(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/RELATIONAL/NPlus/inv".
+
include "NPlus/defs.ma".
-(* primitive generation lemmas proved by elimination and inversion *)
+(* Inversion lemmas *********************************************************)
-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
- ].
+theorem nplus_inv_zero_1: ∀q,r. zero ⊕ q ≍ r → q = r.
+ intros. elim H; clear H q r; autobatch.
qed.
-theorem nplus_gen_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;
- [
- | clear H1.
- decompose.
- subst.
- ]; apply ex_intro; [| auto new timeout=30 || auto new timeout=30 ]. (**)
+ [ autobatch depth = 3
+ | clear H1; decompose; destruct; autobatch depth = 4
+ ]
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.
- 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_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;
- [ destruct H.
- | clear H1 H3 r.
- destruct H2; clear H2.
- subst.
- apply ex_intro; [| auto new timeout=30 ] (**)
- ].
+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_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.
- 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_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.
- destruct H3. clear H3.
- subst.
- ]; apply ex_intro; [| auto new timeout=30 || auto new timeout=30 ] (**)
+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.
-(*
-(* 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
- ].
+(* Corollaries to inversion lemmas ******************************************)
+
+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.
-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 ]. (**)
+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.
-*)
-(* 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.
- destruct H2. clear H2.
- subst
- ]; auto new timeout=30.
+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
+ ]; autobatch.
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.
- destruct H2. clear H2.
- subst
- ]; auto new timeout=30.
+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
+ ]; autobatch.
qed.
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