X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Flpx_sn_tc.ma;h=e994bf0769202b50d428e8061e56bc30697abac4;hb=dffdece065d12d9961a6c3f1222f6d112030336f;hp=2a3a87cf4f97feb4677c3ff248e4f88ea60c6527;hpb=87fbbf33fcc2ed91cc8b8a08e1c378ef49ac723d;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_tc.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_tc.ma index 2a3a87cf4..e994bf076 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_tc.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_tc.ma @@ -18,9 +18,9 @@ include "basic_2/relocation/lpx_sn.ma". (* Properties on transitive_closure *****************************************) -lemma TC_lpx_sn_pair_refl: ∀R. (∀I,L. reflexive … (R I L)) → +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). + ∀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/ @@ -28,18 +28,18 @@ lemma TC_lpx_sn_pair_refl: ∀R. (∀I,L. reflexive … (R I L)) → ] qed-. -lemma TC_lpx_sn_pair: ∀R. (∀I,L. reflexive … (R I L)) → +lemma TC_lpx_sn_pair: ∀R. (∀L. reflexive … (R L)) → ∀I,L1,L2. TC … (lpx_sn R) L1 L2 → - ∀V1,V2. LTC … (R I) L1 V1 V2 → - TC … (lpx_sn R) (L1.ⓑ{I}V1) (L2. ⓑ{I}V2). + ∀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. (∀I,L. reflexive … (R I L)) → - ∀L1,L2. lpx_sn (λI.LTC … (R I)) L1 L2 → +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/ @@ -54,9 +54,9 @@ lemma TC_lpx_sn_inv_atom2: ∀R,L1. TC … (lpx_sn R) L1 (⋆) → L1 = ⋆. ] qed-. -lemma TC_lpx_sn_inv_pair2: ∀R. (∀I.s_rs_transitive … (R I) (λ_.lpx_sn R)) → +lemma TC_lpx_sn_inv_pair2: ∀R. s_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 I) K1 V1 V2 & L1 = K1.ⓑ{I}V1. + ∃∃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 @@ -65,11 +65,11 @@ lemma TC_lpx_sn_inv_pair2: ∀R. (∀I.s_rs_transitive … (R I) (λ_.lpx_sn R)) ] qed-. -lemma TC_lpx_sn_ind: ∀R. (∀I.s_rs_transitive … (R I) (λ_.lpx_sn R)) → +lemma TC_lpx_sn_ind: ∀R. s_rs_transitive … R (λ_. lpx_sn R) → ∀S:relation lenv. S (⋆) (⋆) → ( ∀I,K1,K2,V1,V2. - TC … (lpx_sn R) K1 K2 → LTC … (R I) K1 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. @@ -88,24 +88,24 @@ lemma TC_lpx_sn_inv_atom1: ∀R,L2. TC … (lpx_sn R) (⋆) L2 → L2 = ⋆. ] qed-. -fact TC_lpx_sn_inv_pair1_aux: ∀R. (∀I.s_rs_transitive … (R I) (λ_.lpx_sn R)) → +fact TC_lpx_sn_inv_pair1_aux: ∀R. s_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 I) K1 V1 V2 & L2 = K2. ⓑ{I} V2. + ∃∃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. (∀I.s_rs_transitive … (R I) (λ_.lpx_sn R)) → +lemma TC_lpx_sn_inv_pair1: ∀R. s_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 I) K1 V1 V2 & L2 = K2. ⓑ{I} V2. + ∃∃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. (∀I.s_rs_transitive … (R I) (λ_.lpx_sn R)) → +lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. s_rs_transitive … R (λ_. lpx_sn R) → ∀L1,L2. TC … (lpx_sn R) L1 L2 → - lpx_sn (λI.LTC … (R I)) 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 *************************************)