - -(* DROPPING *****************************************************************) - -(* Properties on append for local environments ******************************) - -fact drop_O1_append_sn_le_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 → - l = 0 → m ≤ |L1| → - ∀L. ⬇[s, 0, m] L @@ L1 ≡ L @@ L2. -#L1 #L2 #s #l #m #H elim H -L1 -L2 -l -m // -[ #l #m #_ #_ #H >(yle_inv_O2 … H) -m // -| /4 width=1 by drop_drop, yle_inv_succ/ -| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #_ #H elim (ysucc_inv_O_dx … H) -] -qed-. - -lemma drop_O1_append_sn_le: ∀L1,L2,s,m. ⬇[s, yinj 0, m] L1 ≡ L2 → m ≤ |L1| → - ∀L. ⬇[s, 0, m] L @@ L1 ≡ L @@ L2. -/2 width=3 by drop_O1_append_sn_le_aux/ qed. - -(* Inversion lemmas on append for local environments ************************) - -lemma drop_O1_inv_append1_ge: ∀K,L1,L2,s,m. ⬇[s, 0, m] L1 @@ L2 ≡ K → - ∀m0. |L2| + m0 = m → ⬇[s, 0, m0] L1 ≡ K. -#K #L1 #L2 elim L2 -L2 -[ #s #m #H #m0 >yplus_O1 #H0 destruct // -| #L2 #I #V #IHL2 #s #m #H #m0 >yplus_succ1 - #H0 elim (drop_inv_O1_pair1 … H) -H * #Hm #HL12 destruct - [ elim (ysucc_inv_O_dx … Hm) - | /2 width=3 by/ - ] -] -qed-. - -lemma drop_O1_inv_append1_le: ∀K,L1,L2,s,m. ⬇[s, 0, m] L1 @@ L2 ≡ K → m ≤ |L2| → - ∀K2. ⬇[s, 0, m] L2 ≡ K2 → K = L1 @@ K2. -#K #L1 #L2 elim L2 -L2 -[ #s #m #H1 #H2 #K2 #H3 lapply (yle_inv_O2 … H2) -H2 - #H2 elim (drop_inv_atom1 … H3) -H3 #H3 #_ destruct - >(drop_inv_O2 … H1) -H1 // -| #L2 #I #V #IHL2 #s #m @(ynat_ind … m) -m [ -IHL2 || -IHL2 ] - [ #H1 #_ #K2 #H2 - lapply (drop_inv_O2 … H1) -H1 #H1 - lapply (drop_inv_O2 … H2) -H2 #H2 destruct // - | /3 width=7 by drop_inv_drop1, yle_inv_succ/ - | #_ #H lapply (yle_inv_Y1 … H) -H - #H elim (ylt_yle_false (|L2.ⓑ{I}V|) (∞)) // - ] -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/drops/drops_drops.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/drops/drops_drops.etc deleted file mode 100644 index 906c85c73..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/drops/drops_drops.etc +++ /dev/null @@ -1,13 +0,0 @@ -(* Inversion lemmas on equivalence ******************************************) - -lemma drop_O1_inj: ∀i,L1,L2,K. ⬇[i] L1 ≡ K → ⬇[i] L2 ≡ K → L1 ⩬[i, ∞] L2. -#i @(ynat_ind … i) -i -[ #L1 #L2 #K #H <(drop_inv_O2 … H) -K #H <(drop_inv_O2 … H) -L1 // -| #i #IHi * [2: #L1 #I1 #V1 ] * [2,4: #L2 #I2 #V2 ] #K #HLK1 #HLK2 // - lapply (drop_fwd_length … HLK1) - <(drop_fwd_length … HLK2) [ /4 width=5 by drop_inv_drop1, lreq_succ/ ] - #H [ elim (ysucc_inv_O_sn … H) | elim (ysucc_inv_O_dx … H) ] -| #L1 #L2 #K #H1 lapply (drop_fwd_Y2 … H1) -H1 - #H elim (ylt_yle_false … H) // -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/drops/drops_length.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/drops/drops_length.etc deleted file mode 100644 index 667f9689d..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/drops/drops_length.etc +++ /dev/null @@ -1,112 +0,0 @@ -lemma drop_inv_O1_gt: ∀L,K,m,s. ⬇[s, 0, m] L ≡ K → |L| < m → - s = Ⓣ ∧ K = ⋆. -#L elim L -L [| #L #Z #X #IHL ] #K #m #s #H normalize in ⊢ (?%?→?); #H1m -[ elim (drop_inv_atom1 … H) -H elim s -s /2 width=1 by conj/ - #_ #Hs lapply (Hs ?) // -Hs #H destruct elim (ylt_yle_false … H1m) -H1m // -| elim (drop_inv_O1_pair1 … H) -H * #H2m #HLK destruct - [ elim (ylt_yle_false … H1m) -H1m // - | elim (IHL … HLK) -IHL -HLK /2 width=1 by ylt_pred, conj/ - ] -] -qed-. - -lemma drop_O1_le: ∀s,m,L. m ≤ |L| → ∃K. ⬇[s, 0, m] L ≡ K. -#s #m @(ynat_ind … m) -m /2 width=2 by ex_intro/ -[ #m #IHm * - [ #H elim (ylt_yle_false … H) -H // - | #L #I #V #H elim (IHm L) -IHm /3 width=2 by drop_drop, yle_inv_succ, ex_intro/ - ] -| #L #H elim (ylt_yle_false … H) -H // -] -qed-. - -lemma drop_O1_lt: ∀s,L,m. m < |L| → ∃∃I,K,V. ⬇[s, 0, m] L ≡ K.ⓑ{I}V. -#s #L elim L -L -[ #m #H elim (ylt_yle_false … H) -H // -| #L #I #V #IHL #m @(ynat_ind … m) -m /2 width=4 by drop_pair, ex1_3_intro/ - [ #m #_#H elim (IHL m) -IHL /3 width=4 by drop_drop, ylt_inv_succ, ex1_3_intro/ - | #H elim (ylt_yle_false … H) -H // - ] -] -qed-. - -lemma drop_O1_pair: ∀L,K,m,s. ⬇[s, 0, m] L ≡ K → m ≤ |L| → ∀I,V. - ∃∃J,W. ⬇[s, 0, m] L.ⓑ{I}V ≡ K.ⓑ{J}W. -#L elim L -L [| #L #Z #X #IHL ] #K #m #s #H #Hm #I #V -[ elim (drop_inv_atom1 … H) -H #H >(yle_inv_O2 … Hm) -m - #Hs destruct /2 width=3 by ex1_2_intro/ -| elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK destruct /2 width=3 by ex1_2_intro/ - elim (IHL … HLK … Z X) -IHL -HLK - /3 width=3 by yle_pred, drop_drop_lt, ex1_2_intro/ -] -qed-. - -lemma drop_O1_ge: ∀L,m. |L| ≤ m → ⬇[Ⓣ, 0, m] L ≡ ⋆. -#L elim L -L [ #m #_ @drop_atom #H destruct ] -#L #I #V #IHL #m @(ynat_ind … m) -m // -[ #H elim (ylt_yle_false … H) -H /2 width=1 by ylt_inj/ -| /4 width=1 by drop_drop, yle_inv_succ/ -] -qed. - -lemma drop_O1_eq: ∀L,s. ⬇[s, 0, |L|] L ≡ ⋆. -#L elim L -L /2 width=1 by drop_drop/ -qed. - -lemma drop_fwd_length_ge: ∀L1,L2,l,m,s. ⬇[s, l, m] L1 ≡ L2 → |L1| ≤ l → |L2| = |L1|. -#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m // -[ #I #L1 #L2 #V #m #_ #_ #H elim (ylt_yle_false … H) -H // -| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #IH #H - lapply (yle_inv_succ … H) -H #H - >length_pair >length_pair /3 width=1 by eq_f/ -] -qed-. - -lemma drop_fwd_length_le_le: ∀L1,L2,l,m,s. ⬇[s, l, m] L1 ≡ L2 → - ∀l0. l + m + l0 = |L1| → |L2| = l + l0. -#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m // -[ #l #m #Hm #l0 #H elim (yplus_inv_O … H) -H - #H #H0 elim (yplus_inv_O … H) -H - #H1 #_ destruct // -| #I #L1 #L2 #V #m #_ >yplus_O1 >yplus_O1 #IH #l0 - /3 width=1 by ysucc_inv_inj/ -| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #IHL12 #l0 >yplus_succ1 >yplus_succ1 #H - lapply (ysucc_inv_inj … H) -H #Hl1 - >yplus_succ1 /3 width=1 by eq_f/ -] -qed-. - -lemma drop_fwd_length_le_ge: ∀L1,L2,l,m,s. ⬇[s, l, m] L1 ≡ L2 → l ≤ |L1| → |L1| ≤ l + m → |L2| = l. -#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m -[ #l #m #_ #H #_ /2 width=1 by yle_inv_O2/ -| #I #L #V #_ #H elim (ylt_yle_false … H) -H // -| #I #L1 #L2 #V #m #_ >yplus_O1 >yplus_O1 - /3 width=1 by yle_inv_succ/ -| #I #L1 #L2 #V1 #v2 #l #m #_ #_ #IH - >yplus_SO2 >yplus_SO2 - /4 width=1 by yle_inv_succ, eq_f/ -] -qed-. - -lemma drop_fwd_be: ∀L,K,s,l,m,i. ⬇[s, l, m] L ≡ K → |K| ≤ i → i < l → |L| ≤ i. -#L #K #s #l #m #i #HLK #HK #Hl elim (ylt_split i (|L|)) // -#HL elim (drop_O1_lt (Ⓕ) … HL) #I #K0 #V #HLK0 -HL -elim (ylt_inv_plus_sn … Hl) -Hl #l0 #H0 -elim (drop_conf_lt … HLK … HLK0 … H0) -HLK -HLK0 -H0 -#K1 #V1 #HK1 #_ #_ lapply (drop_fwd_length_lt2 … HK1) -I -K1 -V1 -#H elim (ylt_yle_false … H) -H // -qed-. - -lemma drop_O1_ex: ∀K2,i,L1. |L1| = |K2| + i → - ∃∃L2. L1 ⩬[0, i] L2 & ⬇[i] L2 ≡ K2. -#K2 #i @(ynat_ind … i) -i -[ /3 width=3 by lreq_O2, ex2_intro/ -| #i #IHi #Y >yplus_succ2 #Hi - elim (drop_O1_lt (Ⓕ) Y 0) [2: >Hi // ] - #I #L1 #V #H lapply (drop_inv_O2 … H) -H #H destruct - >length_pair in Hi; #H lapply (ysucc_inv_inj … H) -H - #HL1K2 elim (IHi L1) -IHi // -HL1K2 - /3 width=5 by lreq_pair, drop_drop, ex2_intro/ -| #L1 >yplus_Y2 #H elim (ylt_yle_false (|L1|) (∞)) // -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/fqu/fqu.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/fqu/fqu.etc deleted file mode 100644 index b5066d00f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/fqu/fqu.etc +++ /dev/null @@ -1,16 +0,0 @@ -(* Removed theorems *********************************************************) - -include "basic_2/substitution/drop.ma". - -| fqu_drop : ∀G,L,K,T,U,k. - ⬇[⫯k] L ≡ K → ⬆[0, ⫯k] T ≡ U → fqu G L U G K T - -lemma fqu_drop_lt: ∀G,L,K,T,U,k. 0 < k → - ⬇[k] L ≡ K → ⬆[0, k] T ≡ U → ⦃G, L, U⦄ ⊐ ⦃G, K, T⦄. -#G #L #K #T #U #k #Hm lapply (ylt_inv_O1 … Hm) -Hm -#Hm (drop_inv_O2 … HLK) -L >(lift_inv_O2 … HTU) -T // -qed. - -inductive fquq: tri_relation genv lenv term ≝ -| fquq_lref_O : ∀I,G,L,V. fquq G (L.ⓑ{I}V) (#0) G L V -| fquq_pair_sn: ∀I,G,L,V,T. fquq G L (②{I}V.T) G L V -| fquq_bind_dx: ∀a,I,G,L,V,T. fquq G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T -| fquq_flat_dx: ∀I,G, L,V,T. fquq G L (ⓕ{I}V.T) G L T -| fquq_drop : ∀G,L,K,T,U,k. - ⬇[k] L ≡ K → ⬆[0, k] T ≡ U → fquq G L U G K T -. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/frees/frees.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/frees/frees.etc deleted file mode 100644 index a9349e870..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/frees/frees.etc +++ /dev/null @@ -1,5 +0,0 @@ -(* A Basic_A2 lemma we do not need so far *) -axiom frees_pair_flat: ∀L,T,f1,I1,V1. L.ⓑ{I1}V1 ⊢ 𝐅*⦃T⦄ ≡ f1 → - ∀f2,I2,V2. L.ⓑ{I2}V2 ⊢ 𝐅*⦃T⦄ ≡ f2 → - ∀f0. f1 ⋓ f2 ≡ f0 → - ∀I0,I. L.ⓑ{I0}ⓕ{I}V1.V2 ⊢ 𝐅*⦃T⦄ ≡ f0. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/frees/frees_append.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/frees/frees_append.etc deleted file mode 100644 index ee8324cc9..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/frees/frees_append.etc +++ /dev/null @@ -1,54 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/substitution/drop_append.ma". -include "basic_2/multiple/frees.ma". - -(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************) - -(* Properties on append for local environments ******************************) - -lemma frees_append: ∀L2,U,l,i. L2 ⊢ i ϵ 𝐅*[l]⦃U⦄ → i ≤ |L2| → - ∀L1. L1 @@ L2 ⊢ i ϵ 𝐅*[l]⦃U⦄. -#L2 #U #l #i #H elim H -L2 -U -l -i /3 width=2 by frees_eq/ -#I #L2 #K2 #U #W #l #i #j #Hlj #Hji #HnU #HLK2 #_ #IHW #Hi #L1 -lapply (drop_fwd_length_minus2 … HLK2) normalize #H0 -lapply (drop_O1_append_sn_le … HLK2 … L1) -HLK2 -[ -I -L1 -K2 -U -W -l /4 width=3 by ylt_yle_trans, ylt_inv_inj, lt_to_le/ -| #HLK2 @(frees_be … HnU HLK2) // -HnU -HLK2 @IHW -IHW - >(minus_plus_m_m (|K2|) 1) >H0 -H0 yminus_SO2 - /3 width=1 by monotonic_yle_minus_dx, yle_pred/ -] -qed. - -(* Inversion lemmas on append for local environments ************************) - -fact frees_inv_append_aux: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ → ∀L1,L2. L = L1 @@ L2 → - i ≤ |L2| → L2 ⊢ i ϵ 𝐅*[l]⦃U⦄. -#L #U #l #i #H elim H -L -U -l -i /3 width=2 by frees_eq/ -#Z #L #Y #U #X #l #i #j #Hlj #Hji #HnU #HLY #_ #IHW #L1 #L2 #H #Hi destruct -elim (drop_O1_lt (Ⓕ) L2 j) [2: -Z -Y -L1 -X -U -l /3 width=3 by ylt_yle_trans, ylt_inv_inj/ ] -#I #K2 #W #HLK2 lapply (drop_fwd_length_minus2 … HLK2) normalize #H0 -lapply (drop_O1_inv_append1_le … HLY … HLK2) -HLY -[ -Z -I -Y -K2 -L1 -X -U -W -l /4 width=3 by ylt_yle_trans, ylt_inv_inj, lt_to_le/ -| normalize #H destruct - @(frees_be … HnU HLK2) -HnU -HLK2 // @IHW -IHW // - >(minus_plus_m_m (|K2|) 1) >H0 -H0 yminus_SO2 - /3 width=1 by monotonic_yle_minus_dx, yle_pred/ -] -qed-. - -lemma frees_inv_append: ∀L1,L2,U,l,i. L1 @@ L2 ⊢ i ϵ 𝐅*[l]⦃U⦄ → - i ≤ |L2| → L2 ⊢ i ϵ 𝐅*[l]⦃U⦄. -/2 width=4 by frees_inv_append_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/frees/frees_fqus.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/frees/frees_fqus.etc deleted file mode 100644 index 618d858ff..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/frees/frees_fqus.etc +++ /dev/null @@ -1,111 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/relocation/drops_weight.ma". -include "basic_2/s_computation/fqup_weight.ma". -include "basic_2/s_computation/fqus_weight.ma". -include "basic_2/static/frees.ma". - -(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************) - -(* Properties with star-iterated supclosure *********************************) - -(* Note: Restrict fqu_drop on atoms to prove frees_fqus_drops - Counterex: ⦃⋆, ⋆.ⓛ#0, ⓛ#1.#0⦄ ⊐ ⦃⋆, ⋆, ⓛ#0.#0⦄ ⊐ ⦃⋆, ⋆.ⓛ#0, #0⦄ -*) -lemma frees_fqus_drops: ∀G,L1,T1,f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 → - ∀L2,T2. ⦃G, L1, T1⦄ ⊐* ⦃G, L2, T2⦄ → - ∀I,n. ⬇*[n] L1 ≡ L2.ⓑ{I}T2 → - ∃∃f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ ⫱*[⫯n] f1. -#G #L1 #T1 @(fqup_wf_ind_eq … G L1 T1) -G -L1 -T1 -#G0 #L0 #T0 #IH #G #L1 * * -[ #s #HG #HL #HT #f1 #Hf1 #L2 #T2 #H12 #I #n #HL12 destruct - lapply (frees_inv_sort … Hf1) -Hf1 #Hf1 - elim (fqus_inv_sort1 … H12) -H12 * [| elim (nat_split n) ] - [ -IH #H1 #H2 #H3 destruct - lapply (drops_fwd_lw … HL12) -HL12 #HL12 - elim (lt_le_false … HL12) -HL12 // - | -IH #Hn #J #L #V #H12 #H destruct - lapply (drops_fwd_isid … HL12 ?) -HL12 // #H destruct - <(fqus_inv_refl_atom3 … H12) -H12 - /3 width=3 by frees_sort_gen, sle_isid_sn, ex2_intro/ - | * #m #Hn #J #L #V #H12 #H destruct - lapply (drops_inv_drop1 … HL12) -HL12 #HL12 - elim (IH … H12 … HL12) -IH -H12 -HL12 - /3 width=3 by frees_sort_gen, fqu_fqup, fqu_drop, lifts_sort, isid_tl, ex2_intro/ - ] -| #i #HG #HL #HT #f1 #Hf1 #L2 #T2 #H12 #I #n #HL12 destruct - elim (fqus_inv_lref1 … H12) -H12 * [|2,3: elim (nat_split n) ] - [ -IH #H1 #H2 #H3 destruct - lapply (drops_fwd_lw … HL12) -HL12 #HL12 - elim (lt_le_false … HL12) -HL12 // - | -IH #Hn #J #L #V #H12 #H1 #H2 destruct - elim (frees_inv_zero_pair … Hf1) -Hf1 #g1 #Hg1 #H destruct - lapply (drops_fwd_isid … HL12 ?) -HL12 -H12 // #H destruct - /2 width=3 by sle_refl, ex2_intro/ - | * #m #Hn #J #L #V #H12 #H1 #H2 destruct - elim (frees_inv_zero_pair … Hf1) -Hf1 #g1 #Hg1 #H destruct - lapply (drops_inv_drop1 … HL12) -HL12 #HL12 - elim (IH … H12 … HL12) -IH -H12 -HL12 - /2 width=3 by fqu_fqup, fqu_lref_O, ex2_intro/ - | -IH #Hn #J #L #V #j #H12 #H1 #H2 destruct - elim (frees_inv_lref_pair … Hf1) -Hf1 #g1 #Hg1 #H destruct - lapply (drops_fwd_isid … HL12 ?) -HL12 // #H destruct - <(fqus_inv_refl_atom3 … H12) -H12 /2 width=3 by sle_refl, ex2_intro/ - | * #m #Hn #J #L #V #j #H12 #H1 #H2 destruct - elim (frees_inv_lref_pair … Hf1) -Hf1 #g1 #Hg1 #H destruct - lapply (drops_inv_drop1 … HL12) -HL12 #HL12 - elim (IH … H12 … HL12) -IH -H12 -HL12 /2 width=3 by fqu_fqup, ex2_intro/ - ] -| #l #HG #HL #HT #f1 #Hf1 #L2 #T2 #H12 #I #n #HL12 destruct - lapply (frees_inv_gref … Hf1) -Hf1 #Hf1 - elim (fqus_inv_gref1 … H12) -H12 * [| elim (nat_split n) ] - [ -IH #H1 #H2 #H3 destruct - lapply (drops_fwd_lw … HL12) -HL12 #HL12 - elim (lt_le_false … HL12) -HL12 // - | -IH #Hn #J #L #V #H12 #H destruct - lapply (drops_fwd_isid … HL12 ?) -HL12 // #H destruct - <(fqus_inv_refl_atom3 … H12) -H12 - /3 width=3 by frees_gref_gen, sle_isid_sn, ex2_intro/ - | * #m #Hn #J #L #V #H12 #H destruct - lapply (drops_inv_drop1 … HL12) -HL12 #HL12 - elim (IH … H12 … HL12) -IH -H12 -HL12 - /3 width=3 by frees_gref_gen, fqu_fqup, fqu_drop, lifts_sort, isid_tl, ex2_intro/ - ] -| #p #I #V1 #T1 #HG #HL #HT #f1 #Hf1 #L2 #T2 #H12 #I #n #HL12 destruct - elim (frees_inv_bind … Hf1) -Hf1 #f1V #f1T #Hf1V #Hf1T #Hf1 - elim (fqus_inv_bind1 … H12) -H12 - [ -IH -f1V -f1T * #H1 #H2 #H3 destruct - lapply (drops_fwd_lw … HL12) -HL12 #HL12 - elim (lt_le_false … HL12) -HL12 // - | #H12 elim (IH … H12 … HL12) -IH -H12 -HL12 - /4 width=6 by sor_tls, sor_inv_sle_sn_trans, ex2_intro/ - | #H12 elim (IH … H12 I (⫯n)) -IH -H12 [|*: /2 width=3 by drops_drop, drops_refl/ ] -HL12 - H -H >commutative_plus #H (**) (* lemma needed here *) - lapply (le_plus_to_le_r … 0 H) -H #H - lapply (le_n_O_to_eq … H) -H #H destruct -| #I #G1 #G2 #V #H1 #_ #H2 - lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 normalize in ⊢ (? % ? → ?); >commutative_plus #H - lapply (lt_plus_to_lt_l … 0 H) -H #H - elim (lt_zero_false … H) -] -qed-. - -lemma gget_inv_eq: ∀G1,G2,m. ⬇[m] G1 ≡ G2 → |G1| = m + 1 → G1 = G2. -#G1 #G2 #m * -G1 -G2 // -[ #G #H1 #H2 >H2 in H1; -H2 >commutative_plus #H (**) (* lemma needed here *) - lapply (le_plus_to_le_r … 0 H) -H #H - lapply (le_n_O_to_eq … H) -H #H destruct -| #I #G1 #G2 #V #H1 #_ normalize #H2 - <(injective_plus_l … H2) in H1; -H2 #H - elim (lt_refl_false … H) -] -qed-. - -fact gget_inv_lt_aux: ∀I,G,G1,G2,V,m. ⬇[m] G ≡ G2 → G = G1. ⓑ{I} V → - m < |G1| → ⬇[m] G1 ≡ G2. -#I #G #G1 #G2 #V #m * -G -G2 -[ #G #H1 #H destruct #H2 - lapply (le_to_lt_to_lt … H1 H2) -H1 -H2 normalize in ⊢ (? % ? → ?); >commutative_plus #H - lapply (lt_plus_to_lt_l … 0 H) -H #H - elim (lt_zero_false … H) -| #G #H1 #H2 destruct >(injective_plus_l … H1) -H1 #H - elim (lt_refl_false … H) -| #J #G #G2 #W #_ #HG2 #H destruct // -] -qed-. - -lemma gget_inv_lt: ∀I,G1,G2,V,m. - ⬇[m] G1. ⓑ{I} V ≡ G2 → m < |G1| → ⬇[m] G1 ≡ G2. -/2 width=5 by gget_inv_lt_aux/ qed-. - -(* Basic properties *********************************************************) - -lemma gget_total: ∀m,G1. ∃G2. ⬇[m] G1 ≡ G2. -#m #G1 elim G1 -G1 /3 width=2 by gget_gt, ex_intro/ -#I #V #G1 * #G2 #HG12 -elim (lt_or_eq_or_gt m (|G1|)) #Hm -[ /3 width=2 by gget_lt, ex_intro/ -| destruct /3 width=2 by gget_eq, ex_intro/ -| @ex_intro [2: @gget_gt normalize /2 width=1 by/ | skip ] (**) (* explicit constructor *) -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/gget/gget_gget.