X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_computation%2Flsubsx.ma;h=e2fb3664846dc1231bbd9dbf17861cef8812806c;hp=d976e52d1c62b410405edae0f54083f41c8c00dc;hb=f308429a0fde273605a2330efc63268b4ac36c99;hpb=87f57ddc367303c33e19c83cd8989cd561f3185b diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx.ma index d976e52d1..e2fb36648 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx.ma @@ -25,7 +25,7 @@ inductive lsubsx (h) (G): rtmap → relation lenv ≝ lsubsx h G (⫯f) (K1.ⓘ{I}) (K2.ⓘ{I}) | lsubsx_unit: ∀f,I,K1,K2. lsubsx h G f K1 K2 → lsubsx h G (↑f) (K1.ⓤ{I}) (K2.ⓧ) -| lsubsx_pair: ∀f,I,K1,K2,V. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ → +| lsubsx_pair: ∀f,I,K1,K2,V. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ → lsubsx h G f K1 K2 → lsubsx h G (↑f) (K1.ⓑ{I}V) (K2.ⓧ) . @@ -35,7 +35,7 @@ interpretation (* Basic inversion lemmas ***************************************************) -fact lsubsx_inv_atom_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 → +fact lsubsx_inv_atom_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 → L1 = ⋆ → L2 = ⋆. #h #g #G #L1 #L2 * -g -L1 -L2 // [ #f #I #K1 #K2 #_ #H destruct @@ -44,12 +44,12 @@ fact lsubsx_inv_atom_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 → ] qed-. -lemma lsubsx_inv_atom_sn: ∀h,g,G,L2. G ⊢ ⋆ ⊆ⓧ[h, g] L2 → L2 = ⋆. +lemma lsubsx_inv_atom_sn: ∀h,g,G,L2. G ⊢ ⋆ ⊆ⓧ[h,g] L2 → L2 = ⋆. /2 width=7 by lsubsx_inv_atom_sn_aux/ qed-. -fact lsubsx_inv_push_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 → +fact lsubsx_inv_push_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 → ∀f,I,K1. g = ⫯f → L1 = K1.ⓘ{I} → - ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓘ{I}. + ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓘ{I}. #h #g #G #L1 #L2 * -g -L1 -L2 [ #f #g #J #L1 #_ #H destruct | #f #I #K1 #K2 #HK12 #g #J #L1 #H1 #H2 destruct @@ -61,13 +61,13 @@ fact lsubsx_inv_push_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 → ] qed-. -lemma lsubsx_inv_push_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⊆ⓧ[h, ⫯f] L2 → - ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓘ{I}. +lemma lsubsx_inv_push_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⊆ⓧ[h,⫯f] L2 → + ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓘ{I}. /2 width=5 by lsubsx_inv_push_sn_aux/ qed-. -fact lsubsx_inv_unit_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 → +fact lsubsx_inv_unit_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 → ∀f,I,K1. g = ↑f → L1 = K1.ⓤ{I} → - ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ. + ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ. #h #g #G #L1 #L2 * -g -L1 -L2 [ #f #g #J #L1 #_ #H destruct | #f #I #K1 #K2 #_ #g #J #L1 #H @@ -78,14 +78,14 @@ fact lsubsx_inv_unit_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 → ] qed-. -lemma lsubsx_inv_unit_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓤ{I} ⊆ⓧ[h, ↑f] L2 → - ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ. +lemma lsubsx_inv_unit_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓤ{I} ⊆ⓧ[h,↑f] L2 → + ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ. /2 width=6 by lsubsx_inv_unit_sn_aux/ qed-. -fact lsubsx_inv_pair_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 → +fact lsubsx_inv_pair_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 → ∀f,I,K1,V. g = ↑f → L1 = K1.