|
(majority_seq
(is_majority_TCC1 0
(is_majority_TCC1-1 nil 3410692309
("" (skosimp*) (("" (rewrite "finite_below") nil)) nil)
((finite_below formula-decl nil finite_sets_below "finite_sets/")
(number nonempty-type-decl nil numbers nil)
(boolean nonempty-type-decl nil booleans nil)
(number_field_pred const-decl "[number -> boolean]" number_fields
nil)
(number_field nonempty-type-from-decl nil number_fields nil)
(real_pred const-decl "[number_field -> boolean]" reals nil)
(real nonempty-type-from-decl nil 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)
(int nonempty-type-eq-decl nil integers nil)
(bool nonempty-type-eq-decl nil booleans nil)
(>= const-decl "bool" reals nil) (< const-decl "bool" reals nil)
(below type-eq-decl nil naturalnumbers nil)
(set type-eq-decl nil sets nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(below type-eq-decl nil nat_types nil)
(T formal-nonempty-type-decl nil majority_seq nil)
(finite_sequence type-eq-decl nil finite_sequences nil))
nil))
(maj_TCC1 0
(maj_TCC1-1 nil 3410692309
(""
(inst 1 "(LAMBDA (fs: finite_sequence[T]):
IF maj_exists(fs) THEN
choose({mv: T | is_majority(mv, fs)})
ELSE
epsilon({mv: T | TRUE})
ENDIF)")
(("1" (skosimp*) nil) ("2" (skosimp*) (("2" (assert) nil)))
("3" (skosimp*)
(("3" (assert)
(("3" (expand "nonempty?")
(("3" (expand "empty?")
(("3" (expand "maj_exists")
(("3" (skosimp*)
(("3" (inst -2 "mv!1")
(("3" (expand "member")
(("3" (propax) nil))))))))))))))))))
nil)
((member const-decl "bool" sets nil)
(empty? const-decl "bool" sets nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(TRUE const-decl "bool" booleans nil)
(epsilon const-decl "T" epsilons nil)
(pred type-eq-decl nil defined_types nil)
(choose const-decl "(p)" sets nil)
(nonempty? const-decl "bool" sets nil)
(set type-eq-decl nil sets nil)
(IF const-decl "[boolean, T, T -> T]" if_def nil)
(is_majority const-decl "bool" majority_seq nil)
(maj_exists const-decl "bool" majority_seq nil)
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil)
(finite_sequence type-eq-decl nil finite_sequences nil)
(T formal-nonempty-type-decl nil majority_seq nil)
(below type-eq-decl nil nat_types nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil))
nil))
(is_majority_unique 0
(is_majority_unique-1 nil 3410692309
("" (skosimp*)
(("" (expand "is_majority")
(("" (name "A" "{i: below(length(fs!1)) | seq(fs!1)(i) = mv1!1}")
((""
(name "B" "{i: below(length(fs!1)) | seq(fs!1)(i) = mv2!1}")
(("" (replace -1)
(("" (hide -1)
(("" (replace -1)
(("" (hide -1)
(("" (case "disjoint?(A,B)")
(("1"
(lemma "card_disj_union[below(length(fs!1))]")
(("1" (inst?)
(("1" (assert)
(("1" (hide -2)
(("1"
(case "card(union(A, B)) <= length(fs!1)")
(("1"
(replace -2)
(("1"
(hide -2 -3 -4 1)
(("1" (assert) nil)))))
("2"
(hide -1 -2 -3)
(("2" (rewrite "card_below") nil)))
("3"
(lemma
"finite_subset[below(length(fs!1))]")
(("3"
(inst
-1
"{x: below(length(fs!1)) | TRUE}"
"union[below(length(fs!1))](A, B)")
(("1"
(assert)
(("1"
(hide -1 -2 -3 -4 2 3)
(("1"
(expand "subset?")