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/gget/gget_gget.etc deleted file mode 100644 index 7bd39f09b..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/gget/gget_gget.etc +++ /dev/null @@ -1,40 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/substitution/gget.ma". - -(* GLOBAL ENVIRONMENT READING ***********************************************) - -(* Main properties **********************************************************) - -theorem gget_mono: ∀G,G1,m. ⬇[m] G ≡ G1 → ∀G2. ⬇[m] G ≡ G2 → G1 = G2. -#G #G1 #m #H elim H -G -G1 -[ #G #Hm #G2 #H - >(gget_inv_gt … H Hm) -H -Hm // -| #G #Hm #G2 #H - >(gget_inv_eq … H Hm) -H -Hm // -| #I #G #G1 #V #Hm #_ #IHG1 #G2 #H - lapply (gget_inv_lt … H Hm) -H -Hm /2 width=1 by/ -] -qed-. - -lemma gget_dec: ∀G1,G2,m. Decidable (⬇[m] G1 ≡ G2). -#G1 #G2 #m -elim (gget_total m G1) #G #HG1 -elim (eq_genv_dec G G2) #HG2 -[ destruct /2 width=1 by or_introl/ -| @or_intror #HG12 - lapply (gget_mono … HG1 … HG12) -HG1 -HG12 /2 width=1 by/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/gget/rdrop_3.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/gget/rdrop_3.etc deleted file mode 100644 index a761d47c6..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/gget/rdrop_3.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( ⬇ [ term 46 m ] break term 46 L1 ≡ break term 46 L2 )" - non associative with precedence 45 - for @{ 'RDrop $m $L1 $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lazyor_5.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/lazyor_5.etc deleted file mode 100644 index 70d3a1c66..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lazyor_5.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( L1 ⋓ break [ term 46 T , break term 46 f ] break term 46 L2 ≡ break term 46 L )" - non associative with precedence 45 - for @{ 'LazyOr $L1 $T $f $L2 $L }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lfpr_main.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/lfpr_main.etc deleted file mode 100644 index bfe23e721..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lfpr_main.etc +++ /dev/null @@ -1,80 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/rt_transition/lfpr_lfpr.ma". - -(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************) - -definition lfxs_confluent_R: relation2 … ≝ - λRP1,RP2. - ∀L0,T0,T1. RP1 L0 T0 T1 → ∀T2. RP2 L0 T0 T2 → - ∀L1. L0 ⦻*[RP1, T0] L1 → ∀L2. L0 ⦻*[RP2, T0] L2 → - ∃∃L. L1 ⦻*[RP2, T1] L & L2 ⦻*[RP1, T2] L. - -(* Main properties **********************************************************) - -fact lfpr_conf_cpr_atom_atom: - ∀h,I,G,L0. ( - ∀L,T. ⦃G, L0, ⓪{I}⦄ ⊐+ ⦃G, L, T⦄ → - ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → - ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 → - ∃∃K0. ⦃G, L1⦄ ⊢ ➡[h, T1] K0 & ⦃G, L2⦄ ⊢ ➡[h, T2] K0 - ) → - ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓪{I}] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓪{I}] L2 → - ∃∃L. ⦃G, L1⦄ ⊢ ➡[h, ⓪{I}] L & ⦃G, L2⦄ ⊢ ➡[h, ⓪{I}] L. -#h #I #G * [ | #K0 #J #V0 cases I -I [ | * | ] ] -[ #_ #L1 #HL01 #L2 #HL02 - lapply (lfpr_inv_atom_sn … HL01) -HL01 #H destruct - lapply (lfpr_inv_atom_sn … HL02) -HL02 #H destruct - /2 width=3 by ex2_intro/ -| #s #IH #L1 #HL01 #L2 #HL02 - elim (lfxs_inv_sort_pair_sn … HL01) -HL01 #K1 #V1 #HK01 #H destruct - elim (lfxs_inv_sort_pair_sn … HL02) -HL02 #K2 #V2 #HK02 #H destruct - elim (IH … HK01 … HK02) -IH -HK01 -HK02 - /3 width=5 by lfpr_sort, fqu_fqup, fqu_drop, ex2_intro/ -| #IH #L1 #HL01 #L2 #HL02 - elim (lfpr_inv_zero_pair_sn … HL01) -HL01 #K1 #V1 #HK01 #HV01 #H destruct - elim (lfpr_inv_zero_pair_sn … HL02) -HL02 #K2 #V2 #HK02 #HV02 #H destruct - elim (cpr_conf_lfpr … HV01 … HV02 … HK01 … HK02) #V #HV1 #HV2 - elim (IH … HV01 … HV02 … HK01 … HK02) -IH -HV01 -HV02 -HK01 -HK02 - /3 width=5 by lfpr_zero, fqu_fqup, fqu_drop, ex2_intro/ -| #i #IH #L1 #HL01 #L2 #HL02 - elim (lfxs_inv_lref_pair_sn … HL01) -HL01 #K1 #V1 #HK01 #H destruct - elim (lfxs_inv_lref_pair_sn … HL02) -HL02 #K2 #V2 #HK02 #H destruct - elim (IH … HK01 … HK02) -IH -HK01 -HK02 - /3 width=5 by lfpr_lref, fqu_fqup, fqu_drop, ex2_intro/ -| #l #IH #L1 #HL01 #L2 #HL02 - elim (lfxs_inv_gref_pair_sn … HL01) -HL01 #K1 #V1 #HK01 #H destruct - elim (lfxs_inv_gref_pair_sn … HL02) -HL02 #K2 #V2 #HK02 #H destruct - elim (IH … HK01 … HK02) -IH -HK01 -HK02 - /3 width=5 by lfpr_gref, fqu_fqup, fqu_drop, ex2_intro/ -] -qed-. - -theorem lfpr_conf_cpr: ∀h,G. lfxs_confluent_R (cpm 0 h G) (cpm 0 h G). -#h #G #L0 #T0 @(fqup_wf_ind_eq … G L0 T0) -G -L0 -T0 #G #L #T #IH #G0 #L0 * [| * ] -[ #I0 #HG #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct - elim (cpr_inv_atom1_drops … H1) -H1 - elim (cpr_inv_atom1_drops … H2) -H2 - [ #H2 #H1 destruct - /3 width=7 by lfpr_conf_cpr_atom_atom/ - | * #K0 #V0 #V2 * [2: #i2 ] #HLK0 #HV02 #HVT2 #H2 #H1 destruct - -(* - -theorem lpr_conf: ∀G. confluent … (lpr G). -/3 width=6 by lpx_sn_conf, cpr_conf_lpr/ -qed-. - -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lifts/lifts_neg.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/lifts/lifts_neg.etc deleted file mode 100644 index f4b8a70c1..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lifts/lifts_neg.etc +++ /dev/null @@ -1,75 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/substitution/lift.ma". - -(* BASIC TERM RELOCATION ****************************************************) - -(* Properties on negated basic relocation ***********************************) - -lemma nlift_lref_be_SO: ∀X,j. j < ∞ → ⬆[j, 1] X ≡ #j → ⊥. -#X #j #Hj #H elim (lift_inv_lref2 … H) -H * -[ #H elim (ylt_yle_false … H) -H // -| #i #Hij #_ #H1 #H2 destruct - elim (ylt_inv_plus_Y … Hj) -Hj #Hi #_ - elim (ylt_yle_false … Hij) -Hij /2 width=1 by monotonic_ylt_plus_sn/ -] -qed-. - -lemma nlift_bind_sn: ∀W,l,m. (∀V. ⬆[l, m] V ≡ W → ⊥) → - ∀a,I,U. (∀X. ⬆[l, m] X ≡ ⓑ{a,I}W.U → ⊥). -#W #l #m #HW #a #I #U #X #H elim (lift_inv_bind2 … H) -H /2 width=2 by/ -qed-. - -lemma nlift_bind_dx: ∀U,l,m. (∀T. ⬆[⫯l, m] T ≡ U → ⊥) → - ∀a,I,W. (∀X. ⬆[l, m] X ≡ ⓑ{a,I}W.U → ⊥). -#U #l #m #HU #a #I #W #X #H elim (lift_inv_bind2 … H) -H /2 width=2 by/ -qed-. - -lemma nlift_flat_sn: ∀W,l,m. (∀V. ⬆[l, m] V ≡ W → ⊥) → - ∀I,U. (∀X. ⬆[l, m] X ≡ ⓕ{I}W.U → ⊥). -#W #l #m #HW #I #U #X #H elim (lift_inv_flat2 … H) -H /2 width=2 by/ -qed-. - -lemma nlift_flat_dx: ∀U,l,m. (∀T. ⬆[l, m] T ≡ U → ⊥) → - ∀I,W. (∀X. ⬆[l, m] X ≡ ⓕ{I}W.U → ⊥). -#U #l #m #HU #I #W #X #H elim (lift_inv_flat2 … H) -H /2 width=2 by/ -qed-. - -(* Inversion lemmas on negated basic relocation *****************************) - -lemma nlift_inv_lref_be_SO: ∀i,j. (∀X. ⬆[i, 1] X ≡ #j → ⊥) → j = i ∧ j < ∞. -#i #j elim (ylt_split_eq i j) #Hij #H destruct -[ elim (H (#⫰j)) -H /2 width=1 by lift_lref_pred/ -| elim (yle_split_eq i (∞)) /2 width=1 by conj/ #H0 destruct - elim (H (#∞)) -H /2 width=1 by lift_lref_plus, ylt_Y/ -| elim (H (#j)) -H /2 width=1 by lift_lref_lt/ -] -qed-. - -lemma nlift_inv_bind: ∀a,I,W,U,l,m. (∀X. ⬆[l, m] X ≡ ⓑ{a,I}W.U → ⊥) → - (∀V. ⬆[l, m] V ≡ W → ⊥) ∨ (∀T. ⬆[⫯l, m] T ≡ U → ⊥). -#a #I #W #U #l #m #H elim (is_lift_dec W l m) -[ * /4 width=2 by lift_bind, or_intror/ -| /4 width=2 by ex_intro, or_introl/ -] -qed-. - -lemma nlift_inv_flat: ∀I,W,U,l,m. (∀X. ⬆[l, m] X ≡ ⓕ{I}W.U → ⊥) → - (∀V. ⬆[l, m] V ≡ W → ⊥) ∨ (∀T. ⬆[l, m] T ≡ U → ⊥). -#I #W #U #l #m #H elim (is_lift_dec W l m) -[ * /4 width=2 by lift_flat, or_intror/ -| /4 width=2 by ex_intro, or_introl/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq.etc deleted file mode 100644 index ac55a2c8a..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq.etc +++ /dev/null @@ -1,72 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/notation/relations/lazyeq_4.ma". -include "basic_2/multiple/llpx_sn.ma". - -(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) - -(* Basic inversion lemmas ***************************************************) - -lemma lleq_ind: ∀R:relation4 ynat term lenv lenv. ( - ∀L1,L2,l,s. |L1| = |L2| → R l (⋆s) L1 L2 - ) → ( - ∀L1,L2,l,i. |L1| = |L2| → yinj i < l → R l (#i) L1 L2 - ) → ( - ∀I,L1,L2,K1,K2,V,l,i. l ≤ yinj i → - ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V → - K1 ≡[V, yinj O] K2 → R (yinj O) V K1 K2 → R l (#i) L1 L2 - ) → ( - ∀L1,L2,l,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → R l (#i) L1 L2 - ) → ( - ∀L1,L2,l,p. |L1| = |L2| → R l (§p) L1 L2 - ) → ( - ∀a,I,L1,L2,V,T,l. - L1 ≡[V, l]L2 → L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V → - R l V L1 L2 → R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → R l (ⓑ{a,I}V.T) L1 L2 - ) → ( - ∀I,L1,L2,V,T,l. - L1 ≡[V, l]L2 → L1 ≡[T, l] L2 → - R l V L1 L2 → R l T L1 L2 → R l (ⓕ{I}V.T) L1 L2 - ) → - ∀l,T,L1,L2. L1 ≡[T, l] L2 → R l T L1 L2. -#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #l #T #L1 #L2 #H elim H -L1 -L2 -T -l /2 width=8 by/ -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lleq_fwd_lref: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → - ∨∨ |L1| ≤ i ∧ |L2| ≤ i - | yinj i < l - | ∃∃I,K1,K2,V. ⬇[i] L1 ≡ K1.ⓑ{I}V & - ⬇[i] L2 ≡ K2.ⓑ{I}V & - K1 ≡[V, yinj 0] K2 & l ≤ yinj i. -#L1 #L2 #l #i #H elim (llpx_sn_fwd_lref … H) /2 width=1 by or3_intro0, or3_intro1/ -* /3 width=7 by or3_intro2, ex4_4_intro/ -qed-. - -lemma lleq_fwd_drop_sn: ∀L1,L2,T,l. L1 ≡[l, T] L2 → ∀K1,i. ⬇[i] L1 ≡ K1 → - ∃K2. ⬇[i] L2 ≡ K2. -/2 width=7 by llpx_sn_fwd_drop_sn/ qed-. - -lemma lleq_fwd_drop_dx: ∀L1,L2,T,l. L1 ≡[l, T] L2 → ∀K2,i. ⬇[i] L2 ≡ K2 → - ∃K1. ⬇[i] L1 ≡ K1. -/2 width=7 by llpx_sn_fwd_drop_dx/ qed-. - -(* Basic properties *********************************************************) - -lemma lleq_lref: ∀I,L1,L2,K1,K2,V,l,i. l ≤ yinj i → - ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V → - K1 ≡[V, 0] K2 → L1 ≡[#i, l] L2. -/2 width=9 by llpx_sn_lref/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq_alt.etc deleted file mode 100644 index d2919f5b6..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq_alt.etc +++ /dev/null @@ -1,41 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/multiple/llpx_sn_alt.ma". -include "basic_2/multiple/lleq.ma". - -(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) - -(* Alternative definition (not recursive) ***********************************) - -theorem lleq_intro_alt: ∀L1,L2,T,l. |L1| = |L2| → - (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - I1 = I2 ∧ V1 = V2 - ) → L1 ≡[T, l] L2. -#L1 #L2 #T #l #HL12 #IH @llpx_sn_alt_inv_llpx_sn @conj // -HL12 -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 -@(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 // -qed. - -theorem lleq_inv_alt: ∀L1,L2,T,l. L1 ≡[T, l] L2 → - |L1| = |L2| ∧ - ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - I1 = I2 ∧ V1 = V2. -#L1 #L2 #T #l #H elim (llpx_sn_llpx_sn_alt … H) -H -#HL12 #IH @conj // -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 -@(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 // -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq_alt_rec.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq_alt_rec.etc deleted file mode 100644 index 440e0f510..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq_alt_rec.etc +++ /dev/null @@ -1,54 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/multiple/llpx_sn_alt_rec.ma". -include "basic_2/multiple/lleq.ma". - -(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) - -(* Alternative definition (recursive) ***************************************) - -theorem lleq_intro_alt_r: ∀L1,L2,T,l. |L1| = |L2| → - (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2 - ) → L1 ≡[T, l] L2. -#L1 #L2 #T #l #HL12 #IH @llpx_sn_intro_alt_r // -HL12 -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 -elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/ -qed. - -theorem lleq_ind_alt_r: ∀S:relation4 ynat term lenv lenv. - (∀L1,L2,T,l. |L1| = |L2| → ( - ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2 & S 0 V1 K1 K2 - ) → S l T L1 L2) → - ∀L1,L2,T,l. L1 ≡[T, l] L2 → S l T L1 L2. -#S #IH1 #L1 #L2 #T #l #H @(llpx_sn_ind_alt_r … H) -L1 -L2 -T -l -#L1 #L2 #T #l #HL12 #IH2 @IH1 -IH1 // -HL12 -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 -elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /2 width=1 by and4_intro/ -qed-. - -theorem lleq_inv_alt_r: ∀L1,L2,T,l. L1 ≡[T, l] L2 → - |L1| = |L2| ∧ - ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2. -#L1 #L2 #T #l #H elim (llpx_sn_inv_alt_r … H) -H -#HL12 #IH @conj // -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 -elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq_drop.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq_drop.etc deleted file mode 100644 index aa4fa9e8c..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq_drop.etc +++ /dev/null @@ -1,150 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/multiple/llpx_sn_drop.ma". -include "basic_2/multiple/lleq.ma". - -(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) - -(* Advanced properties ******************************************************) - -lemma lleq_bind_repl_O: ∀I,L1,L2,V,T. L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V → - ∀J,W. L1 ≡[W, 0] L2 → L1.ⓑ{J}W ≡[T, 0] L2.ⓑ{J}W. -/2 width=7 by llpx_sn_bind_repl_O/ qed-. - -lemma lleq_dec: ∀T,L1,L2,l. Decidable (L1 ≡[T, l] L2). -/3 width=1 by llpx_sn_dec, eq_term_dec/ qed-. - -lemma lleq_llpx_sn_trans: ∀R. lleq_transitive R → - ∀L1,L2,T,l. L1 ≡[T, l] L2 → - ∀L. llpx_sn R l T L2 L → llpx_sn R l T L1 L. -#R #HR #L1 #L2 #T #l #H @(lleq_ind … H) -L1 -L2 -T -l -[1,2,5: /4 width=6 by llpx_sn_fwd_length, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, trans_eq/ -|4: /4 width=6 by llpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux, trans_eq/ -| #I #L1 #L2 #K1 #K2 #V #l #i #Hli #HLK1 #HLK2 #HK12 #IHK12 #L #H elim (llpx_sn_inv_lref_ge_sn … H … HLK2) -H -HLK2 - /3 width=11 by llpx_sn_lref/ -| #a #I #L1 #L2 #V #T #l #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_bind … H) -H - /3 width=1 by llpx_sn_bind/ -| #I #L1 #L2 #V #T #l #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_flat … H) -H - /3 width=1 by llpx_sn_flat/ -] -qed-. - -lemma lleq_llpx_sn_conf: ∀R. lleq_transitive R → - ∀L1,L2,T,l. L1 ≡[T, l] L2 → - ∀L. llpx_sn R l T L1 L → llpx_sn R l T L2 L. -/3 width=3 by lleq_llpx_sn_trans, lleq_sym/ qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma lleq_inv_lref_ge_dx: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i → - ∀I,K2,V. ⬇[i] L2 ≡ K2.ⓑ{I}V → - ∃∃K1. ⬇[i] L1 ≡ K1.ⓑ{I}V & K1 ≡[V, 0] K2. -#L1 #L2 #l #i #H #Hli #I #K2 #V #HLK2 elim (llpx_sn_inv_lref_ge_dx … H … HLK2) -L2 -/2 width=3 by ex2_intro/ -qed-. - -lemma lleq_inv_lref_ge_sn: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i → - ∀I,K1,V. ⬇[i] L1 ≡ K1.ⓑ{I}V → - ∃∃K2. ⬇[i] L2 ≡ K2.ⓑ{I}V & K1 ≡[V, 0] K2. -#L1 #L2 #l #i #H #Hli #I1 #K1 #V #HLK1 elim (llpx_sn_inv_lref_ge_sn … H … HLK1) -L1 -/2 width=3 by ex2_intro/ -qed-. - -lemma lleq_inv_lref_ge_bi: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i → - ∀I1,I2,K1,K2,V1,V2. - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - ∧∧ I1 = I2 & K1 ≡[V1, 0] K2 & V1 = V2. -/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-. - -lemma lleq_inv_lref_ge: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i → - ∀I,K1,K2,V. ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V → - K1 ≡[V, 0] K2. -#L1 #L2 #l #i #HL12 #Hli #I #K1 #K2 #V #HLK1 #HLK2 -elim (lleq_inv_lref_ge_bi … HL12 … HLK1 HLK2) // -qed-. - -lemma lleq_inv_S: ∀L1,L2,T,l. L1 ≡[T, l+1] L2 → - ∀I,K1,K2,V. ⬇[l] L1 ≡ K1.ⓑ{I}V → ⬇[l] L2 ≡ K2.ⓑ{I}V → - K1 ≡[V, 0] K2 → L1 ≡[T, l] L2. -/2 width=9 by llpx_sn_inv_S/ qed-. - -lemma lleq_inv_bind_O: ∀a,I,L1,L2,V,T. L1 ≡[ⓑ{a,I}V.T, 0] L2 → - L1 ≡[V, 0] L2 ∧ L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V. -/2 width=2 by llpx_sn_inv_bind_O/ qed-. - -(* Advanced forward lemmas **************************************************) - -lemma lleq_fwd_lref_dx: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → - ∀I,K2,V. ⬇[i] L2 ≡ K2.ⓑ{I}V → - i < l ∨ - ∃∃K1. ⬇[i] L1 ≡ K1.