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(eq_mod_equiv 0
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(("" (expand "equivalence?")
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("3" (expand "transitive?")
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nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((equivalence? const-decl "bool" relations nil)
(transitive? const-decl "bool" relations nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(symmetric? const-decl "bool" relations nil)
(minus_int_is_int application-judgement "int" integers nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(- const-decl "[numfield -> numfield]" number_fields nil)
(int_plus_int_is_int application-judgement "int" integers nil)
(reflexive? const-decl "bool" relations nil)
(mult_divides1 application-judgement "(divides(n))" divides nil)
(mult_divides2 application-judgement "(divides(m))" divides nil)
(eq_mod const-decl "bool" eq_mod nil)
(even_times_int_is_even application-judgement "even_int" integers
nil)
(nnint_times_nnint_is_nnint application-judgement "nonneg_int"
integers nil)
(int nonempty-type-eq-decl nil integers nil)
(integer_pred const-decl "[rational -> boolean]" integers nil)
(rational nonempty-type-from-decl nil rationals nil)
(rational_pred const-decl "[real -> boolean]" rationals nil)
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(real_pred const-decl "[number_field -> boolean]" reals nil)
(number_field nonempty-type-from-decl nil number_fields nil)
(number_field_pred const-decl "[number -> boolean]" number_fields
nil)
(boolean nonempty-type-decl nil booleans nil)
(number nonempty-type-decl nil numbers nil))
shostak))
(eq_mod_mult 0
(eq_mod_mult-2 nil 3411746929
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(("" (expand "eq_mod")
(("" (skosimp*)
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(("" (inst + "(k!1 * k!2 * p!1 + b!1 * k!2 + d!1 * k!1)")
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nil))
nil))
nil))
nil)
((mult_divides1 application-judgement "(divides(n))" divides nil)
(mult_divides2 application-judgement "(divides(m))" divides nil)
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(nat nonempty-type-eq-decl nil naturalnumbers nil)
(>= const-decl "bool" reals nil)
(bool nonempty-type-eq-decl nil booleans nil)
(* const-decl "[numfield, numfield -> numfield]" number_fields nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(int nonempty-type-eq-decl nil integers nil)
(integer_pred const-decl "[rational -> boolean]" integers nil)
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(real_pred const-decl "[number_field -> boolean]" reals nil)
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(number_field_pred const-decl "[number -> boolean]" number_fields
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nil)
(eq_mod_mult-1 nil 3411734172
("" (skosimp*)
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((""
(name "AA"
" k!1 * k!3 * p!1 * p!1 + k!2 * k!3 * p!1 * p!1 +
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+ d!1 * k!1 * p!1
+ d!1 * k!3 * p!1
")
(("" (factor -1)
((""
(inst +
" (k!1 * k!3 * p!1 + k!2 * k!3 * p!1 + k!3 * k!3 * p!1 + d!1 * k!1 +
d!1 * k!3)")
(("" (assert) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
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nil shostak))
(eq_mod_rem 0
(eq_mod_rem-1 nil 3411813690
("" (skosimp*)
(("" (expand "eq_mod")
(("" (typepred "rem(p!1)(j!1)")
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nil))
nil))
nil)
((eq_mod const-decl "bool" eq_mod nil)
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(real_gt_is_strict_total_order name-judgement
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(boolean nonempty-type-decl nil booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(NOT const-decl "[bool -> bool]" booleans nil)
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nil)
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(real nonempty-type-from-decl nil reals nil)
(< const-decl "bool" reals nil)
(rational_pred const-decl "[real -> boolean]" rationals nil)
(rational nonempty-type-from-decl nil rationals nil)
(integer_pred const-decl "[rational -> boolean]" integers nil)
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(>= const-decl "bool" reals nil)
(nonneg_int nonempty-type-eq-decl nil integers nil)
(> const-decl "bool" reals nil)
(posnat nonempty-type-eq-decl nil integers nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(mod nonempty-type-eq-decl nil euclidean_division nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(* const-decl "[numfield, numfield -> numfield]" number_fields nil)
(rem const-decl "{r: mod(b) | EXISTS q: x = b * q + r}"
modulo_arithmetic nil)
(mult_divides2 application-judgement "(divides(m))" divides nil)
(mult_divides1 application-judgement "(divides(n))" divides nil))
shostak))
(gcd_is_1_TCC1 0
(gcd_is_1_TCC1-1 nil 3411812631 ("" (subtype-tcc) nil nil)
((boolean nonempty-type-decl nil booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(number nonempty-type-decl nil numbers nil)
(number_field_pred const-decl "[number -> boolean]" number_fields
nil)
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(real_pred const-decl "[number_field -> boolean]" reals nil)
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(rational_pred const-decl "[real -> boolean]" rationals nil)
(rational nonempty-type-from-decl nil rationals nil)
(integer_pred const-decl "[rational -> boolean]" integers nil)
(int nonempty-type-eq-decl nil integers nil)
(>= const-decl "bool" reals nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(divides const-decl "bool" divides nil)
(/= const-decl "boolean" notequal nil)
(prime? const-decl "bool" primes "ints/"))
nil))
(gcd_is_1 0
(gcd_is_1-1 nil 3411812637
("" (skosimp*)
(("" (lemma "gcd_def")
(("" (inst?)
