X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fs_transition%2Ffqu.ma;h=0377df1a1af7d2e76d36b7090547b62a220283cb;hp=f5ef91e0cf714e9e755b0afe05aa6a4fe5fc47b7;hb=222044da28742b24584549ba86b1805a87def070;hpb=09b4420070d6a71990e16211e499b51dbb0742cb diff --git a/matita/matita/contribs/lambdadelta/basic_2/s_transition/fqu.ma b/matita/matita/contribs/lambdadelta/basic_2/s_transition/fqu.ma index f5ef91e0c..0377df1a1 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/s_transition/fqu.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/s_transition/fqu.ma @@ -13,6 +13,7 @@ (**************************************************************************) include "basic_2/notation/relations/supterm_6.ma". +include "basic_2/notation/relations/supterm_7.ma". include "basic_2/syntax/lenv.ma". include "basic_2/syntax/genv.ma". include "basic_2/relocation/lifts.ma". @@ -24,148 +25,175 @@ include "basic_2/relocation/lifts.ma". fqu_cpx_trans requires fqu_drop for all terms frees_fqus_drops requires fqu_drop restricted on atoms *) -inductive fqu: tri_relation genv lenv term ≝ -| fqu_lref_O : ∀I,G,L,V. fqu G (L.ⓑ{I}V) (#0) G L V -| fqu_pair_sn: ∀I,G,L,V,T. fqu G L (②{I}V.T) G L V -| fqu_bind_dx: ∀p,I,G,L,V,T. fqu G L (ⓑ{p,I}V.T) G (L.ⓑ{I}V) T -| fqu_flat_dx: ∀I,G,L,V,T. fqu G L (ⓕ{I}V.T) G L T -| fqu_drop : ∀I,G,L,V,T,U. ⬆*[1] T ≡ U → fqu G (L.ⓑ{I}V) U G L T +inductive fqu (b:bool): tri_relation genv lenv term ≝ +| fqu_lref_O : ∀I,G,L,V. fqu b G (L.ⓑ{I}V) (#0) G L V +| fqu_pair_sn: ∀I,G,L,V,T. fqu b G L (②{I}V.T) G L V +| fqu_bind_dx: ∀p,I,G,L,V,T. fqu b G L (ⓑ{p,I}V.T) G (L.ⓑ{I}V) T +| fqu_clear : ∀p,I,G,L,V,T. b = Ⓕ → fqu b G L (ⓑ{p,I}V.T) G (L.ⓧ) T +| fqu_flat_dx: ∀I,G,L,V,T. fqu b G L (ⓕ{I}V.T) G L T +| fqu_drop : ∀I,G,L,T,U. ⬆*[1] T ≘ U → fqu b G (L.ⓘ{I}) U G L T . +interpretation + "extended structural successor (closure)" + 'SupTerm b G1 L1 T1 G2 L2 T2 = (fqu b G1 L1 T1 G2 L2 T2). + interpretation "structural successor (closure)" - 'SupTerm G1 L1 T1 G2 L2 T2 = (fqu G1 L1 T1 G2 L2 T2). + 'SupTerm G1 L1 T1 G2 L2 T2 = (fqu true G1 L1 T1 G2 L2 T2). (* Basic properties *********************************************************) -lemma fqu_lref_S: ∀I,G,L,V,i. ⦃G, L.ⓑ{I}V, #⫯i⦄ ⊐ ⦃G, L, #i⦄. +lemma fqu_sort: ∀b,I,G,L,s. ⦃G, L.ⓘ{I}, ⋆s⦄ ⊐[b] ⦃G, L, ⋆s⦄. +/2 width=1 by fqu_drop/ qed. + +lemma fqu_lref_S: ∀b,I,G,L,i. ⦃G, L.ⓘ{I}, #↑i⦄ ⊐[b] ⦃G, L, #i⦄. +/2 width=1 by fqu_drop/ qed. + +lemma fqu_gref: ∀b,I,G,L,l. ⦃G, L.ⓘ{I}, §l⦄ ⊐[b] ⦃G, L, §l⦄. /2 width=1 by fqu_drop/ qed. (* Basic inversion lemmas ***************************************************) -fact fqu_inv_sort1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → +fact fqu_inv_sort1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → ∀s. T1 = ⋆s → - ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = ⋆s. -#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2 + ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = ⋆s. +#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2 [ #I #G #L #T #s #H destruct | #I #G #L #V #T #s #H destruct | #p #I #G #L #V #T #s #H destruct +| #p #I #G #L #V #T #_ #s #H destruct | #I #G #L #V #T #s #H destruct -| #I #G #L #V #T #U #HI12 #s #H destruct - lapply (lifts_inv_sort2 … HI12) -HI12 /2 width=3 by ex3_2_intro/ +| #I #G #L #T #U #HI12 #s #H destruct + lapply (lifts_inv_sort2 … HI12) -HI12 /2 width=2 by ex3_intro/ ] qed-. -lemma fqu_inv_sort1: ∀G1,G2,L1,L2,T2,s. ⦃G1, L1, ⋆s⦄ ⊐ ⦃G2, L2, T2⦄ → - ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = ⋆s. -/2 width=3 by fqu_inv_sort1_aux/ qed-. +lemma fqu_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1, L1, ⋆s⦄ ⊐[b] ⦃G2, L2, T2⦄ → + ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = ⋆s. +/2 width=4 by fqu_inv_sort1_aux/ qed-. -fact fqu_inv_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → +fact fqu_inv_lref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → ∀i. T1 = #i → (∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = V & i = 0) ∨ - ∃∃J,V,j. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = #j & i = ⫯j. -#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2 + ∃∃J,j. G1 = G2 & L1 = L2.ⓘ{J} & T2 = #j & i = ↑j. +#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2 [ #I #G #L #T #i #H destruct /3 width=4 by ex4_2_intro, or_introl/ | #I #G #L #V #T #i #H destruct | #p #I #G #L #V #T #i #H destruct +| #p #I #G #L #V #T #_ #i #H destruct | #I #G #L #V #T #i #H destruct -| #I #G #L #V #T #U #HI12 #i #H destruct - elim (lifts_inv_lref2_uni … HI12) -HI12 /3 width=3 by ex4_3_intro, or_intror/ +| #I #G #L #T #U #HI12 #i #H destruct + elim (lifts_inv_lref2_uni … HI12) -HI12 /3 width=3 by ex4_2_intro, or_intror/ ] qed-. -lemma fqu_inv_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐ ⦃G2, L2, T2⦄ → +lemma fqu_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐[b] ⦃G2, L2, T2⦄ → (∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = V & i = 0) ∨ - ∃∃J,V,j. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = #j & i = ⫯j. -/2 width=3 by fqu_inv_lref1_aux/ qed-. + ∃∃J,j. G1 = G2 & L1 = L2.ⓘ{J} & T2 = #j & i = ↑j. +/2 width=4 by fqu_inv_lref1_aux/ qed-. -fact fqu_inv_gref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → +fact fqu_inv_gref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → ∀l. T1 = §l → - ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = §l. -#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2 + ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = §l. +#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2 [ #I #G #L #T #l #H destruct | #I #G #L #V #T #l #H destruct | #p #I #G #L #V #T #l #H destruct +| #p #I #G #L #V #T #_ #l #H destruct | #I #G #L #V #T #s #H destruct -| #I #G #L #V #T #U #HI12 #l #H destruct - lapply (lifts_inv_gref2 … HI12) -HI12 /2 width=3 by ex3_2_intro/ +| #I #G #L #T #U #HI12 #l #H destruct + lapply (lifts_inv_gref2 … HI12) -HI12 /2 width=3 by ex3_intro/ ] qed-. -lemma fqu_inv_gref1: ∀G1,G2,L1,L2,T2,l. ⦃G1, L1, §l⦄ ⊐ ⦃G2, L2, T2⦄ → - ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = §l. -/2 width=3 by fqu_inv_gref1_aux/ qed-. +lemma fqu_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1, L1, §l⦄ ⊐[b] ⦃G2, L2, T2⦄ → + ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = §l. +/2 width=4 by fqu_inv_gref1_aux/ qed-. -fact fqu_inv_bind1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → +fact fqu_inv_bind1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → ∀p,I,V1,U1. T1 = ⓑ{p,I}V1.U1 → ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2 | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2 - | ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & ⬆*[1] T2 ≡ ⓑ{p,I}V1.U1. -#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2 + | ∧∧ G1 = G2 & L1.ⓧ = L2 & U1 = T2 & b = Ⓕ + | ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓑ{p,I}V1.U1. +#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2 [ #I #G #L #T #q #J #V0 #U0 #H destruct -| #I #G #L #V #T #q #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or3_intro0/ -| #p #I #G #L #V #T #q #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or3_intro1/ +| #I #G #L #V #T #q #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or4_intro0/ +| #p #I #G #L #V #T #q #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or4_intro1/ +| #p #I #G #L #V #T #Hb #q #J #V0 #U0 #H destruct /3 width=1 by and4_intro, or4_intro2/ | #I #G #L #V #T #q #J #V0 #U0 #H destruct -| #I #G #L #V #T #U #HTU #q #J #V0 #U0 #H destruct /3 width=3 by or3_intro2, ex3_2_intro/ +| #I #G #L #T #U #HTU #q #J #V0 #U0 #H destruct /3 width=2 by or4_intro3, ex3_intro/ ] qed-. -lemma fqu_inv_bind1: ∀p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓑ{p,I}V1.U1⦄ ⊐ ⦃G2, L2, T2⦄ → +lemma fqu_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓑ{p,I}V1.U1⦄ ⊐[b] ⦃G2, L2, T2⦄ → ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2 | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2 - | ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & ⬆*[1] T2 ≡ ⓑ{p,I}V1.U1. + | ∧∧ G1 = G2 & L1.ⓧ = L2 & U1 = T2 & b = Ⓕ + | ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓑ{p,I}V1.U1. /2 width=4 by fqu_inv_bind1_aux/ qed-. -fact fqu_inv_flat1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → +lemma fqu_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓑ{p,I}V1.U1⦄ ⊐ ⦃G2, L2, T2⦄ → + ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2 + | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2 + | ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓑ{p,I}V1.U1. +#p #I #G1 #G2 #L1 #L2 #V1 #U1 #T2 #H elim (fqu_inv_bind1 … H) -H +/3 width=1 by or3_intro0, or3_intro1, or3_intro2/ +* #_ #_ #_ #H destruct +qed-. + +fact fqu_inv_flat1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → ∀I,V1,U1. T1 = ⓕ{I}V1.U1 → ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2 | ∧∧ G1 = G2 & L1 = L2 & U1 = T2 - | ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & ⬆*[1] T2 ≡ ⓕ{I}V1.U1. -#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2 + | ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓕ{I}V1.U1. +#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2 [ #I #G #L #T #J #V0 #U0 #H destruct | #I #G #L #V #T #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or3_intro0/ | #p #I #G #L #V #T #J #V0 #U0 #H