ⓑ{I}V & K1 ≡[V, 0] K2 & l ≤ i. -#L1 #L2 #l #i #H #I #K2 #V #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2 -[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/ -qed-. - -lemma lleq_fwd_lref_sn: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → - ∀I,K1,V. ⬇[i] L1 ≡ K1.ⓑ{I}V → - i < l ∨ - ∃∃K2. ⬇[i] L2 ≡ K2.ⓑ{I}V & K1 ≡[V, 0] K2 & l ≤ i. -#L1 #L2 #l #i #H #I #K1 #V #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1 -[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/ -qed-. - -lemma lleq_fwd_bind_O_dx: ∀a,I,L1,L2,V,T. L1 ≡[ⓑ{a,I}V.T, 0] L2 → - L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V. -/2 width=2 by llpx_sn_fwd_bind_O_dx/ qed-. - -(* Properties on relocation *************************************************) - -lemma lleq_lift_le: ∀K1,K2,T,lt. K1 ≡[T, lt] K2 → - ∀L1,L2,l,k. ⬇[Ⓕ, l, k] L1 ≡ K1 → ⬇[Ⓕ, l, k] L2 ≡ K2 → - ∀U. ⬆[l, k] T ≡ U → lt ≤ l → L1 ≡[U, lt] L2. -/3 width=10 by llpx_sn_lift_le, lift_mono/ qed-. - -lemma lleq_lift_ge: ∀K1,K2,T,lt. K1 ≡[T, lt] K2 → - ∀L1,L2,l,k. ⬇[Ⓕ, l, k] L1 ≡ K1 → ⬇[Ⓕ, l, k] L2 ≡ K2 → - ∀U. ⬆[l, k] T ≡ U → l ≤ lt → L1 ≡[U, lt+k] L2. -/2 width=9 by llpx_sn_lift_ge/ qed-. - -(* Inversion lemmas on relocation *******************************************) - -lemma lleq_inv_lift_le: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 → - ∀K1,K2,l,k. ⬇[Ⓕ, l, k] L1 ≡ K1 → ⬇[Ⓕ, l, k] L2 ≡ K2 → - ∀T. ⬆[l, k] T ≡ U → lt ≤ l → K1 ≡[T, lt] K2. -/3 width=10 by llpx_sn_inv_lift_le, ex2_intro/ qed-. - -lemma lleq_inv_lift_be: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 → - ∀K1,K2,l,k. ⬇[Ⓕ, l, k] L1 ≡ K1 → ⬇[Ⓕ, l, k] L2 ≡ K2 → - ∀T. ⬆[l, k] T ≡ U → l ≤ lt → lt ≤ l + k → K1 ≡[T, l] K2. -/2 width=11 by llpx_sn_inv_lift_be/ qed-. - -lemma lleq_inv_lift_ge: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 → - ∀K1,K2,l,k. ⬇[Ⓕ, l, k] L1 ≡ K1 → ⬇[Ⓕ, l, k] L2 ≡ K2 → - ∀T. ⬆[l, k] T ≡ U → l + k ≤ lt → K1 ≡[T, lt-k] K2. -/2 width=9 by llpx_sn_inv_lift_ge/ qed-. - -(* Inversion lemmas on negated lazy quivalence for local environments *******) - -lemma nlleq_inv_bind: ∀a,I,L1,L2,V,T,l. (L1 ≡[ⓑ{a,I}V.T, l] L2 → ⊥) → - (L1 ≡[V, l] L2 → ⊥) ∨ (L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V → ⊥). -/3 width=2 by nllpx_sn_inv_bind, eq_term_dec/ qed-. - -lemma nlleq_inv_flat: ∀I,L1,L2,V,T,l. (L1 ≡[ⓕ{I}V.T, l] L2 → ⊥) → - (L1 ≡[V, l] L2 → ⊥) ∨ (L1 ≡[T, l] L2 → ⊥). -/3 width=2 by nllpx_sn_inv_flat, eq_term_dec/ qed-. - -lemma nlleq_inv_bind_O: ∀a,I,L1,L2,V,T. (L1 ≡[ⓑ{a,I}V.T, 0] L2 → ⊥) → - (L1 ≡[V, 0] L2 → ⊥) ∨ (L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V → ⊥). -/3 width=2 by nllpx_sn_inv_bind_O, eq_term_dec/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq_fqus.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq_fqus.etc deleted file mode 100644 index bf0cec6d2..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq_fqus.etc +++ /dev/null @@ -1,75 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/multiple/fqus_alt.ma". -include "basic_2/multiple/lleq_drop.ma". - -(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) - -(* Properties on supclosure *************************************************) - -lemma lleq_fqu_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐ ⦃G2, K2, U⦄ → - ∀L1. L1 ≡[T, 0] L2 → - ∃∃K1. ⦃G1, L1, T⦄ ⊐ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2. -#G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U -[ #I #G #L2 #V #L1 #H elim (lleq_inv_lref_ge_dx … H … I L2 V) -H // - #K1 #H1 #H2 lapply (drop_inv_O2 … H1) -H1 - #H destruct /2 width=3 by fqu_lref_O, ex2_intro/ -| * [ #a ] #I #G #L2 #V #T #L1 #H - [ elim (lleq_inv_bind … H) - | elim (lleq_inv_flat … H) - ] -H - /2 width=3 by fqu_pair_sn, ex2_intro/ -| #a #I #G #L2 #V #T #L1 #H elim (lleq_inv_bind_O … H) -H - #H3 #H4 /2 width=3 by fqu_bind_dx, ex2_intro/ -| #I #G #L2 #V #T #L1 #H elim (lleq_inv_flat … H) -H - /2 width=3 by fqu_flat_dx, ex2_intro/ -| #G #L2 #K2 #T #U #k #HLK2 #HTU #L1 #HL12 - elim (drop_O1_le (Ⓕ) (k+1) L1) - [ /3 width=12 by fqu_drop, lleq_inv_lift_le, ex2_intro/ - | lapply (drop_fwd_length_le2 … HLK2) -K2 - lapply (lleq_fwd_length … HL12) -T -U // - ] -] -qed-. - -lemma lleq_fquq_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐⸮ ⦃G2, K2, U⦄ → - ∀L1. L1 ≡[T, 0] L2 → - ∃∃K1. ⦃G1, L1, T⦄ ⊐⸮ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2. -#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fquq_inv_gen … H) -H -[ #H elim (lleq_fqu_trans … H … HL12) -L2 /3 width=3 by fqu_fquq, ex2_intro/ -| * #HG #HL #HT destruct /2 width=3 by ex2_intro/ -] -qed-. - -lemma lleq_fqup_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+ ⦃G2, K2, U⦄ → - ∀L1. L1 ≡[T, 0] L2 → - ∃∃K1. ⦃G1, L1, T⦄ ⊐+ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2. -#G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U -[ #G2 #K2 #U #HTU #L1 #HL12 elim (lleq_fqu_trans … HTU … HL12) -L2 - /3 width=3 by fqu_fqup, ex2_intro/ -| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12 elim (IHTU … HL12) -L2 - #K1 #HTU #HK1 elim (lleq_fqu_trans … HU2 … HK1) -K - /3 width=5 by fqup_strap1, ex2_intro/ -] -qed-. - -lemma lleq_fqus_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐* ⦃G2, K2, U⦄ → - ∀L1. L1 ≡[T, 0] L2 → - ∃∃K1. ⦃G1, L1, T⦄ ⊐* ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2. -#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_gen … H) -H -[ #H elim (lleq_fqup_trans … H … HL12) -L2 /3 width=3 by fqup_fqus, ex2_intro/ -| * #HG #HL #HT destruct /2 width=3 by ex2_intro/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq_llor.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq_llor.etc deleted file mode 100644 index e702043ea..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq_llor.etc +++ /dev/null @@ -1,41 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/multiple/llor.ma". -include "basic_2/multiple/llpx_sn_frees.ma". -include "basic_2/multiple/lleq_alt.ma". - -(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) - -(* Properties on pointwise union for local environments **********************) - -lemma llpx_sn_llor_dx: ∀R. (c_r_confluent1 … R (llpx_sn R 0)) → (frees_trans R) → - ∀L1,L2,T,l. llpx_sn R l T L1 L2 → ∀L. L1 ⋓[T, l] L2 ≡ L → L2 ≡[T, l] L. -#R #H1R #H2R #L1 #L2 #T #l #H1 #L #H2 -lapply (llpx_sn_frees_trans … H1R H2R … H1) -H1R -H2R #HR -elim (llpx_sn_llpx_sn_alt … H1) -H1 #HL12 #IH1 -elim H2 -H2 #_ #HL1 #IH2 -@lleq_intro_alt // #I2 #I #K2 #K #V2 #V #i #Hi #HnT #HLK2 #HLK -lapply (drop_fwd_length_lt2 … HLK) #HiL -elim (drop_O1_lt (Ⓕ) L1 i) // -HiL #I1 #K1 #V1 #HLK1 -elim (IH1 … HLK1 HLK2) -IH1 /2 width=1 by/ #H #_ destruct -elim (IH2 … HLK1 HLK2 HLK) -IH2 -HLK1 -HLK2 -HLK * /2 width=1 by conj/ #H -[ elim (ylt_yle_false … H) -H // -| elim H -H /2 width=1 by/ -] -qed. - -lemma llpx_sn_llor_dx_sym: ∀R. (c_r_confluent1 … R (llpx_sn R 0)) → (frees_trans R) → - ∀L1,L2,T,l. llpx_sn R l T L1 L2 → ∀L. L1 ⋓[T, l] L2 ≡ L → L ≡[T, l] L2. -/3 width=6 by llpx_sn_llor_dx, lleq_sym/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq_lreq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq_lreq.etc deleted file mode 100644 index 8fa21a5da..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lleq/lleq_lreq.etc +++ /dev/null @@ -1,36 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/multiple/llpx_sn_lreq.ma". -include "basic_2/multiple/lleq.ma". - -(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************) - -(* Properties on equivalence for local environments *************************) - -lemma lreq_lleq_trans: ∀L2,L,T,l. L2 ≡[T, l] L → - ∀L1. L1 ⩬[l, ∞] L2 → L1 ≡[T, l] L. -/2 width=3 by lreq_llpx_sn_trans/ qed-. - -lemma lleq_lreq_trans: ∀L,L1,T,l. L ≡[T, l] L1 → - ∀L2. L1 ⩬[l, ∞] L2 → L ≡[T, l] L2. -/2 width=3 by llpx_sn_lreq_trans/ qed-. - -lemma lleq_lreq_repl: ∀L1,L2,T,l. L1 ≡[T, l] L2 → ∀K1. K1 ⩬[l, ∞] L1 → - ∀K2. L2 ⩬[l, ∞] K2 → K1 ≡[T, l] K2. -/2 width=5 by llpx_sn_lreq_repl/ qed-. - -lemma lleq_bind_repl_SO: ∀I1,I2,L1,L2,V1,V2,T. L1.ⓑ{I1}V1 ≡[T, 0] L2.ⓑ{I2}V2 → - ∀J1,J2,W1,W2. L1.ⓑ{J1}W1 ≡[T, 1] L2.ⓑ{J2}W2. -/2 width=5 by llpx_sn_bind_repl_SO/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/llpx_sn/llpx_sn.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/llpx_sn/llpx_sn.etc deleted file mode 100644 index 5a1124b5c..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/llpx_sn/llpx_sn.etc +++ /dev/null @@ -1,45 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "ground_2/ynat/ynat_plus.ma". -include "basic_2/substitution/drop.ma". - -(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) - -| llpx_sn_lref: ∀I,L1,L2,K1,K2,V1,V2,l,i. l ≤ yinj i → - ⬇[i] L1 ≡ K1.ⓑ{I}V1 → ⬇[i] L2 ≡ K2.ⓑ{I}V2 → - llpx_sn R (yinj 0) V1 K1 K2 → R K1 V1 V2 → llpx_sn R l (#i) L1 L2 - -(* Basic forward lemmas *****************************************************) - -lemma llpx_sn_fwd_drop_sn: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → - ∀K1,i. ⬇[i] L1 ≡ K1 → ∃K2. ⬇[i] L2 ≡ K2. -#R #L1 #L2 #T #l #H #K1 #i #HLK1 lapply (llpx_sn_fwd_length … H) -H -#HL12 lapply (drop_fwd_length_le2 … HLK1) -HLK1 /2 width=1 by drop_O1_le/ -qed-. - -lemma llpx_sn_fwd_drop_dx: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → - ∀K2,i. ⬇[i] L2 ≡ K2 → ∃K1. ⬇[i] L1 ≡ K1. -#R #L1 #L2 #T #l #H #K2 #i #HLK2 lapply (llpx_sn_fwd_length … H) -H -#HL12 lapply (drop_fwd_length_le2 … HLK2) -HLK2 /2 width=1 by drop_O1_le/ -qed-. - -lemma llpx_sn_fwd_lref: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 → - ∨∨ |L1| ≤ i ∧ |L2| ≤ i - | yinj i < l - | ∃∃I,K1,K2,V1,V2. ⬇[i] L1 ≡ K1.ⓑ{I}V1 & - ⬇[i] L2 ≡ K2.ⓑ{I}V2 & - llpx_sn R (yinj 0) V1 K1 K2 & - R K1 V1 V2 & l ≤ yinj i. -/2 width=3 by llpx_sn_fwd_lref_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/llpx_sn/llpx_sn_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/llpx_sn/llpx_sn_alt.etc deleted file mode 100644 index ac0d12224..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/llpx_sn/llpx_sn_alt.etc +++ /dev/null @@ -1,63 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/multiple/frees.ma". -include "basic_2/multiple/llpx_sn_alt_rec.ma". - -(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) - -(* alternative definition of llpx_sn (not recursive) *) -definition llpx_sn_alt: relation3 lenv term term → relation4 ynat term lenv lenv ≝ - λR,l,T,L1,L2. |L1| = |L2| ∧ - (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - I1 = I2 ∧ R K1 V1 V2 - ). - -(* Main properties **********************************************************) - -theorem llpx_sn_llpx_sn_alt: ∀R,T,L1,L2,l. llpx_sn R l T L1 L2 → llpx_sn_alt R l T L1 L2. -#R #U #L1 @(f2_ind … rfw … L1 U) -L1 -U -#x #IHx #L1 #U #Hx #L2 #l #H elim (llpx_sn_inv_alt_r … H) -H -#HL12 #IHU @conj // -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2 elim (frees_inv … H) -H -[ -x #HnU elim (IHU … HnU HLK1 HLK2) -IHU -HnU -HLK1 -HLK2 /2 width=1 by conj/ -| * #J1 #K10 #W10 #j #Hlj #Hji #HnU #HLK10 yminus_inj >yminus_inj #HnW10 destruct - lapply (drop_fwd_drop2 … HLK10) #H - lapply (drop_conf_ge … H … HLK1 ?) -H /2 width=1 by ylt_fwd_le_succ1/ (minus_plus_k_k j (i+1)) in ⊢ (%→?); >commutative_plus yminus_SO2 -#HnV1 #HKY1 #HKY2 (**) (* full auto too slow *) -lapply (drop_trans_ge … H1 … HKY1 ?) -H1 -HKY1 // #HLY1 -lapply (drop_trans_ge … H2 … HKY2 ?) -H2 -HKY2 // #HLY2 -/4 width=9 by frees_be, yle_plus_dx2_trans, yle_succ_dx, ylt_inj/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/llpx_sn/llpx_sn_alt_rec.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/llpx_sn/llpx_sn_alt_rec.etc deleted file mode 100644 index 172ccf38a..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/llpx_sn/llpx_sn_alt_rec.etc +++ /dev/null @@ -1,249 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/substitution/lift_neg.ma". -include "basic_2/substitution/drop_drop.ma". -include "basic_2/multiple/llpx_sn.ma". - -(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) - -(* alternative definition of llpx_sn (recursive) *) -inductive llpx_sn_alt_r (R:relation3 lenv term term): relation4 ynat term lenv lenv ≝ -| llpx_sn_alt_r_intro: ∀L1,L2,T,l. - (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R K1 V1 V2 - ) → - (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → llpx_sn_alt_r R 0 V1 K1 K2 - ) → |L1| = |L2| → llpx_sn_alt_r R l T L1 L2 -. - -(* Compact definition of llpx_sn_alt_r **************************************) - -lemma llpx_sn_alt_r_intro_alt: ∀R,L1,L2,T,l. |L1| = |L2| → - (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2 - ) → llpx_sn_alt_r R l T L1 L2. -#R #L1 #L2 #T #l #HL12 #IH @llpx_sn_alt_r_intro // -HL12 -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 -elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/ -qed. - -lemma llpx_sn_alt_r_ind_alt: ∀R. ∀S:relation4 ynat term lenv lenv. - (∀L1,L2,T,l. |L1| = |L2| → ( - ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2 & S 0 V1 K1 K2 - ) → S l T L1 L2) → - ∀L1,L2,T,l. llpx_sn_alt_r R l T L1 L2 → S l T L1 L2. -#R #S #IH #L1 #L2 #T #l #H elim H -L1 -L2 -T -l -#L1 #L2 #T #l #H1 #H2 #HL12 #IH2 @IH -IH // -HL12 -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 -elim (H1 … HnT HLK1 HLK2) -H1 /4 width=8 by and4_intro/ -qed-. - -lemma llpx_sn_alt_r_inv_alt: ∀R,L1,L2,T,l. llpx_sn_alt_r R l T L1 L2 → - |L1| = |L2| ∧ - ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) → - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2. -#R #L1 #L2 #T #l #H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -l -#L1 #L2 #T #l #HL12 #IH @conj // -HL12 -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2 -elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/ -qed-. - -(* Basic inversion lemmas ***************************************************) - -lemma llpx_sn_alt_r_inv_flat: ∀R,I,L1,L2,V,T,l. llpx_sn_alt_r R l (ⓕ{I}V.T) L1 L2 → - llpx_sn_alt_r R l V L1 L2 ∧ llpx_sn_alt_r R l T L1 L2. -#R #I #L1 #L2 #V #T #l #H elim (llpx_sn_alt_r_inv_alt … H) -H -#HL12 #IH @conj @llpx_sn_alt_r_intro_alt // -HL12 -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2 -elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 // -/3 width=8 by nlift_flat_sn, nlift_flat_dx, and3_intro/ -qed-. - -lemma llpx_sn_alt_r_inv_bind: ∀R,a,I,L1,L2,V,T,l. llpx_sn_alt_r R l (ⓑ{a,I}V.T) L1 L2 → - llpx_sn_alt_r R l V L1 L2 ∧ llpx_sn_alt_r R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V). -#R #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_alt_r_inv_alt … H) -H -#HL12 #IH @conj @llpx_sn_alt_r_intro_alt [1,3: normalize // ] -HL12 -#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2 -[ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 - /3 width=9 by nlift_bind_sn, and3_intro/ -| lapply (yle_inv_succ1 … Hli) -Hli * #Hli #Hi yminus_Y_inj #K1 #HK12 #HLK1 - lapply (lreq_inv_O_Y … HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/ -| /4 width=5 by llpx_sn_free, lreq_fwd_length, le_repl_sn_trans_aux, trans_eq/ -| /4 width=1 by llpx_sn_bind, lreq_succ/ -] -qed-. - -lemma llpx_sn_lreq_trans: ∀R,L,L1,T,l. llpx_sn R l T L L1 → - ∀L2. L1 ⩬[l, ∞] L2 → llpx_sn R l T L L2. -#R #L #L1 #T #l #H elim H -L -L1 -T -l -/4 width=5 by llpx_sn_flat, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, lreq_fwd_length, trans_eq/ -[ #I #L #L1 #K #K1 #V #V1 #l #i #Hli #HLK #HLK1 #HK1 #HV1 #_ #L2 #HL12 - elim (lreq_drop_conf_be … HL12 … HLK1) -L1 // >yminus_Y_inj #K2 #HK12 #HLK2 - lapply (lreq_inv_O_Y … HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/ -| /4 width=5 by llpx_sn_free, lreq_fwd_length, le_repl_sn_conf_aux, trans_eq/ -| /4 width=1 by llpx_sn_bind, lreq_succ/ -] -qed-. - -lemma llpx_sn_lreq_repl: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → ∀K1. K1 ⩬[l, ∞] L1 → - ∀K2. L2 ⩬[l, ∞] K2 → llpx_sn R l T K1 K2. -/3 width=4 by llpx_sn_lreq_trans, lreq_llpx_sn_trans/ qed-. - -lemma llpx_sn_bind_repl_SO: ∀R,I1,I2,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) → - ∀J1,J2,W1,W2. llpx_sn R 1 T (L1.ⓑ{J1}W1) (L2.ⓑ{J2}W2). -#R #I1 #I2 #L1 #L2 #V1 #V2 #T #HT #J1 #J2 #W1 #W2 lapply (llpx_sn_ge R … 1 … HT) -HT -/3 width=7 by llpx_sn_lreq_repl, lreq_succ/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/llpx_sn/llpx_sn_tc.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/llpx_sn/llpx_sn_tc.etc deleted file mode 100644 index 13fc14e82..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/llpx_sn/llpx_sn_tc.etc +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/multiple/llpx_sn_drop.ma". - -(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) - -(* Properties about transitive closure **************************************) - -lemma llpx_sn_TC_pair_dx: ∀R. (∀L. reflexive … (R L)) → - ∀I,L,V1,V2,T. LTC … R L V1 V2 → - LTC … (llpx_sn R 0) T (L.