From fa50dfd7a837067b5694c46e8b34663a71312056 Mon Sep 17 00:00:00 2001
From: Claudio Sacerdoti Coen <claudio.sacerdoticoen@unibo.it>
Date: Fri, 18 Apr 2008 18:27:43 +0000
Subject: [PATCH] Inversion lemma for Forall.

---
 .../POPLmark/Fsub/part1a_inversion.ma         | 68 ++++++++++++-------
 1 file changed, 44 insertions(+), 24 deletions(-)

diff --git a/helm/software/matita/contribs/POPLmark/Fsub/part1a_inversion.ma b/helm/software/matita/contribs/POPLmark/Fsub/part1a_inversion.ma
index decaf1b74..b538f4d3a 100644
--- a/helm/software/matita/contribs/POPLmark/Fsub/part1a_inversion.ma
+++ b/helm/software/matita/contribs/POPLmark/Fsub/part1a_inversion.ma
@@ -89,9 +89,7 @@ lemma JSubtype_Arrow_inv:
     G ⊢ T1 ⊴ (Arrow T2 T3) → P G T1.
  intros;
  generalize in match (refl_eq ? (Arrow T2 T3));
- change in ⊢ (% → ?) with (Arrow T2 T3 = Arrow T2 T3);
  generalize in match (refl_eq ? G);
- change in ⊢ (% → ?) with (G = G);
  elim H2 in ⊢ (? ? ? % → ? ? ? % → %);
   [1,2: destruct H6
   |5: destruct H8
@@ -105,37 +103,59 @@ lemma JSubtype_Arrow_inv:
   ]
 qed.
 
+lemma JSubtype_Forall_inv:
+ ∀G:list bound.∀T1,T2,T3:Typ.
+  ∀P:list bound → Typ → Prop.
+   (∀n,t1.
+    (mk_bound true n t1) ∈ G → G ⊢ t1 ⊴ (Forall T2 T3) → P G t1 → P G (TFree n)) →
+   (∀t,t1. G ⊢ T2 ⊴ t → (∀X. ¬(X ∈ fv_env G) → (mk_bound true X T2)::G ⊢ subst_type_nat t1 (TFree X) O ⊴ subst_type_nat T3 (TFree X) O)
+     → P G (Forall t t1)) →
+      G ⊢ T1 ⊴ (Forall T2 T3) → P G T1.
+ intros;
+ generalize in match (refl_eq ? (Forall T2 T3));
+ generalize in match (refl_eq ? G);
+ elim H2 in ⊢ (? ? ? % → ? ? ? % → %);
+  [1,2: destruct H6
+  |4: destruct H8
+  | lapply (H5 H6 H7); destruct; clear H5;
+    apply H;
+    assumption
+  | destruct;
+    clear H4 H6;
+    apply H1;
+    assumption
+  ]
+qed.
+
+
 lemma JS_trans_prova: ∀T,G1.WFType G1 T →
 ∀G,R,U.incl ? (fv_env G1) (fv_env G) → G ⊢ R ⊴ T → G ⊢ T ⊴ U → G ⊢ R ⊴ U.
 intros 3;elim H;clear H; try autobatch;
   [rewrite > (JSubtype_Top ? ? H3);autobatch
   |apply (JSubtype_Arrow_inv ? ? ? ? ? ? ? H6); intros;
     [ autobatch
-    | inversion H7;intros; destruct; autobatch depth=4 width=4 size=9
-    ]
-  |generalize in match H7;generalize in match H4;generalize in match H2;
-   generalize in match H5;clear H7 H4 H2 H5;
-   generalize in match (refl_eq ? (Forall t t1));elim H6 in ⊢ (? ? ? %→%);destruct;
-     [apply (SA_Trans_TVar ? ? ? ? H);apply (H4 ? H7 H8 H9 H10);reflexivity
-     |inversion H11;intros;destruct;
-        [apply SA_Top
-           [autobatch
-              |apply WFT_Forall
-                 [autobatch
-                 |intros;lapply (H4 ? H13);autobatch]]
-           |apply SA_All
-              [autobatch paramodulation
-              |intros;apply (H10 X)
-                 [intro;apply H15;apply H8;assumption
-                 |intro;apply H15;apply H8;apply (WFT_to_incl ? ? ? H3);
+    | inversion H7;intros; destruct; autobatch depth=4 width=4 size=9]
+  |apply (JSubtype_Forall_inv ? ? ? ? ? ? ? H6); intros;
+    [ autobatch
+    | inversion H7;intros; destruct;
+       [ apply SA_Top
+          [ assumption
+          | apply WFT_Forall;
+             [ autobatch
+             | intros;lapply (H8 ? H11);
+               autobatch]]
+       | apply SA_All
+          [ autobatch
+          | intros;apply (H4 X);
+             [intro;apply H13;apply H5;assumption
+                 |intro;apply H13;apply H5;apply (WFT_to_incl ? ? ? H3);
                   assumption
                  |simplify;autobatch
-                 |apply (narrowing X (mk_bound true X t::l1) 
-                         ? ? ? ? ? H7 ? ? [])
-                    [intros;apply H9
-                       [unfold;intros;lapply (H8 ? H17);rewrite > fv_append;
+                 |apply (narrowing X (mk_bound true X t::G) ? ? ? ? ? H9 ? ? [])
+                    [intros;apply H2
+                       [unfold;intros;lapply (H5 ? H15);rewrite > fv_append;
                         autobatch
-                       |apply (JS_weakening ? ? ? H7)
+                       |apply (JS_weakening ? ? ? H9)
                           [autobatch
                           |unfold;intros;autobatch]
                        |assumption]
-- 
2.39.2