partial update in static_2

[helm.git] / matita / matita / contribs / lambdadelta / static_2 / syntax / lveq.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma
index e127fa23654f19aab07b5687cb781ae25b8b4fe8..43d4e636f0266eccf465a92a29d150c639ff7edd 100644 (file)
--- a/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma
@@ -12,21 +12,23 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground/xoa/ex_3_1.ma".
 include "ground/xoa/ex_3_4.ma".
 include "ground/xoa/ex_4_1.ma".
+include "ground/arith/nat_succ.ma".
 include "static_2/notation/relations/voidstareq_4.ma".
 include "static_2/syntax/lenv.ma".
 
 (* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************)
 
 inductive lveq: bi_relation nat lenv ≝
-| lveq_atom   : lveq 0 (⋆) 0 (⋆)
-| lveq_bind   : ∀I1,I2,K1,K2. lveq 0 K1 0 K2 →
-                lveq 0 (K1.ⓘ[I1]) 0 (K2.ⓘ[I2])
-| lveq_void_sn: ∀K1,K2,n1. lveq n1 K1 0 K2 →
-                lveq (↑n1) (K1.ⓧ) 0 K2
-| lveq_void_dx: ∀K1,K2,n2. lveq 0 K1 n2 K2 →
-                lveq 0 K1 (↑n2) (K2.ⓧ)
+| lveq_atom   : lveq 𝟎 (⋆) 𝟎 (⋆)
+| lveq_bind   : ∀I1,I2,K1,K2. lveq 𝟎 K1 𝟎 K2 →
+                lveq 𝟎 (K1.ⓘ[I1]) 𝟎 (K2.ⓘ[I2])
+| lveq_void_sn: ∀K1,K2,n1. lveq n1 K1 𝟎 K2 →
+                lveq (↑n1) (K1.ⓧ) 𝟎 K2
+| lveq_void_dx: ∀K1,K2,n2. lveq 𝟎 K1 n2 K2 →
+                lveq 𝟎 K1 (↑n2) (K2.ⓧ)
 .
 
 interpretation "equivalence up to exclusion binders (local environment)"
@@ -34,7 +36,7 @@ interpretation "equivalence up to exclusion binders (local environment)"
 
 (* Basic properties *********************************************************)
 
-lemma lveq_refl: ∀L. L ≋ⓧ*[0,0] L.
+lemma lveq_refl: ∀L. L ≋ⓧ*[𝟎,𝟎] L.
 #L elim L -L /2 width=1 by lveq_atom, lveq_bind/
 qed.
 
@@ -46,37 +48,41 @@ qed-.
 (* Basic inversion lemmas ***************************************************)
 
 fact lveq_inv_zero_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
-                        0 = n1 → 0 = n2 →
+                        (𝟎 = n1) → 𝟎 = n2 →
                         ∨∨ ∧∧ ⋆ = L1 & ⋆ = L2
-                            | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0,0] K2 & K1.ⓘ[I1] = L1 & K2.ⓘ[I2] = L2.
+                         | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[𝟎,𝟎] K2 & K1.ⓘ[I1] = L1 & K2.ⓘ[I2] = L2.
 #L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
 [1: /3 width=1 by or_introl, conj/
 |2: /3 width=7 by ex3_4_intro, or_intror/
-|*: #K1 #K2 #n #_ #H1 #H2 destruct
+|*: #K1 #K2 #n #_ [ #H #_ | #_ #H ]
+    elim (eq_inv_zero_nsucc … H)
 ]
 qed-.
 
-lemma lveq_inv_zero: ∀L1,L2. L1 ≋ⓧ*[0,0] L2 →
+lemma lveq_inv_zero: ∀L1,L2. L1 ≋ⓧ*[𝟎,𝟎] L2 →
                      ∨∨ ∧∧ ⋆ = L1 & ⋆ = L2
-                      | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0,0] K2 & K1.ⓘ[I1] = L1 & K2.ⓘ[I2] = L2.
+                      | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[𝟎,𝟎] K2 & K1.ⓘ[I1] = L1 & K2.ⓘ[I2] = L2.
 /2 width=5 by lveq_inv_zero_aux/ qed-.
 
 fact lveq_inv_succ_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
                            ∀m1. ↑m1 = n1 →
-                           ∃∃K1. K1 ≋ⓧ*[m1,0] L2 & K1.ⓧ = L1 & 0 = n2.
+                           ∃∃K1. K1 ≋ⓧ*[m1,𝟎] L2 & K1.ⓧ = L1 & 𝟎 = n2.
 #L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
-[1: #m #H destruct
-|2: #I1 #I2 #K1 #K2 #_ #m #H destruct
-|*: #K1 #K2 #n #HK #m #H destruct /2 width=3 by ex3_intro/
+[1: #m #H elim (eq_inv_nsucc_zero … H)
+|2: #I1 #I2 #K1 #K2 #_ #m #H elim (eq_inv_nsucc_zero … H)
+|*: #K1 #K2 #n #HK #m #H
+    [ >(eq_inv_nsucc_bi … H) -m /2 width=3 by ex3_intro/
+    | elim (eq_inv_nsucc_zero … H)
+    ]
 ]
 qed-.
 
 lemma lveq_inv_succ_sn: ∀L1,K2,n1,n2. L1 ≋ⓧ*[↑n1,n2] K2 →
-                        ∃∃K1. K1 ≋ⓧ*[n1,0] K2 & K1.ⓧ = L1 & 0 = n2.
+                        ∃∃K1. K1 ≋ⓧ*[n1,𝟎] K2 & K1.ⓧ = L1 & 𝟎 = n2.
 /2 width=3 by lveq_inv_succ_sn_aux/ qed-.
 
 lemma lveq_inv_succ_dx: ∀K1,L2,n1,n2. K1 ≋ⓧ*[n1,↑n2] L2 →
-                        ∃∃K2. K1 ≋ⓧ*[0,n2] K2 & K2.ⓧ = L2 & 0 = n1.
+                        ∃∃K2. K1 ≋ⓧ*[𝟎,n2] K2 & K2.ⓧ = L2 & 𝟎 = n1.
 #K1 #L2 #n1 #n2 #H
 lapply (lveq_sym … H) -H #H
 elim (lveq_inv_succ_sn … H) -H /3 width=3 by lveq_sym, ex3_intro/
@@ -85,9 +91,10 @@ qed-.
 fact lveq_inv_succ_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
                         ∀m1,m2. ↑m1 = n1 → ↑m2 = n2 → ⊥.
 #L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
-[1: #m1 #m2 #H1 #H2 destruct
-|2: #I1 #I2 #K1 #K2 #_ #m1 #m2 #H1 #H2 destruct
-|*: #K1 #K2 #n #_ #m1 #m2 #H1 #H2 destruct
+[1: #m1 #m2 #H #_ elim (eq_inv_nsucc_zero … H)
+|2: #I1 #I2 #K1 #K2 #_ #m1 #m2 #H #_ elim (eq_inv_nsucc_zero … H)
+|*: #K1 #K2 #n #_ #m1 #m2 [ #_ #H | #H #_ ]
+    elim (eq_inv_nsucc_zero … H)
 ]
 qed-.
 
@@ -96,13 +103,15 @@ lemma lveq_inv_succ: ∀L1,L2,n1,n2. L1 ≋ⓧ*[↑n1,↑n2] L2 → ⊥.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma lveq_inv_bind_O: ∀I1,I2,K1,K2. K1.ⓘ[I1] ≋ⓧ*[0,0] K2.ⓘ[I2] → K1 ≋ⓧ*[0,0] K2.
+lemma lveq_inv_bind_O: ∀I1,I2,K1,K2. K1.ⓘ[I1] ≋ⓧ*[𝟎,𝟎] K2.ⓘ[I2] → K1 ≋ⓧ*[𝟎,𝟎] K2.
 #I1 #I2 #K1 #K2 #H
 elim (lveq_inv_zero … H) -H * [| #Z1 #Z2 #Y1 #Y2 #HY ] #H1 #H2 destruct //
 qed-.
 
-lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1,n2] ⋆ → ∧∧ 0 = n1 & 0 = n2.
-* [2: #n1 ] * [2,4: #n2 ] #H
+lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1,n2] ⋆ → ∧∧ 𝟎 = n1 & 𝟎 = n2.
+#n1 @(nat_ind_succ …  n1) -n1 [2: #n1 #_ ]
+#n2 @(nat_ind_succ …  n2) -n2 [2,4: #n2 #_ ]
+#H
 [ elim (lveq_inv_succ … H)
 | elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
 | elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct
@@ -111,8 +120,11 @@ lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1,n2] ⋆ → ∧∧ 0 = n1 & 0
 qed-.
 
 lemma lveq_inv_bind_atom: ∀I1,K1,n1,n2. K1.ⓘ[I1] ≋ⓧ*[n1,n2] ⋆ →
-                          ∃∃m1. K1 ≋ⓧ*[m1,0] ⋆ & BUnit Void = I1 & ↑m1 = n1 & 0 = n2.
-#I1 #K1 * [2: #n1 ] * [2,4: #n2 ] #H
+                          ∃∃m1. K1 ≋ⓧ*[m1,𝟎] ⋆ & BUnit Void = I1 & ↑m1 = n1 & 𝟎 = n2.
+#I1 #K1
+#n1 @(nat_ind_succ …  n1) -n1 [2: #n1 #_ ]
+#n2 @(nat_ind_succ …  n2) -n2 [2,4: #n2 #_ ]
+#H
 [ elim (lveq_inv_succ … H)
 | elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
 | elim (lveq_inv_succ_sn … H) -H #Y #HY #H1 #H2 destruct /2 width=3 by ex4_intro/
@@ -124,7 +136,7 @@ lemma lveq_inv_bind_atom: ∀I1,K1,n1,n2. K1.ⓘ[I1] ≋ⓧ*[n1,n2] ⋆ →
 qed-.
 