destruct +| #p #I #G #L #V #T #_ #J #V0 #U0 #H destruct | #I #G #L #V #T #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or3_intro1/ -| #I #G #L #V #T #U #HTU #J #V0 #U0 #H destruct /3 width=3 by or3_intro2, ex3_2_intro/ +| #I #G #L #T #U #HTU #J #V0 #U0 #H destruct /3 width=2 by or3_intro2, ex3_intro/ ] qed-. -lemma fqu_inv_flat1: ∀I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓕ{I}V1.U1⦄ ⊐ ⦃G2, L2, T2⦄ → +lemma fqu_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓕ{I}V1.U1⦄ ⊐[b] ⦃G2, L2, T2⦄ → ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2 | ∧∧ G1 = G2 & L1 = L2 & U1 = T2 - | ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & ⬆*[1] T2 ≡ ⓕ{I}V1.U1. + | ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓕ{I}V1.U1. /2 width=4 by fqu_inv_flat1_aux/ qed-. (* Advanced inversion lemmas ************************************************) -lemma fqu_inv_atom1: ∀I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐ ⦃G2, L2, T2⦄ → ⊥. -* #x #G1 #G2 #L2 #T2 #H +lemma fqu_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐[b] ⦃G2, L2, T2⦄ → ⊥. +#b * #x #G1 #G2 #L2 #T2 #H [ elim (fqu_inv_sort1 … H) | elim (fqu_inv_lref1 … H) * | elim (fqu_inv_gref1 … H) ] -H -#I #V [3: #i ] #_ #H destruct +#I [2: #V |3: #i ] #_ #H destruct qed-. -lemma fqu_inv_sort1_pair: ∀I,G1,G2,K,L2,V,T2,s. ⦃G1, K.ⓑ{I}V, ⋆s⦄ ⊐ ⦃G2, L2, T2⦄ → +lemma fqu_inv_sort1_bind: ∀b,I,G1,G2,K,L2,T2,s. ⦃G1, K.ⓘ{I}, ⋆s⦄ ⊐[b] ⦃G2, L2, T2⦄ → ∧∧ G1 = G2 & L2 = K & T2 = ⋆s. -#I #G1 #G2 #K #L2 #V #T2 #s #H elim (fqu_inv_sort1 … H) -H -#Z #X #H1 #H2 #H3 destruct /2 width=1 by and3_intro/ +#b #I #G1 #G2 #K #L2 #T2 #s #H elim (fqu_inv_sort1 … H) -H +#Z #X #H1 #H2 destruct /2 width=1 by and3_intro/ qed-. -lemma fqu_inv_zero1_pair: ∀I,G1,G2,K,L2,V,T2. ⦃G1, K.ⓑ{I}V, #0⦄ ⊐ ⦃G2, L2, T2⦄ → +lemma fqu_inv_zero1_pair: ∀b,I,G1,G2,K,L2,V,T2. ⦃G1, K.ⓑ{I}V, #0⦄ ⊐[b] ⦃G2, L2, T2⦄ → ∧∧ G1 = G2 & L2 = K & T2 = V. -#I #G1 #G2 #K #L2 #V #T2 #H elim (fqu_inv_lref1 … H) -H * -#Z #X [2: #x ] #H1 #H2 #H3 #H4 destruct /2 width=1 by and3_intro/ +#b #I #G1 #G2 #K #L2 #V #T2 #H elim (fqu_inv_lref1 … H) -H * +#Z #X #H1 #H2 #H3 #H4 destruct /2 width=1 by and3_intro/ qed-. -lemma fqu_inv_lref1_pair: ∀I,G1,G2,K,L2,V,T2,i. ⦃G1, K.ⓑ{I}V, #(⫯i)⦄ ⊐ ⦃G2, L2, T2⦄ → +lemma fqu_inv_lref1_bind: ∀b,I,G1,G2,K,L2,T2,i. ⦃G1, K.ⓘ{I}, #(↑i)⦄ ⊐[b] ⦃G2, L2, T2⦄ → ∧∧ G1 = G2 & L2 = K & T2 = #i. -#I #G1 #G2 #K #L2 #V #T2 #i #H elim (fqu_inv_lref1 … H) -H * -#Z #X [2: #x ] #H1 #H2 #H3 #H4 destruct /2 width=1 by and3_intro/ +#b #I #G1 #G2 #K #L2 #T2 #i #H elim (fqu_inv_lref1 … H) -H * +#Z #X #H1 #H2 #H3 #H4 destruct /2 width=1 by and3_intro/ qed-. -lemma fqu_inv_gref1_pair: ∀I,G1,G2,K,L2,V,T2,l. ⦃G1, K.ⓑ{I}V, §l⦄ ⊐ ⦃G2, L2, T2⦄ → +lemma fqu_inv_gref1_bind: ∀b,I,G1,G2,K,L2,T2,l. ⦃G1, K.ⓘ{I}, §l⦄ ⊐[b] ⦃G2, L2, T2⦄ → ∧∧ G1 = G2 & L2 = K & T2 = §l. -#I #G1 #G2 #K #L2 #V #T2 #l #H elim (fqu_inv_gref1 … H) -H -#Z #X #H1 #H2 #H3 destruct /2 width=1 by and3_intro/ +#b #I #G1 #G2 #K #L2 #T2 #l #H elim (fqu_inv_gref1 … H) -H +#Z #H1 #H2 #H3 destruct /2 width=1 by and3_intro/ qed-. (* Basic_2A1: removed theorems 3: