Theory "prelim"

Signature

Definitions

ordering_TY_DEF

|- ∃rep. TYPE_DEFINITION (λn. n < 3) rep

ordering_BIJ

|- (∀a. num2ordering (ordering2num a) = a) ∧
   ∀r. (λn. n < 3) r ⇔ (ordering2num (num2ordering r) = r)

ordering_size_def

|- ∀x. ordering_size x = 0

ordering_CASE

|- ∀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)

compare_def

|- (∀lt eq gt. compare LESS lt eq gt = lt) ∧
   (∀lt eq gt. compare EQUAL lt eq gt = eq) ∧
   ∀lt eq gt. compare GREATER lt eq gt = gt

list_compare_tupled_primitive_def

|- list_compare_tupled =
   WFREC (@R. WF R ∧ ∀y x l2 l1 cmp. R (cmp,l1,l2) (cmp,x::l1,y::l2))
     (λlist_compare_tupled a.
        case a of
          (cmp,[],[]) => I EQUAL
        | (cmp,[],v10::v11) => I LESS
        | (cmp,x::l1,[]) => I GREATER
        | (cmp,x::l1,y::l2) =>
            I
              (compare (cmp x y) LESS (list_compare_tupled (cmp,l1,l2))
                 GREATER))

list_compare_curried_def

|- ∀x x1 x2. list_compare x x1 x2 = list_compare_tupled (x,x1,x2)

list_merge_tupled_primitive_def

|- list_merge_tupled =
   WFREC
     (@R.
        WF R ∧
        (∀l2 l1 y x a_lt. ¬a_lt x y ⇒ R (a_lt,x::l1,l2) (a_lt,x::l1,y::l2)) ∧
        ∀l2 l1 y x a_lt. a_lt x y ⇒ R (a_lt,l1,y::l2) (a_lt,x::l1,y::l2))
     (λlist_merge_tupled a.
        case a of
          (a_lt,l1,[]) => I l1
        | (a_lt,[],y::l2) => I (y::l2)
        | (a_lt,x::l1',y::l2) =>
            I
              (if a_lt x y then x::list_merge_tupled (a_lt,l1',y::l2)
               else y::list_merge_tupled (a_lt,x::l1',l2)))

list_merge_curried_def

|- ∀x x1 x2. list_merge x x1 x2 = list_merge_tupled (x,x1,x2)

Theorems

num2ordering_ordering2num

|- ∀a. num2ordering (ordering2num a) = a

ordering2num_num2ordering

|- ∀r. r < 3 ⇔ (ordering2num (num2ordering r) = r)

num2ordering_11

|- ∀r r'. r < 3 ⇒ r' < 3 ⇒ ((num2ordering r = num2ordering r') ⇔ (r = r'))

ordering2num_11

|- ∀a a'. (ordering2num a = ordering2num a') ⇔ (a = a')

num2ordering_ONTO

|- ∀a. ∃r. (a = num2ordering r) ∧ r < 3

ordering2num_ONTO

|- ∀r. r < 3 ⇔ ∃a. r = ordering2num a

num2ordering_thm

|- (num2ordering 0 = LESS) ∧ (num2ordering 1 = EQUAL) ∧
   (num2ordering 2 = GREATER)

ordering2num_thm

|- (ordering2num LESS = 0) ∧ (ordering2num EQUAL = 1) ∧
   (ordering2num GREATER = 2)

ordering_EQ_ordering

|- ∀a a'. (a = a') ⇔ (ordering2num a = ordering2num a')

ordering_case_def

|- (∀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

datatype_ordering

|- DATATYPE (ordering LESS EQUAL GREATER)

ordering_distinct

|- LESS ≠ EQUAL ∧ LESS ≠ GREATER ∧ EQUAL ≠ GREATER

ordering_case_cong

|- ∀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_nchotomy

|- ∀a. (a = LESS) ∨ (a = EQUAL) ∨ (a = GREATER)

ordering_Axiom

|- ∀x0 x1 x2. ∃f. (f LESS = x0) ∧ (f EQUAL = x1) ∧ (f GREATER = x2)

ordering_induction

|- ∀P. P EQUAL ∧ P GREATER ∧ P LESS ⇒ ∀a. P a

ordering_eq_dec

|- (∀x. (x = x) ⇔ T) ∧ ((LESS = EQUAL) ⇔ F) ∧ ((LESS = GREATER) ⇔ F) ∧
   ((EQUAL = GREATER) ⇔ F) ∧ ((EQUAL = LESS) ⇔ F) ∧ ((GREATER = LESS) ⇔ F) ∧
   ((GREATER = EQUAL) ⇔ F)

list_compare_ind

|- ∀P.
     (∀cmp. P cmp [] []) ∧ (∀cmp v8 v9. P cmp [] (v8::v9)) ∧
     (∀cmp v4 v5. P cmp (v4::v5) []) ∧
     (∀cmp x l1 y l2. P cmp l1 l2 ⇒ P cmp (x::l1) (y::l2)) ⇒
     ∀v v1 v2. P v v1 v2

list_compare_def

|- (∀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) =
     compare (cmp x y) LESS (list_compare cmp l1 l2) GREATER

compare_equal

|- (∀x y. (cmp x y = EQUAL) ⇔ (x = y)) ⇒
   ∀l1 l2. (list_compare cmp l1 l2 = EQUAL) ⇔ (l1 = l2)

list_merge_ind

|- ∀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

list_merge_def

|- (∀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