--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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-ARITHMETICS/NPlus".
+
+include "logic/equality.ma".
+
+include "Nat.ma".
+
+inductive NPlus (p:Nat): Nat \to Nat \to Prop \def
+ | NPlus_zero_2: NPlus p zero p
+ | NPlus_succ_2: \forall q, r. NPlus p q r \to NPlus p (succ q) (succ r).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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-ARITHMETICS/NPlus_fwd".
+
+include "Nat_fwd.ma".
+include "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.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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-ARITHMETICS/NPlus_props".
+
+include "NPlus_fwd.ma".
+
+theorem NPlus_zero_1: \forall q. NPlus zero q q.
+ intros. elim q; clear q; auto.
+qed.
+
+theorem NPlus_succ_1: \forall p,q,r. NPlus p q r \to NPlus (succ p) q (succ r).
+ intros 2. elim q; clear q;
+ [ 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.
+ rewrite > H2. clear H2 r
+ ]; auto.
+qed.
+
+theorem NPlus_sym: \forall p,q,r. NPlus p q r \to NPlus q p r.
+ intros 2. elim q; clear q;
+ [ 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.
+ rewrite > H2. clear H2 r
+ ]; auto.
+qed.
+
+theorem NPlus_shift_succ_sx: \forall p,q,r.
+ NPlus p (succ q) r \to NPlus (succ p) q r.
+ intros.
+ lapply linear NPlus_gen_succ_2 to H as H0.
+ decompose.
+ rewrite > H1. clear H1 r.
+ auto.
+qed.
+
+theorem NPlus_shift_succ_dx: \forall p,q,r.
+ NPlus (succ p) q r \to NPlus p (succ q) r.
+ intros.
+ lapply linear NPlus_gen_succ_1 to H as H0.
+ decompose.
+ rewrite > H1. clear H1 r.
+ auto.
+qed.
+
+theorem NPlus_trans_1: \forall p,q1,r1. NPlus p q1 r1 \to
+ \forall q2,r2. NPlus r1 q2 r2 \to
+ \exists q. NPlus q1 q2 q \land NPlus p q r2.
+ intros 2; elim q1; clear q1; 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.
+ rewrite > H3. rewrite > H3 in H2. clear H3 r1.
+ lapply linear NPlus_gen_succ_1 to H2 as H0.
+ decompose.
+ rewrite > H2. clear H2 r2.
+ lapply linear H to H4, H3 as H0.
+ decompose.
+ ]; apply ex_intro; [| auto || auto ]. (**)
+qed.
+
+theorem NPlus_trans_2: \forall p1,q,r1. NPlus p1 q r1 \to
+ \forall p2,r2. NPlus p2 r1 r2 \to
+ \exists p. NPlus p1 p2 p \land NPlus p q r2.
+ intros 2; elim q; clear q; intros;
+ [ lapply linear NPlus_gen_zero_2 to H as H0.
+ rewrite > H0. clear H0 p1
+ | lapply linear NPlus_gen_succ_2 to H1 as H0.
+ decompose.
+ rewrite > H3. rewrite > H3 in H2. clear H3 r1.
+ lapply linear NPlus_gen_succ_2 to H2 as H0.
+ decompose.
+ rewrite > H2. clear H2 r2.
+ lapply linear H to H4, H3 as H0.
+ decompose.
+ ]; apply ex_intro; [| auto || auto ]. (**)
+qed.
+
+theorem NPlus_conf: \forall p,q,r1. NPlus p q r1 \to
+ \forall r2. NPlus p q r2 \to r1 = r2.
+ intros 2. elim q; clear q; intros;
+ [ lapply linear NPlus_gen_zero_2 to H as H0.
+ rewrite > H0 in H1. clear H0 p
+ | lapply linear NPlus_gen_succ_2 to H1 as H0.
+ decompose.
+ rewrite > H3. clear H3 r1.
+ lapply linear NPlus_gen_succ_2 to H2 as H0.
+ decompose.
+ rewrite > H2. clear H2 r2.
+ ]; auto.
+qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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-ARITHMETICS/Plus".
-
-include "logic/equality.ma".
-
-include "Nat.ma".
-
-inductive Plus (p:Nat): Nat \to Nat \to Prop \def
- | Plus_zero_2: Plus p zero p
- | Plus_succ_2: \forall q, r. Plus p q r \to Plus p (succ q) (succ r).
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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-ARITHMETICS/Plus_fwd".
-
-include "Nat_fwd.ma".
-include "Plus.ma".
-
-(* primitive generation lemmas proved by elimination and inversion *)
-
-theorem Plus_gen_zero_1: \forall q,r. Plus zero q r \to q = r.
- intros. elim H; clear H q r; intros;
- [ reflexivity
- | clear H1. auto
- ].
-qed.
-
-theorem Plus_gen_succ_1: \forall p,q,r. Plus (succ p) q r \to
- \exists s. r = (succ s) \land Plus 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 Plus_gen_zero_2: \forall p,r. Plus 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 Plus_gen_succ_2: \forall p,q,r. Plus p (succ q) r \to
- \exists s. r = (succ s) \land Plus 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 Plus_gen_zero_3: \forall p,q. Plus 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 Plus_gen_succ_3: \forall p,q,r. Plus p q (succ r) \to
- \exists s. p = succ s \land Plus s q r \lor
- q = succ s \land Plus 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 Plus_gen_2 *)
-
-variant Plus_gen_zero_3_alt: \forall p,q. Plus p q zero \to p = zero \land q = zero.
