-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/RELATIONAL/NPlus/fwd".
-
-include "Nat/Nat_fwd.ma".
-include "NPlus/NPlus.ma".
-
-(* primitive generation lemmas proved by elimination and inversion *)
-
-theorem nplus_gen_zero_1: \forall q,r. NPlus 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.
- intros. elim H; clear H q r; intros;
- [
- | clear H1.
- decompose.
- rewrite > H1. clear H1 n2
- ]; apply ex_intro; [| auto || auto ]. (**)
-qed.
-
-theorem nplus_gen_zero_2: \forall p,r. NPlus p zero r \to p = r.
- intros. inversion H; clear H; intros;
- [ auto
- | clear H H1.
- lapply eq_gen_zero_succ to H2 as H0. apply H0
- ].
-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.
- 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.
- apply ex_intro; [| auto ] (**)
- ].
-qed.
-
-theorem nplus_gen_zero_3: \forall p,q. NPlus p q zero \to p = zero \land q = zero.
- intros. inversion H; clear H; intros;
- [ rewrite < H1. clear H1 p.
- 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.
- intros. inversion H; clear H; intros;
- [ rewrite < H1. clear H1 p
- | clear H1.
- lapply linear eq_gen_succ_succ to H3 as H0.
- rewrite > H0. clear H0 r.
- ]; 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.
- intros 2. elim q; clear q; intros;
- [ lapply linear nplus_gen_zero_2 to H as H0.
- rewrite > H0. clear H0 p.
- auto
- | 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.
-
-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.
- intros 2. elim q; clear q; intros;
- [ lapply linear nplus_gen_zero_2 to H as H0.
- rewrite > H0. clear H0 p
- | 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.
- ]; 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.
- intros 2. elim q; clear q; intros;
- [ lapply linear nplus_gen_zero_2 to H as H0.
- rewrite > H0. clear H0 p
- | 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
- ]; auto.
-qed.
-
-theorem nplus_gen_eq_1_3: \forall p,q. NPlus 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
- | 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
- ]; auto.
-qed.