Structure ternaryComparisonsTheory

signature ternaryComparisonsTheory =
sig
  type thm = Thm.thm
  
  (*  Definitions  *)
    val invert_comparison_def : thm
    val num_compare_def : thm
    val ordering_BIJ : thm
    val ordering_CASE : thm
    val ordering_TY_DEF : thm
    val ordering_size_def : thm
    val pair_compare_def : thm
  
  (*  Theorems  *)
    val bool_compare_def : thm
    val bool_compare_ind : thm
    val compare_equal : thm
    val datatype_ordering : thm
    val invert_eq_EQUAL : thm
    val list_compare_def : thm
    val list_compare_ind : thm
    val list_merge_def : thm
    val list_merge_ind : thm
    val num2ordering_11 : thm
    val num2ordering_ONTO : thm
    val num2ordering_ordering2num : thm
    val num2ordering_thm : thm
    val option_compare_def : thm
    val option_compare_ind : thm
    val ordering2num_11 : thm
    val ordering2num_ONTO : thm
    val ordering2num_num2ordering : thm
    val ordering2num_thm : thm
    val ordering_Axiom : thm
    val ordering_EQ_ordering : thm
    val ordering_case_cong : thm
    val ordering_case_def : thm
    val ordering_case_eq : thm
    val ordering_distinct : thm
    val ordering_distinct1 : thm
    val ordering_distinct2 : thm
    val ordering_eq_dec : thm
    val ordering_induction : thm
    val ordering_nchotomy : thm
  
  val ternaryComparisons_grammars : type_grammar.grammar * term_grammar.grammar
(*
   [indexedLists] Parent theory of "ternaryComparisons"
   
   [patternMatches] Parent theory of "ternaryComparisons"
   
   [invert_comparison_def]  Definition
      
      ⊢ invert_comparison GREATER = LESS ∧
        invert_comparison LESS = GREATER ∧ invert_comparison EQUAL = EQUAL
   
   [num_compare_def]  Definition
      
      ⊢ ∀n1 n2.
          num_compare n1 n2 =
          if n1 = n2 then EQUAL else if n1 < n2 then LESS else GREATER
   
   [ordering_BIJ]  Definition
      
      ⊢ (∀a. num2ordering (ordering2num a) = a) ∧
        ∀r. (λn. n < 3) r ⇔ ordering2num (num2ordering r) = r
   
   [ordering_CASE]  Definition
      
      ⊢ ∀x v0 v1 v2.
          (case x of LESS => v0 | EQUAL => v1 | GREATER => v2) =
          (λm. if m < 1 then v0 else if m = 1 then v1 else v2)
            (ordering2num x)
   
   [ordering_TY_DEF]  Definition
      
      ⊢ ∃rep. TYPE_DEFINITION (λn. n < 3) rep
   
   [ordering_size_def]  Definition
      
      ⊢ ∀x. ordering_size x = 0
   
   [pair_compare_def]  Definition
      
      ⊢ ∀c1 c2 a b x y.
          pair_compare c1 c2 (a,b) (x,y) =
          case c1 a x of
            LESS => LESS
          | EQUAL => c2 b y
          | GREATER => GREATER
   
   [bool_compare_def]  Theorem
      
      ⊢ bool_compare T T = EQUAL ∧ bool_compare F F = EQUAL ∧
        bool_compare T F = GREATER ∧ bool_compare F T = LESS
   
   [bool_compare_ind]  Theorem
      
      ⊢ ∀P. P T T ∧ P F F ∧ P T F ∧ P F T ⇒ ∀v v1. P v v1
   
   [compare_equal]  Theorem
      
      ⊢ (∀x y. cmp x y = EQUAL ⇔ x = y) ⇒
        ∀l1 l2. list_compare cmp l1 l2 = EQUAL ⇔ l1 = l2
   
   [datatype_ordering]  Theorem
      
      ⊢ DATATYPE (ordering LESS EQUAL GREATER)
   
   [invert_eq_EQUAL]  Theorem
      
      ⊢ ∀x. invert_comparison x = EQUAL ⇔ x = EQUAL
   
   [list_compare_def]  Theorem
      
      ⊢ (∀cmp. list_compare cmp [] [] = EQUAL) ∧
        (∀v9 v8 cmp. list_compare cmp [] (v8::v9) = LESS) ∧
        (∀v5 v4 cmp. list_compare cmp (v4::v5) [] = GREATER) ∧
        ∀y x l2 l1 cmp.
          list_compare cmp (x::l1) (y::l2) =
          case cmp x y of
            LESS => LESS
          | EQUAL => list_compare cmp l1 l2
          | GREATER => GREATER
   
   [list_compare_ind]  Theorem
      
      ⊢ ∀P. (∀cmp. P cmp [] []) ∧ (∀cmp v8 v9. P cmp [] (v8::v9)) ∧
            (∀cmp v4 v5. P cmp (v4::v5) []) ∧
            (∀cmp x l1 y l2.
               (cmp x y = EQUAL ⇒ P cmp l1 l2) ⇒ P cmp (x::l1) (y::l2)) ⇒
            ∀v v1 v2. P v v1 v2
   
   [list_merge_def]  Theorem
      
      ⊢ (∀l1 a_lt. list_merge a_lt l1 [] = l1) ∧
        (∀v5 v4 a_lt. list_merge a_lt [] (v4::v5) = v4::v5) ∧
        ∀y x l2 l1 a_lt.
          list_merge a_lt (x::l1) (y::l2) =
          if a_lt x y then x::list_merge a_lt l1 (y::l2)
          else y::list_merge a_lt (x::l1) l2
   