(("1"
(expand "union")
(("1"
(expand "member")
(("1"
(propax)
nil)))))))))))
("2"
(rewrite "finite_below")
nil)))))))))))
("2" (reveal -3)
(("2" (hide -2 -3 -4 2)
(("2" (replace -1 1 rl)
(("2"
(hide -1)
(("2"
(rewrite "finite_below")
nil)))))))))
("3" (rewrite "finite_below") nil)))))
("2" (reveal -1 -2)
(("2" (hide -3 -4)
(("2" (replace -1 + rl)
(("2" (hide -1)
(("2" (replace -1 + rl)
(("2"
(hide -1)
(("2"
(expand "disjoint?")
(("2"
(expand "intersection")
(("2"
(expand "empty?")
(("2"
(expand "member")
(("2"
(skosimp*)
(("2"
(assert)
nil))))))))))))))))))))))))))))))))))))))))
nil)
((is_majority const-decl "bool" majority_seq nil)
(empty? const-decl "bool" sets nil)
(member const-decl "bool" sets nil)
(intersection const-decl "set" sets nil)
(card_disj_union formula-decl nil finite_sets nil)
(mult_divides2 application-judgement "(divides(m))" divides nil)
(mult_divides1 application-judgement "(divides(n))" divides nil)
(even_times_int_is_even application-judgement "even_int" integers
nil)
(nnint_times_nnint_is_nnint application-judgement "nonneg_int"
integers nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(<= const-decl "bool" reals nil)
(Card const-decl "nat" finite_sets nil)
(card const-decl "{n: nat | n = Card(S)}" finite_sets nil)
(union const-decl "set" sets nil)
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
integers nil)
(card_below formula-decl nil finite_sets_below "finite_sets/")
(is_finite const-decl "bool" finite_sets nil)
(finite_set type-eq-decl nil finite_sets nil)
(set type-eq-decl nil sets nil)
(disjoint? const-decl "bool" sets nil)
(number nonempty-type-decl nil numbers nil)
(boolean nonempty-type-decl nil booleans nil)
(number_field_pred const-decl "[number -> boolean]" number_fields
nil)
(number_field nonempty-type-from-decl nil number_fields nil)
(real_pred const-decl "[number_field -> boolean]" reals nil)
(real nonempty-type-from-decl nil 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)
(int nonempty-type-eq-decl nil integers nil)
(bool nonempty-type-eq-decl nil booleans nil)
(>= const-decl "bool" reals nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(< const-decl "bool" reals nil)
(below type-eq-decl nil nat_types nil)
(T formal-nonempty-type-decl nil majority_seq nil)
(finite_sequence type-eq-decl nil finite_sequences nil)
(below type-eq-decl nil naturalnumbers nil)
(= const-decl "[T, T -> boolean]" equalities nil))
nil))
(maj_lem 0
(maj_lem-1 nil 3410692309
("" (skosimp*)
(("" (prop)
(("1" (expand "maj_exists") (("1" (inst?) nil)))
("2" (typepred "maj(fs!1)")
(("2" (split -1)
(("1" (lemma "is_majority_unique")
(("1" (inst -1 "fs!1" "mv!1" "maj(fs!1)")
(("1" (assert) nil)))))
("2" (expand "maj_exists") (("2" (inst?) nil)))))))
("3" (typepred "maj(fs!1)") (("3" (assert) nil))))))
nil)
((T formal-nonempty-type-decl nil majority_seq nil)
(maj_exists const-decl "bool" majority_seq nil)
(is_majority_unique formula-decl nil majority_seq nil)
(boolean nonempty-type-decl nil booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(below type-eq-decl nil nat_types nil)
(finite_sequence type-eq-decl nil finite_sequences nil)
(is_majority const-decl "bool" majority_seq nil)
(maj const-decl "{mv | maj_exists(fs) => is_majority(mv, fs)}"
majority_seq nil))
nil))
(maj_subset 0
(maj_subset-1 nil 3410692309
("" (skosimp*)
(("" (expand "is_majority")
((""
(case "subset?(A!1,{i: below(length(fs!1)) | seq(fs!1)(i) = mv!1})")
(("1" (lemma "card_subset[below(length(fs!1))]")
(("1" (inst?)
