(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/RELATIONAL/NPlus/monoid".
+
include "NPlus/fun.ma".
(* Monoidal properties ******************************************************)
-theorem nplus_zero_1: \forall q. zero + q == q.
- intros. elim q; clear q; auto.
+theorem nplus_zero_1: ∀q. zero ⊕ q ≍ q.
+ intros; elim q; clear q; autobatch.
qed.
-theorem nplus_succ_1: \forall p,q,r. NPlus p q r \to
- (succ p) + q == (succ r).
- intros. elim H; clear H q r; auto.
+theorem nplus_succ_1: ∀p,q,r. p ⊕ q ≍ r → succ p ⊕ q ≍ succ r.
+ intros; elim H; clear H q r; autobatch.
qed.
-theorem nplus_comm: \forall p, q, x. (p + q == x) \to
- \forall y. (q + p == y) \to x = y.
+theorem nplus_comm: ∀p, q, x. p ⊕ q ≍ x → ∀y. q ⊕ p ≍ y → x = y.
intros 4; elim H; clear H q x;
[ lapply linear nplus_inv_zero_1 to H1
| lapply linear nplus_inv_succ_1 to H3. decompose
- ]; subst; auto.
+ ]; destruct; autobatch.
qed.
-theorem nplus_comm_rew: \forall p,q,r. (p + q == r) \to q + p == r.
- intros. elim H; clear H q r; auto.
+theorem nplus_comm_rew: ∀p,q,r. p ⊕ q ≍ r → q ⊕ p ≍ r.
+ intros; elim H; clear H q r; autobatch.
qed.
+theorem nplus_ass: ∀p1, p2, r1. p1 ⊕ p2 ≍ r1 → ∀p3, s1. r1 ⊕ p3 ≍ s1 →
+ ∀r3. p2 ⊕ p3 ≍ r3 → ∀s3. p1 ⊕ r3 ≍ s3 → s1 = s3.
+ intros 4; elim H; clear H p2 r1;
+ [ lapply linear nplus_inv_zero_1 to H2. destruct.
+ lapply nplus_mono to H1, H3. destruct. autobatch
+ | lapply linear nplus_inv_succ_1 to H3. decompose. destruct.
+ lapply linear nplus_inv_succ_1 to H4. decompose. destruct.
+ lapply linear nplus_inv_succ_2 to H5. decompose. destruct. autobatch
+ ].
+qed.
+
(* Corollaries of functional properties **************************************)
-theorem nplus_inj_2: \forall p, q1, r. (p + q1 == r) \to
- \forall q2. (p + q2 == r) \to q1 = q2.
- intros. auto.
+theorem nplus_inj_2: ∀p, q1, r. p ⊕ q1 ≍ r → ∀q2. p ⊕ q2 ≍ r → q1 = q2.
+ intros. autobatch.
qed.
(* Corollaries of nonoidal properties ***************************************)
-theorem nplus_comm_1: \forall p1, q, r1. (p1 + q == r1) \to
- \forall p2, r2. (p2 + q == r2) \to
- \forall x. (p2 + r1 == x) \to
- \forall y. (p1 + r2 == y) \to
- x = y.
- intros 4. elim H; clear H q r1;
+theorem nplus_comm_1: ∀p1, q, r1. p1 ⊕ q ≍ r1 → ∀p2, r2. p2 ⊕ q ≍ r2 →
+ ∀x. p2 ⊕ r1 ≍ x → ∀y. p1 ⊕ r2 ≍ y → x = y.
+ intros 4; elim H; clear H q r1;
[ lapply linear nplus_inv_zero_2 to H1
| lapply linear nplus_inv_succ_2 to H3.
- lapply linear nplus_inv_succ_2 to H4. decompose. subst.
+ lapply linear nplus_inv_succ_2 to H4. decompose. destruct.
lapply linear nplus_inv_succ_2 to H5. decompose
- ]; subst; auto.
+ ]; destruct; autobatch.
qed.
