Structure prim_recTheory

signature prim_recTheory =
sig
  type thm = Thm.thm
  
  (*  Definitions  *)
    val LESS_DEF : thm
    val PRE_DEF : thm
    val PRIM_REC : thm
    val PRIM_REC_FUN : thm
    val SIMP_REC : thm
    val SIMP_REC_REL : thm
    val measure_def : thm
    val wellfounded_def : thm
  
  (*  Theorems  *)
    val DC : thm
    val EQ_LESS : thm
    val INV_SUC_EQ : thm
    val LESS_0 : thm
    val LESS_0_0 : thm
    val LESS_ALT : thm
    val LESS_LEMMA1 : thm
    val LESS_LEMMA2 : thm
    val LESS_MONO : thm
    val LESS_MONO_EQ : thm
    val LESS_MONO_REV : thm
    val LESS_NOT_EQ : thm
    val LESS_REFL : thm
    val LESS_SUC : thm
    val LESS_SUC_IMP : thm
    val LESS_SUC_REFL : thm
    val LESS_SUC_SUC : thm
    val LESS_THM : thm
    val NOT_LESS_0 : thm
    val NOT_LESS_EQ : thm
    val PRE : thm
    val PRIM_REC_EQN : thm
    val PRIM_REC_THM : thm
    val RTC_IM_TC : thm
    val SIMP_REC_EXISTS : thm
    val SIMP_REC_REL_UNIQUE : thm
    val SIMP_REC_REL_UNIQUE_RESULT : thm
    val SIMP_REC_THM : thm
    val SUC_ID : thm
    val SUC_LESS : thm
    val TC_IM_RTC_SUC : thm
    val WF_IFF_WELLFOUNDED : thm
    val WF_LESS : thm
    val WF_PRED : thm
    val WF_measure : thm
    val measure_thm : thm
    val num_Axiom : thm
    val num_Axiom_old : thm
  
  val prim_rec_grammars : type_grammar.grammar * term_grammar.grammar
(*
   [num] Parent theory of "prim_rec"
   
   [relation] Parent theory of "prim_rec"
   
   [LESS_DEF]  Definition
      
      ⊢ ∀m n. m < n ⇔ ∃P. (∀n. P (SUC n) ⇒ P n) ∧ P m ∧ ¬P n
   
   [PRE_DEF]  Definition
      
      ⊢ ∀m. PRE m = if m = 0 then 0 else @n. m = SUC n
   
   [PRIM_REC]  Definition
      
      ⊢ ∀x f m. PRIM_REC x f m = PRIM_REC_FUN x f m (PRE m)
   
   [PRIM_REC_FUN]  Definition
      
      ⊢ ∀x f.
          PRIM_REC_FUN x f = SIMP_REC (λn. x) (λfun n. f (fun (PRE n)) n)
   
   [SIMP_REC]  Definition
      
      ⊢ ∀x f' n. ∃g. SIMP_REC_REL g x f' (SUC n) ∧ SIMP_REC x f' n = g n
   
   [SIMP_REC_REL]  Definition
      
      ⊢ ∀fun x f n.
          SIMP_REC_REL fun x f n ⇔
          fun 0 = x ∧ ∀m. m < n ⇒ fun (SUC m) = f (fun m)
   
   [measure_def]  Definition
      
      ⊢ measure = inv_image $<
   
   [wellfounded_def]  Definition
      
      ⊢ ∀R. Wellfounded R ⇔ ¬∃f. ∀n. R (f (SUC n)) (f n)
   
   [DC]  Theorem
      
      ⊢ ∀P R a.
          P a ∧ (∀x. P x ⇒ ∃y. P y ∧ R x y) ⇒
          ∃f. f 0 = a ∧ ∀n. P (f n) ∧ R (f n) (f (SUC n))
   
   [EQ_LESS]  Theorem
      
      ⊢ ∀n. SUC m = n ⇒ m < n
   
   [INV_SUC_EQ]  Theorem
      
      ⊢ ∀m n. SUC m = SUC n ⇔ m = n
   
   [LESS_0]  Theorem
      
      ⊢ ∀n. 0 < SUC n
   
   [LESS_0_0]  Theorem
      
      ⊢ 0 < SUC 0
   
   [LESS_ALT]  Theorem
      
      ⊢ $< = (λx y. y = SUC x)⁺
   
   [LESS_LEMMA1]  Theorem
      
      ⊢ ∀m n. m < SUC n ⇒ m = n ∨ m < n
   
   [LESS_LEMMA2]  Theorem
      
      ⊢ ∀m n. m = n ∨ m < n ⇒ m < SUC n
   
   [LESS_MONO]  Theorem
      
      ⊢ ∀m n. m < n ⇒ SUC m < SUC n
   
   [LESS_MONO_EQ]  Theorem
      
      ⊢ ∀m n. SUC m < SUC n ⇔ m < n
   
   [LESS_MONO_REV]  Theorem
      
      ⊢ ∀m n. SUC m < SUC n ⇒ m < n
   
   [LESS_NOT_EQ]  Theorem
      
      ⊢ ∀m n. m < n ⇒ m ≠ n
   
   [LESS_REFL]  Theorem
      
      ⊢ ∀n. ¬(n < n)
   