(("1" (assert) nil) ("2" (rewrite "finite_below") nil)))))
("2" (hide -1 2)
(("2" (expand "subset?")
(("2" (skosimp*)
(("2" (expand "member")
(("2" (inst -2 "x!1")
(("2" (assert) nil))))))))))))))))
nil)
((is_majority const-decl "bool" majority_seq nil)
(member const-decl "bool" sets nil)
(card_subset formula-decl nil finite_sets nil)
(finite_below formula-decl nil finite_sets_below "finite_sets/")
(mult_divides2 application-judgement "(divides(m))" divides nil)
(mult_divides1 application-judgement "(divides(n))" divides nil)
(even_times_int_is_even application-judgement "even_int" integers
nil)
(nnint_times_nnint_is_nnint application-judgement "nonneg_int"
integers nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(subset_is_partial_order name-judgement "(partial_order?[set[T]])"
sets_lemmas nil)
(mv!1 skolem-const-decl "T" majority_seq nil)
(fs!1 skolem-const-decl "finite_sequence[T]" majority_seq nil)
(number nonempty-type-decl nil numbers nil)
(boolean nonempty-type-decl nil booleans nil)
(number_field_pred const-decl "[number -> boolean]" number_fields
nil)
(number_field nonempty-type-from-decl nil number_fields nil)
(real_pred const-decl "[number_field -> boolean]" reals nil)
(real nonempty-type-from-decl nil 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)
(int nonempty-type-eq-decl nil integers nil)
(bool nonempty-type-eq-decl nil booleans nil)
(>= const-decl "bool" reals nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(< const-decl "bool" reals nil)
(below type-eq-decl nil nat_types nil)
(T formal-nonempty-type-decl nil majority_seq nil)
(finite_sequence type-eq-decl nil finite_sequences nil)
(below type-eq-decl nil naturalnumbers nil)
(set type-eq-decl nil sets nil)
(subset? const-decl "bool" sets nil)
(is_finite const-decl "bool" finite_sets nil)
(finite_set type-eq-decl nil finite_sets nil)
(= const-decl "[T, T -> boolean]" equalities nil))
nil))
(maj_in_seq 0
(maj_in_seq-1 nil 3410692309
("" (skosimp*)
(("" (expand "in_seq")
(("" (expand "is_majority")
(("" (lemma "card_empty?[below(length(fs!1))]")
(("" (inst?)
(("1" (expand "empty?")
(("1" (expand "member")
(("1" (iff -1)
(("1" (flatten)
(("1" (split -1)
(("1" (assert) nil)
("2" (split -1)
(("1" (propax) nil)
("2" (skosimp*)
(("2" (inst 2 "x!1") nil)))))))))))))))
("2" (rewrite "finite_below") nil))))))))))
nil)
((in_seq const-decl "bool" majority_seq nil)
(below type-eq-decl nil naturalnumbers nil)
(finite_sequence type-eq-decl nil finite_sequences nil)
(T formal-nonempty-type-decl nil majority_seq nil)
(below type-eq-decl nil nat_types nil)
(< const-decl "bool" reals nil)
(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)
(boolean nonempty-type-decl nil booleans nil)
(number nonempty-type-decl nil numbers nil)
(card_empty? formula-decl nil finite_sets nil)
(finite_below formula-decl nil finite_sets_below "finite_sets/")
(empty? const-decl "bool" sets nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(nnint_times_nnint_is_nnint application-judgement "nonneg_int"
integers nil)
(even_times_int_is_even application-judgement "even_int" integers
nil)
(mult_divides1 application-judgement "(divides(n))" divides nil)
(mult_divides2 application-judgement "(divides(m))" divides nil)
(member const-decl "bool" sets nil)
(finite_set type-eq-decl nil finite_sets nil)
(fs!1 skolem-const-decl "finite_sequence[T]" majority_seq nil)
(set type-eq-decl nil sets nil)
(is_finite const-decl "bool" finite_sets nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(mv!1 skolem-const-decl "T" majority_seq nil)
(is_majority const-decl "bool" majority_seq nil))
nil))
(length_eq_1 0
(length_eq_1-1 nil 3410692309
("" (skosimp*)
(("" (lemma "maj_lem")
(("" (inst -1 "(# length := 1, seq := f1!1 #)" "f1!1(0)")
(("" (flatten)
(("" (hide -2)
(("" (assert)
(("" (lemma "maj_lem")
(("" (inst?)