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(expand "divides")
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(assert)
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nil))
nil))
nil))
nil))
nil))
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nil))
nil)
("4" (flatten)
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(("4" (expand "divides")
(("4" (inst + "0") (("4" (assert) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((gcd_def formula-decl nil gcd "ints/")
(int_times_even_is_even application-judgement "even_int" integers
nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(prime? const-decl "bool" primes "ints/")
(nnint_times_nnint_is_nnint application-judgement "nonneg_int"
integers nil)
(divides const-decl "bool" divides nil)
(mult_divides2 application-judgement "(divides(m))" divides nil)
(mult_divides1 application-judgement "(divides(n))" divides nil)
(posnat nonempty-type-eq-decl nil integers nil)
(> const-decl "bool" reals nil)
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(nat nonempty-type-eq-decl nil naturalnumbers nil)
(>= const-decl "bool" reals nil)
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(integer_pred const-decl "[rational -> boolean]" integers nil)
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(real_pred const-decl "[number_field -> boolean]" reals nil)
(number_field nonempty-type-from-decl nil number_fields nil)
(number_field_pred const-decl "[number -> boolean]" number_fields
nil)
(boolean nonempty-type-decl nil booleans nil)
(number nonempty-type-decl nil numbers nil))
shostak))
(Euclids_30 0
(Euclids_30-1 nil 3411755092
("" (skeep)
(("" (expand "divides" -2)
(("" (skosimp*)
(("" (lemma "gcd_is_1")
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(("" (assert)
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(assert)
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(replace -5)
((""
(assert)
((""
(factor -1)
((""
(expand "divides" 2)
((""
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+
"(b * n!1 + m!1 * x!1)")
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nil))
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nil))
nil))
nil))
nil))
nil))
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nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((divides const-decl "bool" divides nil)
(gcd_is_1 formula-decl nil eq_mod nil)
(mult_divides2 application-judgement "(divides(m))" divides nil)
(mult_divides1 application-judgement "(divides(n))" divides nil)
(rel_prime const-decl "bool" gcd "ints/")
(= const-decl "[T, T -> boolean]" equalities nil)
(int_plus_int_is_int application-judgement "int" integers nil)
(* const-decl "[numfield, numfield -> numfield]" number_fields nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(both_sides_times1_imp formula-decl nil extra_real_props nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(prime? const-decl "bool" primes "ints/")
(rel_prime_lem formula-decl nil gcd "ints/")
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(>= const-decl "bool" reals nil)
(bool nonempty-type-eq-decl nil booleans nil)
(int nonempty-type-eq-decl nil integers nil)
(integer_pred const-decl "[rational -> boolean]" integers nil)
(rational nonempty-type-from-decl nil rationals nil)
(rational_pred const-decl "[real -> boolean]" rationals nil)
(real nonempty-type-from-decl nil reals nil)
(real_pred const-decl "[number_field -> boolean]" reals nil)
(number_field nonempty-type-from-decl nil number_fields nil)
(number_field_pred const-decl "[number -> boolean]" number_fields
nil)
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shostak))
(eq_mod_cancel 0
(eq_mod_cancel-1 nil 3411754307
("" (skeep)
(("" (case "divides(p,a*c-b*c)")
(("1" (case-replace "a * c - b * c = (a-b)*c")
(("1" (hide -1)
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(("1" (inst?)
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((* const-decl "[numfield, numfield -> numfield]" number_fields nil)
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(number_field nonempty-type-from-decl nil number_fields nil)
(number_field_pred const-decl "[number -> boolean]" number_fields
nil)
(boolean nonempty-type-decl nil booleans nil)
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(int_minus_int_is_int application-judgement "int" integers nil)
(int_plus_int_is_int application-judgement "int" integers nil)
(eq_mod const-decl "bool" eq_mod nil)
(Euclids_30 formula-decl nil eq_mod nil)
(minus_odd_is_odd application-judgement "odd_int" integers nil)
(mult_divides1 application-judgement "(divides(n))" divides nil)
(mult_divides2 application-judgement "(divides(m))" divides nil)
(= const-decl "[T, T -> boolean]" equalities nil))
shostak)))
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