- intros 2. elim q; clear q; intros;
- [ lapply linear Plus_gen_zero_2 to H as H0.
- rewrite > H0. clear H0 p.
- auto
- | clear H.
- lapply linear Plus_gen_succ_2 to H1 as H0.
- decompose.
- lapply linear eq_gen_zero_succ to H1 as H0. apply H0
- ].
-qed.
-
-variant Plus_gen_succ_3_alt: \forall p,q,r. Plus p q (succ r) \to
- \exists s. p = succ s \land Plus s q r \lor
- q = succ s \land Plus p s r.
- intros 2. elim q; clear q; intros;
- [ lapply linear Plus_gen_zero_2 to H as H0.
- rewrite > H0. clear H0 p
- | clear H.
- lapply linear Plus_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 Plus_gen_eq_2_3: \forall p,q. Plus p q q \to p = zero.
- intros 2. elim q; clear q; intros;
- [ lapply linear Plus_gen_zero_2 to H as H0.
- rewrite > H0. clear H0 p
- | lapply linear Plus_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 Plus_gen_eq_1_3: \forall p,q. Plus p q p \to q = zero.
- intros 1. elim p; clear p; intros;
- [ lapply linear Plus_gen_zero_1 to H as H0.
- rewrite > H0. clear H0 q
- | lapply linear Plus_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.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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-ARITHMETICsucc/Plus_props".
-
-include "Plus_fwd.ma".
-
-theorem Plus_zero_1: \forall q. Plus zero q q.
- intros. elim q; clear q; auto.
-qed.
-
-theorem Plus_succ_1: \forall p,q,r. Plus p q r \to Plus (succ p) q (succ r).
- intros 2. elim q; clear q;
- [ lapply linear Plus_gen_zero_2 to H as H0.
- rewrite > H0. clear H0 p
- | lapply linear Plus_gen_succ_2 to H1 as H0.
- decompose.
- rewrite > H2. clear H2 r
- ]; auto.
-qed.
-
-theorem Plus_sym: \forall p,q,r. Plus p q r \to Plus q p r.
- intros 2. elim q; clear q;
- [ lapply linear Plus_gen_zero_2 to H as H0.
- rewrite > H0. clear H0 p
- | lapply linear Plus_gen_succ_2 to H1 as H0.
- decompose.
- rewrite > H2. clear H2 r
- ]; auto.
-qed.
-
-theorem Plus_shift_succ_sx: \forall p,q,r.
- Plus p (succ q) r \to Plus (succ p) q r.
- intros.
- lapply linear Plus_gen_succ_2 to H as H0.
- decompose.
- rewrite > H1. clear H1 r.
- auto.
-qed.
-
-theorem Plus_shift_succ_dx: \forall p,q,r.
- Plus (succ p) q r \to Plus p (succ q) r.
- intros.
- lapply linear Plus_gen_succ_1 to H as H0.
- decompose.
- rewrite > H1. clear H1 r.
- auto.
-qed.
-
-theorem Plus_trans_1: \forall p,q1,r1. Plus p q1 r1 \to
- \forall q2,r2. Plus r1 q2 r2 \to
- \exists q. Plus q1 q2 q \land Plus p q r2.
- intros 2; elim q1; clear q1; intros;
- [ lapply linear Plus_gen_zero_2 to H as H0.
- rewrite > H0. clear H0 p
- | lapply linear Plus_gen_succ_2 to H1 as H0.
- decompose.
- rewrite > H3. rewrite > H3 in H2. clear H3 r1.
- lapply linear Plus_gen_succ_1 to H2 as H0.
- decompose.
- rewrite > H2. clear H2 r2.
- lapply linear H to H4, H3 as H0.
- decompose.
- ]; apply ex_intro; [| auto || auto ]. (**)
-qed.
-
-theorem Plus_trans_2: \forall p1,q,r1. Plus p1 q r1 \to
- \forall p2,r2. Plus p2 r1 r2 \to
- \exists p. Plus p1 p2 p \land Plus p q r2.
- intros 2; elim q; clear q; intros;
- [ lapply linear Plus_gen_zero_2 to H as H0.
- rewrite > H0. clear H0 p1
- | lapply linear Plus_gen_succ_2 to H1 as H0.
- decompose.
- rewrite > H3. rewrite > H3 in H2. clear H3 r1.
- lapply linear Plus_gen_succ_2 to H2 as H0.
- decompose.
- rewrite > H2. clear H2 r2.
- lapply linear H to H4, H3 as H0.
- decompose.
- ]; apply ex_intro; [| auto || auto ]. (**)
-qed.
-
-theorem Plus_conf: \forall p,q,r1. Plus p q r1 \to
- \forall r2. Plus p q r2 \to r1 = r2.
- intros 2. elim q; clear q; intros;
- [ lapply linear Plus_gen_zero_2 to H as H0.
- rewrite > H0 in H1. clear H0 p
- | lapply linear Plus_gen_succ_2 to H1 as H0.
- decompose.
- rewrite > H3. clear H3 r1.
- lapply linear Plus_gen_succ_2 to H2 as H0.
- decompose.
- rewrite > H2. clear H2 r2.
- ]; auto.
-qed.
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