inductive Lift (l: Nat): Nat \to Term \to Term \to Prop \def
| lift_sort : \forall i,h.
- Lift l i (leaf (sort h)) (leaf (sort h))
+ Lift l i (sort h) (sort h)
| lift_lref_gt: \forall i,j. i > j \to
- Lift l i (leaf (lref j)) (leaf (lref j))
+ Lift l i (lref j) (lref j)
| lift_lref_le: \forall i,j1. i <= j1 \to
\forall j2. (l + j1 == j2) \to
- Lift l i (leaf (lref j1)) (leaf (lref j2))
+ Lift l i (lref j1) (lref j2)
| lift_bind : \forall i,u1,u2. Lift l i u1 u2 \to
\forall t1,t2. Lift l (succ i) t1 t2 \to
\forall r. Lift l i (intb r u1 t1) (intb r u2 t2)
include "Lift/inv.ma".
(* Functional properties ****************************************************)
-(*
+
theorem lift_total: \forall l, t, i. \exists u. Lift l i t u.
intros 2. elim t; clear t;
- [
-*)
+ [ auto
+ | lapply (nle_gt_or_le n i). decompose;
+ [ auto
+ | lapply (nplus_total l n). decompose. auto
+ ]
+ | lapply (H i1). lapply (H1 (succ i1)). decompose. auto
+ | lapply (H i1). lapply (H1 i1). decompose. auto
+ ].
+qed.
+
theorem lift_mono: \forall l,i,t,t1. Lift l i t t1 \to
\forall t2. Lift l i t t2 \to
t1 = t2.
intros 5. elim H; clear H i t t1;
[ lapply linear lift_inv_sort_1 to H1
| lapply linear lift_inv_lref_1_gt to H2, H1
- | lapply linear lift_inv_lref_1_le to H3, H1, H2
+ | lapply linear lift_inv_lref_1_le_nplus to H3, H1, H2
| lapply linear lift_inv_bind_1 to H5. decompose
| lapply linear lift_inv_flat_1 to H5. decompose
]; subst; auto.
intros 5. elim H; clear H i t1 t;
[ lapply linear lift_inv_sort_2 to H1
| lapply linear lift_inv_lref_2_gt to H2, H1
- | lapply nle_nplus to H2 as H. (**)
+ | lapply nle_nplus to H2 as H.
lapply linear nle_trans to H1, H as H0.
- lapply lift_inv_lref_2_le to H3, H0, H2
+ lapply lift_inv_lref_2_le_nplus to H3, H0, H2
| lapply linear lift_inv_bind_2 to H5. decompose
| lapply linear lift_inv_flat_2 to H5. decompose
]; subst; auto.
(* Inversion properties *****************************************************)
theorem lift_inv_sort_1: \forall l, i, h, x.
- Lift l i (leaf (sort h)) x \to
- x = leaf (sort h).
+ Lift l i (sort h) x \to
+ x = sort h.
intros. inversion H; clear H; intros;
[ auto
| destruct H2
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
qed.
theorem lift_inv_sort_2: \forall l, i, x, h.
- Lift l i x (leaf (sort h)) \to
- x = leaf (sort h).
+ Lift l i x (sort h) \to
+ x = sort h.
intros. inversion H; clear H; intros;
[ auto
| destruct H3
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
(* 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.
+ Lift l i (lref j1) x \to
+ i > j1 \to x = lref j1.
+ intros.
lapply linear lift_inv_lref_1 to H. decompose; subst;
[ auto
| 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.
+ 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; subst;
+ [ lapply linear nle_false to H1, H2. decompose
+ | auto
+ ].
+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; 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.
+ Lift l i x (lref j2) \to
+ i > j2 \to x = lref j2.
+ intros.
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.
+ 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; subst;
+ [ lapply linear nle_false to H1, H2. decompose
+ | auto
+ ].
+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; subst;
[ lapply linear nle_false to H1, H3. decompose
| lapply linear nplus_inj_2 to H2, H4. subst. auto
include "Lift/fun.ma".
