update in ground_2, static_2, basic_2, apps_2, alpha_1

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / dynamic / lsubv.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubv.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubv.ma
index 20b5259d07c989cb8d5d8e2a799e5241c2a44535..50d1bd0e531208f79ce50045ed3c0cd92b8dd656 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubv.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubv.ma
@@ -17,21 +17,22 @@ include "basic_2/dynamic/cnv.ma".
 
 (* LOCAL ENVIRONMENT REFINEMENT FOR NATIVE VALIDITY *************************)
 
-inductive lsubv (a) (h) (G): relation lenv ≝
-| lsubv_atom: lsubv a h G (⋆) (⋆)
-| lsubv_bind: ∀I,L1,L2. lsubv a h G L1 L2 → lsubv a h G (L1.ⓘ{I}) (L2.ⓘ{I})
-| lsubv_beta: â\88\80L1,L2,W,V. â¦\83G, L1â¦\84 â\8a¢ â\93\9dW.V ![a,h] →
-              lsubv a h G L1 L2 → lsubv a h G (L1.ⓓⓝW.V) (L2.ⓛW)
+inductive lsubv (h) (a) (G): relation lenv ≝
+| lsubv_atom: lsubv h a G (⋆) (⋆)
+| lsubv_bind: ∀I,L1,L2. lsubv h a G L1 L2 → lsubv h a G (L1.ⓘ[I]) (L2.ⓘ[I])
+| lsubv_beta: â\88\80L1,L2,W,V. â\9dªG,L1â\9d« â\8a¢ â\93\9dW.V ![h,a] →
+              lsubv h a G L1 L2 → lsubv h a G (L1.ⓓⓝW.V) (L2.ⓛW)
 .
 
 interpretation
   "local environment refinement (native validity)"
-  'LRSubEqV a h G L1 L2 = (lsubv a h G L1 L2).
+  'LRSubEqV h a G L1 L2 = (lsubv h a G L1 L2).
 
 (* Basic inversion lemmas ***************************************************)
 
-fact lsubv_inv_atom_sn_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 → L1 = ⋆ → L2 = ⋆.
-#a #h #G #L1 #L2 * -L1 -L2
+fact lsubv_inv_atom_sn_aux (h) (a) (G):
+     ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 → L1 = ⋆ → L2 = ⋆.
+#h #a #G #L1 #L2 * -L1 -L2
 [ //
 | #I #L1 #L2 #_ #H destruct
 | #L1 #L2 #W #V #_ #_ #H destruct
@@ -39,16 +40,16 @@ fact lsubv_inv_atom_sn_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 → L1 =
 qed-.
 
 (* Basic_2A1: uses: lsubsv_inv_atom1 *)
-lemma lsubv_inv_atom_sn (a) (h) (G): ∀L2. G ⊢ ⋆ ⫃![a,h] L2 → L2 = ⋆.
+lemma lsubv_inv_atom_sn (h) (a) (G):
+      ∀L2. G ⊢ ⋆ ⫃![h,a] L2 → L2 = ⋆.
 /2 width=6 by lsubv_inv_atom_sn_aux/ qed-.
 
-fact lsubv_inv_bind_sn_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 →
-                           ∀I,K1. L1 = K1.ⓘ{I} →
-                           ∨∨ ∃∃K2. G ⊢ K1 ⫃![a,h] K2 & L2 = K2.ⓘ{I}
-                            | ∃∃K2,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] &
-                                        G ⊢ K1 ⫃![a,h] K2 &
-                                        I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
-#a #h #G #L1 #L2 * -L1 -L2
+fact lsubv_inv_bind_sn_aux (h) (a) (G): ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 →
+     ∀I,K1. L1 = K1.ⓘ[I] →
+     ∨∨ ∃∃K2. G ⊢ K1 ⫃![h,a] K2 & L2 = K2.ⓘ[I]
+      | ∃∃K2,W,V. ❪G,K1❫ ⊢ ⓝW.V ![h,a] & G ⊢ K1 ⫃![h,a] K2
+                & I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
+#h #a #G #L1 #L2 * -L1 -L2
 [ #J #K1 #H destruct
 | #I #L1 #L2 #HL12 #J #K1 #H destruct /3 width=3 by ex2_intro, or_introl/
 | #L1 #L2 #W #V #HWV #HL12 #J #K1 #H destruct /3 width=7 by ex4_3_intro, or_intror/
@@ -56,15 +57,16 @@ fact lsubv_inv_bind_sn_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 →
 qed-.
 
 (* Basic_2A1: uses: lsubsv_inv_pair1 *)
-lemma lsubv_inv_bind_sn (a) (h) (G): ∀I,K1,L2. G ⊢ K1.ⓘ{I} ⫃![a,h] L2 →
-                        ∨∨ ∃∃K2. G ⊢ K1 ⫃![a,h] K2 & L2 = K2.ⓘ{I}
-                         | ∃∃K2,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] &
-                                     G ⊢ K1 ⫃![a,h] K2 &
-                                     I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
+lemma lsubv_inv_bind_sn (h) (a) (G):
+      ∀I,K1,L2. G ⊢ K1.ⓘ[I] ⫃![h,a] L2 →
+      ∨∨ ∃∃K2. G ⊢ K1 ⫃![h,a] K2 & L2 = K2.ⓘ[I]
+       | ∃∃K2,W,V. ❪G,K1❫ ⊢ ⓝW.V ![h,a] & G ⊢ K1 ⫃![h,a] K2
+                 & I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
 /2 width=3 by lsubv_inv_bind_sn_aux/ qed-.
 
