-(**************************************************************************)
-(* ___ *)
-(* ||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/fwd.ma".
-include "NPlus/defs.ma".
-
-(* primitive generation lemmas proved by elimination and inversion *)
-
-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
- ].
-qed.
-
-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.
- subst.
- ]; apply ex_intro; [| auto new timeout=30 || auto new timeout=30 ]. (**)
-qed.
-
-theorem nplus_gen_zero_2: \forall p,r. (p + zero == r) \to p = r.
- intros. inversion H; clear H; intros;
- [ auto new timeout=30
- | clear H H1.
- lapply eq_gen_zero_succ to H2 as H0. decompose.
- ].
-qed.
-
-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. decompose.
- | clear H1 H3 r.
- lapply linear eq_gen_succ_succ to H2 as H0.
- subst.
- apply ex_intro; [| auto new timeout=30 ] (**)
- ].
-qed.
-
-theorem nplus_gen_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.
- lapply eq_gen_zero_succ to H3 as H0. decompose.
- ].
-qed.
-
-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;
- [ subst.
- | clear H1.
- lapply linear eq_gen_succ_succ to H3 as H0.
- subst.
- ]; apply ex_intro; [| auto new timeout=30 || auto new timeout=30 ] (**)
-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.
-
-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 ]. (**)
-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.
- lapply linear eq_gen_succ_succ to H2 as H0.
- subst
- ]; auto new timeout=30.
-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.
- lapply linear eq_gen_succ_succ to H2 as H0.
- subst
- ]; auto new timeout=30.
-qed.