   [list_merge_ind]  Theorem
      
      ⊢ ∀P. (∀a_lt l1. P a_lt l1 []) ∧ (∀a_lt v4 v5. P a_lt [] (v4::v5)) ∧
            (∀a_lt x l1 y l2.
               (¬a_lt x y ⇒ P a_lt (x::l1) l2) ∧
               (a_lt x y ⇒ P a_lt l1 (y::l2)) ⇒
               P a_lt (x::l1) (y::l2)) ⇒
            ∀v v1 v2. P v v1 v2
   
   [num2ordering_11]  Theorem
      
      ⊢ ∀r r'. r < 3 ⇒ r' < 3 ⇒ (num2ordering r = num2ordering r' ⇔ r = r')
   
   [num2ordering_ONTO]  Theorem
      
      ⊢ ∀a. ∃r. a = num2ordering r ∧ r < 3
   
   [num2ordering_ordering2num]  Theorem
      
      ⊢ ∀a. num2ordering (ordering2num a) = a
   
   [num2ordering_thm]  Theorem
      
      ⊢ num2ordering 0 = LESS ∧ num2ordering 1 = EQUAL ∧
        num2ordering 2 = GREATER
   
   [option_compare_def]  Theorem
      
      ⊢ option_compare c NONE NONE = EQUAL ∧
        option_compare c NONE (SOME v0) = LESS ∧
        option_compare c (SOME v3) NONE = GREATER ∧
        option_compare c (SOME v1) (SOME v2) = c v1 v2
   
   [option_compare_ind]  Theorem
      
      ⊢ ∀P. (∀c. P c NONE NONE) ∧ (∀c v0. P c NONE (SOME v0)) ∧
            (∀c v3. P c (SOME v3) NONE) ∧
            (∀c v1 v2. P c (SOME v1) (SOME v2)) ⇒
            ∀v v1 v2. P v v1 v2
   
   [ordering2num_11]  Theorem
      
      ⊢ ∀a a'. ordering2num a = ordering2num a' ⇔ a = a'
   
   [ordering2num_ONTO]  Theorem
      
      ⊢ ∀r. r < 3 ⇔ ∃a. r = ordering2num a
   
   [ordering2num_num2ordering]  Theorem
      
      ⊢ ∀r. r < 3 ⇔ ordering2num (num2ordering r) = r
   
   [ordering2num_thm]  Theorem
      
      ⊢ ordering2num LESS = 0 ∧ ordering2num EQUAL = 1 ∧
        ordering2num GREATER = 2
   
   [ordering_Axiom]  Theorem
      
      ⊢ ∀x0 x1 x2. ∃f. f LESS = x0 ∧ f EQUAL = x1 ∧ f GREATER = x2
   
   [ordering_EQ_ordering]  Theorem
      
      ⊢ ∀a a'. a = a' ⇔ ordering2num a = ordering2num a'
   
   [ordering_case_cong]  Theorem
      
      ⊢ ∀M M' v0 v1 v2.
          M = M' ∧ (M' = LESS ⇒ v0 = v0') ∧ (M' = EQUAL ⇒ v1 = v1') ∧
          (M' = GREATER ⇒ v2 = v2') ⇒
          (case M of LESS => v0 | EQUAL => v1 | GREATER => v2) =
          case M' of LESS => v0' | EQUAL => v1' | GREATER => v2'
   
   [ordering_case_def]  Theorem
      
      ⊢ (∀v0 v1 v2.
           (case LESS of LESS => v0 | EQUAL => v1 | GREATER => v2) = v0) ∧
        (∀v0 v1 v2.
           (case EQUAL of LESS => v0 | EQUAL => v1 | GREATER => v2) = v1) ∧
        ∀v0 v1 v2.
          (case GREATER of LESS => v0 | EQUAL => v1 | GREATER => v2) = v2
   
   [ordering_case_eq]  Theorem
      
      ⊢ (case x of LESS => v0 | EQUAL => v1 | GREATER => v2) = v ⇔
        x = LESS ∧ v0 = v ∨ x = EQUAL ∧ v1 = v ∨ x = GREATER ∧ v2 = v
   
   [ordering_distinct]  Theorem
      
      ⊢ LESS ≠ EQUAL ∧ LESS ≠ GREATER ∧ EQUAL ≠ GREATER
   
   [ordering_distinct1]  Theorem
      
      ⊢ EQUAL ≠ LESS
   
   [ordering_distinct2]  Theorem
      
      ⊢ GREATER ≠ EQUAL
   
   [ordering_eq_dec]  Theorem
      
      ⊢ (∀x. x = x ⇔ T) ∧ (LESS = EQUAL ⇔ F) ∧ (LESS = GREATER ⇔ F) ∧
        (EQUAL = GREATER ⇔ F) ∧ (EQUAL = LESS ⇔ F) ∧ (GREATER = LESS ⇔ F) ∧
        (GREATER = EQUAL ⇔ F) ∧
        (ordering2num LESS = 0 ∧ ordering2num EQUAL = 1 ∧
         ordering2num GREATER = 2) ∧ num2ordering 0 = LESS ∧
        num2ordering 1 = EQUAL ∧ num2ordering 2 = GREATER
   
   [ordering_induction]  Theorem
      
      ⊢ ∀P. P EQUAL ∧ P GREATER ∧ P LESS ⇒ ∀a. P a
   
   [ordering_nchotomy]  Theorem
      
      ⊢ ∀a. a = LESS ∨ a = EQUAL ∨ a = GREATER
   
   
*)
end

HOL 4, Trindemossen-1