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).
| destruct H5. clear H5 H1 H3. subst. auto depth = 5
].
qed.
+
+theorem lift_inv_sort_2: \forall l, i, x, h.
+ Lift l i x (leaf (sort h)) \to
+ x = leaf (sort h).
+ intros. inversion H; clear H; intros;
+ [ auto
+ | destruct H3
+ | destruct H4
+ | destruct H6
+ | destruct H6
+ ].
+qed.
+
+theorem lift_inv_lref_2: \forall l, i, x, j2.
+ Lift l i x (leaf (lref j2)) \to
+ (i > j2 \land x = leaf (lref j2)) \lor
+ (i <= j2 \land
+ \exists j1. (l + j1 == j2) \land x = leaf (lref j1)
+ ).
+ intros. inversion H; clear H; intros;
+ [ destruct H2
+ | destruct H3. clear H3. subst. auto
+ | destruct H4. clear H4. subst. auto depth = 5
+ | destruct H6
+ | destruct H6
+ ].
+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 H2
+ | destruct H3
+ | destruct H4
+ | destruct H6. clear H5 H1 H3. subst. auto depth = 5
+ | destruct H6
+ ].
+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 H2
+ | destruct H3
+ | destruct H4
+ | destruct H6
+ | destruct H6. clear H6 H1 H3. subst. auto depth = 5
+ ].
+qed.
+
+(* Corollaries of inversion properties ***************************************)
+
+theorem lift_inv_lref_1_gt: \forall l, i, j1, x.
+ Lift l i (leaf (lref j1)) x \to
+ i > j1 \to x = leaf (lref j1).
+ intros 5.
+ lapply linear lift_inv_lref_1 to H. decompose; subst;
+ [ auto
+ | lapply linear nle_false to H2, H1. decompose
+ ].
+ qed.
+
+theorem lift_inv_lref_1_le: \forall l, i, j1, x.
+ Lift l i (leaf (lref j1)) x \to
+ i <= j1 \to \forall j2. (l + j1 == j2) \to
+ x = leaf (lref j2).
+ intros 5.
+ lapply linear lift_inv_lref_1 to H. decompose; subst;
+ [ lapply linear nle_false to H1, H3. decompose
+ | lapply linear nplus_mono to H2, H4. subst. auto
+ ].
+qed.
+
+theorem lift_inv_lref_2_gt: \forall l, i, x, j2.
+ Lift l i x (leaf (lref j2)) \to
+ i > j2 \to x = leaf (lref j2).
+ intros 5.
+ lapply linear lift_inv_lref_2 to H. decompose; subst;
+ [ auto
+ | lapply linear nle_false to H2, H1. decompose
+ ].
+ qed.
+
+theorem lift_inv_lref_2_le: \forall l, i, x, j2.
+ Lift l i x (leaf (lref j2)) \to
+ i <= j2 \to \forall j1. (l + j1 == j2) \to
+ x = leaf (lref j1).
+ intros 5.
+ lapply linear lift_inv_lref_2 to H. decompose; subst;
+ [ lapply linear nle_false to H1, H3. decompose
+ | lapply linear nplus_inj_2 to H2, H4. subst. auto
+ ].
+qed.