ⓑ{I}V → - ∃∃K2. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ & - G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ. + ∃∃K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ & + G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ. #h #g #G #L1 #L2 * -g -L1 -L2 [ #f #g #J #L1 #W #_ #H destruct | #f #I #K1 #K2 #_ #g #J #L1 #W #H @@ -97,17 +97,17 @@ fact lsubsx_inv_pair_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 → qed-. (* Basic_2A1: uses: lcosx_inv_pair *) -lemma lsubsx_inv_pair_sn: ∀h,f,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h, ↑f] L2 → - ∃∃K2. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ & - G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ. +lemma lsubsx_inv_pair_sn: ∀h,f,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h,↑f] L2 → + ∃∃K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ & + G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ. /2 width=6 by lsubsx_inv_pair_sn_aux/ qed-. (* Advanced inversion lemmas ************************************************) -lemma lsubsx_inv_pair_sn_gen: ∀h,g,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h, g] L2 → - ∨∨ ∃∃f,K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & g = ⫯f & L2 = K2.ⓑ{I}V - | ∃∃f,K2. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ & - G ⊢ K1 ⊆ⓧ[h, f] K2 & g = ↑f & L2 = K2.ⓧ. +lemma lsubsx_inv_pair_sn_gen: ∀h,g,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h,g] L2 → + ∨∨ ∃∃f,K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & g = ⫯f & L2 = K2.ⓑ{I}V + | ∃∃f,K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ & + G ⊢ K1 ⊆ⓧ[h,f] K2 & g = ↑f & L2 = K2.ⓧ. #h #g #I #G #K1 #L2 #V #H elim (pn_split g) * #f #Hf destruct [ elim (lsubsx_inv_push_sn … H) -H /3 width=5 by ex3_2_intro, or_introl/ @@ -117,8 +117,8 @@ qed-. (* Advanced forward lemmas **************************************************) -lemma lsubsx_fwd_bind_sn: ∀h,g,I1,G,K1,L2. G ⊢ K1.ⓘ{I1} ⊆ⓧ[h, g] L2 → - ∃∃I2,K2. G ⊢ K1 ⊆ⓧ[h, ⫱g] K2 & L2 = K2.ⓘ{I2}. +lemma lsubsx_fwd_bind_sn: ∀h,g,I1,G,K1,L2. G ⊢ K1.ⓘ{I1} ⊆ⓧ[h,g] L2 → + ∃∃I2,K2. G ⊢ K1 ⊆ⓧ[h,⫱g] K2 & L2 = K2.ⓘ{I2}. #h #g #I1 #G #K1 #L2 elim (pn_split g) * #f #Hf destruct [ #H elim (lsubsx_inv_push_sn … H) -H @@ -132,7 +132,7 @@ qed-. (* Basic properties *********************************************************) -lemma lsubsx_eq_repl_back: ∀h,G,L1,L2. eq_repl_back … (λf. G ⊢ L1 ⊆ⓧ[h, f] L2). +lemma lsubsx_eq_repl_back: ∀h,G,L1,L2. eq_repl_back … (λf. G ⊢ L1 ⊆ⓧ[h,f] L2). #h #G #L1 #L2 #f1 #H elim H -L1 -L2 -f1 // [ #f #I #L1 #L2 #_ #IH #x #H elim (eq_inv_px … H) -H /3 width=3 by lsubsx_push/ @@ -143,7 +143,7 @@ lemma lsubsx_eq_repl_back: ∀h,G,L1,L2. eq_repl_back … (λf. G ⊢ L1 ⊆ⓧ[ ] qed-. -lemma lsubsx_eq_repl_fwd: ∀h,G,L1,L2. eq_repl_fwd … (λf. G ⊢ L1 ⊆ⓧ[h, f] L2). +lemma lsubsx_eq_repl_fwd: ∀h,G,L1,L2. eq_repl_fwd … (λf. G ⊢ L1 ⊆ⓧ[h,f] L2). #h #G #L1 #L2 @eq_repl_sym /2 width=3 by lsubsx_eq_repl_back/ qed-.