(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/RELATIONAL/NPlus/inv".
+
include "NPlus/defs.ma".
(* Inversion lemmas *********************************************************)
-theorem nplus_inv_zero_1: \forall q,r. (zero + q == r) \to q = r.
- intros. elim H; clear H q r; auto.
+theorem nplus_inv_zero_1: ∀q,r. zero ⊕ q ≍ r → q = r.
+ intros. elim H; clear H q r; autobatch.
qed.
-theorem nplus_inv_succ_1: \forall p,q,r. ((succ p) + q == r) \to
- \exists s. r = (succ s) \land p + q == s.
+theorem nplus_inv_succ_1: ∀p,q,r. succ p ⊕ q ≍ r →
+ ∃s. r = succ s ∧ p ⊕ q ≍ s.
intros. elim H; clear H q r; intros;
- [ auto depth = 4
- | clear H1. decompose. subst. auto depth = 4
+ [ autobatch depth = 3
+ | clear H1; decompose; destruct; autobatch depth = 4
]
qed.
-theorem nplus_inv_zero_2: \forall p,r. (p + zero == r) \to p = r.
- intros. inversion H; clear H; intros;
- [ auto.
- | clear H H1. destruct H2.
- ].
+theorem nplus_inv_zero_2: ∀p,r. p ⊕ zero ≍ r → p = r.
+ intros; inversion H; clear H; intros; destruct; autobatch.
qed.
-theorem nplus_inv_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;
- [ destruct H.
- | clear H1 H3 r. destruct H2; clear H2. subst. auto depth = 4.
- ].
+theorem nplus_inv_succ_2: ∀p,q,r. p ⊕ succ q ≍ r →
+ ∃s. r = succ s ∧ p ⊕ q ≍ s.
+ intros; inversion H; clear H; intros; destruct.
+ autobatch depth = 3.
qed.
-theorem nplus_inv_zero_3: \forall p,q. (p + q == zero) \to
- p = zero \land q = zero.
- intros. inversion H; clear H; intros;
- [ subst. auto
- | clear H H1. destruct H3.
- ].
+theorem nplus_inv_zero_3: ∀p,q. p ⊕ q ≍ zero →
+ p = zero ∧ q = zero.
+ intros; inversion H; clear H; intros; destruct; autobatch.
qed.
-theorem nplus_inv_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. destruct H3. clear H3. subst.
- ]; auto depth = 4.
+theorem nplus_inv_succ_3: ∀p,q,r. p ⊕ q ≍ succ r →
+ ∃s. p = succ s ∧ s ⊕ q ≍ r ∨
+ q = succ s ∧ p ⊕ s ≍ r.
+ intros; inversion H; clear H; intros; destruct;
+ autobatch depth = 4.
qed.
(* Corollaries to inversion lemmas ******************************************)
-theorem nplus_inv_succ_2_3: \forall p,q,r.
- (p + (succ q) == (succ r)) \to p + q == r.
- intros.
- lapply linear nplus_inv_succ_2 to H. decompose. subst.
- destruct H1. clear H1. subst. auto.
+theorem nplus_inv_succ_2_3: ∀p,q,r.
+ p ⊕ succ q ≍ succ r → p ⊕ q ≍ r.
+ intros;
+ lapply linear nplus_inv_succ_2 to H; decompose; destruct; autobatch.
qed.
-theorem nplus_inv_succ_1_3: \forall p,q,r.
- ((succ p) + q == (succ r)) \to p + q == r.
- intros.
- lapply linear nplus_inv_succ_1 to H. decompose. subst.
- destruct H1. clear H1. subst. auto.
+theorem nplus_inv_succ_1_3: ∀p,q,r.
+ succ p ⊕ q ≍ succ r → p ⊕ q ≍ r.
+ intros;
+ lapply linear nplus_inv_succ_1 to H; decompose; destruct; autobatch.
qed.
-theorem nplus_inv_eq_2_3: \forall p,q. (p + q == q) \to p = zero.
- intros 2. elim q; clear q;
+theorem nplus_inv_eq_2_3: ∀p,q. p ⊕ q ≍ q → p = zero.
+ intros 2; elim q; clear q;
[ lapply linear nplus_inv_zero_2 to H
| lapply linear nplus_inv_succ_2_3 to H1
- ]; auto.
+ ]; autobatch.
qed.
-theorem nplus_inv_eq_1_3: \forall p,q. (p + q == p) \to q = zero.
- intros 1. elim p; clear p;
+theorem nplus_inv_eq_1_3: ∀p,q. p ⊕ q ≍ p → q = zero.
+ intros 1; elim p; clear p;
[ lapply linear nplus_inv_zero_1 to H
- | lapply linear nplus_inv_succ_1_3 to H1.
- ]; auto.
+ | lapply linear nplus_inv_succ_1_3 to H1
+ ]; autobatch.
qed.