(* 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.
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.
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 (leaf (sort h)) \to
- x = leaf (sort h).
- intros. inversion H; clear H; intros;
- [ auto
- | destruct H3
- | destruct H4
- | destruct H6
- | destruct H6
- ].
+ 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 (leaf (lref j2)) \to
- (i > j2 \land x = leaf (lref j2)) \lor
+ Lift l i x (lref j2) \to
+ (i > j2 \land x = lref j2) \lor
(i <= j2 \land
- \exists j1. (l + j1 == j2) \land x = leaf (lref j1)
+ \exists j1. (l + j1 == j2) \land x = 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
- ].
+ 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 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
- ].
+ 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 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
- ].
+ 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 (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
+ 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.
+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;
+ 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. subst. auto
+ | 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 (leaf (lref j2)) \to
- i > j2 \to x = leaf (lref j2).
- intros 5.
- lapply linear lift_inv_lref_2 to H. decompose; subst;
- [ auto
+ 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 (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;
+ 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. subst. auto
+ | lapply linear nplus_inj_2 to H2, H4. destruct. autobatch
].
qed.
projects / helm.git / blobdiff