ⓑ{I}V1) (L.ⓑ{I}V2). -#R #HR #I #L #V1 #V2 #T #H @(TC_star_ind … V2 H) -V2 -/4 width=9 by llpx_sn_bind_repl_O, llpx_sn_refl, step, inj/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lpx/lpx_fquq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/lpx/lpx_fquq.etc deleted file mode 100644 index 799808fc6..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lpx/lpx_fquq.etc +++ /dev/null @@ -1,47 +0,0 @@ -(* Properties on supclosure *************************************************) - -lemma fqu_lpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → - ∀K2. ⦃G2, L2⦄ ⊢ ➡[h, o] K2 → - ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h, o] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h, o] T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄. -#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 -/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_flat_dx, lpx_pair, ex3_2_intro/ -[ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpx_inv_pair1 … H) -H - #K2 #W2 #HLK2 #HVW2 #H destruct - /3 width=5 by cpx_pair_sn, fqu_bind_dx, ex3_2_intro/ -| #G #L1 #K1 #T1 #U1 #k #HLK1 #HTU1 #K2 #HK12 - elim (drop_lpx_trans … HLK1 … HK12) -HK12 - /3 width=7 by fqu_drop, ex3_2_intro/ -] -qed-. - -lemma fquq_lpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → - ∀K2. ⦃G2, L2⦄ ⊢ ➡[h, o] K2 → - ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h, o] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h, o] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄. -#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 elim (fquq_inv_gen … H) -H -[ #HT12 elim (fqu_lpx_trans … HT12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/ -| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/ -] -qed-. - -lemma lpx_fqu_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → - ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → - ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, o] T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2. -#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 -/3 width=7 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpx_pair, ex3_2_intro/ -[ #I #G1 #L1 #V1 #X #H elim (lpx_inv_pair2 … H) -H - #K1 #W1 #HKL1 #HWV1 #H destruct elim (lift_total V1 0 1) - /4 width=7 by cpx_delta, fqu_drop, drop_drop, ex3_2_intro/ -| #G #L1 #K1 #T1 #U1 #k #HLK1 #HTU1 #L0 #HL01 - elim (lpx_drop_trans_O1 … HL01 … HLK1) -L1 - /3 width=5 by fqu_drop, ex3_2_intro/ -] -qed-. - -lemma lpx_fquq_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → - ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → - ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, o] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2. -#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fquq_inv_gen … H) -H -[ #HT12 elim (lpx_fqu_trans … HT12 … HKL1) /3 width=5 by fqu_fquq, ex3_2_intro/ -| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lpx_sn/lpx_sn_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/lpx_sn/lpx_sn_alt.etc deleted file mode 100644 index 5099ffa1a..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lpx_sn/lpx_sn_alt.etc +++ /dev/null @@ -1,127 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/substitution/drop.ma". -include "basic_2/substitution/lpx_sn.ma". - -(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********) - -(* alternative definition of lpx_sn *) -definition lpx_sn_alt: relation3 lenv term term → relation lenv ≝ - λR,L1,L2. |L1| = |L2| ∧ - (∀I1,I2,K1,K2,V1,V2,i. - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - I1 = I2 ∧ R K1 V1 V2 - ). - -(* Basic forward lemmas ******************************************************) - -lemma lpx_sn_alt_fwd_length: ∀R,L1,L2. lpx_sn_alt R L1 L2 → |L1| = |L2|. -#R #L1 #L2 #H elim H // -qed-. - -(* Basic inversion lemmas ***************************************************) - -lemma lpx_sn_alt_inv_atom1: ∀R,L2. lpx_sn_alt R (⋆) L2 → L2 = ⋆. -#R #L2 #H lapply (lpx_sn_alt_fwd_length … H) -H -normalize /2 width=1 by length_inv_zero_sn/ -qed-. - -lemma lpx_sn_alt_inv_pair1: ∀R,I,L2,K1,V1. lpx_sn_alt R (K1.ⓑ{I}V1) L2 → - ∃∃K2,V2. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2. -#R #I1 #L2 #K1 #V1 #H elim H -H -#H #IH elim (length_inv_pos_sn … H) -H -#I2 #K2 #V2 #HK12 #H destruct -elim (IH I1 I2 K1 K2 V1 V2 0) // -#H #HV12 destruct @(ex3_2_intro … K2 V2) // -HV12 -@conj // -HK12 -#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (⫯i)) -IH -/2 width=1 by drop_drop, conj/ -qed-. - -lemma lpx_sn_alt_inv_atom2: ∀R,L1. lpx_sn_alt R L1 (⋆) → L1 = ⋆. -#R #L1 #H lapply (lpx_sn_alt_fwd_length … H) -H -normalize /2 width=1 by length_inv_zero_dx/ -qed-. - -lemma lpx_sn_alt_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn_alt R L1 (K2.ⓑ{I}V2) → - ∃∃K1,V1. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1. -#R #I2 #L1 #K2 #V2 #H elim H -H -#H #IH elim (length_inv_pos_dx … H) -H -#I1 #K1 #V1 #HK12 #H destruct -elim (IH I1 I2 K1 K2 V1 V2 0) // -#H #HV12 destruct @(ex3_2_intro … K1 V1) // -HV12 -@conj // -HK12 -#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (⫯i)) -IH -/2 width=1 by drop_drop, conj/ -qed-. - -(* Basic properties *********************************************************) - -lemma lpx_sn_alt_atom: ∀R. lpx_sn_alt R (⋆) (⋆). -#R @conj // -#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 elim (drop_inv_atom1 … HLK1) -HLK1 -#H destruct -qed. - -lemma lpx_sn_alt_pair: ∀R,I,L1,L2,V1,V2. - lpx_sn_alt R L1 L2 → R L1 V1 V2 → - lpx_sn_alt R (L1.ⓑ{I}V1) (L2.ⓑ{I}V2). -#R #I #L1 #L2 #V1 #V2 #H #HV12 elim H -H -#HL12 #IH @conj // -#I1 #I2 #K1 #K2 #W1 #W2 #i @(ynat_ind … i) -i -[ #HLK1 #HLK2 - lapply (drop_inv_O2 … HLK1) -HLK1 #H destruct - lapply (drop_inv_O2 … HLK2) -HLK2 #H destruct - /2 width=1 by conj/ -| -HL12 -HV12 /3 width=6 by drop_inv_drop1/ -| #H lapply (drop_fwd_Y2 … H) -H - #H elim (ylt_yle_false … H) -H // -] -qed. - -(* Main properties **********************************************************) - -theorem lpx_sn_lpx_sn_alt: ∀R,L1,L2. lpx_sn R L1 L2 → lpx_sn_alt R L1 L2. -#R #L1 #L2 #H elim H -L1 -L2 -/2 width=1 by lpx_sn_alt_atom, lpx_sn_alt_pair/ -qed. - -(* Main inversion lemmas ****************************************************) - -theorem lpx_sn_alt_inv_lpx_sn: ∀R,L1,L2. lpx_sn_alt R L1 L2 → lpx_sn R L1 L2. -#R #L1 elim L1 -L1 -[ #L2 #H lapply (lpx_sn_alt_inv_atom1 … H) -H // -| #L1 #I #V1 #IH #X #H elim (lpx_sn_alt_inv_pair1 … H) -H - #L2 #V2 #HL12 #HV12 #H destruct /3 width=1 by lpx_sn_pair/ -] -qed-. - -(* alternative definition of lpx_sn *****************************************) - -lemma lpx_sn_intro_alt: ∀R,L1,L2. |L1| = |L2| → - (∀I1,I2,K1,K2,V1,V2,i. - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - I1 = I2 ∧ R K1 V1 V2 - ) → lpx_sn R L1 L2. -/4 width=4 by lpx_sn_alt_inv_lpx_sn, conj/ qed. - -lemma lpx_sn_inv_alt: ∀R,L1,L2. lpx_sn R L1 L2 → - |L1| = |L2| ∧ - ∀I1,I2,K1,K2,V1,V2,i. - ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → - I1 = I2 ∧ R K1 V1 V2. -#R #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H -#H elim H -H /3 width=4 by conj/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lpx_sn/lpx_sn_tc.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/lpx_sn/lpx_sn_tc.etc deleted file mode 100644 index 23eaab6ab..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lpx_sn/lpx_sn_tc.etc +++ /dev/null @@ -1,119 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/substitution/lpx_sn.ma". - -(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********) - -(* Properties on transitive_closure *****************************************) - -lemma TC_lpx_sn_pair_refl: ∀R. (∀L. reflexive … (R L)) → - ∀L1,L2. TC … (lpx_sn R) L1 L2 → - ∀I,V. TC … (lpx_sn R) (L1. ⓑ{I} V) (L2. ⓑ{I} V). -#R #HR #L1 #L2 #H @(TC_star_ind … L2 H) -L2 -[ /2 width=1 by lpx_sn_refl/ -| /3 width=1 by TC_reflexive, lpx_sn_refl/ -| /3 width=5 by lpx_sn_pair, step/ -] -qed-. - -lemma TC_lpx_sn_pair: ∀R. (∀L. reflexive … (R L)) → - ∀I,L1,L2. TC … (lpx_sn R) L1 L2 → - ∀V1,V2. LTC … R L1 V1 V2 → - TC … (lpx_sn R) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2). -#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx … V1 H) -V1 // -[ /2 width=1 by TC_lpx_sn_pair_refl/ -| /4 width=3 by TC_strap, lpx_sn_pair, lpx_sn_refl/ -] -qed-. - -lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀L. reflexive … (R L)) → - ∀L1,L2. lpx_sn (LTC … R) L1 L2 → - TC … (lpx_sn R) L1 L2. -#R #HR #L1 #L2 #H elim H -L1 -L2 -/2 width=1 by TC_lpx_sn_pair, lpx_sn_atom, inj/ -qed-. - -(* Inversion lemmas on transitive closure ***********************************) - -lemma TC_lpx_sn_inv_atom2: ∀R,L1. TC … (lpx_sn R) L1 (⋆) → L1 = ⋆. -#R #L1 #H @(TC_ind_dx … L1 H) -L1 -[ /2 width=2 by lpx_sn_inv_atom2/ -| #L1 #L #HL1 #_ #IHL2 destruct /2 width=2 by lpx_sn_inv_atom2/ -] -qed-. - -lemma TC_lpx_sn_inv_pair2: ∀R. c_rs_transitive … R (λ_. lpx_sn R) → - ∀I,L1,K2,V2. TC … (lpx_sn R) L1 (K2.ⓑ{I}V2) → - ∃∃K1,V1. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L1 = K1. ⓑ{I} V1. -#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx … L1 H) -L1 -[ #L1 #H elim (lpx_sn_inv_pair2 … H) -H /3 width=5 by inj, ex3_2_intro/ -| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct - elim (lpx_sn_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct - lapply (HR … HV2 … HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/ -] -qed-. - -lemma TC_lpx_sn_ind: ∀R. c_rs_transitive … R (λ_. lpx_sn R) → - ∀S:relation lenv. - S (⋆) (⋆) → ( - ∀I,K1,K2,V1,V2. - TC … (lpx_sn R) K1 K2 → LTC … R K1 V1 V2 → - S K1 K2 → S (K1.ⓑ{I}V1) (K2.ⓑ{I}V2) - ) → - ∀L2,L1. TC … (lpx_sn R) L1 L2 → S L1 L2. -#R #HR #S #IH1 #IH2 #L2 elim L2 -L2 -[ #X #H >(TC_lpx_sn_inv_atom2 … H) -X // -| #L2 #I #V2 #IHL2 #X #H - elim (TC_lpx_sn_inv_pair2 … H) // -H -HR - #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/ -] -qed-. - -lemma TC_lpx_sn_inv_atom1: ∀R,L2. TC … (lpx_sn R) (⋆) L2 → L2 = ⋆. -#R #L2 #H elim H -L2 -[ /2 width=2 by lpx_sn_inv_atom1/ -| #L #L2 #_ #HL2 #IHL1 destruct /2 width=2 by lpx_sn_inv_atom1/ -] -qed-. - -fact TC_lpx_sn_inv_pair1_aux: ∀R. c_rs_transitive … R (λ_. lpx_sn R) → - ∀L1,L2. TC … (lpx_sn R) L1 L2 → - ∀I,K1,V1. L1 = K1.ⓑ{I}V1 → - ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2. -#R #HR #L1 #L2 #H @(TC_lpx_sn_ind … H) // -HR -L1 -L2 -[ #J #K #W #H destruct -| #I #L1 #L2 #V1 #V2 #HL12 #HV12 #_ #J #K #W #H destruct /2 width=5 by ex3_2_intro/ -] -qed-. - -lemma TC_lpx_sn_inv_pair1: ∀R. c_rs_transitive … R (λ_. lpx_sn R) → - ∀I,K1,L2,V1. TC … (lpx_sn R) (K1.ⓑ{I}V1) L2 → - ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2. -/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-. - -lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. c_rs_transitive … R (λ_. lpx_sn R) → - ∀L1,L2. TC … (lpx_sn R) L1 L2 → - lpx_sn (LTC … R) L1 L2. -/3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-. - -(* Forward lemmas on transitive closure *************************************) - -lemma TC_lpx_sn_fwd_length: ∀R,L1,L2. TC … (lpx_sn R) L1 L2 → |L1| = |L2|. -#R #L1 #L2 #H elim H -L2 -[ #L2 #HL12 >(lpx_sn_fwd_length … HL12) -HL12 // -| #L #L2 #_ #HL2 #IHL1 - >IHL1 -L1 >(lpx_sn_fwd_length … HL2) -HL2 // -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lrsubeq_4.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/lrsubeq_4.etc deleted file mode 100644 index 6135be4d4..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lrsubeq_4.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( L1 break ⊆ [ term 46 l , break term 46 m ] break term 46 L2 )" - non associative with precedence 45 - for @{ 'LRSubEq $L1 $l $k $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lstas/lstas.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/lstas/lstas.etc deleted file mode 100644 index e9f034116..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lstas/lstas.etc +++ /dev/null @@ -1,190 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/notation/relations/statictypestar_6.ma". -include "basic_2/grammar/genv.ma". -include "basic_2/substitution/drop.ma". -include "basic_2/static/sh.ma". - -(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************) - -(* activate genv *) -inductive lstas (h): nat → relation4 genv lenv term term ≝ -| lstas_sort: ∀G,L,d,s. lstas h d G L (⋆s) (⋆((next h)^d s)) -| lstas_ldef: ∀G,L,K,V,W,U,i,d. ⬇[i] L ≡ K.ⓓV → lstas h d G K V W → - ⬆[0, i+1] W ≡ U → lstas h d G L (#i) U -| lstas_zero: ∀G,L,K,W,V,i. ⬇[i] L ≡ K.ⓛW → lstas h 0 G K W V → - lstas h 0 G L (#i) (#i) -| lstas_succ: ∀G,L,K,W,V,U,i,d. ⬇[i] L ≡ K.ⓛW → lstas h d G K W V → - ⬆[0, i+1] V ≡ U → lstas h (d+1) G L (#i) U -| lstas_bind: ∀a,I,G,L,V,T,U,d. lstas h d G (L.ⓑ{I}V) T U → - lstas h d G L (ⓑ{a,I}V.T) (ⓑ{a,I}V.U) -| lstas_appl: ∀G,L,V,T,U,d. lstas h d G L T U → lstas h d G L (ⓐV.T) (ⓐV.U) -| lstas_cast: ∀G,L,W,T,U,d. lstas h d G L T U → lstas h d G L (ⓝW.T) U -. - -interpretation "nat-iterated static type assignment (term)" - 'StaticTypeStar h G L d T U = (lstas h d G L T U). - -(* Basic inversion lemmas ***************************************************) - -fact lstas_inv_sort1_aux: ∀h,G,L,T,U,d. ⦃G, L⦄ ⊢ T •*[h, d] U → ∀s0. T = ⋆s0 → - U = ⋆((next h)^d s0). -#h #G #L #T #U #d * -G -L -T -U -d -[ #G #L #d #s #s0 #H destruct // -| #G #L #K #V #W #U #i #d #_ #_ #_ #s0 #H destruct -| #G #L #K #W #V #i #_ #_ #s0 #H destruct -| #G #L #K #W #V #U #i #d #_ #_ #_ #s0 #H destruct -| #a #I #G #L #V #T #U #d #_ #s0 #H destruct -| #G #L #V #T #U #d #_ #s0 #H destruct -| #G #L #W #T #U #d #_ #s0 #H destruct -qed-. - -(* Basic_1: was just: sty0_gen_sort *) -lemma lstas_inv_sort1: ∀h,G,L,X,s,d. ⦃G, L⦄ ⊢ ⋆s •*[h, d] X → X = ⋆((next h)^d s). -/2 width=5 by lstas_inv_sort1_aux/ -qed-. - -fact lstas_inv_lref1_aux: ∀h,G,L,T,U,d. ⦃G, L⦄ ⊢ T •*[h, d] U → ∀j. T = #j → ∨∨ - (∃∃K,V,W. ⬇[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, d] W & - ⬆[0, j+1] W ≡ U - ) | - (∃∃K,W,V. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, 0] V & - U = #j & d = 0 - ) | - (∃∃K,W,V,d0. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, d0] V & - ⬆[0, j+1] V ≡ U & d = d0+1 - ). -#h #G #L #T #U #d * -G -L -T -U -d -[ #G #L #d #s #j #H destruct -| #G #L #K #V #W #U #i #d #HLK #HVW #HWU #j #H destruct /3 width=6 by or3_intro0, ex3_3_intro/ -| #G #L #K #W #V #i #HLK #HWV #j #H destruct /3 width=5 by or3_intro1, ex4_3_intro/ -| #G #L #K #W #V #U #i #d #HLK #HWV #HWU #j #H destruct /3 width=8 by or3_intro2, ex4_4_intro/ -| #a #I #G #L #V #T #U #d #_ #j #H destruct -| #G #L #V #T #U #d #_ #j #H destruct -| #G #L #W #T #U #d #_ #j #H destruct -] -qed-. - -lemma lstas_inv_lref1: ∀h,G,L,X,i,d. ⦃G, L⦄ ⊢ #i •*[h, d] X → ∨∨ - (∃∃K,V,W. ⬇[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, d] W & - ⬆[0, i+1] W ≡ X - ) | - (∃∃K,W,V. ⬇[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, 0] V & - X = #i & d = 0 - ) | - (∃∃K,W,V,d0. ⬇[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, d0] V & - ⬆[0, i+1] V ≡ X & d = d0+1 - ). -/2 width=3 by lstas_inv_lref1_aux/ -qed-. - -lemma lstas_inv_lref1_O: ∀h,G,L,X,i. ⦃G, L⦄ ⊢ #i •*[h, 0] X → - (∃∃K,V,W. ⬇[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, 0] W & - ⬆[0, i+1] W ≡ X - ) ∨ - (∃∃K,W,V. ⬇[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, 0] V & - X = #i - ). -#h #G #L #X #i #H elim (lstas_inv_lref1 … H) -H * /3 width=6 by ex3_3_intro, or_introl, or_intror/ -#K #W #V #d #_ #_ #_ (lift_inv_sort1 … H1) -X1 - >(lift_inv_sort1 … H2) -X2 // -| #G #L1 #K1 #V1 #W1 #W #i #d #HLK1 #_ #HW1 #IHVW1 #L2 #c #l #k #HL21 #X #H #U2 #HWU2 - elim (lift_inv_lref1 … H) * #Hil #H destruct - [ elim (lift_trans_ge … HW1 … HWU2) -W /2 width=1 by ylt_fwd_le_succ1/ #W2 #HW12 #HWU2 - elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by ylt_fwd_le/ #X #HLK2 #H - elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K2 #V2 #HK21 #HV12 #H destruct - /3 width=9 by lstas_ldef/ - | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W /2 width=1 by yle_succ_dx/ #HW1U2 - lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 /3 width=9 by lstas_ldef, drop_inv_gen/ - ] -| #G #L1 #K1 #V1 #W1 #i #HLK1 #_ #IHVW1 #L2 #c #l #k #HL21 #X #H #U2 #HWU2 - >(lift_mono … HWU2 … H) -U2 - elim (lift_inv_lref1 … H) * #Hil #H destruct - [ elim (lift_total W1 (l-i-1) k) #W2 #HW12 - elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by ylt_fwd_le/ #X #HLK2 #H - elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K2 #V2 #HK21 #HV12 #H destruct - /3 width=10 by lstas_zero/ - | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 - /3 width=10 by lstas_zero, drop_inv_gen/ - ] -| #G #L1 #K1 #W1 #V1 #W #i #d #HLK1 #_ #HW1 #IHWV1 #L2 #c #l #k #HL21 #X #H #U2 #HWU2 - elim (lift_inv_lref1 … H) * #Hil #H destruct - [ elim (lift_trans_ge … HW1 … HWU2) -W /2 width=1 by ylt_fwd_le_succ1/ #W #HW1 #HWU2 - elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by ylt_fwd_le/ #X #HLK2 #H - elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K2 #W2 #HK21 #HW12 #H destruct - /3 width=9 by lstas_succ/ - | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W /2 width=1 by yle_succ_dx/ #HW1U2 - lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 /3 width=9 by lstas_succ, drop_inv_gen/ - ] -| #a #I #G #L1 #V1 #T1 #U1 #d #_ #IHTU1 #L2 #c #l #k #HL21 #X1 #H1 #X2 #H2 - elim (lift_inv_bind1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct - elim (lift_inv_bind1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct - lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=6 by lstas_bind, drop_skip/ -| #G #L1 #V1 #T1 #U1 #d #_ #IHTU1 #L2 #c #l #k #HL21 #X1 #H1 #X2 #H2 - elim (lift_inv_flat1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct - elim (lift_inv_flat1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct - lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=6 by lstas_appl/ -| #G #L1 #W1 #T1 #U1 #d #_ #IHTU1 #L2 #c #l #k #HL21 #X #H #U2 #HU12 - elim (lift_inv_flat1 … H) -H #W2 #T2 #_ #HT12 #H destruct /3 width=6 by lstas_cast/ -] -qed. - -(* Inversion lemmas on relocation *******************************************) - -(* Note: apparently this was missing in basic_1 *) -lemma lstas_inv_lift1: ∀h,G,d. d_deliftable_sn (lstas h G d). -#h #G #d #L2 #T2 #U2 #H elim H -G -L2 -T2 -U2 -d -[ #G #L2 #d #s #L1 #c #l #k #_ #X #H - >(lift_inv_sort2 … H) -X /2 width=3 by lstas_sort, lift_sort, ex2_intro/ -| #G #L2 #K2 #V2 #W2 #W #i #d #HLK2 #HVW2 #HW2 #IHVW2 #L1 #c #l #k #HL21 #X #H - elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HVW2 | -IHVW2 ] - [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #V1 #HLK1 #HK21 #HV12 - elim (IHVW2 … HK21 … HV12) -K2 -V2 #W1 #HW12 #HVW1 - elim (lift_trans_le … HW12 … HW2) -W2 // yplus_SO2 >ymax_pre_sn /3 width=8 by lstas_ldef, ylt_fwd_le_succ1, ex2_intro/ - | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2 - elim (yle_inv_plus_inj2 … Hil) -Hil #Hlim #mi - elim (lift_split … HW2 l (i-k+1)) -HW2 /2 width=1 by yle_succ_dx, le_S_S/ - #W0 #HW20 minus_minus_k_k /3 width=8 by lstas_ldef, yle_inv_inj, le_S, ex2_intro/ - ] -| #G #L2 #K2 #W2 #V2 #i #HLK2 #HWV2 #IHWV2 #L1 #c #l #k #HL21 #X #H - elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HWV2 | -IHWV2 ] - [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12 - elim (IHWV2 … HK21 … HW12) -K2 - /3 width=5 by lstas_zero, lift_lref_lt, ex2_intro/ - | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 - /3 width=5 by lstas_zero, lift_lref_ge_minus, ex2_intro/ - ] -| #G #L2 #K2 #W2 #V2 #W #i #d #HLK2 #HWV2 #HW2 #IHWV2 #L1 #c #l #k #HL21 #X #H - elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HWV2 | -IHWV2 ] - [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12 - elim (IHWV2 … HK21 … HW12) -K2 #V1 #HV12 #HWV1 - elim (lift_trans_le … HV12 … HW2) -W2 // yplus_SO2 >ymax_pre_sn /3 width=8 by lstas_succ, ylt_fwd_le_succ1, ex2_intro/ - | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2 - elim (yle_inv_plus_inj2 … Hil) -Hil #Hlim #mi - elim (lift_split … HW2 l (i-k+1)) -HW2 /2 width=1 by yle_succ_dx, le_S_S/ - #W0 #HW20 minus_minus_k_k /3 width=8 by lstas_succ, yle_inv_inj, le_S, ex2_intro/ - ] -| #a #I #G #L2 #V2 #T2 #U2 #d #_ #IHTU2 #L1 #c #l #k #HL21 #X #H - elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct - elim (IHTU2 (L1.ⓑ{I}V1) … HT12) -IHTU2 -HT12 /3 width=5 by lstas_bind, drop_skip, lift_bind, ex2_intro/ -| #G #L2 #V2 #T2 #U2 #d #_ #IHTU2 #L1 #c #l #k #HL21 #X #H - elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct - elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=5 by lstas_appl, lift_flat, ex2_intro/ -| #G #L2 #W2 #T2 #U2 #d #_ #IHTU2 #L1 #c #l #k #HL21 #X #H - elim (lift_inv_flat2 … H) -H #W1 #T1 #_ #HT12 #H destruct - elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=3 by lstas_cast, ex2_intro/ -] -qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma lstas_split_aux: ∀h,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*[h, d] T2 → ∀d1,d2. d = d1 + d2 → - ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, d1] T & ⦃G, L⦄ ⊢ T •*[h, d2] T2. -#h #G #L #T1 #T2 #d #H elim H -G -L -T1 -T2 -d -[ #G #L #d #s #d1 #d2 #H destruct - >commutative_plus >iter_plus /2 width=3 by lstas_sort, ex2_intro/ -| #G #L #K #V1 #V2 #U2 #i #d #HLK #_ #VU2 #IHV12 #d1 #d2 #H destruct - elim (IHV12 d1 d2) -IHV12 // #V - elim (lift_total V 0 (i+1)) - lapply (drop_fwd_drop2 … HLK) - /3 width=12 by lstas_lift, lstas_ldef, ex2_intro/ -| #G #L #K #W1 #W2 #i #HLK #HW12 #_ #d1 #d2 #H - elim (zero_eq_plus … H) -H #H1 #H2 destruct - /3 width=5 by lstas_zero, ex2_intro/ -| #G #L #K #W1 #W2 #U2 #i #d #HLK #HW12 #HWU2 #IHW12 #d1 @(nat_ind_plus … d1) -d1 - [ #d2 normalize #H destruct - elim (IHW12 0 d) -IHW12 // - lapply (drop_fwd_drop2 … HLK) - /3 width=8 by lstas_succ, lstas_zero, ex2_intro/ - | #d1 #_ #d2 (lstas_inv_sort1 … H) -X - (lstas_inv_sort1 … H) -X // -| #G #L #K #V #V1 #U1 #i #d #HLK #_ #HVU1 #IHV1 #X #H - elim (lstas_inv_lref1 … H) -H * - #K0 #V0 #W0 [3: #d0 ] #HLK0 - lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct - #HVW0 #HX lapply (IHV1 … HVW0) -IHV1 -HVW0 #H destruct - /2 width=5 by lift_mono/ -| #G #L #K #W #W1 #i #HLK #_ #_ #X #H - elim (lstas_inv_lref1_O … H) -H * - #K0 #V0 #W0 #HLK0 - lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct // -| #G #L #K #W #W1 #U1 #i #d #HLK #_ #HWU1 #IHWV #X #H - elim (lstas_inv_lref1_S … H) -H * #K0 #W0 #V0 #HLK0 - lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct - #HW0 #HX lapply (IHWV … HW0) -IHWV -HW0 #H destruct - /2 width=5 by lift_mono/ -| #a #I #G #L #V #T #U1 #d #_ #IHTU1 #X #H - elim (lstas_inv_bind1 … H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/ -| #G #L #V #T #U1 #d #_ #IHTU1 #X #H - elim (lstas_inv_appl1 … H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/ -| #G #L #W #T #U1 #d #_ #IHTU1 #U2 #H - lapply (lstas_inv_cast1 … H) -H /2 width=1 by/ -] -qed-. - -(* Advanced inversion lemmas ************************************************) - -(* Basic_1: was just: sty0_correct *) -lemma lstas_correct: ∀h,G,L,T1,T,d1. ⦃G, L⦄ ⊢ T1 •*[h, d1] T → - ∀d2. ∃T2. ⦃G, L⦄ ⊢ T •*[h, d2] T2. -#h #G #L #T1 #T #d1 #H elim H -G -L -T1 -T -d1 -[ /2 width=2 by lstas_sort, ex_intro/ -| #G #L #K #V1 #V #U #i #d #HLK #_ #HVU #IHV1 #d2 - elim (IHV1 d2) -IHV1 #V2 - elim (lift_total V2 0 (i+1)) - lapply (drop_fwd_drop2 … HLK) -HLK - /3 width=11 by ex_intro, lstas_lift/ -| #G #L #K #W1 #W #i #HLK #HW1 #IHW1 #d2 - @(nat_ind_plus … d2) -d2 /3 width=5 by lstas_zero, ex_intro/ - #d2 #_ elim (IHW1 d2) -IHW1 #W2 #HW2 - lapply (lstas_trans … HW1 … HW2) -W - elim (lift_total W2 0 (i+1)) - /3 width=7 by lstas_succ, ex_intro/ -| #G #L #K #W1 #W #U #i #d #HLK #_ #HWU #IHW1 #d2 - elim (IHW1 d2) -IHW1 #W2 - elim (lift_total W2 0 (i+1)) - lapply (drop_fwd_drop2 … HLK) -HLK - /3 width=11 by ex_intro, lstas_lift/ -| #a #I #G #L #V #T #U #d #_ #IHTU #d2 - elim (IHTU d2) -IHTU /3 width=2 by lstas_bind, ex_intro/ -| #G #L #V #T #U #d #_ #IHTU #d2 - elim (IHTU d2) -IHTU /3 width=2 by lstas_appl, ex_intro/ -| #G #L #W #T #U #d #_ #IHTU #d2 - elim (IHTU d2) -IHTU /2 width=2 by ex_intro/ -] -qed-. - -(* more main properties *****************************************************) - -theorem lstas_conf_le: ∀h,G,L,T,U1,d1. ⦃G, L⦄ ⊢ T •*[h, d1] U1 → - ∀U2,d2. d1 ≤ d2 → ⦃G, L⦄ ⊢ T •*[h, d2] U2 → - ⦃G, L⦄ ⊢ U1 •*[h, d2-d1] U2. -#h #G #L #T #U1 #d1 #HTU1 #U2 #d2 #Hd12 ->(plus_minus_k_k … Hd12) in ⊢ (%→?); -Hd12 >commutative_plus #H -elim (lstas_split … H) -H #U #HTU ->(lstas_mono … HTU … HTU1) -T // -qed-. - -theorem lstas_conf: ∀h,G,L,T0,T1,d1. ⦃G, L⦄ ⊢ T0 •*[h, d1] T1 → - ∀T2,d2. ⦃G, L⦄ ⊢ T0 •*[h, d2] T2 → - ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, d2] T & ⦃G, L⦄ ⊢ T2 •*[h, d1] T. -#h #G #L #T0 #T1 #d1 #HT01 #T2 #d2 #HT02 -elim (lstas_lstas … HT01 (d1+d2)) #T #HT0 -lapply (lstas_conf_le … HT01 … HT0) // -HT01 (lsubd_inv_atom1 … H) -H // -| #I #L1 #L #Y #HL1 #IHL1 #X #H - elim (lsubd_inv_pair1 … H) -H * #L2 - [ #HL2 #H destruct /3 width=1 by lsubd_pair/ - | #W #V #d #HV #HW #HL2 #H1 #H2 #H3 destruct - /3 width=3 by lsubd_beta, lsubd_da_trans/ - ] -| #L1 #L #W #V #d #HV #HW #HL1 #IHL1 #X #H - elim (lsubd_inv_pair1 … H) -H * #L2 - [ #HL2 #H destruct /3 width=5 by lsubd_beta, lsubd_da_conf/ - | #W0 #V0 #d0 #_ #_ #_ #H destruct - ] -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lsuby/lsuby.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/lsuby/lsuby.etc deleted file mode 100644 index e2ee7c7c2..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/lsuby/lsuby.etc +++ /dev/null @@ -1,236 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/notation/relations/lrsubeq_4.ma". -include "basic_2/substitution/drop.ma". - -(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************) - -inductive lsuby: relation4 ynat ynat lenv lenv ≝ -| lsuby_atom: ∀L,l,m. lsuby l m L (⋆) -| lsuby_zero: ∀I1,I2,L1,L2,V1,V2. - lsuby 0 0 L1 L2 → lsuby 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) -| lsuby_pair: ∀I1,I2,L1,L2,V,m. lsuby 0 m L1 L2 → - lsuby 0 (⫯m) (L1.ⓑ{I1}V) (L2.ⓑ{I2}V) -| lsuby_succ: ∀I1,I2,L1,L2,V1,V2,l,m. - lsuby l m L1 L2 → lsuby (⫯l) m (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) -. - -interpretation - "local environment refinement (extended substitution)" - 'LRSubEq L1 l m L2 = (lsuby l m L1 L2). - -(* Basic properties *********************************************************) - -lemma lsuby_pair_lt: ∀I1,I2,L1,L2,V,m. L1 ⊆[0, ⫰m] L2 → 0 < m → - L1.ⓑ{I1}V ⊆[0, m] L2.ⓑ{I2}V. -#I1 #I2 #L1 #L2 #V #m #HL12 #Hm <(ylt_inv_O1 … Hm) /2 width=1 by lsuby_pair/ -qed. - -lemma lsuby_succ_lt: ∀I1,I2,L1,L2,V1,V2,l,m. L1 ⊆[⫰l, m] L2 → 0 < l → - L1.ⓑ{I1}V1 ⊆[l, m] L2. ⓑ{I2}V2. -#I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #Hl <(ylt_inv_O1 … Hl) /2 width=1 by lsuby_succ/ -qed. - -lemma lsuby_pair_O_Y: ∀L1,L2. L1 ⊆[0, ∞] L2 → - ∀I1,I2,V. L1.ⓑ{I1}V ⊆[0,∞] L2.ⓑ{I2}V. -#L1 #L2 #HL12 #I1 #I2 #V lapply (lsuby_pair I1 I2 … V … HL12) -HL12 // -qed. - -lemma lsuby_refl: ∀L,l,m. L ⊆[l, m] L. -#L elim L -L // -#L #I #V #IHL #l elim (ynat_cases … l) [| * #x ] -#Hl destruct /2 width=1 by lsuby_succ/ -#m elim (ynat_cases … m) [| * #x ] -#Hm destruct /2 width=1 by lsuby_zero, lsuby_pair/ -qed. - -lemma lsuby_O2: ∀L2,L1,l. |L2| ≤ |L1| → L1 ⊆[l, 0] L2. -#L2 elim L2 -L2 // #L2 #I2 #V2 #IHL2 * -[ #l #H elim (ylt_yle_false … H) -H // -| #L1 #I1 #V1 #l - #H lapply (yle_inv_succ … H) -H #HL12 - elim (ynat_cases l) /3 width=1 by lsuby_zero/ - * /3 width=1 by lsuby_succ/ -] -qed. - -lemma lsuby_sym: ∀l,m,L1,L2. L1 ⊆[l, m] L2 → |L1| = |L2| → L2 ⊆[l, m] L1. -#l #m #L1 #L2 #H elim H -l -m -L1 -L2 -[ #L1 #l #m #H >(length_inv_zero_dx … H) -L1 // -| /2 width=1 by lsuby_O2/ -| #I1 #I2 #L1 #L2 #V #m #_ #IHL12 #H lapply (ysucc_inv_inj … H) -H - /3 width=1 by lsuby_pair/ -| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #H lapply (ysucc_inv_inj … H) -H - /3 width=1 by lsuby_succ/ -] -qed-. - -(* Basic inversion lemmas ***************************************************) - -fact lsuby_inv_atom1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → L1 = ⋆ → L2 = ⋆. -#L1 #L2 #l #m * -L1 -L2 -l -m // -[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct -| #I1 #I2 #L1 #L2 #V #m #_ #H destruct -| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #H destruct -] -qed-. - -lemma lsuby_inv_atom1: ∀L2,l,m. ⋆ ⊆[l, m] L2 → L2 = ⋆. -/2 width=5 by lsuby_inv_atom1_aux/ qed-. - -fact lsuby_inv_zero1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → - ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → l = 0 → m = 0 → - L2 = ⋆ ∨ - ∃∃J2,K2,W2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{J2}W2. -#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/ -[ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct - /3 width=5 by ex2_3_intro, or_intror/ -| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #_ #H - elim (ysucc_inv_O_dx … H) -| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W1 #_ #H - elim (ysucc_inv_O_dx … H) -] -qed-. - -lemma lsuby_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⊆[0, 0] L2 → - L2 = ⋆ ∨ - ∃∃I2,K2,V2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{I2}V2. -/2 width=9 by lsuby_inv_zero1_aux/ qed-. - -fact lsuby_inv_pair1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → - ∀J1,K1,W. L1 = K1.ⓑ{J1}W → l = 0 → 0 < m → - L2 = ⋆ ∨ - ∃∃J2,K2. K1 ⊆[0, ⫰m] K2 & L2 = K2.ⓑ{J2}W. -#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/ -[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W #_ #_ #H - elim (ylt_yle_false … H) // -| #I1 #I2 #L1 #L2 #V #m #HL12 #J1 #K1 #W #H #_ #_ destruct - /3 width=4 by ex2_2_intro, or_intror/ -| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W #_ #H - elim (ysucc_inv_O_dx … H) -] -qed-. - -lemma lsuby_inv_pair1: ∀I1,K1,L2,V,m. K1.ⓑ{I1}V ⊆[0, m] L2 → 0 < m → - L2 = ⋆ ∨ - ∃∃I2,K2. K1 ⊆[0, ⫰m] K2 & L2 = K2.ⓑ{I2}V. -/2 width=6 by lsuby_inv_pair1_aux/ qed-. - -fact lsuby_inv_succ1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → - ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < l → - L2 = ⋆ ∨ - ∃∃J2,K2,W2. K1 ⊆[⫰l, m] K2 & L2 = K2.ⓑ{J2}W2. -#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/ -[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H - elim (ylt_yle_false … H) // -| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #H - elim (ylt_yle_false … H) // -| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J1 #K1 #W1 #H #_ destruct - /3 width=5 by ex2_3_intro, or_intror/ -] -qed-. - -lemma lsuby_inv_succ1: ∀I1,K1,L2,V1,l,m. K1.ⓑ{I1}V1 ⊆[l, m] L2 → 0 < l → - L2 = ⋆ ∨ - ∃∃I2,K2,V2. K1 ⊆[⫰l, m] K2 & L2 = K2.ⓑ{I2}V2. -/2 width=5 by lsuby_inv_succ1_aux/ qed-. - -fact lsuby_inv_zero2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → - ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → l = 0 → m = 0 → - ∃∃J1,K1,W1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{J1}W1. -#L1 #L2 #l #m * -L1 -L2 -l -m -[ #L1 #l #m #J2 #K2 #W1 #H destruct -| #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J2 #K2 #W2 #H #_ #_ destruct - /2 width=5 by ex2_3_intro/ -| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K2 #W2 #_ #_ #H - elim (ysucc_inv_O_dx … H) -| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W2 #_ #H - elim (ysucc_inv_O_dx … H) -] -qed-. - -lemma lsuby_inv_zero2: ∀I2,K2,L1,V2. L1 ⊆[0, 0] K2.ⓑ{I2}V2 → - ∃∃I1,K1,V1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{I1}V1. -/2 width=9 by lsuby_inv_zero2_aux/ qed-. - -fact lsuby_inv_pair2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → - ∀J2,K2,W. L2 = K2.ⓑ{J2}W → l = 0 → 0 < m → - ∃∃J1,K1. K1 ⊆[0, ⫰m] K2 & L1 = K1.ⓑ{J1}W. -#L1 #L2 #l #m * -L1 -L2 -l -m -[ #L1 #l #m #J2 #K2 #W #H destruct -| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W #_ #_ #H - elim (ylt_yle_false … H) // -| #I1 #I2 #L1 #L2 #V #m #HL12 #J2 #K2 #W #H #_ #_ destruct - /2 width=4 by ex2_2_intro/ -| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W #_ #H - elim (ysucc_inv_O_dx … H) -] -qed-. - -lemma lsuby_inv_pair2: ∀I2,K2,L1,V,m. L1 ⊆[0, m] K2.ⓑ{I2}V → 0 < m → - ∃∃I1,K1. K1 ⊆[0, ⫰m] K2 & L1 = K1.ⓑ{I1}V. -/2 width=6 by lsuby_inv_pair2_aux/ qed-. - -fact lsuby_inv_succ2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → - ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → 0 < l → - ∃∃J1,K1,W1. K1 ⊆[⫰l, m] K2 & L1 = K1.ⓑ{J1}W1. -#L1 #L2 #l #m * -L1 -L2 -l -m -[ #L1 #l #m #J2 #K2 #W2 #H destruct -| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W2 #_ #H - elim (ylt_yle_false … H) // -| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K1 #W2 #_ #H - elim (ylt_yle_false … H) // -| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J2 #K2 #W2 #H #_ destruct - /2 width=5 by ex2_3_intro/ -] -qed-. - -lemma lsuby_inv_succ2: ∀I2,K2,L1,V2,l,m. L1 ⊆[l, m] K2.ⓑ{I2}V2 → 0 < l → - ∃∃I1,K1,V1. K1 ⊆[⫰l, m] K2 & L1 = K1.ⓑ{I1}V1. -/2 width=5 by lsuby_inv_succ2_aux/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lsuby_fwd_length: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → |L2| ≤ |L1|. -#L1 #L2 #l #m #H elim H -L1 -L2 -l -m /2 width=1 by yle_succ/ -qed-. - -(* Properties on basic slicing **********************************************) - -lemma lsuby_drop_trans_be: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → - ∀I2,K2,W,s,i. ⬇[s, 0, i] L2 ≡ K2.ⓑ{I2}W → - l ≤ i → ∀m0. i + ⫯m0 = l + m → - ∃∃I1,K1. K1 ⊆[0, m0] K2 & ⬇[s, 0, i] L1 ≡ K1.ⓑ{I1}W. -#L1 #L2 #l #m #H elim H -L1 -L2 -l -m -[ #L1 #l #m #J2 #K2 #W #s #i #H - elim (drop_inv_atom1 … H) -H #H destruct -| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #s #i #_ #_ #m0 - >yplus_O2 >yplus_succ2 #H elim (ysucc_inv_O_dx … H) -| #I1 #I2 #L1 #L2 #V #m #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ #m0 - >yplus_succ2 >yplus_succ2 #H0 lapply (ysucc_inv_inj … H0) -H0 - elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ] - [ destruct -I2 /2 width=4 by drop_pair, ex2_2_intro/ - | lapply (ylt_inv_O1 … Hi) - #H yplus_succ1 >yplus_succ2 #H lapply (ysucc_inv_inj … H) -H - tls_S >S_pred // -qed-. - -definition ltls (f): lenv → lenv → rtmap ≝ λL,K. ⫱*[|L|-|K|] f. - -interpretation "ltls (rtmap)" 'DropPreds L K f = (ltls f L K). - -lemma ltls_refl: ∀f,L1,L2. |L1| ≤ |L2| → ⫱*[L1, L2] f = f. -#f #L1 #L2 #HL12 whd in ⊢ (??%?); >(eq_minus_O … HL12) -HL12 // -qed. - -lemma ltls_pair2: ∀f,I,L1,L2,V. |L2| < |L1| → ⫱⫱*[L1, L2.ⓑ{I}V] f = ⫱*[L1, L2] f. -#f #I #L1 #L2 #V #HL12 whd in ⊢ (??(?%)%); minus_Sn_m // -qed. - -lemma ltls_pair1_next: ∀f,I,L1,L2,V. |L2| ≤ |L1| → ⫱*[L1.ⓑ{I}V, L2] ⫯f = ⫱*[L1, L2] f. -#f #I #L1 #L2 #V #HL12 whd in ⊢ (??%%); >minus_Sn_m // -qed. - -lemma ltls_sle_pair: ∀f1,f2,L1,L2. ⫱*[L2, L1] f2 ⊆ ⫱*[L1, L2] f1 → - ∀I,V1. ⫱*[L2, L1.ⓑ{I}V1] f2 ⊆ ⫱*[L1.ⓑ{I}V1, L2] ⫯f1. -#f1 #f2 #L1 #L2 elim (lt_or_ge (|L1|) (|L2|)) -[ #HL12 >ltls_refl in ⊢ (??%→?); /2 width=1 by lt_to_le/ - #Hf21 #I #V1 >ltls_refl in ⊢ (??%); // - <(ltls_pair2 … I … V1 HL12) in Hf21; -HL12 /2 width=1 by sle_inv_tl1/ -| #HL21 >ltls_refl // #Hf21 #I #V1 >ltls_refl /2 width=1 by le_S/ - >ltls_pair1_next // -] -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/psubst_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/psubst_6.etc deleted file mode 100644 index 56cb72ec4..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/psubst_6.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ [ term 46 l , break term 46 m ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'PSubst $G $L $T1 $l $k $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/psubststar_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/psubststar_6.etc deleted file mode 100644 index 90362edb6..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/psubststar_6.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ * [ term 46 l , break term 46 m ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'PSubstStar $G $L $T1 $l $k $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/psubststaralt_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/psubststaralt_6.etc deleted file mode 100644 index 80a29a841..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/psubststaralt_6.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ ▶ * [ term 46 l , break term 46 m ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'PSubstStarAlt $G $L $T1 $l $k $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/unfold/unfold.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/unfold/unfold.etc deleted file mode 100644 index 445421935..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/unfold/unfold.etc +++ /dev/null @@ -1,35 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/notation/relations/unfold_4.ma". -include "basic_2/grammar/lenv_append.ma". -include "basic_2/grammar/genv.ma". -include "basic_2/substitution/drop.ma". - -(* CONTEXT-SENSITIVE UNFOLD FOR TERMS ***************************************) - -(* activate genv *) -inductive unfold: relation4 genv lenv term lenv ≝ -| unfold_sort: ∀G,L,s. unfold G L (⋆s) L -| unfold_lref: ∀I,G,L1,L2,K1,K2,V,i. ⬇[i] L1 ≡ K1. ⓑ{I}V → - unfold G K1 V K2 → ⬇[Ⓣ, |L2|, i] L2 ≡ K2 → - unfold G L1 (#i) (L1@@L2) -| unfold_bind: ∀a,I,G,L1,L2,V,T. - unfold G (L1.ⓑ{I}V) T L2 → unfold G L1 (ⓑ{a,I}V.T) L2 -| unfold_flat: ∀I,G,L1,L2,V,T. - unfold G L1 T L2 → unfold G L1 (ⓕ{I}V.T) L2 -. - -interpretation "context-sensitive unfold (term)" - 'Unfold G L1 T L2 = (unfold G L1 T L2). diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/unfold/unfold_4.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/unfold/unfold_4.etc deleted file mode 100644 index 3381c5632..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/unfold/unfold_4.etc +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( ⦃ term 46 G, break term 46 L1 ⦄ ⊢ ⧫ * break term 46 T ≡ break term 46 L2 )" - non associative with precedence 45 - for @{ 'Unfold $G $L1 $T $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeq_5.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeq_5.ma new file mode 100644 index 000000000..dd6a6046a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeq_5.ma @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +notation "hvbox( L1 ≡ break [ term 46 h , term 46 o , term 46 T ] break term 46 L2 )" + non associative with precedence 45 + for @{ 'LazyEq $h $o $T $L1 $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeq_6.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeq_6.ma deleted file mode 100644 index eb25da48b..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeq_6.ma +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≡ break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )" - non associative with precedence 45 - for @{ 'LazyEq $G1 $L1 $T1 $G2 $L2 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeq_8.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeq_8.ma new file mode 100644 index 000000000..b4b796901 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeq_8.ma @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≡ [ break term 46 h , break term 46 o ] ⦃ break term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )" + non associative with precedence 45 + for @{ 'LazyEq $h $o $G1 $L1 $T1 $G2 $L2 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma index e99ea0393..4e0c82132 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma @@ -173,16 +173,23 @@ lemma lexs_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⦻*[RN, RP, f] #RN #RP #L1 #L2 @eq_repl_sym /2 width=3 by lexs_eq_repl_back/ (**) (* full auto fails *) qed-. -(* Note: fexs_sym and fexs_trans hold, but lexs_sym and lexs_trans do not *) (* Basic_2A1: includes: lpx_sn_refl *) -lemma lexs_refl: ∀RN,RP,f. +lemma lexs_refl: ∀RN,RP. (∀L. reflexive … (RN L)) → (∀L. reflexive … (RP L)) → - reflexive … (lexs RN RP f). -#RN #RP #f #HRN #HRP #L generalize in match f; -f elim L -L // + ∀f.reflexive … (lexs RN RP f). +#RN #RP #HRN #HRP #f #L generalize in match f; -f elim L -L // #L #I #V #IH * * /2 width=1 by lexs_next, lexs_push/ qed. +lemma lexs_sym: ∀RN,RP. + (∀L1,L2,T1,T2. RN L1 T1 T2 → RN L2 T2 T1) → + (∀L1,L2,T1,T2. RP L1 T1 T2 → RP L2 T2 T1) → + ∀f. symmetric … (lexs RN RP f). +#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -L1 -L2 -f +/3 width=2 by lexs_next, lexs_push/ +qed-. + lemma lexs_pair_repl: ∀RN,RP,f,I,L1,L2,V1,V2. L1.ⓑ{I}V1 ⦻*[RN, RP, f] L2.ⓑ{I}V2 → ∀W1,W2. RN L1 W1 W2 → RP L1 W1 W2 → diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lfdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lfdeq.ma new file mode 100644 index 000000000..cb6b1d180 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lfdeq.ma @@ -0,0 +1,45 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/static/lfdeq.ma". +include "basic_2/static/aaa.ma". + +(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************) + +(* Properties with degree-based equivalence on referred entries *************) + +lemma aaa_tdeq_conf_fldeq: ∀h,o,G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀T2. T1 ≡[h, o] T2 → + ∀L2. L1 ≡[h, o, T1] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A. +#h #o #G #L1 #T1 #A #H elim H -G -L1 -T1 -A +[ #G #L1 #s1 #X #H1 elim (tdeq_inv_sort1 … H1) -H1 // +| #I #G #L1 #V1 #B #_ #IH #X #H1 >(tdeq_inv_lref1 … H1) -H1 + #Y #H2 elim (lfdeq_inv_zero_pair_sn … H2) -H2 + #L2 #V2 #HL12 #HV12 #H2 destruct /3 width=1 by aaa_zero/ +| #I #G #L1 #V1 #A #i #_ #IH #X #H1 >(tdeq_inv_lref1 … H1) -H1 + #Y #H2 elim (lfdeq_inv_lref_pair_sn … H2) -H2 + #L2 #V2 #HL12 #H2 destruct /3 width=1 by aaa_lref/ +| #p #G #L1 #V1 #T1 #B #A #_ #_ #IHV #IHT #X #H1 elim (tdeq_inv_pair1 … H1) -H1 + #V2 #T2 #HV12 #HT12 #H1 #L2 #H2 elim (lfdeq_inv_bind … H2) -H2 + /4 width=2 by aaa_abbr, lfdeq_pair_repl_dx/ +| #p #G #L1 #V1 #T1 #B #A #_ #_ #IHV #IHT #X #H1 elim (tdeq_inv_pair1 … H1) -H1 + #V2 #T2 #HV12 #HT12 #H1 #L2 #H2 elim (lfdeq_inv_bind … H2) -H2 + /4 width=2 by aaa_abst, lfdeq_pair_repl_dx/ +| #G #L1 #V1 #T1 #B #A #_ #_ #IHV #IHT #X #H1 elim (tdeq_inv_pair1 … H1) -H1 + #V2 #T2 #HV12 #HT12 #H1 #L2 #H2 elim (lfdeq_inv_flat … H2) -H2 + /3 width=3 by aaa_appl/ +| #G #L1 #V1 #T1 #A #_ #_ #IHV #IHT #X #H1 elim (tdeq_inv_pair1 … H1) -H1 + #V2 #T2 #HV12 #HT12 #H1 #L2 #H2 elim (lfdeq_inv_flat … H2) -H2 + /3 width=1 by aaa_cast/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lfeq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lfeq.ma deleted file mode 100644 index e5d7d0b5b..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lfeq.ma +++ /dev/null @@ -1,42 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/static/lfeq_lreq.ma". -include "basic_2/static/aaa.ma". - -(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************) - -(* Properties with equivalence on referred entries **************************) - -lemma lfeq_aaa_trans: ∀G,L2,T,A. ⦃G, L2⦄ ⊢ T ⁝ A → - ∀L1. L1 ≡[T] L2 → ⦃G, L1⦄ ⊢ T ⁝ A. -#G #L2 #T #A #H elim H -G -L2 -T -A /2 width=1 by aaa_sort/ -[ #I #G #L2 #V2 #A #_ #IH #L1 #H - elim (lfeq_inv_zero_pair_dx … H) -H /3 width=1 by aaa_zero/ -| #I #G #L2 #V2 #A #i #_ #IH #L1 #H - elim (lfeq_inv_lref_pair_dx … H) -H /3 width=1 by aaa_lref/ -| #p #G #L2 #V #T #B #A #_ #_ #IHV #IHT #L1 #H - elim (lfeq_inv_bind … H) -H /3 width=2 by aaa_abbr/ -| #p #G #L2 #V #T #B #A #_ #_ #IHV #IHT #L1 #H - elim (lfeq_inv_bind … H) -H /3 width=1 by aaa_abst/ -| #G #L2 #V #T #B #A #_ #_ #IHV #IHT #L1 #H - elim (lfeq_inv_flat … H) -H /3 width=3 by aaa_appl/ -| #G #L2 #V #T #A #_ #_ #IHV #IHT #L1 #H - elim (lfeq_inv_flat … H) -H /3 width=1 by aaa_cast/ -] -qed-. - -lemma aaa_lfeq_conf: ∀G,L2,T,A. ⦃G, L2⦄ ⊢ T ⁝ A → - ∀L1. L2 ≡[T] L1 → ⦃G, L1⦄ ⊢ T ⁝ A. -/3 width=3 by lfeq_aaa_trans, lfeq_sym/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/ffdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/ffdeq.ma new file mode 100644 index 000000000..207e1e2d3 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/ffdeq.ma @@ -0,0 +1,45 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/notation/relations/lazyeq_8.ma". +include "basic_2/syntax/genv.ma". +include "basic_2/static/lfdeq_fqup.ma". + +(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************) + +inductive ffdeq (h) (o) (G) (L1) (T): relation3 genv lenv term ≝ +| ffdeq_intro: ∀L2. L1 ≡[h, o, T] L2 → ffdeq h o G L1 T G L2 T +. + +interpretation + "degree-based equivalence on referred entries (closure)" + 'LazyEq h o G1 L1 T1 G2 L2 T2 = (ffdeq h o G1 L1 T1 G2 L2 T2). + +(* Basic properties *********************************************************) + +lemma ffdeq_sym: ∀h,o. tri_symmetric … (ffdeq h o). +#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -L1 -T1 /3 width=1 by ffdeq_intro, lfdeq_sym/ +qed-. + +(* Basic inversion lemmas ***************************************************) + +lemma ffdeq_inv_gen: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≡[h, o] ⦃G2, L2, T2⦄ → + ∧∧ G1 = G2 & L1 ≡[h, o, T1] L2 & T1 = T2. +#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=1 by and3_intro/ +qed-. + +(* Basic_2A1: removed theorems 6: + fleq_refl fleq_sym fleq_inv_gen + fleq_trans fleq_canc_sn fleq_canc_dx +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/ffdeq_ffdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/ffdeq_ffdeq.ma new file mode 100644 index 000000000..46400a913 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/ffdeq_ffdeq.ma @@ -0,0 +1,33 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/static/lfdeq_lfdeq.ma". +include "basic_2/static/ffdeq.ma". + +(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************) + +(* Main properties **********************************************************) + +theorem ffdeq_trans: ∀h,o. tri_transitive … (ffdeq h o). +#h #o #G1 #G #L1 #L #T1 #T * -G -L -T +#L #HL1 #G2 #L2 #T2 * -G2 -L2 -T2 /3 width=3 by ffdeq_intro, lfdeq_trans/ +qed-. + +theorem ffdeq_canc_sn: ∀h,o,G,G1,G2,L,L1,L2,T,T1,T2. + ⦃G, L, T⦄ ≡[h, o] ⦃G1, L1, T1⦄→ ⦃G, L, T⦄ ≡[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≡[h, o] ⦃G2, L2, T2⦄. +/3 width=5 by ffdeq_trans, ffdeq_sym/ qed-. + +theorem ffdeq_canc_dx: ∀h,o,G1,G2,G,L1,L2,L,T1,T2,T. + ⦃G1, L1, T1⦄ ≡[h, o] ⦃G, L, T⦄ → ⦃G2, L2, T2⦄ ≡[h, o] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≡[h, o] ⦃G2, L2, T2⦄. +/3 width=5 by ffdeq_trans, ffdeq_sym/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/ffdeq_fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/static/ffdeq_fqup.ma new file mode 100644 index 000000000..ab06d6d90 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/ffdeq_fqup.ma @@ -0,0 +1,23 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/static/lfdeq_fqup.ma". +include "basic_2/static/ffdeq.ma". + +(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************) + +(* Advanced properties ******************************************************) + +lemma ffdeq_refl: ∀h,o. tri_reflexive … (ffdeq h o). +/2 width=1 by ffdeq_intro/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/ffeq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/ffeq.ma deleted file mode 100644 index 48eb1ec0f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/static/ffeq.ma +++ /dev/null @@ -1,48 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/notation/relations/lazyeq_6.ma". -include "basic_2/static/lfeq_lreq.ma". -include "basic_2/static/lfeq_fqup.ma". - -(* EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES *****************************) - -inductive ffeq (G) (L1) (T): relation3 genv lenv term ≝ -| fleq_intro: ∀L2. L1 ≡[T] L2 → ffeq G L1 T G L2 T -. - -interpretation - "equivalence on referred entries (closure)" - 'LazyEq G1 L1 T1 G2 L2 T2 = (ffeq G1 L1 T1 G2 L2 T2). - -(* Basic properties *********************************************************) - -lemma ffeq_refl: tri_reflexive … ffeq. -/2 width=1 by fleq_intro/ qed. - -lemma ffeq_sym: tri_symmetric … ffeq. -#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -L1 -T1 /3 width=1 by fleq_intro, lfeq_sym/ -qed-. - -(* Basic inversion lemmas ***************************************************) - -lemma ffeq_inv_gen: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≡ ⦃G2, L2, T2⦄ → - ∧∧ G1 = G2 & L1 ≡[T1] L2 & T1 = T2. -#G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=1 by and3_intro/ -qed-. - -(* Basic_2A1: removed theorems 6: - fleq_refl fleq_sym fleq_inv_gen - fleq_trans fleq_canc_sn fleq_canc_dx -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/ffeq_ffeq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/ffeq_ffeq.ma deleted file mode 100644 index e4a4cdd87..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/static/ffeq_ffeq.ma +++ /dev/null @@ -1,33 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/static/lfeq_lfeq.ma". -include "basic_2/static/ffeq.ma". - -(* EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES *****************************) - -(* Main properties **********************************************************) - -theorem ffeq_trans: tri_transitive … ffeq. -#G1 #G #L1 #L #T1 #T * -G -L -T -#L #HL1 #G2 #L2 #T2 * -G2 -L2 -T2 /3 width=3 by fleq_intro, lfeq_trans/ -qed-. - -theorem ffeq_canc_sn: ∀G,G1,G2,L,L1,L2,T,T1,T2. - ⦃G, L, T⦄ ≡ ⦃G1, L1, T1⦄→ ⦃G, L, T⦄ ≡ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≡ ⦃G2, L2, T2⦄. -/3 width=5 by ffeq_trans, ffeq_sym/ qed-. - -theorem ffeq_canc_dx: ∀G1,G2,G,L1,L2,L,T1,T2,T. - ⦃G1, L1, T1⦄ ≡ ⦃G, L, T⦄ → ⦃G2, L2, T2⦄ ≡ ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≡ ⦃G2, L2, T2⦄. -/3 width=5 by ffeq_trans, ffeq_sym/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/frees_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/frees_lreq.ma deleted file mode 100644 index 279ad0ca1..