X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=matita%2Fmatita%2Fcontribs%2Flambda_delta%2Fbasic_2%2Fgrammar%2Flenv_px.ma;fp=matita%2Fmatita%2Fcontribs%2Flambda_delta%2Fbasic_2%2Fgrammar%2Flenv_px.ma;h=0000000000000000000000000000000000000000;hb=e8998d29ab83e7b6aa495a079193705b2f6743d3;hp=41710626a8cf70bf594baad61604c14f207cb9f7;hpb=bde429ac54e48de74b3d8b1df72dfcb86aa9bae5;p=helm.git diff --git a/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_px.ma b/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_px.ma deleted file mode 100644 index 41710626a..000000000 --- a/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_px.ma +++ /dev/null @@ -1,166 +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". - -(* POINTWISE EXTENSION OF A CONTEXT-FREE REALTION FOR TERMS *****************) - -inductive lpx (R:relation term): relation lenv ≝ -| lpx_stom: lpx R (⋆) (⋆) -| lpx_pair: ∀I,K1,K2,V1,V2. - lpx R K1 K2 → R V1 V2 → lpx R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2) -. - -(* Basic inversion lemmas ***************************************************) - -fact lpx_inv_atom1_aux: ∀R,L1,L2. lpx R L1 L2 → L1 = ⋆ → L2 = ⋆. -#R #L1 #L2 * -L1 -L2 -[ // -| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct -] -qed-. - -lemma lpx_inv_atom1: ∀R,L2. lpx R (⋆) L2 → L2 = ⋆. -/2 width=4 by lpx_inv_atom1_aux/ qed-. - -fact lpx_inv_pair1_aux: ∀R,L1,L2. lpx R L1 L2 → ∀I,K1,V1. L1 = K1. ⓑ{I} V1 → - ∃∃K2,V2. lpx R K1 K2 & R V1 V2 & L2 = K2. ⓑ{I} V2. -#R #L1 #L2 * -L1 -L2 -[ #J #K1 #V1 #H destruct -| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5/ -] -qed-. - -lemma lpx_inv_pair1: ∀R,I,K1,V1,L2. lpx R (K1. ⓑ{I} V1) L2 → - ∃∃K2,V2. lpx R K1 K2 & R V1 V2 & L2 = K2. ⓑ{I} V2. -/2 width=3 by lpx_inv_pair1_aux/ qed-. - -fact lpx_inv_atom2_aux: ∀R,L1,L2. lpx R L1 L2 → L2 = ⋆ → L1 = ⋆. -#R #L1 #L2 * -L1 -L2 -[ // -| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct -] -qed-. - -lemma lpx_inv_atom2: ∀R,L1. lpx R L1 (⋆) → L1 = ⋆. -/2 width=4 by lpx_inv_atom2_aux/ qed-. - -fact lpx_inv_pair2_aux: ∀R,L1,L2. lpx R L1 L2 → ∀I,K2,V2. L2 = K2. ⓑ{I} V2 → - ∃∃K1,V1. lpx R K1 K2 & R V1 V2 & L1 = K1. ⓑ{I} V1. -#R #L1 #L2 * -L1 -L2 -[ #J #K2 #V2 #H destruct -| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5/ -] -qed-. - -lemma lpx_inv_pair2: ∀R,I,L1,K2,V2. lpx R L1 (K2. ⓑ{I} V2) → - ∃∃K1,V1. lpx R K1 K2 & R V1 V2 & L1 = K1. ⓑ{I} V1. -/2 width=3 by lpx_inv_pair2_aux/ qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lpx_fwd_length: ∀R,L1,L2. lpx R L1 L2 → |L1| = |L2|. -#R #L1 #L2 #H elim H -L1 -L2 normalize // -qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma lpx_inv_append1: ∀R,L1,K1,L. lpx R (K1 @@ L1) L → - ∃∃K2,L2. lpx R K1 K2 & lpx R L1 L2 & L = K2 @@ L2. -#R #L1 elim L1 -L1 normalize -[ #K1 #K2 #HK12 - @(ex3_2_intro … K2 (⋆)) // (**) (* explicit constructor, /2 width=5/ does not work *) -| #L1 #I #V1 #IH #K1 #X #H - elim (lpx_inv_pair1 … H) -H #L #V2 #H1 #HV12 #H destruct - elim (IH … H1) -IH -H1 #K2 #L2 #HK12 #HL12 #H destruct - @(ex3_2_intro … HK12) [2: /2 width=2/ | skip | // ] (* explicit constructor, /3 width=5/ does not work *) -] -qed-. - -lemma lpx_inv_append2: ∀R,L2,K2,L. lpx R L (K2 @@ L2) → - ∃∃K1,L1. lpx R K1 K2 & lpx R L1 L2 & L = K1 @@ L1. -#R #L2 elim L2 -L2 normalize -[ #K2 #K1 #HK12 - @(ex3_2_intro … K1 (⋆)) // (**) (* explicit constructor, /2 width=5/ does not work *) -| #L2 #I #V2 #IH #K2 #X #H - elim (lpx_inv_pair2 … H) -H #L #V1 #H1 #HV12 #H destruct - elim (IH … H1) -IH -H1 #K1 #L1 #HK12 #HL12 #H destruct - @(ex3_2_intro … HK12) [2: /2 width=2/ | skip | // ] (* explicit constructor, /3 width=5/ does not work *) -] -qed-. - -(* Basic properties *********************************************************) - -lemma lpx_refl: ∀R. reflexive ? R → reflexive … (lpx R). -#R #HR #L elim L -L // /2 width=1/ -qed. - -lemma lpx_sym: ∀R. symmetric ? R → symmetric … (lpx R). -#R #HR #L1 #L2 #H elim H -H // /3 width=1/ -qed. - -lemma lpx_trans: ∀R. Transitive ? R → Transitive … (lpx R). -#R #HR #L1 #L #H elim H -L // -#I #K1 #K #V1 #V #_ #HV1 #IHK1 #X #H -elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK2 #HV2 #H destruct /3 width=3/ -qed. - -lemma lpx_conf: ∀R. Confluent ? R → Confluent … (lpx R). -#R #HR #L0 #L1 #H elim H -L1 -[ #X #H >(lpx_inv_atom1 … H) -X /2 width=3/ -| #I #K0 #K1 #V0 #V1 #_ #HV01 #IHK01 #X #H - elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK02 #HV02 #H destruct - elim (IHK01 … HK02) -K0 #K #HK1 #HK2 - elim (HR … HV01 … HV02) -HR -V0 /3 width=5/ -] -qed. - -lemma lpx_TC_inj: ∀R,L1,L2. lpx R L1 L2 → lpx (TC … R) L1 L2. -#R #L1 #L2 #H elim H -L1 -L2 // /3 width=1/ -qed. - -lemma lpx_TC_step: ∀R,L1,L. lpx (TC … R) L1 L → - ∀L2. lpx R L L2 → lpx (TC … R) L1 L2. -#R #L1 #L #H elim H -L /2 width=1/ -#I #K1 #K #V1 #V #_ #HV1 #IHK1 #X #H -elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK2 #HV2 #H destruct /3 width=3/ -qed. - -lemma TC_lpx_pair_dx: ∀R. reflexive ? R → - ∀I,K,V1,V2. TC … R V1 V2 → - TC … (lpx R) (K.ⓑ{I}V1) (K.ⓑ{I}V2). -#R #HR #I #K #V1 #V2 #H elim H -V2 -/4 width=5 by lpx_refl, lpx_pair, inj, step/ (**) (* too slow without trace *) -qed. - -lemma TC_lpx_pair_sn: ∀R. reflexive ? R → - ∀I,V,K1,K2. TC … (lpx R) K1 K2 → - TC … (lpx R) (K1.ⓑ{I}V) (K2.ⓑ{I}V). -#R #HR #I #V #K1 #K2 #H elim H -K2 -/4 width=5 by lpx_refl, lpx_pair, inj, step/ (**) (* too slow without trace *) -qed. - -lemma lpx_TC: ∀R,L1,L2. TC … (lpx R) L1 L2 → lpx (TC … R) L1 L2. -#R #L1 #L2 #H elim H -L2 /2 width=1/ /2 width=3/ -qed. - -lemma lpx_inv_TC: ∀R. reflexive ? R → - ∀L1,L2. lpx (TC … R) L1 L2 → TC … (lpx R) L1 L2. -#R #HR #L1 #L2 #H elim H -L1 -L2 /3 width=1/ /3 width=3/ -qed. - -lemma lpx_append: ∀R,K1,K2. lpx R K1 K2 → ∀L1,L2. lpx R L1 L2 → - lpx R (L1 @@ K1) (L2 @@ K2). -#R #K1 #K2 #H elim H -K1 -K2 // /3 width=1/ -qed.