X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fs_computation%2Ffqus.ma;h=32555f1f6d3ab551bc8669dafc82490d56e5ab16;hp=851ac510848688caa32b2bedf2151310ac11a01c;hb=222044da28742b24584549ba86b1805a87def070;hpb=5275f55f5ec528edbb223834f3ec2cf1d3ce9b84 diff --git a/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus.ma index 851ac5108..32555f1f6 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus.ma @@ -12,46 +12,155 @@ (* *) (**************************************************************************) +include "ground_2/lib/star.ma". include "basic_2/notation/relations/suptermstar_6.ma". +include "basic_2/notation/relations/suptermstar_7.ma". include "basic_2/s_transition/fquq.ma". (* STAR-ITERATED SUPCLOSURE *************************************************) -definition fqus: tri_relation genv lenv term ≝ tri_TC … fquq. +definition fqus: bool → tri_relation genv lenv term ≝ + λb. tri_TC … (fquq b). + +interpretation "extended star-iterated structural successor (closure)" + 'SupTermStar b G1 L1 T1 G2 L2 T2 = (fqus b G1 L1 T1 G2 L2 T2). interpretation "star-iterated structural successor (closure)" - 'SupTermStar G1 L1 T1 G2 L2 T2 = (fqus G1 L1 T1 G2 L2 T2). + 'SupTermStar G1 L1 T1 G2 L2 T2 = (fqus true G1 L1 T1 G2 L2 T2). (* Basic eliminators ********************************************************) -lemma fqus_ind: ∀G1,L1,T1. ∀R:relation3 …. R G1 L1 T1 → - (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮ ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) → - ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → R G2 L2 T2. -#G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H +lemma fqus_ind: ∀b,G1,L1,T1. ∀Q:relation3 …. Q G1 L1 T1 → + (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → Q G L T → Q G2 L2 T2) → + ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2. +#b #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H @(tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) // qed-. -lemma fqus_ind_dx: ∀G2,L2,T2. ∀R:relation3 …. R G2 L2 T2 → - (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) → - ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → R G1 L1 T1. -#G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H +lemma fqus_ind_dx: ∀b,G2,L2,T2. ∀Q:relation3 …. Q G2 L2 T2 → + (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G L T → Q G1 L1 T1) → + ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1. +#b #G2 #L2 #T2 #Q #IH1 #IH2 #G1 #L1 #T1 #H @(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) // qed-. (* Basic properties *********************************************************) -lemma fqus_refl: tri_reflexive … fqus. +lemma fqus_refl: ∀b. tri_reflexive … (fqus b). /2 width=1 by tri_inj/ qed. -lemma fquq_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄. +lemma fquq_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → + ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄. /2 width=1 by tri_inj/ qed. -lemma fqus_strap1: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮ ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄. +lemma fqus_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → + ⦃G, L, T⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄. /2 width=5 by tri_step/ qed-. -lemma fqus_strap2: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄. +lemma fqus_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G, L, T⦄ → + ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄. /2 width=5 by tri_TC_strap/ qed-. +(* Basic inversion lemmas ***************************************************) + +lemma fqus_inv_fqu_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → + (∧∧ G1 = G2 & L1 = L2 & T1 = T2) ∨ + ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄. +#b #G1 #G2 #L1 #L2 #T1 #T2 #H12 @(fqus_ind_dx … H12) -G1 -L1 -T1 /3 width=1 by and3_intro, or_introl/ +#G1 #G #L1 #L #T1 #T * /3 width=5 by ex2_3_intro, or_intror/ +* #HG #HL #HT #_ destruct // +qed-. + +lemma fqus_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1, L1, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ → + (∧∧ G1 = G2 & L1 = L2 & ⋆s = T2) ∨ + ∃∃J,L. ⦃G1, L, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J}. +#b #G1 #G2 #L1 #L2 #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/ +#G #L #T #H elim (fqu_inv_sort1 … H) -H /3 width=4 by ex2_2_intro, or_intror/ +qed-. + +lemma fqus_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐*[b] ⦃G2, L2, T2⦄ → + ∨∨ ∧∧ G1 = G2 & L1 = L2 & #i = T2 + | ∃∃J,L,V. ⦃G1, L, V⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓑ{J}V & i = 0 + | ∃∃J,L,j. ⦃G1, L, #j⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J} & i = ↑j. +#b #G1 #G2 #L1 #L2 #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or3_intro0/ +#G #L #T #H elim (fqu_inv_lref1 … H) -H * /3 width=7 by or3_intro1, or3_intro2, ex3_4_intro, ex3_3_intro/ +qed-. + +lemma fqus_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1, L1, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ → + (∧∧ G1 = G2 & L1 = L2 & §l = T2) ∨ + ∃∃J,L. ⦃G1, L, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J}. +#b #G1 #G2 #L1 #L2 #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/ +#G #L #T #H elim (fqu_inv_gref1 … H) -H /3 width=4 by ex2_2_intro, or_intror/ +qed-. + +lemma fqus_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → + ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓑ{p,I}V1.T1 = T2 + | ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄ + | ⦃G1, L1.ⓑ{I}V1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ + | ⦃G1, L1.ⓧ, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ ∧ b = Ⓕ + | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}. +#b #p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or5_intro0/ +#G #L #T #H elim (fqu_inv_bind1 … H) -H * +[4: #J ] #H1 #H2 #H3 [4: #Hb ] #H destruct +/3 width=6 by or5_intro1, or5_intro2, or5_intro3, or5_intro4, ex3_3_intro, conj/ +qed-. + + +lemma fqus_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ ⊐* ⦃G2, L2, T2⦄ → + ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓑ{p,I}V1.T1 = T2 + | ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄ + | ⦃G1, L1.ⓑ{I}V1, T1⦄ ⊐* ⦃G2, L2, T2⦄ + | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐* ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}. +#p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_bind1 … H) -H [1,4: * ] +/3 width=1 by and3_intro, or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex3_3_intro/ +#_ #H destruct +qed-. + +lemma fqus_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓕ{I}V1.T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → + ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓕ{I}V1.T1 = T2 + | ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄ + | ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ + | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓕ{I}V1.T1 & L1 = L.ⓘ{J}. +#b #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or4_intro0/ +#G #L #T #H elim (fqu_inv_flat1 … H) -H * +[3: #J ] #H1 #H2 #H3 #H destruct +/3 width=6 by or4_intro1, or4_intro2, or4_intro3, ex3_3_intro/ +qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma fqus_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐*[b] ⦃G2, L2, T2⦄ → + ∧∧ G1 = G2 & ⋆ = L2 & ⓪{I} = T2. +#b #I #G1 #G2 #L2 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /2 width=1 by and3_intro/ +#G #L #T #H elim (fqu_inv_atom1 … H) +qed-. + +lemma fqus_inv_sort1_bind: ∀b,I,G1,G2,L1,L2,T2,s. ⦃G1, L1.ⓘ{I}, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ → + (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & ⋆s = T2) ∨ ⦃G1, L1, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄. +#b #I #G1 #G2 #L1 #L2 #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/ +#G #L #T #H elim (fqu_inv_sort1_bind … H) -H +#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/ +qed-. + +lemma fqus_inv_zero1_pair: ∀b,I,G1,G2,L1,L2,V1,T2. ⦃G1, L1.ⓑ{I}V1, #0⦄ ⊐*[b] ⦃G2, L2, T2⦄ → + (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #0 = T2) ∨ ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄. +#b #I #G1 #G2 #L1 #L2 #V1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/ +#G #L #T #H elim (fqu_inv_zero1_pair … H) -H +#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/ +qed-. + +lemma fqus_inv_lref1_bind: ∀b,I,G1,G2,L1,L2,T2,i. ⦃G1, L1.ⓘ{I}, #↑i⦄ ⊐*[b] ⦃G2, L2, T2⦄ → + (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & #(↑i) = T2) ∨ ⦃G1, L1, #i⦄ ⊐*[b] ⦃G2, L2, T2⦄. +#b #I #G1 #G2 #L1 #L2 #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/ +#G #L #T #H elim (fqu_inv_lref1_bind … H) -H +#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/ +qed-. + +lemma fqus_inv_gref1_bind: ∀b,I,G1,G2,L1,L2,T2,l. ⦃G1, L1.ⓘ{I}, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ → + (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & §l = T2) ∨ ⦃G1, L1, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄. +#b #I #G1 #G2 #L1 #L2 #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/ +#G #L #T #H elim (fqu_inv_gref1_bind … H) -H +#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/ +qed-. + (* Basic_2A1: removed theorems 1: fqus_drop *)