some restyling ...

[helm.git] / matita / matita / contribs / lambdadelta / apps_2 / models / veq.ma

diff --git a/matita/matita/contribs/lambdadelta/apps_2/models/veq.ma b/matita/matita/contribs/lambdadelta/apps_2/models/veq.ma
index 11f07c404293f8df145f2828635c0885659af203..3d9e440a7f42980bf63ca0bd8e7aafae74927249 100644 (file)
--- a/matita/matita/contribs/lambdadelta/apps_2/models/veq.ma
+++ b/matita/matita/contribs/lambdadelta/apps_2/models/veq.ma
@@ -1,3 +1,4 @@
+
 (**************************************************************************)
 (*       ___                                                              *)
 (*      ||M||                                                             *)
@@ -14,7 +15,7 @@
 
 include "apps_2/models/model_props.ma".
 
-(* EVALUATION EQUIVALENCE  **************************************************)
+(* EVALUATION EQUIVALENCE ***************************************************)
 
 definition veq (M): relation (evaluation M) ≝
                     λv1,v2. ∀d. v1 d ≗ v2 d.
@@ -26,42 +27,70 @@ interpretation "evaluation equivalence (model)"
 
 lemma veq_refl (M): is_model M →
                     reflexive … (veq M).
-/2 width=1 by mq/ qed.
+/2 width=1 by mr/ qed.
 
 lemma veq_repl (M): is_model M →
                     replace_2 … (veq M) (veq M) (veq M).
-/2 width=5 by mr/ qed-.
+/2 width=5 by mq/ qed-.
+
+lemma veq_sym (M): is_model M → symmetric … (veq M).
+/3 width=5 by veq_repl, veq_refl/ qed-.
+
+lemma veq_trans (M): is_model M → Transitive … (veq M).
+/3 width=5 by veq_repl, veq_refl/ qed-.
+
+lemma veq_canc_sn (M): is_model M → left_cancellable … (veq M).
+/3 width=3 by veq_trans, veq_sym/ qed-.
+
+lemma veq_canc_dx (M): is_model M → right_cancellable … (veq M).
+/3 width=3 by veq_trans, veq_sym/ qed-.
 
 (* Properties with evaluation push ******************************************)
 
-lemma push_comp (M): ∀i. compatible_3 … (push M i) (sq M) (veq M) (veq M).
-#m #i #d1 #d2 #Hd12 #lv1 #lv2 #HLv12 #j
-elim (lt_or_eq_or_gt j i) #Hij destruct
-[ >(push_lt … Hij) >(push_lt … Hij) //
-| >(push_eq …) >(push_eq …) //
-| >(push_gt … Hij) >(push_gt … Hij) //
+theorem vpush_swap (M): is_model M →
+                        ∀i1,i2. i1 ≤ i2 →
+                        ∀lv,d1,d2. ⫯[i1←d1] ⫯[i2←d2] lv ≗{M} ⫯[↑i2←d2] ⫯[i1←d1] lv.
+#M #HM #i1 #i2 #Hi12 #lv #d1 #d2 #j
+elim (lt_or_eq_or_gt j i1) #Hji1 destruct
+[ lapply (lt_to_le_to_lt … Hji1 Hi12) #Hji2
+  >vpush_lt // >vpush_lt // >vpush_lt /2 width=1 by lt_S/ >vpush_lt //
+  /2 width=1 by veq_refl/
+| >vpush_eq >vpush_lt /2 width=1 by monotonic_le_plus_l/ >vpush_eq
+  /2 width=1 by mr/
+| >vpush_gt // elim (lt_or_eq_or_gt (↓j) i2) #Hji2 destruct
+  [ >vpush_lt // >vpush_lt /2 width=1 by lt_minus_to_plus/ >vpush_gt //
+    /2 width=1 by veq_refl/
+  | >vpush_eq <(lt_succ_pred … Hji1) >vpush_eq
+    /2 width=1 by mr/
+  | lapply (le_to_lt_to_lt … Hi12 Hji2) #Hi1j
+    >vpush_gt // >vpush_gt /2 width=1 by lt_minus_to_plus_r/ >vpush_gt //
+    /2 width=1 by veq_refl/
+  ]
 ]
 qed.
 
-(* Inversion lemmas with evaluation push *************************************)
+lemma vpush_comp (M): is_model M →
+                      ∀i. compatible_3 … (vpush M i) (sq M) (veq M) (veq M).
+#M #HM #i #d1 #d2 #Hd12 #lv1 #lv2 #HLv12 #j
+elim (lt_or_eq_or_gt j i) #Hij destruct
+[ >vpush_lt // >vpush_lt //
+| >vpush_eq >vpush_eq //
+| >vpush_gt // >vpush_gt //
+]
+qed-.
 
-axiom veq_inv_push_sn: ∀M,lv1,y2,d1,i. ⫯[i←d1]lv1 ≗{M} y2 →
-                       ∃∃lv2,d2. lv1 ≗ lv2 & d1 ≗ d2 & ⫯[i←d2]lv2 = y2.   
-(*
-#M #lv1 #y2 #d1 #i #H 
-*)
 (* Properies with term interpretation ***************************************) 
 
-lemma ti_comp_l (M): is_model M →
-                     ∀T,gv,lv1,lv2. lv1 ≗{M} lv2 →
-                     ⟦T⟧[gv, lv1] ≗ ⟦T⟧[gv, lv2].
+lemma ti_comp (M): is_model M →
+                   ∀T,gv1,gv2. gv1 ≗ gv2 → ∀lv1,lv2. lv1 ≗ lv2 →
+                   ⟦T⟧[gv1,lv1] ≗{M} ⟦T⟧[gv2,lv2].
 #M #HM #T elim T -T * [||| #p * | * ]
-[ /4 width=3 by seq_trans, seq_sym, ms/
-| /4 width=5 by seq_sym, ml, mr/
+[ /4 width=5 by seq_trans, seq_sym, ms/
+| /4 width=5 by seq_sym, ml, mq/
 | /4 width=3 by seq_trans, seq_sym, mg/
-| /5 width=5 by push_comp, seq_sym, md, mr/
-| /5 width=1 by push_comp, mi, mq/
-| /4 width=5 by seq_sym, ma, mc, mr/
-| /4 width=5 by seq_sym, me, mr/
+| /6 width=5 by vpush_comp, seq_sym, md, mc, mq/
+| /5 width=1 by vpush_comp, mi, mr/
+| /4 width=5 by seq_sym, ma, mp, mq/
+| /4 width=5 by seq_sym, me, mq/
 ]
 qed.