-theorem lift_comp_ge_1: \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.
+(* 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_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 lift_mono to H1, H2. clear H2. subst.
+ lapply linear lift_inv_sort_1 to H1. subst.
+ lapply linear lift_inv_sort_1 to H3. subst. auto
+ | lapply lift_mono 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.
- ]
- ]
+ decompose; subst; clear H2 H5;
+ lapply linear lift_inv_lref_1_gt to H4; subst; auto width = 4
+ | lapply lift_inv_lref_1_le to H3; [ 2: auto ]. clear H3.
+ lapply lift_inv_lref_1_le to H4; [ 2: auto ]. clear H4.
+ decompose. subst. clear H6 i2.
+ lapply lift_inv_lref_1_le to H5; [ 2: auto depth = 4 width = 4 ].
+ decompose. subst. clear H5 H1 H7 i. auto depth = 4
| clear H1 H3.
lapply linear lift_inv_bind_1 to H5.
- lapply linear lift_inv_bind_1 to H6.
- decompose. subst. auto width = 4
+ lapply linear lift_inv_bind_1 to H6. decompose. subst.
+ lapply linear lift_inv_bind_1 to H7. decompose. subst.
+ auto width = 5
| clear H1 H3.
lapply linear lift_inv_flat_1 to H5.
- lapply linear lift_inv_flat_1 to H6.
- decompose. subst. auto width = 4
+ lapply linear lift_inv_flat_1 to H6. decompose. subst.
+ lapply linear lift_inv_flat_1 to H7. decompose. subst.
+ auto width = 5
].
qed.
-theorem lift_comp_ge_2: \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 5. elim H; clear H i1 t1 t2;
- [ lapply linear lift_inv_sort_1 to H1.
- lapply linear lift_inv_sort_1 to H2.
- subst. auto
- | lapply linear lift_inv_lref_1 to H2.
- lapply linear lift_inv_lref_1 to H3.
- decompose; subst; (* clear H2 H4 i2; *)
- [ clear H H3 H4 H5. auto
- | clear H1 H4 H7.
- lapply linear nle_nplus to H5 as H0. (**)
- lapply linear nle_trans to H3, H0 as H2.
- lapply nle_false to H, H2. decompose
- | clear H H5 H6.
- lapply linear nle_trans to H4, H3 as H.
- lapply nle_false to H, H1. decompose
- | clear H H2 H5 H7.
- lapply linear nle_trans to H4, H3 as H.
- lapply nle_false to H, H1. decompose
- ]
-(*
- | 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.
- ]
- ]
- | clear H1 H3.
- lapply linear lift_inv_bind_1 to H5.
- lapply linear lift_inv_bind_1 to H6.
- decompose. subst. auto width = 4
- | clear H1 H3.
- lapply linear lift_inv_flat_1 to H5.
- lapply linear lift_inv_flat_1 to H6.
- decompose. subst. auto width = 4
- ].
+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. subst. auto.
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. subst. auto.
+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
- Naming policy:
- natural numbers : sorts h k, local references i j, lengths l o
- boolean values : p q
- - generic leaf items : m n
- generic binding items: r s
- generic flat items : r s
- generic head items : m n
include "preamble.ma".
-inductive Leaf: Set \def
- | sort: Nat \to Leaf
- | lref: Nat \to Leaf
-.
-
inductive Bind: Set \def
| abbr: Bind
| abst: Bind
.
inductive Term: Set \def
- | leaf: Leaf \to Term
+ | sort: Nat \to Term
+ | lref: Nat \to Term
| intb: IntB \to Term \to Term \to Term
| intf: IntF \to Term \to Term \to Term
.
include "NLE/defs.ma".
theorem nle_inv_succ_1: \forall x,y. x < y \to
- \exists z. y = succ z \land x <= z.
+ \exists z. y = succ z \land x <= z.
intros. elim H.
- lapply linear nplus_gen_succ_2 to H1.
+ lapply linear nplus_inv_succ_2 to H1.
decompose. subst. auto depth = 4.
qed.
].
qed.
-theorem nat_dec_lt_ge: \forall x,y. x < y \lor y <= x.
+theorem nle_gt_or_le: \forall x,y. y > x \lor y <= x.
intros 2; elim y; clear y;
[ auto
| decompose;
theorem nplus_mono: \forall p,q,r1. (p + q == r1) \to
\forall r2. (p + q == r2) \to r1 = r2.
intros 4. elim H; clear H q r1;
- [ lapply linear nplus_gen_zero_2 to H1
- | lapply linear nplus_gen_succ_2 to H3. decompose
+ [ lapply linear nplus_inv_zero_2 to H1
+ | lapply linear nplus_inv_succ_2 to H3. decompose
]; subst; auto.
qed.
theorem nplus_inj_1: \forall p1, q, r. (p1 + q == r) \to
\forall p2. (p2 + q == r) \to p2 = p1.
intros 4. elim H; clear H q r;
- [ lapply linear nplus_gen_zero_2 to H1
- | lapply linear nplus_gen_succ_2_3 to H3
+ [ lapply linear nplus_inv_zero_2 to H1
+ | lapply linear nplus_inv_succ_2_3 to H3
]; auto.
qed.
include "NPlus/defs.ma".
-(* primitive generation lemmas proved by elimination and inversion *)
+(* Inversion lemmas *********************************************************)
-theorem nplus_gen_zero_1: \forall q,r. (zero + q == r) \to q = r.
- intros. elim H; clear H q r; intros;
- [ reflexivity
- | clear H1. auto new timeout=30
- ].
+theorem nplus_inv_zero_1: \forall q,r. (zero + q == r) \to q = r.
+ intros. elim H; clear H q r; auto.
qed.
-theorem nplus_gen_succ_1: \forall p,q,r. ((succ p) + q == r) \to
+theorem nplus_inv_succ_1: \forall p,q,r. ((succ p) + q == r) \to
\exists s. r = (succ s) \land p + q == s.
intros. elim H; clear H q r; intros;
- [
- | clear H1.
- decompose.
- subst.
- ]; apply ex_intro; [| auto new timeout=30 || auto new timeout=30 ]. (**)
+ [ auto depth = 4
+ | clear H1. decompose. subst. auto depth = 4
+ ]
qed.
-theorem nplus_gen_zero_2: \forall p,r. (p + zero == r) \to p = r.
+theorem nplus_inv_zero_2: \forall p,r. (p + zero == r) \to p = r.
intros. inversion H; clear H; intros;
- [ auto new timeout=30
- | clear H H1.
- destruct H2.
+ [ auto.
+ | clear H H1. destruct H2.
].
qed.
-theorem nplus_gen_succ_2: \forall p,q,r. (p + (succ q) == r) \to
+theorem nplus_inv_succ_2: \forall p,q,r. (p + (succ q) == r) \to
\exists s. r = (succ s) \land p + q == s.
intros. inversion H; clear H; intros;
[ destruct H.
- | clear H1 H3 r.
- destruct H2; clear H2.
- subst.
- apply ex_intro; [| auto new timeout=30 ] (**)
+ | clear H1 H3 r. destruct H2; clear H2. subst. auto depth = 4.
].
qed.
-theorem nplus_gen_zero_3: \forall p,q. (p + q == zero) \to
+theorem nplus_inv_zero_3: \forall p,q. (p + q == zero) \to
p = zero \land q = zero.
intros. inversion H; clear H; intros;
- [ subst. auto new timeout=30
- | clear H H1.
- destruct H3.
+ [ subst. auto
+ | clear H H1. destruct H3.
].
qed.
-theorem nplus_gen_succ_3: \forall p,q,r. (p + q == (succ r)) \to
+theorem nplus_inv_succ_3: \forall p,q,r. (p + q == (succ r)) \to
\exists s. p = succ s \land (s + q == r) \lor
q = succ s \land p + s == r.
intros. inversion H; clear H; intros;
- [ subst.
- | clear H1.
- destruct H3. clear H3.
- subst.
- ]; apply ex_intro; [| auto new timeout=30 || auto new timeout=30 ] (**)
+ [ subst
+ | clear H1. destruct H3. clear H3. subst.
+ ]; auto depth = 4.
qed.
-(*
-(* alternative proofs invoking nplus_gen_2 *)
-variant nplus_gen_zero_3_alt: \forall p,q. (p + q == zero) \to
- p = zero \land q = zero.
- intros 2. elim q; clear q; intros;
- [ lapply linear nplus_gen_zero_2 to H as H0.
- subst. auto new timeout=30
- | clear H.
- lapply linear nplus_gen_succ_2 to H1 as H0.
- decompose.
- lapply linear eq_gen_zero_succ to H1 as H0. apply H0
- ].
-qed.
+(* Corollaries to inversion lemmas ******************************************)
-variant nplus_gen_succ_3_alt: \forall p,q,r. (p + q == (succ r)) \to
- \exists s. p = succ s \land (s + q == r) \lor
- q = succ s \land p + s == r.
- intros 2. elim q; clear q; intros;
- [ lapply linear nplus_gen_zero_2 to H as H0.
- subst
- | clear H.
- lapply linear nplus_gen_succ_2 to H1 as H0.
- decompose.
- lapply linear eq_gen_succ_succ to H1 as H0.
- subst
- ]; apply ex_intro; [| auto new timeout=30 || auto new timeout=30 ]. (**)
+theorem nplus_inv_succ_2_3: \forall p,q,r.
+ (p + (succ q) == (succ r)) \to p + q == r.
+ intros.
+ lapply linear nplus_inv_succ_2 to H. decompose. subst.
+ destruct H1. clear H1. subst. auto.
qed.
-*)
-(* other simplification lemmas *)
-theorem nplus_gen_eq_2_3: \forall p,q. (p + q == q) \to p = zero.
- intros 2. elim q; clear q; intros;
- [ lapply linear nplus_gen_zero_2 to H as H0.
- subst
- | lapply linear nplus_gen_succ_2 to H1 as H0.
- decompose.
- destruct H2. clear H2.
- subst
- ]; auto new timeout=30.
+theorem nplus_inv_succ_1_3: \forall p,q,r.
+ ((succ p) + q == (succ r)) \to p + q == r.
+ intros.
+ lapply linear nplus_inv_succ_1 to H. decompose. subst.
+ destruct H1. clear H1. subst. auto.
qed.
-theorem nplus_gen_eq_1_3: \forall p,q. (p + q == p) \to q = zero.
- intros 1. elim p; clear p; intros;
- [ lapply linear nplus_gen_zero_1 to H as H0.
- subst
- | lapply linear nplus_gen_succ_1 to H1 as H0.
- decompose.
- destruct H2. clear H2.
- subst
- ]; auto new timeout=30.
+theorem nplus_inv_eq_2_3: \forall p,q. (p + q == q) \to p = zero.
+ intros 2. elim q; clear q;
+ [ lapply linear nplus_inv_zero_2 to H
+ | lapply linear nplus_inv_succ_2_3 to H1
+ ]; auto.
qed.
-theorem nplus_gen_succ_2_3: \forall p,q,r.
- (p + (succ q) == (succ r)) \to p + q == r.
- intros.
- lapply linear nplus_gen_succ_2 to H. decompose. subst.
- destruct H1. clear H1. subst. auto.
+theorem nplus_inv_eq_1_3: \forall p,q. (p + q == p) \to q = zero.
+ intros 1. elim p; clear p;
+ [ lapply linear nplus_inv_zero_1 to H
+ | lapply linear nplus_inv_succ_1_3 to H1.
+ ]; auto.
qed.
intros. elim H; clear H q r; auto.
qed.
+theorem nplus_comm: \forall p, q, x. (p + q == x) \to
+ \forall y. (q + p == y) \to x = y.