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/static/frees_lreq.ma +++ /dev/null @@ -1,40 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/relocation/lreq.ma". -include "basic_2/static/frees.ma". - -(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************) - -(* Properties with ranged equivalence for local environments ****************) - -lemma frees_lreq_conf: ∀f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f → ∀L2. L1 ≡[f] L2 → L2 ⊢ 𝐅*⦃T⦄ ≡ f. -#f #L1 #T #H elim H -f -L1 -T -[ #f #I #Hf #X #H lapply (lreq_inv_atom1 … H) -H - #H destruct /2 width=1 by frees_atom/ -| #f #I #L1 #V1 #s #_ #IH #X #H elim (lreq_inv_push1 … H) -H - /3 width=1 by frees_sort/ -| #f #I #L1 #V1 #_ #IH #X #H elim (lreq_inv_next1 … H) -H - /3 width=1 by frees_zero/ -| #f #I #L1 #V1 #i #_ #IH #X #H elim (lreq_inv_push1 … H) -H - /3 width=1 by frees_lref/ -| #f #I #L1 #V1 #l #_ #IH #X #H elim (lreq_inv_push1 … H) -H - /3 width=1 by frees_gref/ -| /6 width=5 by frees_bind, lreq_inv_tl, sle_lreq_trans, sor_inv_sle_dx, sor_inv_sle_sn/ -| /5 width=5 by frees_flat, sle_lreq_trans, sor_inv_sle_dx, sor_inv_sle_sn/ -] -qed-. - -lemma lreq_frees_trans: ∀f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f → ∀L2. L2 ≡[f] L1 → L2 ⊢ 𝐅*⦃T⦄ ≡ f. -/3 width=3 by frees_lreq_conf, lreq_sym/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq.ma new file mode 100644 index 000000000..e75c4e734 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq.ma @@ -0,0 +1,170 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/notation/relations/lazyeq_5.ma". +include "basic_2/syntax/tdeq.ma". +include "basic_2/static/lfxs.ma". + +(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******) + +definition lfdeq: ∀h. sd h → relation3 term lenv lenv ≝ + λh,o. lfxs (cdeq h o). + +interpretation + "degree-based equivalence on referred entries (local environment)" + 'LazyEq h o T L1 L2 = (lfdeq h o T L1 L2). + +interpretation + "degree-based ranged equivalence (local environment)" + 'LazyEq h o f L1 L2 = (lexs (cdeq h o) cfull f L1 L2). +(* +definition lfdeq_transitive: predicate (relation3 lenv term term) ≝ + λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[h, o, T1] L2 → R L1 T1 T2. +*) +(* Basic properties ***********************************************************) + +lemma frees_tdeq_conf_lexs: ∀h,o,f,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f → ∀T2. T1 ≡[h, o] T2 → + ∀L2. L1 ≡[h, o, f] L2 → L2 ⊢ 𝐅*⦃T2⦄ ≡ f. +#h #o #f #L1 #T1 #H elim H -f -L1 -T1 +[ #f #I1 #Hf #X #H1 elim (tdeq_fwd_atom1 … H1) -H1 + #I2 #H1 #Y #H2 lapply (lexs_inv_atom1 … H2) -H2 + #H2 destruct /2 width=1 by frees_atom/ +| #f #I #L1 #V1 #s1 #_ #IH #X #H1 elim (tdeq_inv_sort1 … H1) -H1 + #s2 #d #Hs1 #Hs2 #H1 #Y #H2 elim (lexs_inv_push1 … H2) -H2 + #L2 #V2 #HL12 #_ #H2 destruct /4 width=3 by frees_sort, tdeq_sort/ +| #f #I #L1 #V1 #_ #IH #X #H1 >(tdeq_inv_lref1 … H1) -H1 + #Y #H2 elim (lexs_inv_next1 … H2) -H2 + #L2 #V2 #HL12 #HV12 #H2 destruct /3 width=1 by frees_zero/ +| #f #I #L1 #V1 #i #_ #IH #X #H1 >(tdeq_inv_lref1 … H1) -H1 + #Y #H2 elim (lexs_inv_push1 … H2) -H2 + #L2 #V2 #HL12 #_ #H2 destruct /3 width=1 by frees_lref/ +| #f #I #L1 #V1 #l #_ #IH #X #H1 >(tdeq_inv_gref1 … H1) -H1 + #Y #H2 elim (lexs_inv_push1 … H2) -H2 + #L2 #V2 #HL12 #_ #H2 destruct /3 width=1 by frees_gref/ +| #f1V #f1T #f1 #p #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #X #H1 elim (tdeq_inv_pair1 … H1) -H1 + #V2 #T2 #HV12 #HT12 #H1 #L2 #HL12 destruct + /6 width=5 by frees_bind, lexs_inv_tl, sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn/ +| #f1V #f1T #f1 #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #X #H1 elim (tdeq_inv_pair1 … H1) -H1 + #V2 #T2 #HV12 #HT12 #H1 #L2 #HL12 destruct + /5 width=5 by frees_flat, sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn/ +] +qed-. + +lemma lfdeq_sym: ∀h,o,T. symmetric … (lfdeq h o T). +#h #o #T #L1 #L2 * +/4 width=7 by frees_tdeq_conf_lexs, lfxs_sym, tdeq_sym, ex2_intro/ +qed-. + +lemma lfdeq_atom: ∀h,o,I. ⋆ ≡[h, o, ⓪{I}] ⋆. +/2 width=1 by lfxs_atom/ qed. + +lemma lfdeq_sort: ∀h,o,I,L1,L2,V1,V2,s. + L1 ≡[h, o, ⋆s] L2 → L1.ⓑ{I}V1 ≡[h, o, ⋆s] L2.ⓑ{I}V2. +/2 width=1 by lfxs_sort/ qed. + +lemma lfdeq_zero: ∀h,o,I,L1,L2,V. + L1 ≡[h, o, V] L2 → L1.ⓑ{I}V ≡[h, o, #0] L2.ⓑ{I}V. +/2 width=1 by lfxs_zero/ qed. + +lemma lfdeq_lref: ∀h,o,I,L1,L2,V1,V2,i. + L1 ≡[h, o, #i] L2 → L1.ⓑ{I}V1 ≡[h, o, #⫯i] L2.ⓑ{I}V2. +/2 width=1 by lfxs_lref/ qed. + +lemma lfdeq_gref: ∀h,o,I,L1,L2,V1,V2,l. + L1 ≡[h, o, §l] L2 → L1.ⓑ{I}V1 ≡[h, o, §l] L2.ⓑ{I}V2. +/2 width=1 by lfxs_gref/ qed. + +lemma lfdeq_pair_repl_dx: ∀h,o,I,L1,L2.∀T:term.∀V,V1. + L1.ⓑ{I}V ≡[h, o, T] L2.ⓑ{I}V1 → + ∀V2. V ≡[h, o] V2 → + L1.ⓑ{I}V ≡[h, o, T] L2.ⓑ{I}V2. +/2 width=2 by lfxs_pair_repl_dx/ qed-. + +(* Basic inversion lemmas ***************************************************) + +lemma lfdeq_inv_atom_sn: ∀h,o,I,Y2. ⋆ ≡[h, o, ⓪{I}] Y2 → Y2 = ⋆. +/2 width=3 by lfxs_inv_atom_sn/ qed-. + +lemma lfdeq_inv_atom_dx: ∀h,o,I,Y1. Y1 ≡[h, o, ⓪{I}] ⋆ → Y1 = ⋆. +/2 width=3 by lfxs_inv_atom_dx/ qed-. + +lemma lfdeq_inv_zero: ∀h,o,Y1,Y2. Y1 ≡[h, o, #0] Y2 → + (Y1 = ⋆ ∧ Y2 = ⋆) ∨ + ∃∃I,L1,L2,V1,V2. L1 ≡[h, o, V1] L2 & V1 ≡[h, o] V2 & + Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. +#h #o #Y1 #Y2 #H elim (lfxs_inv_zero … H) -H * +/3 width=9 by ex4_5_intro, or_introl, or_intror, conj/ +qed-. + +lemma lfdeq_inv_lref: ∀h,o,Y1,Y2,i. Y1 ≡[h, o, #⫯i] Y2 → + (Y1 = ⋆ ∧ Y2 = ⋆) ∨ + ∃∃I,L1,L2,V1,V2. L1 ≡[h, o, #i] L2 & + Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. +/2 width=1 by lfxs_inv_lref/ qed-. + +lemma lfdeq_inv_bind: ∀h,o,p,I,L1,L2,V,T. L1 ≡[h, o, ⓑ{p,I}V.T] L2 → + L1 ≡[h, o, V] L2 ∧ L1.ⓑ{I}V ≡[h, o, T] L2.ⓑ{I}V. +/2 width=2 by lfxs_inv_bind/ qed-. + +lemma lfdeq_inv_flat: ∀h,o,I,L1,L2,V,T. L1 ≡[h, o, ⓕ{I}V.T] L2 → + L1 ≡[h, o, V] L2 ∧ L1 ≡[h, o, T] L2. +/2 width=2 by lfxs_inv_flat/ qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma lfdeq_inv_zero_pair_sn: ∀h,o,I,Y2,L1,V1. L1.ⓑ{I}V1 ≡[h, o, #0] Y2 → + ∃∃L2,V2. L1 ≡[h, o, V1] L2 & V1 ≡[h, o] V2 & Y2 = L2.ⓑ{I}V2. +#h #o #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero_pair_sn … H) -H /2 width=5 by ex3_2_intro/ +qed-. + +lemma lfdeq_inv_zero_pair_dx: ∀h,o,I,Y1,L2,V2. Y1 ≡[h, o, #0] L2.ⓑ{I}V2 → + ∃∃L1,V1. L1 ≡[h, o, V1] L2 & V1 ≡[h, o] V2 & Y1 = L1.ⓑ{I}V1. +#h #o #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero_pair_dx … H) -H +#L1 #V1 #HL12 #HV12 #H destruct /2 width=5 by ex3_2_intro/ +qed-. + +lemma lfdeq_inv_lref_pair_sn: ∀h,o,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ≡[h, o, #⫯i] Y2 → + ∃∃L2,V2. L1 ≡[h, o, #i] L2 & Y2 = L2.ⓑ{I}V2. +/2 width=2 by lfxs_inv_lref_pair_sn/ qed-. + +lemma lfdeq_inv_lref_pair_dx: ∀h,o,I,Y1,L2,V2,i. Y1 ≡[h, o, #⫯i] L2.ⓑ{I}V2 → + ∃∃L1,V1. L1 ≡[h, o, #i] L2 & Y1 = L1.ⓑ{I}V1. +/2 width=2 by lfxs_inv_lref_pair_dx/ qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lfdeq_fwd_bind_sn: ∀h,o,p,I,L1,L2,V,T. L1 ≡[h, o, ⓑ{p,I}V.T] L2 → L1 ≡[h, o, V] L2. +/2 width=4 by lfxs_fwd_bind_sn/ qed-. + +lemma lfdeq_fwd_bind_dx: ∀h,o,p,I,L1,L2,V,T. + L1 ≡[h, o, ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≡[h, o, T] L2.ⓑ{I}V. +/2 width=2 by lfxs_fwd_bind_dx/ qed-. + +lemma lfdeq_fwd_flat_sn: ∀h,o,I,L1,L2,V,T. L1 ≡[h, o, ⓕ{I}V.T] L2 → L1 ≡[h, o, V] L2. +/2 width=3 by lfxs_fwd_flat_sn/ qed-. + +lemma lfdeq_fwd_flat_dx: ∀h,o,I,L1,L2,V,T. L1 ≡[h, o, ⓕ{I}V.T] L2 → L1 ≡[h, o, T] L2. +/2 width=3 by lfxs_fwd_flat_dx/ qed-. + +lemma lfdeq_fwd_pair_sn: ∀h,o,I,L1,L2,V,T. L1 ≡[h, o, ②{I}V.T] L2 → L1 ≡[h, o, V] L2. +/2 width=3 by lfxs_fwd_pair_sn/ qed-. + +(* Basic_2A1: removed theorems 30: + lleq_ind lleq_inv_bind lleq_inv_flat lleq_fwd_length lleq_fwd_lref + lleq_fwd_drop_sn lleq_fwd_drop_dx + lleq_fwd_bind_sn lleq_fwd_bind_dx lleq_fwd_flat_sn lleq_fwd_flat_dx + lleq_sort lleq_skip lleq_lref lleq_free lleq_gref lleq_bind lleq_flat + lleq_refl lleq_Y lleq_sym lleq_ge_up lleq_ge lleq_bind_O llpx_sn_lrefl + lleq_trans lleq_canc_sn lleq_canc_dx lleq_nlleq_trans nlleq_lleq_div +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_fqup.ma new file mode 100644 index 000000000..b6e2a18f6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_fqup.ma @@ -0,0 +1,23 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/static/lfxs_fqup.ma". +include "basic_2/static/lfdeq.ma". + +(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******) + +(* Advanced properties ******************************************************) + +lemma lfdeq_refl: ∀h,o,T. reflexive … (lfdeq h o T). +/2 width=1 by lfxs_refl/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_length.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_length.ma new file mode 100644 index 000000000..d7c04b6ec --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_length.ma @@ -0,0 +1,23 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/static/lfxs_length.ma". +include "basic_2/static/lfdeq.ma". + +(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******) + +(* Forward lemmas with length for local environments ************************) + +lemma lfdeq_fwd_length: ∀h,o,L1,L2. ∀T:term. L1 ≡[h, o, T] L2 → |L1| = |L2|. +/2 width=3 by lfxs_fwd_length/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_lfdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_lfdeq.ma new file mode 100644 index 000000000..71b1c0d4f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_lfdeq.ma @@ -0,0 +1,54 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/syntax/tdeq_tdeq.ma". +include "basic_2/static/lfxs_lfxs.ma". +include "basic_2/static/lfdeq.ma". + +(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******) + +(* Main properties **********************************************************) + +theorem lfdeq_bind: ∀h,o,p,I,L1,L2,V1,V2,T. + L1 ≡[h, o, V1] L2 → L1.ⓑ{I}V1 ≡[h, o, T] L2.ⓑ{I}V2 → + L1 ≡[h, o, ⓑ{p,I}V1.T] L2. +/2 width=2 by lfxs_bind/ qed. + +theorem lfdeq_flat: ∀h,o,I,L1,L2,V,T. L1 ≡[h, o, V] L2 → L1 ≡[h, o, T] L2 → + L1 ≡[h, o, ⓕ{I}V.T] L2. +/2 width=1 by lfxs_flat/ qed. + +theorem lfdeq_trans: ∀h,o,T. Transitive … (lfdeq h o T). +#h #o #T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2 +lapply (frees_tdeq_conf_lexs … Hf1 T … HL1) // #H0 +lapply (frees_mono … Hf2 … H0) -Hf2 -H0 +/4 width=7 by lexs_trans, lexs_eq_repl_back, tdeq_trans, ex2_intro/ +qed-. + +theorem lfdeq_canc_sn: ∀h,o,T. left_cancellable … (lfdeq h o T). +/3 width=3 by lfdeq_trans, lfdeq_sym/ qed-. + +theorem lfdeq_canc_dx: ∀h,o,T. right_cancellable … (lfdeq h o T). +/3 width=3 by lfdeq_trans, lfdeq_sym/ qed-. + +(* Advanced properies on negated lazy equivalence *****************************) + +(* Note: auto works with /4 width=8/ so lfdeq_canc_sn is preferred ************) +lemma lfdeq_nlfdeq_trans: ∀h,o.∀T:term.∀L1,L. L1 ≡[h, o, T] L → + ∀L2. (L ≡[h, o, T] L2 → ⊥) → (L1 ≡[h, o, T] L2 → ⊥). +/3 width=3 by lfdeq_canc_sn/ qed-. + +lemma nlfdeq_lfdeq_div: ∀h,o.∀T:term.∀L2,L. L2 ≡[h, o, T] L → + ∀L1. (L1 ≡[h, o, T] L → ⊥) → (L1 ≡[h, o, T] L2 → ⊥). +/3 width=3 by lfdeq_trans/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfeq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfeq.ma deleted file mode 100644 index 1386a6a7d..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfeq.ma +++ /dev/null @@ -1,132 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/notation/relations/lazyeq_3.ma". -include "basic_2/static/lfxs.ma". - -(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *******************) - -definition lfeq: relation3 term lenv lenv ≝ lfxs ceq. - -interpretation - "equivalence on referred entries (local environment)" - 'LazyEq T L1 L2 = (lfeq T L1 L2). - -definition lfeq_transitive: predicate (relation3 lenv term term) ≝ - λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[T1] L2 → R L1 T1 T2. - -(* Basic properties ***********************************************************) - -lemma lfeq_atom: ∀I. ⋆ ≡[⓪{I}] ⋆. -/2 width=1 by lfxs_atom/ qed. - -lemma lfeq_sort: ∀I,L1,L2,V1,V2,s. - L1 ≡[⋆s] L2 → L1.ⓑ{I}V1 ≡[⋆s] L2.ⓑ{I}V2. -/2 width=1 by lfxs_sort/ qed. - -lemma lfeq_zero: ∀I,L1,L2,V. - L1 ≡[V] L2 → L1.ⓑ{I}V ≡[#0] L2.ⓑ{I}V. -/2 width=1 by lfxs_zero/ qed. - -lemma lfeq_lref: ∀I,L1,L2,V1,V2,i. - L1 ≡[#i] L2 → L1.ⓑ{I}V1 ≡[#⫯i] L2.ⓑ{I}V2. -/2 width=1 by lfxs_lref/ qed. - -lemma lfeq_gref: ∀I,L1,L2,V1,V2,l. - L1 ≡[§l] L2 → L1.ⓑ{I}V1 ≡[§l] L2.ⓑ{I}V2. -/2 width=1 by lfxs_gref/ qed. - -(* Basic inversion lemmas ***************************************************) - -lemma lfeq_inv_atom_sn: ∀I,Y2. ⋆ ≡[⓪{I}] Y2 → Y2 = ⋆. -/2 width=3 by lfxs_inv_atom_sn/ qed-. - -lemma lfeq_inv_atom_dx: ∀I,Y1. Y1 ≡[⓪{I}] ⋆ → Y1 = ⋆. -/2 width=3 by lfxs_inv_atom_dx/ qed-. - -lemma lfeq_inv_zero: ∀Y1,Y2. Y1 ≡[#0] Y2 → - (Y1 = ⋆ ∧ Y2 = ⋆) ∨ - ∃∃I,L1,L2,V. L1 ≡[V] L2 & - Y1 = L1.ⓑ{I}V & Y2 = L2.ⓑ{I}V. -#Y1 #Y2 #H elim (lfxs_inv_zero … H) -H * -/3 width=7 by ex3_4_intro, or_introl, or_intror, conj/ -qed-. - -lemma lfeq_inv_lref: ∀Y1,Y2,i. Y1 ≡[#⫯i] Y2 → - (Y1 = ⋆ ∧ Y2 = ⋆) ∨ - ∃∃I,L1,L2,V1,V2. L1 ≡[#i] L2 & - Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. -/2 width=1 by lfxs_inv_lref/ qed-. - -lemma lfeq_inv_bind: ∀p,I,L1,L2,V,T. L1 ≡[ⓑ{p,I}V.T] L2 → - L1 ≡[V] L2 ∧ L1.ⓑ{I}V ≡[T] L2.ⓑ{I}V. -/2 width=2 by lfxs_inv_bind/ qed-. - -lemma lfeq_inv_flat: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 → - L1 ≡[V] L2 ∧ L1 ≡[T] L2. -/2 width=2 by lfxs_inv_flat/ qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma lfeq_inv_zero_pair_sn: ∀I,Y2,L1,V. L1.ⓑ{I}V ≡[#0] Y2 → - ∃∃L2. L1 ≡[V] L2 & Y2 = L2.ⓑ{I}V. -#I #Y2 #L1 #V #H elim (lfxs_inv_zero_pair_sn … H) -H /2 width=3 by ex2_intro/ -qed-. - -lemma lfeq_inv_zero_pair_dx: ∀I,Y1,L2,V. Y1 ≡[#0] L2.ⓑ{I}V → - ∃∃L1. L1 ≡[V] L2 & Y1 = L1.ⓑ{I}V. -#I #Y1 #L2 #V #H elim (lfxs_inv_zero_pair_dx … H) -H -#L1 #X #HL12 #HX #H destruct /2 width=3 by ex2_intro/ -qed-. - -lemma lfeq_inv_lref_pair_sn: ∀I,Y2,L1,V1,i. L1.ⓑ{I}V1 ≡[#⫯i] Y2 → - ∃∃L2,V2. L1 ≡[#i] L2 & Y2 = L2.ⓑ{I}V2. -/2 width=2 by lfxs_inv_lref_pair_sn/ qed-. - -lemma lfeq_inv_lref_pair_dx: ∀I,Y1,L2,V2,i. Y1 ≡[#⫯i] L2.ⓑ{I}V2 → - ∃∃L1,V1. L1 ≡[#i] L2 & Y1 = L1.ⓑ{I}V1. -/2 width=2 by lfxs_inv_lref_pair_dx/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lfeq_fwd_bind_sn: ∀p,I,L1,L2,V,T. L1 ≡[ⓑ{p,I}V.T] L2 → L1 ≡[V] L2. -/2 width=4 by lfxs_fwd_bind_sn/ qed-. - -lemma lfeq_fwd_bind_dx: ∀p,I,L1,L2,V,T. - L1 ≡[ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≡[T] L2.ⓑ{I}V. -/2 width=2 by lfxs_fwd_bind_dx/ qed-. - -lemma lfeq_fwd_flat_sn: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 → L1 ≡[V] L2. -/2 width=3 by lfxs_fwd_flat_sn/ qed-. - -lemma lfeq_fwd_flat_dx: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 → L1 ≡[T] L2. -/2 width=3 by lfxs_fwd_flat_dx/ qed-. - -lemma lfeq_fwd_pair_sn: ∀I,L1,L2,V,T. L1 ≡[②{I}V.T] L2 → L1 ≡[V] L2. -/2 width=3 by lfxs_fwd_pair_sn/ qed-. - -(* Advanceded forward lemmas with generic extension on referred entries *****) - -lemma lfex_fwd_lfxs_refl: ∀R. (∀L. reflexive … (R L)) → - ∀L1,L2,T. L1 ≡[T] L2 → L1 ⦻*[R, T] L2. -/2 width=3 by lfxs_co/ qed-. - -(* Basic_2A1: removed theorems 30: - lleq_ind lleq_inv_bind lleq_inv_flat lleq_fwd_length lleq_fwd_lref - lleq_fwd_drop_sn lleq_fwd_drop_dx - lleq_fwd_bind_sn lleq_fwd_bind_dx lleq_fwd_flat_sn lleq_fwd_flat_dx - lleq_sort lleq_skip lleq_lref lleq_free lleq_gref lleq_bind lleq_flat - lleq_refl lleq_Y lleq_sym lleq_ge_up lleq_ge lleq_bind_O llpx_sn_lrefl - lleq_trans lleq_canc_sn lleq_canc_dx lleq_nlleq_trans nlleq_lleq_div -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfeq_fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfeq_fqup.ma deleted file mode 100644 index bb513ab37..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfeq_fqup.