set "baseuri" "cic:/matita/LAMBDA-TYPES/Unified-Sub/Lift/props".
-include "Lift/inv.ma".
+include "Lift/fun.ma".
-theorem lift_conf: \forall l,i,t,x. Lift l i t x \to
- \forall y. Lift l i t y \to
- x = y.
- intros 5. elim H; clear H i t x;
- [ lapply linear lift_inv_sort_1 to H1.
- subst. auto
- | lapply linear lift_inv_lref_1 to H2.
- decompose; subst;
- [ auto | lapply nle_false to H2, H1. decompose ]
- | lapply linear lift_inv_lref_1 to H3.
- decompose; subst;
- [ lapply linear nle_false to H1, H3. decompose
- | lapply linear nplus_conf to H2, H4. subst. auto
- ]
- | lapply linear lift_inv_bind_1 to H5.
- decompose. subst. auto.
- | lapply linear lift_inv_flat_1 to H5.
- decompose. subst. auto.
+(* NOTE: this holds because of nplus_comm_1 *)
+theorem lift_comp: \forall l1,i1,t1,t2. Lift l1 i1 t1 t2 \to
+ \forall l2,i2,u1. Lift l2 i2 t1 u1 \to
+ \forall x. Lift l2 i2 t2 x \to
+ \forall i,y. Lift l1 i u1 y \to
+ i1 >= i2 \to (l2 + i1 == i) \to x = y.
+ intros 5. elim H; clear H i1 t1 t2;
+ [ lapply lift_mono to H1, H2. clear H2. destruct.
+ lapply linear lift_inv_sort_1 to H1. destruct.
+ lapply linear lift_inv_sort_1 to H3. destruct. autobatch
+ | lapply lift_mono to H2, H3. clear H3. destruct.
+ lapply linear lift_inv_lref_1 to H2.
+ decompose; destruct; clear H2 H5;
+ lapply linear lift_inv_lref_1_gt to H4; destruct; autobatch width = 4
+ | lapply lift_inv_lref_1_le to H3; [ 2: autobatch ]. clear H3.
+ lapply lift_inv_lref_1_le to H4; [ 2: autobatch ]. clear H4.
+ decompose. destruct. clear H6 i2.
+ lapply lift_inv_lref_1_le to H5; [ 2: autobatch depth = 4 width = 4 ].
+ decompose. destruct. clear H5 H1 H7 i. autobatch depth = 4 size = 7
+ | clear H1 H3.
+ lapply linear lift_inv_bind_1 to H5.
+ lapply linear lift_inv_bind_1 to H6. decompose. destruct.
+ lapply linear lift_inv_bind_1 to H7. decompose. destruct.
+ autobatch depth = 4 width = 6 size = 15
+ | clear H1 H3.
+ lapply linear lift_inv_flat_1 to H5.
+ lapply linear lift_inv_flat_1 to H6. decompose. destruct.
+ lapply linear lift_inv_flat_1 to H7. decompose. destruct.
+ autobatch depth = 4 width = 6 size = 9
].
qed.
-theorem lift_comp_le: \forall l1,i1,t,y. (Lift l1 i1 t y) \to
- \forall l2,i2,x. (Lift l2 i2 t x) \to
- \forall z. (Lift l2 i2 y z) \to
- i2 <= i1 \to \forall i. (l2 + i1 == i) \to
- (Lift l1 i x z).
- intros 5. elim H; clear H i1 t y;
- [ lapply lift_conf to H1, H2. clear H2. subst.
- lapply linear lift_inv_sort_1 to H1.
- subst. auto
- | lapply lift_conf to H2, H3. clear H3. subst.
- lapply linear lift_inv_lref_1 to H2.
- decompose; subst; clear H2 H4 i2;
- [ lapply linear nle_nplus to H5 as H0. clear l2. (**)
- lapply linear nle_trans to H1, H0.
- auto
- | lapply nle_nplus_comp_lt_2 to H3, H5; auto.
- ]
- | lapply linear lift_inv_lref_1 to H3.
- decompose; subst;
- [ clear H2 H4 H6 n3 l2.
- lapply linear nle_trans to H3, H5 as H0.
- lapply linear nle_false to H1, H0. decompose
- | lapply linear lift_inv_lref_1 to H4.
- decompose; subst;
- [ clear H1 H5 H6 H7 n1.
- lapply linear nle_nplus to H2 as H0. (**)
- lapply linear nle_trans to H3, H0 as H2.
- lapply linear nle_false to H2, H4. decompose
- | clear H3 H4 H5.
- lapply nle_nplus_comp to H6, H7; auto.
- ]
- ]
- |
\ No newline at end of file
+theorem lift_comp_rew_dx: \forall l1,i1,t1,t2. Lift l1 i1 t1 t2 \to
+ \forall l2,i2,u1. Lift l2 i2 t1 u1 \to
+ \forall u2. Lift l2 i2 t2 u2 \to
+ i1 >= i2 \to \forall i. (l2 + i1 == i) \to
+ Lift l1 i u1 u2.
+ intros.
+ lapply (lift_total l1 u1 i). decompose.
+ lapply lift_comp to H, H1, H2, H5, H3, H4. destruct. autobatch.
+qed.
+
+theorem lift_comp_rew_sx: \forall l1,i1,t1,t2. Lift l1 i1 t1 t2 \to
+ \forall l2,i2,u1. Lift l2 i2 t1 u1 \to
+ \forall i,u2. Lift l2 i t2 u2 \to
+ i2 >= i1 \to (l1 + i2 == i) \to
+ Lift l1 i1 u1 u2.
+ intros.
+ lapply (lift_total l1 u1 i1). decompose.
+ lapply lift_comp to H1, H, H5, H2, H3, H4. destruct. autobatch.
+qed.
+(*
+theorem lift_trans_le: \forall l1,i1,t1,t2. Lift l1 i1 t1 t2 \to
+ \forall l2,i2,z. Lift l2 i2 t2 t3 \to
+ i1 <= i2 \to
+ \forall i. \to i2 <= i \to (l1 + i1 == i) \to
+ \forall l. (l1 + l2 == l) \to Lift l i1 t1 t3.
+
+axiom lift_conf_back_ge: \forall l1,i1,u1,u2. Lift l1 i1 u1 u2 \to
+ \forall l2,i,t2. Lift l2 i t2 u2 \to
+ \forall i2. i2 >= i1 \to (l1 + i2 == i) \to
+ \exists t1. | Lift l2 i2 t1 u1 \land
+ Lift l1 i1 t1 t2.
+
+*)