X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Fdrops.ma;h=324660cbf105947b7214a618789ea98423a9052a;hb=e9da8e091898b6e67a2f270581bdc5cdbe80e9b0;hp=e4e11049c910bddca67adfac03ffe6d7a1ef0922;hpb=3a430d712f9d87185e9271b7b0c5188c5f311e4b;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma index e4e11049c..324660cbf 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma @@ -23,102 +23,102 @@ include "basic_2/relocation/lifts.ma". (* Basic_2A1: includes: drop_atom drop_pair drop_drop drop_skip drop_refl_atom_O2 drop_drop_lt drop_skip_lt *) -inductive drops (c:bool): rtmap → relation lenv ≝ -| drops_atom: ∀f. (c = Ⓣ → 𝐈⦃f⦄) → drops c (f) (⋆) (⋆) -| drops_drop: ∀I,L1,L2,V,f. drops c f L1 L2 → drops c (⫯f) (L1.ⓑ{I}V) L2 -| drops_skip: ∀I,L1,L2,V1,V2,f. - drops c f L1 L2 → ⬆*[f] V2 ≡ V1 → - drops c (↑f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2) +inductive drops (b:bool): rtmap → relation lenv ≝ +| drops_atom: ∀f. (b = Ⓣ → 𝐈⦃f⦄) → drops b (f) (⋆) (⋆) +| drops_drop: ∀f,I,L1,L2,V. drops b f L1 L2 → drops b (⫯f) (L1.ⓑ{I}V) L2 +| drops_skip: ∀f,I,L1,L2,V1,V2. + drops b f L1 L2 → ⬆*[f] V2 ≡ V1 → + drops b (↑f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2) . interpretation "uniform slicing (local environment)" 'RDropStar i L1 L2 = (drops true (uni i) L1 L2). interpretation "generic slicing (local environment)" - 'RDropStar c f L1 L2 = (drops c f L1 L2). + 'RDropStar b f L1 L2 = (drops b f L1 L2). definition d_liftable1: relation2 lenv term → predicate bool ≝ - λR,c. ∀L,K,f. ⬇*[c, f] L ≡ K → + λR,b. ∀f,L,K. ⬇*[b, f] L ≡ K → ∀T,U. ⬆*[f] T ≡ U → R K T → R L U. definition d_liftable2: predicate (lenv → relation term) ≝ - λR. ∀K,T1,T2. R K T1 T2 → ∀L,c,f. ⬇*[c, f] L ≡ K → + λR. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b, f] L ≡ K → ∀U1. ⬆*[f] T1 ≡ U1 → ∃∃U2. ⬆*[f] T2 ≡ U2 & R L U1 U2. definition d_deliftable2_sn: predicate (lenv → relation term) ≝ - λR. ∀L,U1,U2. R L U1 U2 → ∀K,c,f. ⬇*[c, f] L ≡ K → + λR. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⬇*[b, f] L ≡ K → ∀T1. ⬆*[f] T1 ≡ U1 → ∃∃T2. ⬆*[f] T2 ≡ U2 & R K T1 T2. definition dropable_sn: predicate (rtmap → relation lenv) ≝ - λR. ∀L1,K1,c,f. ⬇*[c, f] L1 ≡ K1 → ∀L2,f2. R f2 L1 L2 → + λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 → ∀f2,L2. R f2 L1 L2 → ∀f1. f ⊚ f1 ≡ f2 → - ∃∃K2. R f1 K1 K2 & ⬇*[c, f] L2 ≡ K2. + ∃∃K2. R f1 K1 K2 & ⬇*[b, f] L2 ≡ K2. definition dropable_dx: predicate (rtmap → relation lenv) ≝ - λR. ∀L1,L2,f2. R f2 L1 L2 → - ∀K2,c,f. ⬇*[c, f] L2 ≡ K2 → 𝐔⦃f⦄ → + λR. ∀f2,L1,L2. R f2 L1 L2 → + ∀b,f,K2. ⬇*[b, f] L2 ≡ K2 → 𝐔⦃f⦄ → ∀f1. f ⊚ f1 ≡ f2 → - ∃∃K1. ⬇*[c, f] L1 ≡ K1 & R f1 K1 K2. + ∃∃K1. ⬇*[b, f] L1 ≡ K1 & R f1 K1 K2. definition dedropable_sn: predicate (rtmap → relation lenv) ≝ - λR. ∀L1,K1,c,f. ⬇*[c, f] L1 ≡ K1 → ∀K2,f1. R f1 K1 K2 → + λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 → ∀f1,K2. R f1 K1 K2 → ∀f2. f ⊚ f1 ≡ f2 → - ∃∃L2. R f2 L1 L2 & ⬇*[c, f] L2 ≡ K2 & L1 ≡[f] L2. + ∃∃L2. R f2 L1 L2 & ⬇*[b, f] L2 ≡ K2 & L1 ≡[f] L2. (* Basic properties *********************************************************) -lemma drops_eq_repl_back: ∀L1,L2,c. eq_repl_back … (λf. ⬇*[c, f] L1 ≡ L2). -#L1 #L2 #c #f1 #H elim H -L1 -L2 -f1 +lemma drops_eq_repl_back: ∀b,L1,L2. eq_repl_back … (λf. ⬇*[b, f] L1 ≡ L2). +#b #L1 #L2 #f1 #H elim H -f1 -L1 -L2 [ /4 width=3 by drops_atom, isid_eq_repl_back/ -| #I #L1 #L2 #V #f1 #_ #IH #f2 #H elim (eq_inv_nx … H) -H +| #f1 #I #L1 #L2 #V #_ #IH #f2 #H elim (eq_inv_nx … H) -H /3 width=3 by drops_drop/ -| #I #L1 #L2 #V1 #v2 #f1 #_ #HV #IH #f2 #H elim (eq_inv_px … H) -H +| #f1 #I #L1 #L2 #V1 #v2 #_ #HV #IH #f2 #H elim (eq_inv_px … H) -H /3 width=3 by drops_skip, lifts_eq_repl_back/ ] qed-. -lemma drops_eq_repl_fwd: ∀L1,L2,c. eq_repl_fwd … (λf. ⬇*[c, f] L1 ≡ L2). -#L1 #L2 #c @eq_repl_sym /2 width=3 by drops_eq_repl_back/ (**) (* full auto fails *) +lemma drops_eq_repl_fwd: ∀b,L1,L2. eq_repl_fwd … (λf. ⬇*[b, f] L1 ≡ L2). +#b #L1 #L2 @eq_repl_sym /2 width=3 by drops_eq_repl_back/ (**) (* full auto fails *) qed-. lemma drops_inv_tls_at: ∀f,i1,i2. @⦃i1,f⦄ ≡ i2 → - ∀c, L1,L2. ⬇*[c,⫱*[i2]f] L1 ≡ L2 → - ⬇*[c,↑⫱*[⫯i2]f] L1 ≡ L2. + ∀b,L1,L2. ⬇*[b,⫱*[i2]f] L1 ≡ L2 → + ⬇*[b,↑⫱*[⫯i2]f] L1 ≡ L2. /3 width=3 by drops_eq_repl_fwd, at_inv_tls/ qed-. (* Basic_2A1: includes: drop_FT *) -lemma drops_TF: ∀L1,L2,f. ⬇*[Ⓣ, f] L1 ≡ L2 → ⬇*[Ⓕ, f] L1 ≡ L2. -#L1 #L2 #f #H elim H -L1 -L2 -f +lemma drops_TF: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≡ L2 → ⬇*[Ⓕ, f] L1 ≡ L2. +#f #L1 #L2 #H elim H -f -L1 -L2 /3 width=1 by drops_atom, drops_drop, drops_skip/ qed. (* Basic_2A1: includes: drop_gen *) -lemma drops_gen: ∀L1,L2,c,f. ⬇*[Ⓣ, f] L1 ≡ L2 → ⬇*[c, f] L1 ≡ L2. -#L1 #L2 * /2 width=1 by drops_TF/ +lemma drops_gen: ∀b,f,L1,L2. ⬇*[Ⓣ, f] L1 ≡ L2 → ⬇*[b, f] L1 ≡ L2. +* /2 width=1 by drops_TF/ qed-. (* Basic_2A1: includes: drop_T *) -lemma drops_F: ∀L1,L2,c,f. ⬇*[c, f] L1 ≡ L2 → ⬇*[Ⓕ, f] L1 ≡ L2. -#L1 #L2 * /2 width=1 by drops_TF/ +lemma drops_F: ∀b,f,L1,L2. ⬇*[b, f] L1 ≡ L2 → ⬇*[Ⓕ, f] L1 ≡ L2. +* /2 width=1 by drops_TF/ qed-. (* Basic_2A1: includes: drop_refl *) -lemma drops_refl: ∀c,L,f. 