(* primitive generation lemmas proved by elimination and inversion *)
-theorem nplus_gen_zero_1: \forall q,r. NPlus zero q r \to q = r.
+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
].
qed.
-theorem nplus_gen_succ_1: \forall p,q,r. NPlus (succ p) q r \to
- \exists s. r = (succ s) \land NPlus p q s.
+theorem nplus_gen_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.
- rewrite > H1. clear H1 n2
+ subst.
]; apply ex_intro; [| auto || auto ]. (**)
qed.
-theorem nplus_gen_zero_2: \forall p,r. NPlus p zero r \to p = r.
+theorem nplus_gen_zero_2: \forall p,r. (p + zero == r) \to p = r.
intros. inversion H; clear H; intros;
[ auto
| clear H H1.
].
qed.
-theorem nplus_gen_succ_2: \forall p,q,r. NPlus p (succ q) r \to
- \exists s. r = (succ s) \land NPlus p q s.
+theorem nplus_gen_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;
[ lapply eq_gen_succ_zero to H as H0. apply H0
| clear H1 H3 r.
lapply linear eq_gen_succ_succ to H2 as H0.
- rewrite > H0. clear H0 q.
+ subst.
apply ex_intro; [| auto ] (**)
].
qed.
-theorem nplus_gen_zero_3: \forall p,q. NPlus p q zero \to p = zero \land q = zero.
+theorem nplus_gen_zero_3: \forall p,q. (p + q == zero) \to
+ p = zero \land q = zero.
intros. inversion H; clear H; intros;
- [ rewrite < H1. clear H1 p.
- auto
+ [ subst. auto
| clear H H1.
lapply eq_gen_zero_succ to H3 as H0. apply H0
].
qed.
-theorem nplus_gen_succ_3: \forall p,q,r. NPlus p q (succ r) \to
- \exists s. p = succ s \land NPlus s q r \lor
- q = succ s \land NPlus p s r.
+theorem nplus_gen_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;
- [ rewrite < H1. clear H1 p
+ [ subst.
| clear H1.
lapply linear eq_gen_succ_succ to H3 as H0.
- rewrite > H0. clear H0 r.
+ subst.
]; apply ex_intro; [| auto || auto ] (**)
qed.
(*
(* alternative proofs invoking nplus_gen_2 *)
-variant nplus_gen_zero_3_alt: \forall p,q. NPlus p q zero \to p = zero \land q = zero.
+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.
- rewrite > H0. clear H0 p.
- auto
+ subst. auto
| clear H.
lapply linear nplus_gen_succ_2 to H1 as H0.
decompose.
].
qed.
-variant nplus_gen_succ_3_alt: \forall p,q,r. NPlus p q (succ r) \to
- \exists s. p = succ s \land NPlus s q r \lor
- q = succ s \land NPlus p s r.
+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.
- rewrite > H0. clear H0 p
+ subst
| clear H.
lapply linear nplus_gen_succ_2 to H1 as H0.
decompose.
lapply linear eq_gen_succ_succ to H1 as H0.
- rewrite > H0. clear H0 r.
+ subst
]; apply ex_intro; [| auto || auto ]. (**)
qed.
*)
(* other simplification lemmas *)
-theorem nplus_gen_eq_2_3: \forall p,q. NPlus p q q \to p = zero.
+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.
- rewrite > H0. clear H0 p
+ subst
| lapply linear nplus_gen_succ_2 to H1 as H0.
decompose.
lapply linear eq_gen_succ_succ to H2 as H0.
- rewrite < H0 in H3. clear H0 a
+ subst
]; auto.
qed.
-theorem nplus_gen_eq_1_3: \forall p,q. NPlus p q p \to q = zero.
+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.
- rewrite > H0. clear H0 q
+ subst
| lapply linear nplus_gen_succ_1 to H1 as H0.
decompose.
lapply linear eq_gen_succ_succ to H2 as H0.
- rewrite < H0 in H3. clear H0 a
+ subst
]; auto.
qed.