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/static/lfxs_fqup.ma". -include "basic_2/static/lfeq.ma". - -(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *******************) - -(* Advanced properties ******************************************************) - -lemma lfeq_refl: ∀T. reflexive … (lfeq T). -/2 width=1 by lfxs_refl/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfeq_length.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfeq_length.ma deleted file mode 100644 index db2217702..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfeq_length.ma +++ /dev/null @@ -1,23 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/static/lfxs_length.ma". -include "basic_2/static/lfeq.ma". - -(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *******************) - -(* Forward lemmas with length for local environments ************************) - -lemma lfeq_fwd_length: ∀L1,L2,T. L1 ≡[T] L2 → |L1| = |L2|. -/2 width=3 by lfxs_fwd_length/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfeq_lfeq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfeq_lfeq.ma deleted file mode 100644 index 677f53973..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfeq_lfeq.ma +++ /dev/null @@ -1,55 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/relocation/lreq_lreq.ma". -include "basic_2/static/lfxs_lfxs.ma". -include "basic_2/static/lfeq_lreq.ma". - -(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *******************) - -(* Main properties **********************************************************) - -theorem lfeq_bind: ∀p,I,L1,L2,V1,V2,T. - L1 ≡[V1] L2 → L1.ⓑ{I}V1 ≡[T] L2.ⓑ{I}V2 → - L1 ≡[ⓑ{p,I}V1.T] L2. -/2 width=2 by lfxs_bind/ qed. - -theorem lfeq_flat: ∀I,L1,L2,V,T. L1 ≡[V] L2 → L1 ≡[T] L2 → - L1 ≡[ⓕ{I}V.T] L2. -/2 width=1 by lfxs_flat/ qed. - -(* Note: /2 width=3 by lfeq_lfxs_trans/ *) -theorem lfeq_trans: ∀T. Transitive … (lfeq T). -#T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2 -lapply (frees_lreq_conf … Hf1 … HL1) #H0 -lapply (frees_mono … Hf2 … H0) -Hf2 -H0 -/4 width=7 by lreq_trans, lexs_eq_repl_back, ex2_intro/ -qed-. - -theorem lfeq_canc_sn: ∀T. left_cancellable … (lfeq T). -/3 width=3 by lfeq_trans, lfeq_sym/ qed-. - -theorem lfeq_canc_dx: ∀T. right_cancellable … (lfeq T). -/3 width=3 by lfeq_trans, lfeq_sym/ qed-. - -(* Advanced properies on negated lazy equivalence *****************************) - -(* Note: for use in auto, works with /4 width=8/ so lfeq_canc_sn is preferred *) -lemma lfeq_nlfeq_trans: ∀T,L1,L. L1 ≡[T] L → - ∀L2. (L ≡[T] L2 → ⊥) → (L1 ≡[T] L2 → ⊥). -/3 width=3 by lfeq_canc_sn/ qed-. - -lemma nlfeq_lfeq_div: ∀T,L2,L. L2 ≡[T] L → - ∀L1. (L1 ≡[T] L → ⊥) → (L1 ≡[T] L2 → ⊥). -/3 width=3 by lfeq_trans/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfeq_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfeq_lreq.ma deleted file mode 100644 index fc902a4a3..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfeq_lreq.ma +++ /dev/null @@ -1,36 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/static/frees_lreq.ma". -include "basic_2/static/lfeq.ma". - -(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *******************) - -(* Inversion lemmas with ranged equivalence for local environments **********) - -lemma lfeq_inv_lreq: ∀L1,L2,T. L1 ≡[T] L2 → ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≡ f & L1 ≡[f] L2. -#L1 #L2 #T * /2 width=3 by ex2_intro/ -qed-. - -(* Properties with ranged equivalence for local environments ****************) - -lemma lreq_lfeq: ∀f,L1,L2,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f → L1 ≡[f] L2 → L1 ≡[T] L2. -/2 width=3 by ex2_intro/ qed. - -(* Advanced properties ******************************************************) - -lemma lfeq_sym: ∀T. symmetric … (lfeq T). -#T #L1 #L2 #H elim (lfeq_inv_lreq … H) -H -/3 width=3 by lreq_lfeq, frees_lreq_conf, lreq_sym/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma index 92888ef46..a2f681634 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma @@ -77,6 +77,13 @@ lemma lfxs_pair_repl_dx: ∀R,I,L1,L2,T,V,V1. /3 width=5 by lexs_pair_repl, ex2_intro/ qed-. +lemma lfxs_sym: ∀R. lexs_frees_confluent R cfull → + (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) → + ∀T. symmetric … (lfxs R T). +#R #H1R #H2R #T #L1 #L2 * #f1 #Hf1 #HL12 elim (H1R … Hf1 … HL12) -Hf1 +/4 width=5 by sle_lexs_trans, lexs_sym, ex2_intro/ +qed-. + lemma lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) → ∀L1,L2,T. L1 ⦻*[R1, T] L2 → L1 ⦻*[R2, T] L2. #R1 #R2 #HR #L1 #L2 #T * /4 width=7 by lexs_co, ex2_intro/ @@ -93,7 +100,7 @@ lemma lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻*[R, ⓪{I}] ⋆ → Y1 = ⋆. qed-. lemma lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻*[R, ⋆s] Y2 → - (Y1 = ⋆ ∧ Y2 = ⋆) ∨ + (Y1 = ⋆ ∧ Y2 = ⋆) ∨ ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. #R * [ | #Y1 #I #V1 ] #Y2 #s * #f #H1 #H2 @@ -106,7 +113,7 @@ lemma lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻*[R, ⋆s] Y2 → qed-. lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻*[R, #0] Y2 → - (Y1 = ⋆ ∧ Y2 = ⋆) ∨ + (Y1 = ⋆ ∧ Y2 = ⋆) ∨ ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 & Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. #R #Y1 #Y2 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 * @@ -117,7 +124,7 @@ lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻*[R, #0] Y2 → qed-. lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 → - (Y1 = ⋆ ∧ Y2 = ⋆) ∨ + (Y1 = ⋆ ∧ Y2 = ⋆) ∨ ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, #i] L2 & Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. #R #Y1 #Y2 #i * #f #H1 #H2 elim (frees_inv_lref … H1) -H1 * diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma index 26289760b..654ab4d0d 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma @@ -35,7 +35,19 @@ theorem lfxs_flat: ∀R,I,L1,L2,V,T. #R #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 elim (sor_isfin_ex f1 f2) /3 width=7 by frees_fwd_isfin, frees_flat, lexs_join, ex2_intro/ qed. +(* +theorem lfxs_trans: ∀R. lexs_frees_confluent R cfull → + ∀T. Transitive … (lfxs R T). +#R #H1R #T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2 +elim (H1R … Hf1 … HL1) #f #H0 #H1 +lapply (frees_mono … Hf2 … H0) -Hf2 -H0 #Hf2 +lapply (lexs_eq_repl_back … HL2 … Hf2) -f2 #HL2 +lapply (sle_lexs_trans … HL1 … H1) -HL1 // #Hl1 +@(ex2_intro … f) +/4 width=7 by lreq_trans, lexs_eq_repl_back, ex2_intro/ +qed-. +*) theorem lfxs_conf: ∀R. lexs_frees_confluent R cfull → R_confluent2_lfxs R R R R → ∀T. confluent … (lfxs R T). diff --git a/matita/matita/contribs/lambdadelta/basic_2/syntax/deq.ma b/matita/matita/contribs/lambdadelta/basic_2/syntax/deq.ma deleted file mode 100644 index a14f7e3d8..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/syntax/deq.ma +++ /dev/null @@ -1,104 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/notation/relations/lazyeq_4.ma". -include "basic_2/syntax/item_sd.ma". -include "basic_2/syntax/lenv.ma". - -(* DEGREE-BASED EQUIVALENCE ON TERMS ****************************************) - -inductive deq (h) (o): relation term ≝ -| deq_sort: ∀s1,s2,d. deg h o s1 d → deg h o s2 d → deq h o (⋆s1) (⋆s2) -| deq_lref: ∀i. deq h o (#i) (#i) -| deq_gref: ∀l. deq h o (§l) (§l) -| deq_pair: ∀I,V1,V2,T1,T2. deq h o V1 V2 → deq h o T1 T2 → deq h o (②{I}V1.T1) (②{I}V2.T2) -. - -interpretation - "degree-based equivalence (terms)" - 'LazyEq h o T1 T2 = (deq h o T1 T2). - -definition cdeq: ∀h. sd h → relation3 lenv term term ≝ - λh,o,L. deq h o. - -(* Basic properties *********************************************************) - -lemma deq_refl: ∀h,o. reflexive … (deq h o). -#h #o #T elim T -T /2 width=1 by deq_pair/ -* /2 width=1 by deq_lref, deq_gref/ -#s elim (deg_total h o s) /2 width=3 by deq_sort/ -qed. - -lemma deq_sym: ∀h,o. symmetric … (deq h o). -#h #o #T1 #T2 #H elim H -T1 -T2 -/2 width=3 by deq_sort, deq_lref, deq_gref, deq_pair/ -qed-. - -(* Basic inversion lemmas ***************************************************) - -fact deq_inv_sort1_aux: ∀h,o,X,Y. X ≡[h, o] Y → ∀s1. X = ⋆s1 → - ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2. -#h #o #X #Y * -X -Y -[ #s1 #s2 #d #Hs1 #Hs2 #s #H destruct /2 width=5 by ex3_2_intro/ -| #i #s #H destruct -| #l #s #H destruct -| #I #V1 #V2 #T1 #T2 #_ #_ #s #H destruct -] -qed-. - -lemma deq_inv_sort1: ∀h,o,Y,s1. ⋆s1 ≡[h, o] Y → - ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2. -/2 width=3 by deq_inv_sort1_aux/ qed-. - -fact deq_inv_lref1_aux: ∀h,o,X,Y. X ≡[h, o] Y → ∀i. X = #i → Y = #i. -#h #o #X #Y * -X -Y // -[ #s1 #s2 #d #_ #_ #j #H destruct -| #I #V1 #V2 #T1 #T2 #_ #_ #j #H destruct -] -qed-. - -lemma deq_inv_lref1: ∀h,o,Y,i. #i ≡[h, o] Y → Y = #i. -/2 width=5 by deq_inv_lref1_aux/ qed-. - -fact deq_inv_gref1_aux: ∀h,o,X,Y. X ≡[h, o] Y → ∀l. X = §l → Y = §l. -#h #o #X #Y * -X -Y // -[ #s1 #s2 #d #_ #_ #k #H destruct -| #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct -] -qed-. - -lemma deq_inv_gref1: ∀h,o,Y,l. §l ≡[h, o] Y → Y = §l. -/2 width=5 by deq_inv_gref1_aux/ qed-. - -fact deq_inv_pair1_aux: ∀h,o,X,Y. X ≡[h, o] Y → ∀I,V1,T1. X = ②{I}V1.T1 → - ∃∃V2,T2. V1 ≡[h, o] V2 & T1 ≡[h, o] T2 & Y = ②{I}V2.T2. -#h #o #X #Y * -X -Y -[ #s1 #s2 #d #_ #_ #J #W1 #U1 #H destruct -| #i #J #W1 #U1 #H destruct -| #l #J #W1 #U1 #H destruct -| #I #V1 #V2 #T1 #T2 #HV #HT #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/ -] -qed-. - -lemma deq_inv_pair1: ∀h,o,I,V1,T1,Y. ②{I}V1.T1 ≡[h, o] Y → - ∃∃V2,T2. V1 ≡[h, o] V2 & T1 ≡[h, o] T2 & Y = ②{I}V2.T2. -/2 width=3 by deq_inv_pair1_aux/ qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma deq_inv_sort1_deg: ∀h,o,Y,s1. ⋆s1 ≡[h, o] Y → ∀d. deg h o s1 d → - ∃∃s2. deg h o s2 d & Y = ⋆s2. -#h #o #Y #s1 #H #d #Hs1 elim (deq_inv_sort1 … H) -H -#s2 #x #Hx <(deg_mono h o … Hx … Hs1) -s1 -d /2 width=3 by ex2_intro/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/syntax/deq_deq.ma b/matita/matita/contribs/lambdadelta/basic_2/syntax/deq_deq.ma deleted file mode 100644 index b69e2dc4e..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/syntax/deq_deq.ma +++ /dev/null @@ -1,30 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/syntax/deq.ma". - -(* DEGREE-BASED EQUIVALENCE ON TERMS ****************************************) - -(* Main properties **********************************************************) - -theorem deq_trans: ∀h,o. Transitive … (deq h o). -#h #o #T1 #T #H elim H -T1 -T -[ #s1 #s #d #Hs1 #Hs #X #H - elim (deq_inv_sort1_deg … H … Hs) -s /2 width=3 by deq_sort/ -| #i1 #i #H <(deq_inv_lref1 … H) -H // -| #l1 #l #H <(deq_inv_gref1 … H) -H // -| #I #V1 #V #T1 #T #_ #_ #IHV #IHT #X #H destruct - elim (deq_inv_pair1 … H) -H /3 width=1 by deq_pair/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/syntax/tdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/syntax/tdeq.ma new file mode 100644 index 000000000..b1658534c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/syntax/tdeq.ma @@ -0,0 +1,111 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/notation/relations/lazyeq_4.ma". +include "basic_2/syntax/item_sd.ma". +include "basic_2/syntax/lenv.ma". + +(* DEGREE-BASED EQUIVALENCE ON TERMS ****************************************) + +inductive tdeq (h) (o): relation term ≝ +| tdeq_sort: ∀s1,s2,d. deg h o s1 d → deg h o s2 d → tdeq h o (⋆s1) (⋆s2) +| tdeq_lref: ∀i. tdeq h o (#i) (#i) +| tdeq_gref: ∀l. tdeq h o (§l) (§l) +| tdeq_pair: ∀I,V1,V2,T1,T2. tdeq h o V1 V2 → tdeq h o T1 T2 → tdeq h o (②{I}V1.T1) (②{I}V2.T2) +. + +interpretation + "degree-based equivalence (terms)" + 'LazyEq h o T1 T2 = (tdeq h o T1 T2). + +definition cdeq: ∀h. sd h → relation3 lenv term term ≝ + λh,o,L. tdeq h o. + +(* Basic properties *********************************************************) + +lemma tdeq_refl: ∀h,o. reflexive … (tdeq h o). +#h #o #T elim T -T /2 width=1 by tdeq_pair/ +* /2 width=1 by tdeq_lref, tdeq_gref/ +#s elim (deg_total h o s) /2 width=3 by tdeq_sort/ +qed. + +lemma tdeq_sym: ∀h,o. symmetric … (tdeq h o). +#h #o #T1 #T2 #H elim H -T1 -T2 +/2 width=3 by tdeq_sort, tdeq_lref, tdeq_gref, tdeq_pair/ +qed-. + +(* Basic inversion lemmas ***************************************************) + +fact tdeq_inv_sort1_aux: ∀h,o,X,Y. X ≡[h, o] Y → ∀s1. X = ⋆s1 → + ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2. +#h #o #X #Y * -X -Y +[ #s1 #s2 #d #Hs1 #Hs2 #s #H destruct /2 width=5 by ex3_2_intro/ +| #i #s #H destruct +| #l #s #H destruct +| #I #V1 #V2 #T1 #T2 #_ #_ #s #H destruct +] +qed-. + +lemma tdeq_inv_sort1: ∀h,o,Y,s1. ⋆s1 ≡[h, o] Y → + ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2. +/2 width=3 by tdeq_inv_sort1_aux/ qed-. + +fact tdeq_inv_lref1_aux: ∀h,o,X,Y. X ≡[h, o] Y → ∀i. X = #i → Y = #i. +#h #o #X #Y * -X -Y // +[ #s1 #s2 #d #_ #_ #j #H destruct +| #I #V1 #V2 #T1 #T2 #_ #_ #j #H destruct +] +qed-. + +lemma tdeq_inv_lref1: ∀h,o,Y,i. #i ≡[h, o] Y → Y = #i. +/2 width=5 by tdeq_inv_lref1_aux/ qed-. + +fact tdeq_inv_gref1_aux: ∀h,o,X,Y. X ≡[h, o] Y → ∀l. X = §l → Y = §l. +#h #o #X #Y * -X -Y // +[ #s1 #s2 #d #_ #_ #k #H destruct +| #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct +] +qed-. + +lemma tdeq_inv_gref1: ∀h,o,Y,l. §l ≡[h, o] Y → Y = §l. +/2 width=5 by tdeq_inv_gref1_aux/ qed-. + +fact tdeq_inv_pair1_aux: ∀h,o,X,Y. X ≡[h, o] Y → ∀I,V1,T1. X = ②{I}V1.T1 → + ∃∃V2,T2. V1 ≡[h, o] V2 & T1 ≡[h, o] T2 & Y = ②{I}V2.T2. +#h #o #X #Y * -X -Y +[ #s1 #s2 #d #_ #_ #J #W1 #U1 #H destruct +| #i #J #W1 #U1 #H destruct +| #l #J #W1 #U1 #H destruct +| #I #V1 #V2 #T1 #T2 #HV #HT #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/ +] +qed-. + +lemma tdeq_inv_pair1: ∀h,o,I,V1,T1,Y. ②{I}V1.T1 ≡[h, o] Y → + ∃∃V2,T2. V1 ≡[h, o] V2 & T1 ≡[h, o] T2 & Y = ②{I}V2.T2. +/2 width=3 by tdeq_inv_pair1_aux/ qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma tdeq_inv_sort1_deg: ∀h,o,Y,s1. ⋆s1 ≡[h, o] Y → ∀d. deg h o s1 d → + ∃∃s2. deg h o s2 d & Y = ⋆s2. +#h #o #Y #s1 #H #d #Hs1 elim (tdeq_inv_sort1 … H) -H +#s2 #x #Hx <(deg_mono h o … Hx … Hs1) -s1 -d /2 width=3 by ex2_intro/ +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma tdeq_fwd_atom1: ∀h,o,I,Y. ⓪{I} ≡[h, o] Y → ∃J. Y = ⓪{J}. +#h #o * #x #Y #H [ elim (tdeq_inv_sort1 … H) -H ] +/3 width=4 by tdeq_inv_gref1, tdeq_inv_lref1, ex_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/syntax/tdeq_tdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/syntax/tdeq_tdeq.ma new file mode 100644 index 000000000..39b06145c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/syntax/tdeq_tdeq.ma @@ -0,0 +1,30 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/syntax/tdeq.ma". + +(* DEGREE-BASED EQUIVALENCE ON TERMS ****************************************) + +(* Main properties **********************************************************) + +theorem tdeq_trans: ∀h,o. Transitive … (tdeq h o). +#h #o #T1 #T #H elim H -T1 -T +[ #s1 #s #d #Hs1 #Hs #X #H + elim (tdeq_inv_sort1_deg … H … Hs) -s /2 width=3 by tdeq_sort/ +| #i1 #i #H <(tdeq_inv_lref1 … H) -H // +| #l1 #l #H <(tdeq_inv_gref1 … H) -H // +| #I #V1 #V #T1 #T #_ #_ #IHV #IHT #X #H destruct + elim (tdeq_inv_pair1 … H) -H /3 width=1 by tdeq_pair/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl index cd88f565a..a0f7ae830 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl +++ b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl @@ -167,15 +167,15 @@ table { } ] [ { "atomic arity assignment" * } { - [ "aaa ( ⦃?,?⦄ ⊢ ? ⁝ ? )" "aaa_drops" + "aaa_fqus" + "aaa_lfeq" + "aaa_aaa" * ] + [ "aaa ( ⦃?,?⦄ ⊢ ? ⁝ ? )" "aaa_drops" + "aaa_fqus" + "aaa_lfdeq" + "aaa_aaa" * ] } ] - [ { "equivalence for closures on referred entries" * } { - [ "ffeq ( ⦃?,?,?⦄ ≡ ⦃?,?,?⦄ )" "ffeq_freq" * ] + [ { "degree-based equivalence for closures on referred entries" * } { + [ "ffdeq ( ⦃?,?,?⦄ ≡[?,?] ⦃?,?,?⦄ )" "ffdeq_fqup" + "ffdeq_ffdeq" * ] } ] - [ { "equivalence for local environments on referred entries" * } { - [ "lfeq ( ? ≡[?] ? )" "lfeq_length" + "lfeq_lreq" + "lfeq_fqup" + "lfeq_lfeq" * ] + [ { "degree-based equivalence for local environments on referred entries" * } { + [ "lfdeq ( ? ≡[?,?,?] ? )" "lfdeq_length" + "lfdeq_fqup" + "lfdeq_lfdeq" * ] } ] [ { "generic extension on referred entries" * } { @@ -187,7 +187,7 @@ table { } ] [ { "context-sensitive free variables" * } { - [ "frees ( ? ⊢ 𝐅*⦃?⦄ ≡ ? )" "frees_weight" + "frees_lreq" + "frees_drops" + "frees_fqup" + "frees_frees" * ] + [ "frees ( ? ⊢ 𝐅*⦃?⦄ ≡ ? )" "frees_weight" + "frees_drops" + "frees_fqup" + "frees_frees" * ] } ] [ { "restricted ref. for local env." * } {