𝐈⦃f⦄ → ⬇*[c, f] L ≡ L. -#c #L elim L -L /2 width=1 by drops_atom/ +lemma drops_refl: ∀b,L,f. 𝐈⦃f⦄ → ⬇*[b, f] L ≡ L. +#b #L elim L -L /2 width=1 by drops_atom/ #L #I #V #IHL #f #Hf elim (isid_inv_gen … Hf) -Hf /3 width=1 by drops_skip, lifts_refl/ qed. (* Basic_2A1: includes: drop_split *) -lemma drops_split_trans: ∀L1,L2,f,c. ⬇*[c, f] L1 ≡ L2 → ∀f1,f2. f1 ⊚ f2 ≡ f → 𝐔⦃f1⦄ → - ∃∃L. ⬇*[c, f1] L1 ≡ L & ⬇*[c, f2] L ≡ L2. -#L1 #L2 #f #c #H elim H -L1 -L2 -f -[ #f #Hc #f1 #f2 #Hf #Hf1 @(ex2_intro … (⋆)) @drops_atom - #H lapply (Hc H) -c +lemma drops_split_trans: ∀b,f,L1,L2. ⬇*[b, f] L1 ≡ L2 → ∀f1,f2. f1 ⊚ f2 ≡ f → 𝐔⦃f1⦄ → + ∃∃L. ⬇*[b, f1] L1 ≡ L & ⬇*[b, f2] L ≡ L2. +#b #f #L1 #L2 #H elim H -f -L1 -L2 +[ #f #H0f #f1 #f2 #Hf #Hf1 @(ex2_intro … (⋆)) @drops_atom + #H lapply (H0f H) -b #H elim (after_inv_isid3 … Hf H) -f // -| #I #L1 #L2 #V #f #HL12 #IHL12 #f1 #f2 #Hf #Hf1 elim (after_inv_xxn … Hf) -Hf [1,3: * |*: // ] +| #f #I #L1 #L2 #V #HL12 #IHL12 #f1 #f2 #Hf #Hf1 elim (after_inv_xxn … Hf) -Hf [1,3: * |*: // ] [ #g1 #g2 #Hf #H1 #H2 destruct lapply (isuni_inv_push … Hf1 ??) -Hf1 [1,2: // ] #Hg1 elim (IHL12 … Hf) -f @@ -126,19 +126,19 @@ lemma drops_split_trans: ∀L1,L2,f,c. ⬇*[c, f] L1 ≡ L2 → ∀f1,f2. f1 ⊚ | #g1 #Hf #H destruct elim (IHL12 … Hf) -f /3 width=5 by ex2_intro, drops_drop, isuni_inv_next/ ] -| #I #L1 #L2 #V1 #V2 #f #_ #HV21 #IHL12 #f1 #f2 #Hf #Hf1 elim (after_inv_xxp … Hf) -Hf [2,3: // ] +| #f #I #L1 #L2 #V1 #V2 #_ #HV21 #IHL12 #f1 #f2 #Hf #Hf1 elim (after_inv_xxp … Hf) -Hf [2,3: // ] #g1 #g2 #Hf #H1 #H2 destruct elim (lifts_split_trans … HV21 … Hf) -HV21 elim (IHL12 … Hf) -f /3 width=5 by ex2_intro, drops_skip, isuni_fwd_push/ ] qed-. -lemma drops_split_div: ∀L1,L,f1,c. ⬇*[c, f1] L1 ≡ L → ∀f2,f. f1 ⊚ f2 ≡ f → 𝐔⦃f2⦄ → +lemma drops_split_div: ∀b,f1,L1,L. ⬇*[b, f1] L1 ≡ L → ∀f2,f. f1 ⊚ f2 ≡ f → 𝐔⦃f2⦄ → ∃∃L2. ⬇*[Ⓕ, f2] L ≡ L2 & ⬇*[Ⓕ, f] L1 ≡ L2. -#L1 #L #f1 #c #H elim H -L1 -L -f1 -[ #f1 #Hc #f2 #f #Hf #Hf2 @(ex2_intro … (⋆)) @drops_atom #H destruct -| #I #L1 #L #V #f1 #HL1 #IH #f2 #f #Hf #Hf2 elim (after_inv_nxx … Hf) -Hf [2,3: // ] +#b #f1 #L1 #L #H elim H -f1 -L1 -L +[ #f1 #Hf1 #f2 #f #Hf #Hf2 @(ex2_intro … (⋆)) @drops_atom #H destruct +| #f1 #I #L1 #L #V #HL1 #IH #f2 #f #Hf #Hf2 elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct elim (IH … Hg) -IH -Hg /3 width=5 by drops_drop, ex2_intro/ -| #I #L1 #L #V1 #V #f1 #HL1 #HV1 #IH #f2 #f #Hf #Hf2 +| #f1 #I #L1 #L #V1 #V #HL1 #HV1 #IH #f2 #f #Hf #Hf2 elim (after_inv_pxx … Hf) -Hf [1,3: * |*: // ] #g2 #g #Hg #H2 #H0 destruct [ lapply (isuni_inv_push … Hf2 ??) -Hf2 [1,2: // ] #Hg2 -IH @@ -154,118 +154,117 @@ qed-. (* Basic_1: includes: drop_gen_refl *) (* Basic_2A1: includes: drop_inv_O2 *) -lemma drops_fwd_isid: ∀L1,L2,c,f. ⬇*[c, f] L1 ≡ L2 → 𝐈⦃f⦄ → L1 = L2. -#L1 #L2 #c #f #H elim H -L1 -L2 -f // -[ #I #L1 #L2 #V #f #_ #_ #H elim (isid_inv_next … H) // +lemma drops_fwd_isid: ∀b,f,L1,L2. ⬇*[b, f] L1 ≡ L2 → 𝐈⦃f⦄ → L1 = L2. +#b #f #L1 #L2 #H elim H -f -L1 -L2 // +[ #f #I #L1 #L2 #V #_ #_ #H elim (isid_inv_next … H) // | /5 width=5 by isid_inv_push, lifts_fwd_isid, eq_f3, sym_eq/ ] qed-. -fact drops_fwd_drop2_aux: ∀X,Y,c,f2. ⬇*[c, f2] X ≡ Y → ∀I,K,V. Y = K.ⓑ{I}V → - ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ⫯f1 ≡ f & ⬇*[c, f] X ≡ K. -#X #Y #c #f2 #H elim H -X -Y -f2 -[ #f2 #Ht2 #J #K #W #H destruct -| #I #L1 #L2 #V #f2 #_ #IHL #J #K #W #H elim (IHL … H) -IHL +fact drops_fwd_drop2_aux: ∀b,f2,X,Y. ⬇*[b, f2] X ≡ Y → ∀I,K,V. Y = K.ⓑ{I}V → + ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ⫯f1 ≡ f & ⬇*[b, f] X ≡ K. +#b #f2 #X #Y #H elim H -f2 -X -Y +[ #f2 #Hf2 #J #K #W #H destruct +| #f2 #I #L1 #L2 #V #_ #IHL #J #K #W #H elim (IHL … H) -IHL /3 width=7 by after_next, ex3_2_intro, drops_drop/ -| #I #L1 #L2 #V1 #V2 #f2 #HL #_ #_ #J #K #W #H destruct +| #f2 #I #L1 #L2 #V1 #V2 #HL #_ #_ #J #K #W #H destruct lapply (isid_after_dx 𝐈𝐝 … f2) /3 width=9 by after_push, ex3_2_intro, drops_drop/ ] qed-. -lemma drops_fwd_drop2: ∀I,X,K,V,c,f2. ⬇*[c, f2] X ≡ K.ⓑ{I}V → - ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ⫯f1 ≡ f & ⬇*[c, f] X ≡ K. +lemma drops_fwd_drop2: ∀b,f2,I,X,K,V. ⬇*[b, f2] X ≡ K.ⓑ{I}V → + ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ⫯f1 ≡ f & ⬇*[b, f] X ≡ K. /2 width=5 by drops_fwd_drop2_aux/ qed-. -lemma drops_after_fwd_drop2: ∀I,X,K,V,c,f2. ⬇*[c, f2] X ≡ K.ⓑ{I}V → - ∀f1,f. 𝐈⦃f1⦄ → f2 ⊚ ⫯f1 ≡ f → ⬇*[c, f] X ≡ K. -#I #X #K #V #c #f2 #H #f1 #f #Hf1 #Hf elim (drops_fwd_drop2 … H) -H +lemma drops_after_fwd_drop2: ∀b,f2,I,X,K,V. ⬇*[b, f2] X ≡ K.ⓑ{I}V → + ∀f1,f. 𝐈⦃f1⦄ → f2 ⊚ ⫯f1 ≡ f → ⬇*[b, f] X ≡ K. +#b #f2 #I #X #K #V #H #f1 #f #Hf1 #Hf elim (drops_fwd_drop2 … H) -H #g1 #g #Hg1 #Hg #HK lapply (after_mono_eq … Hg … Hf ??) -Hg -Hf /3 width=5 by drops_eq_repl_back, isid_inv_eq_repl, eq_next/ qed-. (* Basic_1: was: drop_S *) (* Basic_2A1: was: drop_fwd_drop2 *) -lemma drops_isuni_fwd_drop2: ∀I,X,K,V,c,f. 𝐔⦃f⦄ → ⬇*[c, f] X ≡ K.ⓑ{I}V → ⬇*[c, ⫯f] X ≡ K. +lemma drops_isuni_fwd_drop2: ∀b,f,I,X,K,V. 𝐔⦃f⦄ → ⬇*[b, f] X ≡ K.ⓑ{I}V → ⬇*[b, ⫯f] X ≡ K. /3 width=7 by drops_after_fwd_drop2, after_isid_isuni/ qed-. (* Forward lemmas with test for finite colength *****************************) -lemma drops_fwd_isfin: ∀L1,L2,f. ⬇*[Ⓣ, f] L1 ≡ L2 → 𝐅⦃f⦄. -#L1 #L2 #f #H elim H -L1 -L2 -f +lemma drops_fwd_isfin: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≡ L2 → 𝐅⦃f⦄. +#f #L1 #L2 #H elim H -f -L1 -L2 /3 width=1 by isfin_next, isfin_push, isfin_isid/ qed-. (* Basic inversion lemmas ***************************************************) -fact drops_inv_atom1_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → X = ⋆ → - Y = ⋆ ∧ (c = Ⓣ → 𝐈⦃f⦄). -#X #Y #c #f * -X -Y -f +fact drops_inv_atom1_aux: ∀b,f,X,Y. ⬇*[b, f] X ≡ Y → X = ⋆ → + Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄). +#b #f #X #Y * -f -X -Y [ /3 width=1 by conj/ -| #I #L1 #L2 #V #f #_ #H destruct -| #I #L1 #L2 #V1 #V2 #f #_ #_ #H destruct +| #f #I #L1 #L2 #V #_ #H destruct +| #f #I #L1 #L2 #V1 #V2 #_ #_ #H destruct ] qed-. (* Basic_1: includes: drop_gen_sort *) (* Basic_2A1: includes: drop_inv_atom1 *) -lemma drops_inv_atom1: ∀Y,c,f. ⬇*[c, f] ⋆ ≡ Y → Y = ⋆ ∧ (c = Ⓣ → 𝐈⦃f⦄). +lemma drops_inv_atom1: ∀b,f,Y. ⬇*[b, f] ⋆ ≡ Y → Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄). /2 width=3 by drops_inv_atom1_aux/ qed-. -fact drops_inv_drop1_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → ∀I,K,V,g. X = K.ⓑ{I}V → f = ⫯g → - ⬇*[c, g] K ≡ Y. -#X #Y #c #f * -X -Y -f -[ #f #Ht #J #K #W #g #H destruct -| #I #L1 #L2 #V #f #HL #J #K #W #g #H1 #H2 <(injective_next … H2) -g destruct // -| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K #W #g #_ #H2 elim (discr_push_next … H2) +fact drops_inv_drop1_aux: ∀b,f,X,Y. ⬇*[b, f] X ≡ Y → ∀g,I,K,V. X = K.ⓑ{I}V → f = ⫯g → + ⬇*[b, g] K ≡ Y. +#b #f #X #Y * -f -X -Y +[ #f #Hf #g #J #K #W #H destruct +| #f #I #L1 #L2 #V #HL #g #J #K #W #H1 #H2 <(injective_next … H2) -g destruct // +| #f #I #L1 #L2 #V1 #V2 #_ #_ #g #J #K #W #_ #H2 elim (discr_push_next … H2) ] qed-. (* Basic_1: includes: drop_gen_drop *) (* Basic_2A1: includes: drop_inv_drop1_lt drop_inv_drop1 *) -lemma drops_inv_drop1: ∀I,K,Y,V,c,f. ⬇*[c, ⫯f] K.