old subst tactics removed. New destruct tactic used instead

[helm.git] / matita / contribs / LAMBDA-TYPES / Unified-Sub / Lift / inv.ma

diff --git a/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/inv.ma b/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/inv.ma
index 8c3d27d7c6bb75e362f6b1a641e786376afef526..95630e43e4cd8ae73dc2223d04e1c36a4cc86539 100644 (file)
--- a/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/inv.ma
+++ b/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/inv.ma
@@ -16,31 +16,21 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Unified-Sub/Lift/inv".
 
 include "Lift/defs.ma".
 
+(* Inversion properties *****************************************************)
+
 theorem lift_inv_sort_1: \forall l, i, h, x.
-                         Lift l i (leaf (sort h)) x \to
-                         x = leaf (sort h).
- intros. inversion H; clear H; intros;
- [ auto
- | destruct H2
- | destruct H3
- | destruct H5
- | destruct H5
- ].
+                         Lift l i (sort h) x \to
+                         x = sort h.
+ intros. inversion H; clear H; intros; destruct. autobatch.
 qed.
 
 theorem lift_inv_lref_1: \forall l, i, j1, x.
-                         Lift l i (leaf (lref j1)) x \to
-                         (i > j1 \land x = leaf (lref j1)) \lor
+                         Lift l i (lref j1) x \to
+                         (i > j1 \land x = lref j1) \lor
                          (i <= j1 \land 
-                          \exists j2. (l + j1 == j2) \land x = leaf (lref j2)
+                          \exists j2. (l + j1 == j2) \land x = lref j2
                          ).
- intros. inversion H; clear H; intros;
- [ destruct H1
- | destruct H2. clear H2. subst. auto
- | destruct H3. clear H3. subst. auto depth = 5
- | destruct H5
- | destruct H5
- ].
+ intros. inversion H; clear H; intros; destruct; autobatch depth = 5 size = 7.
 qed.
 
 theorem lift_inv_bind_1: \forall l, i, r, u1, t1, x.
@@ -49,13 +39,7 @@ theorem lift_inv_bind_1: \forall l, i, r, u1, t1, x.
                          Lift l i u1 u2 \land
                          Lift l (succ i) t1 t2 \land
                          x = intb r u2 t2.
- intros. inversion H; clear H; intros;
- [ destruct H1
- | destruct H2
- | destruct H3
- | destruct H5. clear H5 H1 H3. subst. auto depth = 5
- | destruct H5
- ].
+ intros. inversion H; clear H; intros; destruct; autobatch depth = 5 size = 7.
 qed.
 
 theorem lift_inv_flat_1: \forall l, i, r, u1, t1, x.
@@ -64,11 +48,102 @@ theorem lift_inv_flat_1: \forall l, i, r, u1, t1, x.
                          Lift l i u1 u2 \land
                          Lift l i t1 t2 \land
                          x = intf r u2 t2.
- intros. inversion H; clear H; intros;
- [ destruct H1
- | destruct H2
- | destruct H3
- | destruct H5 
- | destruct H5. clear H5 H1 H3. subst. auto depth = 5
+ intros. inversion H; clear H; intros; destruct. autobatch depth = 5 size = 7.
+qed.
+
+theorem lift_inv_sort_2: \forall l, i, x, h.
+                         Lift l i x (sort h) \to
+                         x = sort h.
+ intros. inversion H; clear H; intros; destruct. autobatch.
+qed.
+
+theorem lift_inv_lref_2: \forall l, i, x, j2.
+                         Lift l i x (lref j2) \to
+                         (i > j2 \land x = lref j2) \lor
+                         (i <= j2 \land 
+                          \exists j1. (l + j1 == j2) \land x = lref j1
+                         ).
+ intros. inversion H; clear H; intros; destruct; autobatch depth = 5 size = 10.
+qed.
+
+theorem lift_inv_bind_2: \forall l, i, r, x, u2, t2.
+                         Lift l i x (intb r u2 t2) \to
+                         \exists u1, t1. 
+                         Lift l i u1 u2 \land
+                         Lift l (succ i) t1 t2 \land
+                         x = intb r u1 t1.
+ intros. inversion H; clear H; intros; destruct. autobatch depth = 5 size = 7.
+qed.
+
+theorem lift_inv_flat_2: \forall l, i, r, x, u2, t2.
+                         Lift l i x (intf r u2 t2) \to
+                         \exists u1, t1. 
+                         Lift l i u1 u2 \land
+                         Lift l i t1 t2 \land
+                         x = intf r u1 t1.
+ intros. inversion H; clear H; intros; destruct. autobatch depth = 5 size = 7.
+qed.
+
+(* Corollaries of inversion properties ***************************************)
+
+theorem lift_inv_lref_1_gt: \forall l, i, j1, x.
+                            Lift l i (lref j1) x \to
+                            i > j1 \to x = lref j1.
+ intros.
+ lapply linear lift_inv_lref_1 to H. decompose; destruct;
+ [ autobatch
+ | lapply linear nle_false to H2, H1. decompose
+ ].
+qed.
+
+theorem lift_inv_lref_1_le: \forall l, i, j1, x.
+                            Lift l i (lref j1) x \to i <= j1 \to 
+                            \exists j2. (l + j1 == j2) \land x = lref j2.
+ intros.
+ lapply linear lift_inv_lref_1 to H. decompose; destruct;
+ [ lapply linear nle_false to H1, H2. decompose
+ | autobatch
+ ].
+qed.
+
+theorem lift_inv_lref_1_le_nplus: \forall l, i, j1, x.
+                                 Lift l i (lref j1) x \to
+                                 i <= j1 \to \forall j2. (l + j1 == j2) \to
+                                 x = lref j2.
+ intros.
+ lapply linear lift_inv_lref_1 to H. decompose; destruct;
+ [ lapply linear nle_false to H1, H3. decompose
+ | lapply linear nplus_mono to H2, H4. destruct. autobatch
+ ].
+qed.
+
+theorem lift_inv_lref_2_gt: \forall l, i, x, j2.
+                            Lift l i x (lref j2) \to
+                            i > j2 \to x = lref j2.
+ intros.
+ lapply linear lift_inv_lref_2 to H. decompose; destruct;
+ [ autobatch
+ | lapply linear nle_false to H2, H1. decompose
+ ].
+ qed.
+
+theorem lift_inv_lref_2_le: \forall l, i, x, j2.
+                            Lift l i x (lref j2) \to i <= j2 \to 
+                            \exists j1. (l + j1 == j2) \land x = lref j1.
+ intros.
+ lapply linear lift_inv_lref_2 to H. decompose; destruct;
+ [ lapply linear nle_false to H1, H2. decompose
+ | autobatch
+ ].
+qed.
+
+theorem lift_inv_lref_2_le_nplus: \forall l, i, x, j2.
+                                  Lift l i x (lref j2) \to
+                                  i <= j2 \to \forall j1. (l + j1 == j2) \to
+                                  x = lref j1.
+ intros.
+ lapply linear lift_inv_lref_2 to H. decompose; destruct;
+ [ lapply linear nle_false to H1, H3. decompose
+ | lapply linear nplus_inj_2 to H2, H4. destruct. autobatch
  ].
 qed.