-theorem nplus_comm_1_rew: \forall p1,q,r1. (p1 + q == r1) \to
- \forall p2,r2. (p2 + q == r2) \to
- \forall s. (p1 + r2 == s) \to (p2 + r1 == s).
- intros 4. elim H; clear H q r1;
- [ lapply linear nplus_inv_zero_2 to H1. subst
- | lapply linear nplus_inv_succ_2 to H3. decompose. subst.
- lapply linear nplus_inv_succ_2 to H4. decompose. subst
- ]; auto.
+theorem nplus_comm_1_rew: ∀p1,q,r1. p1 ⊕ q ≍ r1 → ∀p2,r2. p2 ⊕ q ≍ r2 →
+ ∀s. p1 ⊕ r2 ≍ s → p2 ⊕ r1 ≍ s.
+ intros 4; elim H; clear H q r1;
+ [ lapply linear nplus_inv_zero_2 to H1. destruct
+ | lapply linear nplus_inv_succ_2 to H3. decompose. destruct.
+ lapply linear nplus_inv_succ_2 to H4. decompose. destruct
+ ]; autobatch.
qed.
(*
theorem nplus_shift_succ_sx: \forall p,q,r.
- (p + (succ q) == r) \to (succ p) + q == r.
+ (p \oplus (succ q) \asymp r) \to (succ p) \oplus q \asymp r.
intros.
lapply linear nplus_inv_succ_2 to H as H0.
- decompose. subst. auto new timeout=100.
+ decompose. destruct. auto new timeout=100.
qed.
theorem nplus_shift_succ_dx: \forall p,q,r.
- ((succ p) + q == r) \to p + (succ q) == r.
+ ((succ p) \oplus q \asymp r) \to p \oplus (succ q) \asymp r.
intros.
lapply linear nplus_inv_succ_1 to H as H0.
- decompose. subst. auto new timeout=100.
+ decompose. destruct. auto new timeout=100.
qed.
-theorem nplus_trans_1: \forall p,q1,r1. (p + q1 == r1) \to
- \forall q2,r2. (r1 + q2 == r2) \to
- \exists q. (q1 + q2 == q) \land p + q == r2.
+theorem nplus_trans_1: \forall p,q1,r1. (p \oplus q1 \asymp r1) \to
+ \forall q2,r2. (r1 \oplus q2 \asymp r2) \to
+ \exists q. (q1 \oplus q2 \asymp q) \land p \oplus q \asymp r2.
intros 2; elim q1; clear q1; intros;
[ lapply linear nplus_inv_zero_2 to H as H0.
- subst.
+ destruct.
| lapply linear nplus_inv_succ_2 to H1 as H0.
- decompose. subst.
+ decompose. destruct.
lapply linear nplus_inv_succ_1 to H2 as H0.
- decompose. subst.
+ decompose. destruct.
lapply linear H to H4, H3 as H0.
decompose.
]; apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
qed.
-theorem nplus_trans_2: \forall p1,q,r1. (p1 + q == r1) \to
- \forall p2,r2. (p2 + r1 == r2) \to
- \exists p. (p1 + p2 == p) \land p + q == r2.
+theorem nplus_trans_2: ∀p1,q,r1. p1 ⊕ q ≍ r1 → ∀p2,r2. p2 ⊕ r1 ≍ r2 →
+ ∃p. p1 ⊕ p2 ≍ p ∧ p ⊕ q ≍ r2.
intros 2; elim q; clear q; intros;
[ lapply linear nplus_inv_zero_2 to H as H0.
- subst
+ destruct
| lapply linear nplus_inv_succ_2 to H1 as H0.
- decompose. subst.
+ decompose. destruct.
lapply linear nplus_inv_succ_2 to H2 as H0.
- decompose. subst.
+ decompose. destruct.
lapply linear H to H4, H3 as H0.
decompose.
- ]; apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
+ ]; autobatch. apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
qed.
*)
projects / helm.git / blobdiff