ⓑ{I}V ≡ Y → ⬇*[c, f] K ≡ Y. +lemma drops_inv_drop1: ∀b,f,I,K,Y,V. ⬇*[b, ⫯f] K.ⓑ{I}V ≡ Y → ⬇*[b, f] K ≡ Y. /2 width=7 by drops_inv_drop1_aux/ qed-. - -fact drops_inv_skip1_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → ∀I,K1,V1,g. X = K1.ⓑ{I}V1 → f = ↑g → - ∃∃K2,V2. ⬇*[c, g] K1 ≡ K2 & ⬆*[g] V2 ≡ V1 & Y = K2.ⓑ{I}V2. -#X #Y #c #f * -X -Y -f -[ #f #Ht #J #K1 #W1 #g #H destruct -| #I #L1 #L2 #V #f #_ #J #K1 #W1 #g #_ #H2 elim (discr_next_push … H2) -| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K1 #W1 #g #H1 #H2 <(injective_push … H2) -g destruct +fact drops_inv_skip1_aux: ∀b,f,X,Y. ⬇*[b, f] X ≡ Y → ∀g,I,K1,V1. X = K1.ⓑ{I}V1 → f = ↑g → + ∃∃K2,V2. ⬇*[b, g] K1 ≡ K2 & ⬆*[g] V2 ≡ V1 & Y = K2.ⓑ{I}V2. +#b #f #X #Y * -f -X -Y +[ #f #Hf #g #J #K1 #W1 #H destruct +| #f #I #L1 #L2 #V #_ #g #J #K1 #W1 #_ #H2 elim (discr_next_push … H2) +| #f #I #L1 #L2 #V1 #V2 #HL #HV #g #J #K1 #W1 #H1 #H2 <(injective_push … H2) -g destruct /2 width=5 by ex3_2_intro/ ] qed-. (* Basic_1: includes: drop_gen_skip_l *) (* Basic_2A1: includes: drop_inv_skip1 *) -lemma drops_inv_skip1: ∀I,K1,V1,Y,c,f. ⬇*[c, ↑f] K1.ⓑ{I}V1 ≡ Y → - ∃∃K2,V2. ⬇*[c, f] K1 ≡ K2 & ⬆*[f] V2 ≡ V1 & Y = K2.ⓑ{I}V2. +lemma drops_inv_skip1: ∀b,f,I,K1,V1,Y. ⬇*[b, ↑f] K1.ⓑ{I}V1 ≡ Y → + ∃∃K2,V2. ⬇*[b, f] K1 ≡ K2 & ⬆*[f] V2 ≡ V1 & Y = K2.ⓑ{I}V2. /2 width=5 by drops_inv_skip1_aux/ qed-. -fact drops_inv_skip2_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → ∀I,K2,V2,g. Y = K2.ⓑ{I}V2 → f = ↑g → - ∃∃K1,V1. ⬇*[c, g] K1 ≡ K2 & ⬆*[g] V2 ≡ V1 & X = K1.ⓑ{I}V1. -#X #Y #c #f * -X -Y -f -[ #f #Ht #J #K2 #W2 #g #H destruct -| #I #L1 #L2 #V #f #_ #J #K2 #W2 #g #_ #H2 elim (discr_next_push … H2) -| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K2 #W2 #g #H1 #H2 <(injective_push … H2) -g destruct +fact drops_inv_skip2_aux: ∀b,f,X,Y. ⬇*[b, f] X ≡ Y → ∀g,I,K2,V2. Y = K2.ⓑ{I}V2 → f = ↑g → + ∃∃K1,V1. ⬇*[b, g] K1 ≡ K2 & ⬆*[g] V2 ≡ V1 & X = K1.ⓑ{I}V1. +#b #f #X #Y * -f -X -Y +[ #f #Hf #g #J #K2 #W2 #H destruct +| #f #I #L1 #L2 #V #_ #g #J #K2 #W2 #_ #H2 elim (discr_next_push … H2) +| #f #I #L1 #L2 #V1 #V2 #HL #HV #g #J #K2 #W2 #H1 #H2 <(injective_push … H2) -g destruct /2 width=5 by ex3_2_intro/ ] qed-. (* Basic_1: includes: drop_gen_skip_r *) (* Basic_2A1: includes: drop_inv_skip2 *) -lemma drops_inv_skip2: ∀I,X,K2,V2,c,f. ⬇*[c, ↑f] X ≡ K2.ⓑ{I}V2 → - ∃∃K1,V1. ⬇*[c, f] K1 ≡ K2 & ⬆*[f] V2 ≡ V1 & X = K1.