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Specification wellfounded
enrich
   nat
with
   predicates 
       wf : ('a × 'a → bool);
       wfinv : ('a × 'a → bool);
       twf : ('a × 'a → bool);
       trans : ('a × 'a → bool);
       twfinv : ('a × 'a → bool);
   
   variables 
       wf≪ : 'a × 'a → bool;
       wf≫ : 'a × 'a → bool;
       chn : nat → 'a;
       p : 'a → bool;
       q : 'b × 'c → bool;
       a : 'a;
       b : 'b;
       c : 'c;
       wf1≪ : 'b × 'b → bool;
       wf2≪ : 'c × 'c → bool;
       bc : ('b, 'c);
       af : 'a → 'a;
       chngen : nat → nat → 'a;
   
   
   axioms 
      wf-def: wf(wf≪) ↔ ¬(∃ chn. ∀ n. wf≪(chn(n + 1), chn(n)));
      
      wfinv-def: wfinv(wf≫) ↔ wf(λ a, a0. wf≫(a0, a));
      
      trans-def: trans(wf≪) ↔ (∀ a, a0, a1. wf≪(a, a0) ∧ wf≪(a0, a1) → wf≪(a, a1));
      used for: ls;
      
      twf-def: twf(wf≪) ↔ wf(wf≪) ∧ trans(wf≪);
      used for: ls;
      
      twfinv-def: twfinv(wf≫) ↔ wfinv(wf≫) ∧ trans(wf≫);
      used for: ls;
   
   theorems 
      wf-equiv-noethind: wf(wf≪) ↔ (∀ p. (∀ a. (∀ a0. wf≪(a0, a) → p(a0)) → p(a)) → (∀ a. p(a)));
      
      wfinv-equiv-noethind: wfinv(wf≫) ↔ (∀ p. (∀ a. (∀ a0. wf≫(a, a0) → p(a0)) → p(a)) → (∀ a. p(a)));
      
      wf-noethind: ∀ a. (∀ a0. wf≪(a0, a) → p(a0)) → p(a) ⊦ wf(wf≪) → p(a);
      
      wfinv-noethind: ∀ a. (∀ a0. wf≫(a, a0) → p(a0)) → p(a) ⊦ wfinv(wf≫) → p(a);
      
      wf-lexical: wf(wf1≪), wf(wf2≪) ⊦ wf(λ bc0, bc. wf1≪(bc0._1, bc._1) ∨ bc0._1 = bc._1 ∧ wf2≪(bc0._2, bc._2));
      
      wf-lexicalind: ∀ b, c. (∀ b0, c0. wf1≪(b0, b) ∨ b0 = b ∧ wf2≪(c0, c) → q(b0, c0)) → q(b, c) ⊦ wf(wf1≪) ∧ wf(wf2≪) → q(b, c);
   
   
end enrich