 lemma lveq_inv_atom_bind: ∀I2,K2,n1,n2. ⋆ ≋ⓧ*[n1,n2] K2.ⓘ[I2] →
-                          ∃∃m2. ⋆ ≋ⓧ*[0,m2] K2 & BUnit Void = I2 & 0 = n1 & ↑m2 = n2.
+                          ∃∃m2. ⋆ ≋ⓧ*[𝟎,m2] K2 & BUnit Void = I2 & 𝟎 = n1 & ↑m2 = n2.
 #I2 #K2 #n1 #n2 #H
 lapply (lveq_sym … H) -H #H
 elim (lveq_inv_bind_atom … H) -H
@@ -132,8 +144,11 @@ elim (lveq_inv_bind_atom … H) -H
 qed-.
 
 lemma lveq_inv_pair_pair: ∀I1,I2,K1,K2,V1,V2,n1,n2. K1.ⓑ[I1]V1 ≋ⓧ*[n1,n2] K2.ⓑ[I2]V2 →
-                          ∧∧ K1 ≋ⓧ*[0,0] K2 & 0 = n1 & 0 = n2.
-#I1 #I2 #K1 #K2 #V1 #V2 * [2: #n1 ] * [2,4: #n2 ] #H
+                          ∧∧ K1 ≋ⓧ*[𝟎,𝟎] K2 & 𝟎 = n1 & 𝟎 = n2.
+#I1 #I2 #K1 #K2 #V1 #V2
+#n1 @(nat_ind_succ …  n1) -n1 [2: #n1 #_ ]
+#n2 @(nat_ind_succ …  n2) -n2 [2,4: #n2 #_ ]
+#H
 [ elim (lveq_inv_succ … H)
 | elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
 | elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct
@@ -145,13 +160,13 @@ lemma lveq_inv_pair_pair: ∀I1,I2,K1,K2,V1,V2,n1,n2. K1.ⓑ[I1]V1 ≋ⓧ*[n1,n2
 qed-.
 
 lemma lveq_inv_void_succ_sn: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[↑n1,n2] L2 →
-                             ∧∧ L1 ≋ ⓧ*[n1,0] L2 & 0 = n2.
+                             ∧∧ L1 ≋ ⓧ*[n1,𝟎] L2 & 𝟎 = n2.
 #L1 #L2 #n1 #n2 #H
 elim (lveq_inv_succ_sn … H) -H #Y #HY #H1 #H2 destruct /2 width=1 by conj/
 qed-.
 
 lemma lveq_inv_void_succ_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,↑n2] L2.ⓧ →
-                             ∧∧ L1 ≋ ⓧ*[0,n2] L2 & 0 = n1.
+                             ∧∧ L1 ≋ ⓧ*[𝟎,n2] L2 & 𝟎 = n1.
 #L1 #L2 #n1 #n2 #H
 lapply (lveq_sym … H) -H #H
 elim (lveq_inv_void_succ_sn … H) -H
@@ -161,18 +176,25 @@ qed-.
 (* Advanced forward lemmas **************************************************)
 
 lemma lveq_fwd_gen: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
-                    ∨∨ 0 = n1 | 0 = n2.
-#L1 #L2 * [2: #n1 ] * [2,4: #n2 ] #H
+                    ∨∨ 𝟎 = n1 | 𝟎 = n2.
+#L1 #L2
+#n1 @(nat_ind_succ …  n1) -n1 [2: #n1 #_ ]
+#n2 @(nat_ind_succ …  n2) -n2 [2,4: #n2 #_ ]
+#H
 [ elim (lveq_inv_succ … H) ]
 /2 width=1 by or_introl, or_intror/
 qed-.
 
-lemma lveq_fwd_pair_sn: ∀I1,K1,L2,V1,n1,n2. K1.ⓑ[I1]V1 ≋ⓧ*[n1,n2] L2 → 0 = n1.
-#I1 #K1 #L2 #V1 * [2: #n1 ] // * [2: #n2 ] #H
+lemma lveq_fwd_pair_sn:
+      ∀I1,K1,L2,V1,n1,n2. K1.ⓑ[I1]V1 ≋ⓧ*[n1,n2] L2 → 𝟎 = n1.
+#I1 #K1 #L2 #V1
+#n1 @(nat_ind_succ …  n1) -n1 [2: #n1 #_ ] //
+#n2 @(nat_ind_succ …  n2) -n2 [2: #n2 #_ ] #H
 [ elim (lveq_inv_succ … H)
 | elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct
 ]
 qed-.
 
-lemma lveq_fwd_pair_dx: ∀I2,L1,K2,V2,n1,n2. L1 ≋ⓧ*[n1,n2] K2.ⓑ[I2]V2 → 0 = n2.
+lemma lveq_fwd_pair_dx:
+      ∀I2,L1,K2,V2,n1,n2. L1 ≋ⓧ*[n1,n2] K2.ⓑ[I2]V2 → 𝟎 = n2.
 /3 width=6 by lveq_fwd_pair_sn, lveq_sym/ qed-.