-fact lsubv_inv_atom_dx_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 → L2 = ⋆ → L1 = ⋆.
-#a #h #G #L1 #L2 * -L1 -L2
+fact lsubv_inv_atom_dx_aux (h) (a) (G):
+     ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 → L2 = ⋆ → L1 = ⋆.
+#h #a #G #L1 #L2 * -L1 -L2
 [ //
 | #I #L1 #L2 #_ #H destruct
 | #L1 #L2 #W #V #_ #_ #H destruct
@@ -72,15 +74,17 @@ fact lsubv_inv_atom_dx_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 → L2 =
 qed-.
 
 (* Basic_2A1: uses: lsubsv_inv_atom2 *)
-lemma lsubv_inv_atom2 (a) (h) (G): ∀L1. G ⊢ L1 ⫃![a,h] ⋆ → L1 = ⋆.
+lemma lsubv_inv_atom_dx (h) (a) (G):
+      ∀L1. G ⊢ L1 ⫃![h,a] ⋆ → L1 = ⋆.
 /2 width=6 by lsubv_inv_atom_dx_aux/ qed-.
 
-fact lsubv_inv_bind_dx_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 →
-                           ∀I,K2. L2 = K2.ⓘ{I} →
-                           ∨∨ ∃∃K1. G ⊢ K1 ⫃![a,h] K2 & L1 = K1.ⓘ{I}
-                            | ∃∃K1,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] &
-                                        G ⊢ K1 ⫃![a,h] K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
-#a #h #G #L1 #L2 * -L1 -L2
+fact lsubv_inv_bind_dx_aux (h) (a) (G):
+     ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 →
+     ∀I,K2. L2 = K2.ⓘ[I] →
+     ∨∨ ∃∃K1. G ⊢ K1 ⫃![h,a] K2 & L1 = K1.ⓘ[I]
+      | ∃∃K1,W,V. ❪G,K1❫ ⊢ ⓝW.V ![h,a] &
+                  G ⊢ K1 ⫃![h,a] K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
+#h #a #G #L1 #L2 * -L1 -L2
 [ #J #K2 #H destruct
 | #I #L1 #L2 #HL12 #J #K2 #H destruct /3 width=3 by ex2_intro, or_introl/
 | #L1 #L2 #W #V #HWV #HL12 #J #K2 #H destruct /3 width=7 by ex4_3_intro, or_intror/
@@ -88,17 +92,19 @@ fact lsubv_inv_bind_dx_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 →
 qed-.
 
 (* Basic_2A1: uses: lsubsv_inv_pair2 *)
-lemma lsubv_inv_bind_dx (a) (h) (G): ∀I,L1,K2. G ⊢ L1 ⫃![a,h] K2.ⓘ{I} →
-                        ∨∨ ∃∃K1. G ⊢ K1 ⫃![a,h] K2 & L1 = K1.ⓘ{I}
-                         | ∃∃K1,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] &
-                                     G ⊢ K1 ⫃![a,h] K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
+lemma lsubv_inv_bind_dx (h) (a) (G):
+      ∀I,L1,K2. G ⊢ L1 ⫃![h,a] K2.ⓘ[I] →
+      ∨∨ ∃∃K1. G ⊢ K1 ⫃![h,a] K2 & L1 = K1.ⓘ[I]
+       | ∃∃K1,W,V. ❪G,K1❫ ⊢ ⓝW.V ![h,a] &
+                   G ⊢ K1 ⫃![h,a] K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
 /2 width=3 by lsubv_inv_bind_dx_aux/ qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma lsubv_inv_abst_sn (a) (h) (G): ∀K1,L2,W. G ⊢ K1.ⓛW ⫃![a,h] L2 →
-                        ∃∃K2. G ⊢ K1 ⫃![a,h] K2 & L2 = K2.ⓛW.
-#a #h #G #K1 #L2 #W #H
+lemma lsubv_inv_abst_sn (h) (a) (G):
+      ∀K1,L2,W. G ⊢ K1.ⓛW ⫃![h,a] L2 →
+      ∃∃K2. G ⊢ K1 ⫃![h,a] K2 & L2 = K2.ⓛW.
+#h #a #G #K1 #L2 #W #H
 elim (lsubv_inv_bind_sn … H) -H // *
 #K2 #XW #XV #_ #_ #H1 #H2 destruct
 qed-.
@@ -106,8 +112,8 @@ qed-.
 (* Basic properties *********************************************************)
 
 (* Basic_2A1: uses: lsubsv_refl *)
-lemma lsubv_refl (a) (h) (G): reflexive … (lsubv a h G).
-#a #h #G #L elim L -L /2 width=1 by lsubv_atom, lsubv_bind/
+lemma lsubv_refl (h) (a) (G): reflexive … (lsubv h a G).
+#h #a #G #L elim L -L /2 width=1 by lsubv_atom, lsubv_bind/
 qed.
 
 (* Basic_2A1: removed theorems 3: