qed.
theorem plus_reg_l: \forall n,m,p. n + m = n + p \to m = p.
- intros. apply plus_reg_l; auto.
+ intros. apply plus_reg_l; autobatch.
qed.
theorem plus_le_reg_l: \forall p,n,m. p + n <= p + m \to n <= m.
- intros. apply plus_le_reg_l; auto.
+ intros. apply plus_le_reg_l; autobatch.
qed.
theorem lift_inv_sort_1: \forall l, i, h, x.
Lift l i (sort h) x \to
x = sort h.
- intros. inversion H; clear H; intros; subst. auto.
+ intros. inversion H; clear H; intros; subst. autobatch.
qed.
theorem lift_inv_lref_1: \forall l, i, j1, x.
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; subst. auto.
+ intros. inversion H; clear H; intros; subst. autobatch.
qed.
theorem lift_inv_lref_2: \forall l, i, x, j2.
theorem nle_inv_succ_1: \forall x,y. x < y \to
\exists z. y = succ z \land x <= z.
- intros. inversion H; clear H; intros; subst. auto.
+ intros. inversion H; clear H; intros; subst. autobatch.
qed.
theorem nle_inv_succ_succ: \forall x,y. x < succ y \to x <= y.
- intros. inversion H; clear H; intros; subst. auto.
+ intros. inversion H; clear H; intros; subst. autobatch.
qed.
theorem nle_inv_succ_zero: \forall x. x < zero \to False.
qed.
theorem nle_inv_zero_2: \forall x. x <= zero \to x = zero.
- intros. inversion H; clear H; intros; subst. auto.
+ intros. inversion H; clear H; intros; subst. autobatch.
qed.
theorem nle_inv_succ_2: \forall y,x. x <= succ y \to
include "NLE/defs.ma".
theorem nle_nplus: \forall p, q, r. (p + q == r) \to q <= r.
- intros. elim H; clear H q r; auto.
+ intros. elim H; clear H q r; autobatch.
qed.
axiom nle_nplus_comp: \forall x1, x2, x3. (x1 + x2 == x3) \to
(* Inversion lemmas *********************************************************)
theorem nplus_inv_zero_1: \forall q,r. (zero + q == r) \to q = r.
- intros. elim H; clear H q r; auto.
+ intros. elim H; clear H q r; autobatch.
qed.
theorem nplus_inv_succ_1: \forall p,q,r. ((succ p) + q == r) \to
qed.
theorem nplus_inv_zero_2: \forall p,r. (p + zero == r) \to p = r.
- intros. inversion H; clear H; intros; subst. auto.
+ intros. inversion H; clear H; intros; subst. autobatch.
qed.
theorem nplus_inv_succ_2: \forall p,q,r. (p + (succ q) == r) \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.
+ intros. inversion H; clear H; intros; subst. autobatch.
qed.
theorem nplus_inv_succ_3: \forall p,q,r. (p + q == (succ r)) \to
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. auto.
+ lapply linear nplus_inv_succ_2 to H. decompose. subst. autobatch.
qed.
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. auto.
+ lapply linear nplus_inv_succ_1 to H. decompose. subst. autobatch.
qed.
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.
+ ]; autobatch.
qed.
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.
+ ]; autobatch.
qed.
assumption
| unfold y;
simplify;
- auto paramodulation library
+ autobatch paramodulation library
]
].
qed.
intros 4. elim H; clear H x y.
apply eq_gen_S_O. exact m. elim H1. auto paramodulation.
clear H2. cut (n = m).
-elim Hcut. apply ex_intro. exact n1. split; auto.
-apply eq_gen_S_S. auto.
+elim Hcut. apply ex_intro. exact n1. split; autobatch.
+apply eq_gen_S_S. autobatch.
qed.
theorem le_gen_S_x: \forall m,x. (le (S m) x) \to