   [LESS_SUC]  Theorem
      
      ⊢ ∀m n. m < n ⇒ m < SUC n
   
   [LESS_SUC_IMP]  Theorem
      
      ⊢ ∀m n. m < SUC n ⇒ m ≠ n ⇒ m < n
   
   [LESS_SUC_REFL]  Theorem
      
      ⊢ ∀n. n < SUC n
   
   [LESS_SUC_SUC]  Theorem
      
      ⊢ ∀m. m < SUC m ∧ m < SUC (SUC m)
   
   [LESS_THM]  Theorem
      
      ⊢ ∀m n. m < SUC n ⇔ m = n ∨ m < n
   
   [NOT_LESS_0]  Theorem
      
      ⊢ ∀n. ¬(n < 0)
   
   [NOT_LESS_EQ]  Theorem
      
      ⊢ ∀m n. m = n ⇒ ¬(m < n)
   
   [PRE]  Theorem
      
      ⊢ PRE 0 = 0 ∧ ∀m. PRE (SUC m) = m
   
   [PRIM_REC_EQN]  Theorem
      
      ⊢ ∀x f.
          (∀n. PRIM_REC_FUN x f 0 n = x) ∧
          ∀m n.
            PRIM_REC_FUN x f (SUC m) n = f (PRIM_REC_FUN x f m (PRE n)) n
   
   [PRIM_REC_THM]  Theorem
      
      ⊢ ∀x f.
          PRIM_REC x f 0 = x ∧
          ∀m. PRIM_REC x f (SUC m) = f (PRIM_REC x f m) m
   
   [RTC_IM_TC]  Theorem
      
      ⊢ ∀m n. (λx y. y = f x)꙳ (f m) n ⇔ (λx y. y = f x)⁺ m n
   
   [SIMP_REC_EXISTS]  Theorem
      
      ⊢ ∀x f n. ∃fun. SIMP_REC_REL fun x f n
   
   [SIMP_REC_REL_UNIQUE]  Theorem
      
      ⊢ ∀x f g1 g2 m1 m2.
          SIMP_REC_REL g1 x f m1 ∧ SIMP_REC_REL g2 x f m2 ⇒
          ∀n. n < m1 ∧ n < m2 ⇒ g1 n = g2 n
   
   [SIMP_REC_REL_UNIQUE_RESULT]  Theorem
      
      ⊢ ∀x f n. ∃!y. ∃g. SIMP_REC_REL g x f (SUC n) ∧ y = g n
   
   [SIMP_REC_THM]  Theorem
      
      ⊢ ∀x f.
          SIMP_REC x f 0 = x ∧
          ∀m. SIMP_REC x f (SUC m) = f (SIMP_REC x f m)
   
   [SUC_ID]  Theorem
      
      ⊢ ∀n. SUC n ≠ n
   
   [SUC_LESS]  Theorem
      
      ⊢ ∀m n. SUC m < n ⇒ m < n
   
   [TC_IM_RTC_SUC]  Theorem
      
      ⊢ ∀m n. (λx y. y = SUC x)⁺ m (SUC n) ⇔ (λx y. y = SUC x)꙳ m n
   
   [WF_IFF_WELLFOUNDED]  Theorem
      
      ⊢ ∀R. WF R ⇔ Wellfounded R
   
   [WF_LESS]  Theorem
      
      ⊢ WF $<
   
   [WF_PRED]  Theorem
      
      ⊢ WF (λx y. y = SUC x)
   
   [WF_measure]  Theorem
      
      ⊢ ∀m. WF (measure m)
   
   [measure_thm]  Theorem
      
      ⊢ ∀f x y. measure f x y ⇔ f x < f y
   
   [num_Axiom]  Theorem
      
      ⊢ ∀e f. ∃fn. fn 0 = e ∧ ∀n. fn (SUC n) = f n (fn n)
   
   [num_Axiom_old]  Theorem
      
      ⊢ ∀e f. ∃!fn1. fn1 0 = e ∧ ∀n. fn1 (SUC n) = f (fn1 n) n
   
   
*)
end

HOL 4, Kananaskis-14