+ intros 4; elim H; clear H q x;
+ [ lapply linear nplus_inv_zero_1 to H1
+ | lapply linear nplus_inv_succ_1 to H3. decompose
+ ]; subst; auto.
+qed.
+
theorem nplus_comm_rew: \forall p,q,r. (p + q == r) \to q + p == r.
intros. elim H; clear H q r; auto.
qed.
(* Corollaries of nonoidal properties ***************************************)
+theorem nplus_comm_1: \forall p1, q, r1. (p1 + q == r1) \to
+ \forall p2, r2. (p2 + q == r2) \to
+ \forall x. (p2 + r1 == x) \to
+ \forall y. (p1 + r2 == y) \to
+ x = y.
+ intros 4. elim H; clear H q r1;
+ [ lapply linear nplus_inv_zero_2 to H1
+ | lapply linear nplus_inv_succ_2 to H3.
+ lapply linear nplus_inv_succ_2 to H4. decompose. subst.
+ lapply linear nplus_inv_succ_2 to H5. decompose
+ ]; subst; auto.
+qed.
+
theorem nplus_comm_1_rew: \forall p1,q,r1. (p1 + q == r1) \to
\forall p2,r2. (p2 + q == r2) \to
\forall s. (p1 + r2 == s) \to (p2 + r1 == s).
intros 4. elim H; clear H q r1;
- [ lapply linear nplus_gen_zero_2 to H1. subst
- | lapply linear nplus_gen_succ_2 to H3. decompose. subst.
- lapply linear nplus_gen_succ_2 to H4. decompose. subst
+ [ lapply linear nplus_inv_zero_2 to H1. subst
+ | lapply linear nplus_inv_succ_2 to H3. decompose. subst.
+ lapply linear nplus_inv_succ_2 to H4. decompose. subst
]; auto.
qed.
theorem nplus_shift_succ_sx: \forall p,q,r.
(p + (succ q) == r) \to (succ p) + q == r.
intros.
- lapply linear nplus_gen_succ_2 to H as H0.
+ lapply linear nplus_inv_succ_2 to H as H0.
decompose. subst. auto new timeout=100.
qed.
theorem nplus_shift_succ_dx: \forall p,q,r.
((succ p) + q == r) \to p + (succ q) == r.
intros.
- lapply linear nplus_gen_succ_1 to H as H0.
+ lapply linear nplus_inv_succ_1 to H as H0.
decompose. subst. auto new timeout=100.
qed.
\forall q2,r2. (r1 + q2 == r2) \to
\exists q. (q1 + q2 == q) \land p + q == r2.
intros 2; elim q1; clear q1; intros;
- [ lapply linear nplus_gen_zero_2 to H as H0.
+ [ lapply linear nplus_inv_zero_2 to H as H0.
subst.
- | lapply linear nplus_gen_succ_2 to H1 as H0.
+ | lapply linear nplus_inv_succ_2 to H1 as H0.
decompose. subst.
- lapply linear nplus_gen_succ_1 to H2 as H0.
+ lapply linear nplus_inv_succ_1 to H2 as H0.
decompose. subst.
lapply linear H to H4, H3 as H0.
decompose.
\forall p2,r2. (p2 + r1 == r2) \to
\exists p. (p1 + p2 == p) \land p + q == r2.
intros 2; elim q; clear q; intros;
- [ lapply linear nplus_gen_zero_2 to H as H0.
+ [ lapply linear nplus_inv_zero_2 to H as H0.
subst
- | lapply linear nplus_gen_succ_2 to H1 as H0.
+ | lapply linear nplus_inv_succ_2 to H1 as H0.
decompose. subst.
- lapply linear nplus_gen_succ_2 to H2 as H0.
+ lapply linear nplus_inv_succ_2 to H2 as H0.
decompose. subst.
lapply linear H to H4, H3 as H0.
decompose.
]; apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
qed.
*)
-
projects / helm.git / commitdiff