ⓑ{I}V1. +lemma drops_inv_skip2: ∀b,f,I,X,K2,V2. ⬇*[b, ↑f] X ≡ K2.ⓑ{I}V2 → + ∃∃K1,V1. ⬇*[b, f] K1 ≡ K2 & ⬆*[f] V2 ≡ V1 & X = K1.ⓑ{I}V1. /2 width=5 by drops_inv_skip2_aux/ qed-. -fact drops_inv_TF_aux: ∀L1,L2,f. ⬇*[Ⓕ, f] L1 ≡ L2 → 𝐔⦃f⦄ → +fact drops_inv_TF_aux: ∀f,L1,L2. ⬇*[Ⓕ, f] L1 ≡ L2 → 𝐔⦃f⦄ → ∀I,K,V. L2 = K.ⓑ{I}V → ⬇*[Ⓣ, f] L1 ≡ K.ⓑ{I}V. -#L1 #L2 #f #H elim H -L1 -L2 -f +#f #L1 #L2 #H elim H -f -L1 -L2 [ #f #_ #_ #J #K #W #H destruct -| #I #L1 #L2 #V #f #_ #IH #Hf #J #K #W #H destruct +| #f #I #L1 #L2 #V #_ #IH #Hf #J #K #W #H destruct /4 width=3 by drops_drop, isuni_inv_next/ -| #I #L1 #L2 #V1 #V2 #f #HL12 #HV21 #_ #Hf #J #K #W #H destruct +| #f #I #L1 #L2 #V1 #V2 #HL12 #HV21 #_ #Hf #J #K #W #H destruct lapply (isuni_inv_push … Hf ??) -Hf [1,2: // ] #Hf <(drops_fwd_isid … HL12) -K // <(lifts_fwd_isid … HV21) -V1 /3 width=3 by drops_refl, isid_push/ @@ -273,30 +272,30 @@ fact drops_inv_TF_aux: ∀L1,L2,f. ⬇*[Ⓕ, f] L1 ≡ L2 → 𝐔⦃f⦄ → qed-. (* Basic_2A1: includes: drop_inv_FT *) -lemma drops_inv_TF: ∀I,L,K,V,f. ⬇*[Ⓕ, f] L ≡ K.ⓑ{I}V → 𝐔⦃f⦄ → +lemma drops_inv_TF: ∀f,I,L,K,V. ⬇*[Ⓕ, f] L ≡ K.ⓑ{I}V → 𝐔⦃f⦄ → ⬇*[Ⓣ, f] L ≡ K.ⓑ{I}V. /2 width=3 by drops_inv_TF_aux/ qed-. (* Advanced inversion lemmas ************************************************) (* Basic_2A1: includes: drop_inv_gen *) -lemma drops_inv_gen: ∀I,L,K,V,c,f. ⬇*[c, f] L ≡ K.ⓑ{I}V → 𝐔⦃f⦄ → +lemma drops_inv_gen: ∀b,f,I,L,K,V. ⬇*[b, f] L ≡ K.ⓑ{I}V → 𝐔⦃f⦄ → ⬇*[Ⓣ, f] L ≡ K.ⓑ{I}V. -#I #L #K #V * /2 width=1 by drops_inv_TF/ +* /2 width=1 by drops_inv_TF/ qed-. (* Basic_2A1: includes: drop_inv_T *) -lemma drops_inv_F: ∀I,L,K,V,c,f. ⬇*[Ⓕ, f] L ≡ K.ⓑ{I}V → 𝐔⦃f⦄ → - ⬇*[c, f] L ≡ K.ⓑ{I}V. -#I #L #K #V * /2 width=1 by drops_inv_TF/ +lemma drops_inv_F: ∀b,f,I,L,K,V. ⬇*[Ⓕ, f] L ≡ K.ⓑ{I}V → 𝐔⦃f⦄ → + ⬇*[b, f] L ≡ K.ⓑ{I}V. +* /2 width=1 by drops_inv_TF/ qed-. (* Inversion lemmas with test for uniformity ********************************) -lemma drops_inv_isuni: ∀L1,L2,f. ⬇*[Ⓣ, f] L1 ≡ L2 → 𝐔⦃f⦄ → +lemma drops_inv_isuni: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≡ L2 → 𝐔⦃f⦄ → (𝐈⦃f⦄ ∧ L1 = L2) ∨ - ∃∃I,K,V,g. ⬇*[Ⓣ, g] K ≡ L2 & 𝐔⦃g⦄ & L1 = K.ⓑ{I}V & f = ⫯g. -#L1 #L2 #f * -L1 -L2 -f + ∃∃g,I,K,V. ⬇*[Ⓣ, g] K ≡ L2 & 𝐔⦃g⦄ & L1 = K.ⓑ{I}V & f = ⫯g. +#f #L1 #L2 * -f -L1 -L2 [ /4 width=1 by or_introl, conj/ | /4 width=8 by isuni_inv_next, ex4_4_intro, or_intror/ | /7 width=6 by drops_fwd_isid, lifts_fwd_isid, isuni_inv_push, isid_push, or_introl, conj, eq_f3, sym_eq/ @@ -304,10 +303,10 @@ lemma drops_inv_isuni: ∀L1,L2,f. ⬇*[Ⓣ, f] L1 ≡ L2 → 𝐔⦃f⦄ → qed-. (* Basic_2A1: was: drop_inv_O1_pair1 *) -lemma drops_inv_pair1_isuni: ∀I,K,L2,V,c,f. 𝐔⦃f⦄ → ⬇*[c, f] K.ⓑ{I}V ≡ L2 → +lemma drops_inv_pair1_isuni: ∀b,f,I,K,L2,V. 𝐔⦃f⦄ → ⬇*[b, f] K.ⓑ{I}V ≡ L2 → (𝐈⦃f⦄ ∧ L2 = K.ⓑ{I}V) ∨ - ∃∃g. 𝐔⦃g⦄ & ⬇*[c, g] K ≡ L2 & f = ⫯g. -#I #K #L2 #V #c #f #Hf #H elim (isuni_split … Hf) -Hf * #g #Hg #H0 destruct + ∃∃g. 𝐔⦃g⦄ & ⬇*[b, g] K ≡ L2 & f = ⫯g. +#b #f #I #K #L2 #V #Hf #H elim (isuni_split … Hf) -Hf * #g #Hg #H0 destruct [ lapply (drops_inv_skip1 … H) -H * #Y #X #HY #HX #H destruct <(drops_fwd_isid … HY Hg) -Y >(lifts_fwd_isid … HX Hg) -X /4 width=3 by isid_push, or_introl, conj/ @@ -316,10 +315,10 @@ lemma drops_inv_pair1_isuni: ∀I,K,L2,V,c,f. 𝐔⦃f⦄ → ⬇*[c, f] K.ⓑ{I qed-. (* Basic_2A1: was: drop_inv_O1_pair2 *) -lemma drops_inv_pair2_isuni: ∀I,K,V,c,f,L1. 𝐔⦃f⦄ → ⬇*[c, f] L1 ≡ K.ⓑ{I}V → +lemma drops_inv_pair2_isuni: ∀b,f,I,K,V,L1. 𝐔⦃f⦄ → ⬇*[b, f] L1 ≡ K.ⓑ{I}V → (𝐈⦃f⦄ ∧ L1 = K.ⓑ{I}V) ∨ - ∃∃I1,K1,V1,g. 𝐔⦃g⦄ & ⬇*[c, g] K1 ≡ K.ⓑ{I}V & L1 = K1.ⓑ{I1}V1 & f = ⫯g. -#I #K #V #c #f * + ∃∃g,I1,K1,V1. 𝐔⦃g⦄ & ⬇*[b, g] K1 ≡ K.ⓑ{I}V & L1 = K1.ⓑ{I1}V1 & f = ⫯g. +#b #f #I #K #V * [ #Hf #H elim (drops_inv_atom1 … H) -H #H destruct | #L1 #I1 #V1 #Hf #H elim (drops_inv_pair1_isuni … Hf H) -Hf -H * [ #Hf #H destruct /3 width=1 by or_introl, conj/ @@ -328,9 +327,9 @@ lemma drops_inv_pair2_isuni: ∀I,K,V,c,f,L1. 𝐔⦃f⦄ → ⬇*[c, f] L1 ≡ ] qed-. -lemma drops_inv_pair2_isuni_next: ∀I,K,V,c,f,L1. 𝐔⦃f⦄ → ⬇*[c, ⫯f] L1 ≡ K.ⓑ{I}V → - ∃∃I1,K1,V1. ⬇*[c, f] K1 ≡ K.ⓑ{I}V & L1 = K1.ⓑ{I1}V1. -#I #K #V #c #f #L1 #Hf #H elim (drops_inv_pair2_isuni … H) -H /2 width=3 by isuni_next/ -Hf * +lemma drops_inv_pair2_isuni_next: ∀b,f,I,K,V,L1. 𝐔⦃f⦄ → ⬇*[b, ⫯f] L1 ≡ K.ⓑ{I}V → + ∃∃I1,K1,V1. ⬇*[b, f] K1 ≡ K.ⓑ{I}V & L1 = K1.ⓑ{I1}V1. +#b #f #I #K #V #L1 #Hf #H elim (drops_inv_pair2_isuni … H) -H /2 width=3 by isuni_next/ -Hf * [ #H elim (isid_inv_next … H) -H // | /2 width=5 by ex2_3_intro/ ] @@ -338,9 +337,9 @@ qed-. (* Inversion lemmas with uniform relocations ********************************) -lemma drops_inv_succ: ∀L1,L2,l. ⬇*[⫯l] L1 ≡ L2 → +lemma drops_inv_succ: ∀l,L1,L2. ⬇*[⫯l] L1 ≡ L2 → ∃∃I,K,V. ⬇*[l] K ≡ L2 & L1 = K.ⓑ{I}V. -#L1 #L2 #l #H elim (drops_inv_isuni … H) -H // * +#l #L1 #L2 #H elim (drops_inv_isuni … H) -H // * [ #H elim (isid_inv_next … H) -H // | /2 width=5 by ex2_3_intro/ ]