From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Tue, 4 Feb 2014 20:16:21 +0000 (+0000)
Subject: evaluation for context-sensitive extended substitution (cpye) completed!
X-Git-Tag: make_still_working~980
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=e2527c6784c2593ca67af35fafaf0b3725d80a60;p=helm.git

evaluation for context-sensitive extended substitution (cpye) completed!
---

diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubstevalalt_6.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubstevalalt_6.ma
index dc32bedc3..18cdf608f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubstevalalt_6.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubstevalalt_6.ma
@@ -16,4 +16,4 @@
 
 notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ▶ ▶ * break [ term 46 d , break term 46 e ] break 𝐍 ⦃ term 46 T2 ⦄ )"
    non associative with precedence 45
-   for @{ 'PRedEval $G $L $T1 $T2 $d $e }.
+   for @{ 'PSubstEvalAlt $G $L $T1 $T2 $d $e }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/cny_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/cny_lift.ma
index b1acaafaf..2d9b36856 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/cny_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/cny_lift.ma
@@ -85,3 +85,25 @@ lemma cny_subst: ∀I,G,L,K,V,W,i,d,e. d ≤ yinj i → i < d + e →
                  ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ ▶[O, ⫰(d+e-i)] 𝐍⦃V⦄ →
                  ⇧[O, i+1] V ≡ W → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃W⦄.
 /3 width=13 by cny_subst_aux, ldrop_fwd_drop2/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+fact cny_inv_subst_aux: ∀G,L,K,V,W,i,d,e. d ≤ yinj i → i < d + e →
+                        ⇩[i+1] L ≡ K → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃W⦄ →
+                        ⇧[O, i+1] V ≡ W → ⦃G, K⦄ ⊢ ▶[O, ⫰(d+e-i)] 𝐍⦃V⦄.
+#G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HW #HVW
+lapply (cny_narrow … HW (i+1) (⫰(d+e-i)) ? ?) -HW
+[ >yplus_SO2 <yplus_succ_swap >ylt_inv_O1
+  [ >ymax_pre_sn_comm /2 width=2 by ylt_fwd_le/
+  | lapply (monotonic_ylt_minus_dx … Hide i ?) //
+  ]
+| /2 width=3 by yle_trans/
+| #HW lapply (cny_lift_inv_ge … HW … HLK … HVW ?) // -L -W
+  >yplus_inj >yminus_refl //
+]
+qed-.
+
+lemma cny_inv_subst: ∀I,G,L,K,V,W,i,d,e. d ≤ yinj i → i < d + e →
+                     ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃W⦄ →
+                     ⇧[O, i+1] V ≡ W →  ⦃G, K⦄ ⊢ ▶[O, ⫰(d+e-i)] 𝐍⦃V⦄.
+/3 width=13 by cny_inv_subst_aux, ldrop_fwd_drop2/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpye.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpye.ma
index c93e64972..82a1e3b46 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpye.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpye.ma
@@ -57,4 +57,26 @@ qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma 
\ No newline at end of file
+lemma cpye_inv_sort1: ∀G,L,X,d,e,k. ⦃G, L⦄ ⊢ ⋆k ▶*[d, e] 𝐍⦃X⦄ → X = ⋆k.
+#G #L #X #d #e #k * /2 width=5 by cpys_inv_sort1/
+qed-.
+
+lemma cpye_inv_gref1: ∀G,L,X,d,e,p. ⦃G, L⦄ ⊢ §p ▶*[d, e] 𝐍⦃X⦄ → X = §p.
+#G #L #X #d #e #p * /2 width=5 by cpys_inv_gref1/
+qed-.
+
+lemma cpye_inv_bind1: ∀a,I,G,L,V1,T1,X,d,e. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[d, e] 𝐍⦃X⦄ →
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] 𝐍⦃T2⦄ &
+                               X = ⓑ{a,I}V2.T2.
+#a #I #G #L #V1 #T1 #X #d #e * #H1 #H2 elim (cpys_inv_bind1 … H1) -H1
+#V2 #T2 #HV12 #HT12 #H destruct elim (cny_inv_bind … H2) -H2
+/5 width=8 by lsuby_cny_conf, lsuby_succ, ex3_2_intro, conj/
+qed-.
+
+lemma cpye_inv_flat1: ∀I,G,L,V1,T1,X,d,e. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[d, e] 𝐍⦃X⦄ →
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ & ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ &
+                               X = ⓕ{I}V2.T2.
+#I #G #L #V1 #T1 #X #d #e * #H1 #H2 elim (cpys_inv_flat1 … H1) -H1
+#V2 #T2 #HV12 #HT12 #H destruct elim (cny_inv_flat … H2) -H2
+/3 width=5 by ex3_2_intro, conj/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpye_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpye_alt.ma
new file mode 100644
index 000000000..a9dc53cf5
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpye_alt.ma
@@ -0,0 +1,89 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/psubstevalalt_6.ma".
+include "basic_2/substitution/cpye_lift.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION ON TERMS **********)
+
+(* Note: alternative definition of cpye *)
+inductive cpyea: ynat → ynat → relation4 genv lenv term term ≝
+| cpyea_sort : ∀G,L,d,e,k. cpyea d e G L (⋆k) (⋆k)
+| cpyea_free : ∀G,L,d,e,i. |L| ≤ i → cpyea d e G L (#i) (#i)
+| cpyea_top  : ∀G,L,d,e,i. d + e ≤ yinj i → cpyea d e G L (#i) (#i)
+| cpyea_skip : ∀G,L,d,e,i. yinj i < d → cpyea d e G L (#i) (#i)
+| cpyea_subst: ∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → yinj i < d+e →
+               ⇩[i] L ≡ K.ⓑ{I}V1 → cpyea (yinj 0) (⫰(d+e-yinj i)) G K V1 V2 →
+               ⇧[0, i+1] V2 ≡ W2 → cpyea d e G L (#i) W2
+| cpyea_gref : ∀G,L,d,e,p. cpyea d e G L (§p) (§p)
+| cpyea_bind : ∀a,I,G,L,V1,V2,T1,T2,d,e.
+               cpyea d e G L V1 V2 → cpyea (⫯d) e G (L.ⓑ{I}V1) T1 T2 →
+               cpyea d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+| cpyea_flat : ∀I,G,L,V1,V2,T1,T2,d,e.
+               cpyea d e G L V1 V2 → cpyea d e G L T1 T2 →
+               cpyea d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+.
+
+interpretation
+   "evaluation for context-sensitive extended substitution (term) alternative"
+   'PSubstEvalAlt G L T1 T2 d e = (cpyea d e G L T1 T2).
+
+(* Main properties **********************************************************)
+
+theorem cpye_cpyea: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ → ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] 𝐍⦃T2⦄.
+#G #L #T1 @(fqup_wf_ind_eq … G L T1) -G -L -T1
+#Z #Y #X #IH #G #L * *
+[ #k #_ #_ #_ #T2 #d #e #H -X -Y -Z >(cpye_inv_sort1 … H) -H //
+| #i #HG #HL #HT #T2 #d #e #H destruct
+  elim (cpye_inv_lref1 … H) -H *
+  /4 width=7 by cpyea_subst, cpyea_free, cpyea_top, cpyea_skip, fqup_lref/
+| #p #_ #_ #_ #T2 #d #e #H -X -Y -Z >(cpye_inv_gref1 … H) -H //
+| #a #I #V1 #T1 #HG #HL #HT #T #d #e #H destruct
+  elim (cpye_inv_bind1 … H) -H /3 width=1 by cpyea_bind/
+| #I #V1 #T1 #HG #HL #HT #T #d #e #H destruct
+  elim (cpye_inv_flat1 … H) -H /3 width=1 by cpyea_flat/
+]
+qed.
+
+(* Main inversion properties ************************************************)
+
+theorem cpyea_inv_cpye: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] 𝐍⦃T2⦄ → ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄.
+#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
+/2 width=7 by cpye_subst, cpye_flat, cpye_bind, cpye_skip, cpye_top, cpye_free/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma cpye_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term.
+                    (∀G,L,d,e,k. R d e G L (⋆k) (⋆k)) →
+                    (∀G,L,d,e,i. |L| ≤ i → R d e G L (#i) (#i)) →
+                    (∀G,L,d,e,i. d + e ≤ yinj i → R d e G L (#i) (#i)) →
+                    (∀G,L,d,e,i. yinj i < d → R d e G L (#i) (#i)) →
+                    (∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → yinj i < d + e →
+                     ⇩[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[yinj O, ⫰(d+e-yinj i)] 𝐍⦃V2⦄ →
+                     ⇧[O, i+1] V2 ≡ W2 → R (yinj O) (⫰(d+e-yinj i)) G K V1 V2 → R d e G L (#i) W2
+                    ) →
+                    (∀G,L,d,e,p. R d e G L (§p) (§p)) →
+                    (∀a,I,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ →
+                     ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] 𝐍⦃T2⦄ → R d e G L V1 V2 →
+                     R (⫯d) e G (L.ⓑ{I}V1) T1 T2 → R d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+                    ) →
+                    (∀I,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ →
+                     ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ → R d e G L V1 V2 →
+                     R d e G L T1 T2 → R d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+                    ) →
+                    ∀d,e,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ → R d e G L T1 T2.
+#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #d #e #G #L #T1 #T2 #H elim (cpye_cpyea … H) -G -L -T1 -T2 -d -e
+/3 width=8 by cpyea_inv_cpye/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpye_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpye_lift.ma
index a14708dfc..1d494871c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpye_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpye_lift.ma
@@ -19,6 +19,8 @@ include "basic_2/substitution/cpye.ma".
 
 (* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION ON TERMS **********)
 
+(* Advanced properties ******************************************************)
+
 lemma cpye_subst: ∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → i < d + e →
                   ⇩[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(d+e-i)] 𝐍⦃V2⦄ →
                   ⇧[O, i+1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃W2⦄.
@@ -46,3 +48,46 @@ lemma cpys_total: ∀G,L,T1,d,e. ∃T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2
   /3 width=2 by cpye_flat, ex_intro/
 ]
 qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpye_inv_lref1: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
+                      ∨∨ |L| ≤ i ∧ T2 = #i
+                       | d + e ≤ yinj i ∧ T2 = #i
+                       | yinj i < d ∧ T2 = #i
+                       | ∃∃I,K,V1,V2. d ≤ yinj i & yinj i < d + e &
+                                      ⇩[i] L ≡ K.ⓑ{I}V1 &
+                                      ⦃G, K⦄ ⊢ V1 ▶*[yinj 0, ⫰(d+e-yinj i)]  𝐍⦃V2⦄ &
+                                      ⇧[O, i+1] V2 ≡ T2.
+#G #L #T2 #i #d #e * #H1 #H2 elim (cpys_inv_lref1 … H1) -H1
+[ #H destruct elim (cny_inv_lref … H2) -H2
+  /3 width=1 by or4_intro0, or4_intro1, or4_intro2, conj/
+| * #I #K #V1 #V2 #Hdi #Hide #HLK #HV12 #HVT2
+    @or4_intro3 @(ex5_4_intro … HLK … HVT2) (**) (* explicit constructor *)
+    /4 width=13 by cny_inv_subst_aux, ldrop_fwd_drop2, conj/
+]
+qed-.
+
+lemma cpye_inv_lref1_free: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
+                           (∨∨ |L| ≤ i | d + e ≤ yinj i | yinj i < d) → T2 = #i.
+#G #L #T2 #d #e #i #H * elim (cpye_inv_lref1 … H) -H * //
+#I #K #V1 #V2 #Hdi #Hide #HLK #_ #_ #H
+[ elim (lt_refl_false i) -d
+  @(lt_to_le_to_lt … H) -H /2 width=5 by ldrop_fwd_length_lt2/ (**) (* full auto slow: 19s *)
+]
+elim (ylt_yle_false … H) //
+qed-.
+
+lemma cpye_inv_lref1_subst: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
+                            ∀I,K,V1,V2. d ≤ yinj i → yinj i < d + e →
+                            ⇩[i] L ≡ K.ⓑ{I}V1 → ⇧[O, i+1] V2 ≡ T2 →
+                            ⦃G, K⦄ ⊢ V1 ▶*[yinj 0, ⫰(d+e-yinj i)]  𝐍⦃V2⦄.
+#G #L #T2 #d #e #i #H #I #K #V1 #V2 #Hdi #Hide #HLK #HVT2 elim (cpye_inv_lref1 … H) -H *
+[ #H elim (lt_refl_false i) -V2 -T2 -d
+  @(lt_to_le_to_lt … H) -H /2 width=5 by ldrop_fwd_length_lt2/
+|2,3: #H elim (ylt_yle_false … H) //
+| #Z #Y #X1 #X2 #_ #_ #HLY #HX12 #HXT2
+  lapply (ldrop_mono … HLY … HLK) -HLY -HLK #H destruct
+  lapply (lift_inj … HXT2 … HVT2) -HXT2 -HVT2 #H destruct //
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_alt.ma
index 02f291022..6a5e2694c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_alt.ma
@@ -75,11 +75,13 @@ lemma cpys_cpysa: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ⦃G, L
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma cpysa_cpys: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
+lemma cpysa_inv_cpys: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
 #G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
 /2 width=7 by cpys_subst, cpys_flat, cpys_bind, cpy_cpys/
 qed-.
 
+(* Advanced eliminators *****************************************************)
+
 lemma cpys_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term.
                     (∀I,G,L,d,e. R d e G L (⓪{I}) (⓪{I})) →
                     (∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → i < d + e →
@@ -96,5 +98,5 @@ lemma cpys_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term.
                     ) →
                     ∀d,e,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → R d e G L T1 T2.
 #R #H1 #H2 #H3 #H4 #d #e #G #L #T1 #T2 #H elim (cpys_cpysa … H) -G -L -T1 -T2 -d -e
-/3 width=8 by cpysa_cpys/
+/3 width=8 by cpysa_inv_cpys/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
index 43e2aa669..7dccdcc14 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
+++ b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
@@ -71,11 +71,11 @@ table {
    ]
    class "cyan"
    [ { "computation" * } {
-        [ { "context-sensitive extended evaluation" * } {
+        [ { "evaluation for context-sensitive extended reduction" * } {
              [ "cpxe ( ⦃?,?⦄ ⊢ ➡*[?,?] 𝐍⦃?⦄ )" * ]
           }
         ]
-        [ { "context-sensitive evaluation" * } {
+        [ { "evaluation for context-sensitive reduction" * } {
              [ "cpre ( ⦃?,?⦄ ⊢ ➡* 𝐍⦃?⦄ )" "cpre_cpre" * ]
           }
         ]
@@ -212,6 +212,10 @@ table {
              [ "lleq ( ? ⋕[?,?] ? )" "lleq_alt ( ? ⋕⋕[?,?] ? )" "lleq_ldrop" + "lleq_fqus" + "lleq_lleq" + "lleq_ext" * ]
           }
         ]
+        [ { "evaluation for contxt-sensitive extended substitution" * } {
+             [ "cpye ( ⦃?,?⦄ ⊢ ? ▶*[?,?] 𝐍⦃?⦄ )" "cpye_alt ( ⦃?,?⦄ ⊢ ? ▶▶*[?,?] 𝐍⦃?⦄ )" "cpye_lift" * ]
+          }
+        ]        
         [ { "contxt-sensitive extended multiple substitution" * } {
              [ "cpys ( ⦃?,?⦄ ⊢ ? ▶*[?,?] ? )" "cpys_alt ( ⦃?,?⦄ ⊢ ? ▶▶*[?,?] ? )" "cpys_lift" + "cpys_cpys" * ]
           }