(("" (flatten)
(("" (hide -2)
(("" (assert)
(("" (hide 2)
((""
(case "length((# length := 1, seq := f1!1 #)) = 1")
(("1" (expand "is_majority")
(("1"
(lemma
"card_below_fullset[length((# length := 1, seq := f1!1 #))]")
(("1"
(expand "fullset")
(("1"
(replace -1)
(("1" (assert) nil)))))))))
("2" (assert)
nil))))))))))))))))))))))))))
nil)
((maj_lem formula-decl nil majority_seq nil)
(is_majority const-decl "bool" majority_seq nil)
(fullset const-decl "set" sets nil)
(mult_divides2 application-judgement "(divides(m))" divides nil)
(mult_divides1 application-judgement "(divides(n))" divides nil)
(even_times_int_is_even application-judgement "even_int" integers
nil)
(posint_times_posint_is_posint application-judgement "posint"
integers nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(card_below_fullset formula-decl nil finite_sets_below
"finite_sets/")
(= const-decl "[T, T -> boolean]" equalities nil)
(< const-decl "bool" reals 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)
(boolean nonempty-type-decl nil booleans nil)
(number nonempty-type-decl nil numbers nil)
(finite_sequence type-eq-decl nil finite_sequences nil)
(T formal-nonempty-type-decl nil majority_seq nil)
(below type-eq-decl nil nat_types nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil))
nil))
(is_majority_const 0
(is_majority_const-2 nil 3410692328
("" (skosimp*)
(("" (expand "is_majority")
(("" (expand "const_seq")
(("" (lemma "card_below_fullset[length(const_seq(n!1, c!1))]")
(("" (expand "fullset")
(("" (expand "const_seq") (("" (assert) nil nil)) nil))
nil))
nil))
nil))
nil))
nil)
((is_majority const-decl "bool" majority_seq nil)
(posnat nonempty-type-eq-decl nil integers nil)
(> const-decl "bool" reals nil)
(nonneg_int nonempty-type-eq-decl nil integers nil)
(finite_sequence type-eq-decl nil finite_sequences nil)
(below type-eq-decl nil nat_types nil)
(T formal-nonempty-type-decl nil majority_seq nil)
(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)
(boolean nonempty-type-decl nil booleans nil)
(number nonempty-type-decl nil numbers nil)
(card_below_fullset formula-decl nil finite_sets_below
"finite_sets/")
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(nnint_times_nnint_is_nnint application-judgement "nonneg_int"
integers nil)
(even_times_int_is_even application-judgement "even_int" integers
nil)
(mult_divides1 application-judgement "(divides(n))" divides nil)
(mult_divides2 application-judgement "(divides(m))" divides nil)
(fullset const-decl "set" sets nil)
(const_seq const-decl "finite_sequence[T]" seqs nil))
nil)
(is_majority_const-1 nil 3410692309
("" (skosimp*)
(("" (expand "is_majority")
(("" (expand "constant_seq")
((""
(lemma "card_below_fullset[length(constant_seq(n!1, c!1))]")
(("" (expand "fullset")
(("" (expand "constant_seq") (("" (assert) nil))))))))))))
nil)
nil nil))
(maj_const 0
(maj_const-1 nil 3410692309
("" (skosimp*)
(("" (lemma "is_majority_const")
(("" (inst?)
(("" (lemma "maj_lem") (("" (inst?) (("" (ground) nil))))))))))
nil)
((is_majority_const formula-decl nil majority_seq nil)
(maj_lem formula-decl nil majority_seq nil)
(const_seq const-decl "finite_sequence[T]" seqs nil)
(finite_sequence type-eq-decl nil finite_sequences nil)
(below type-eq-decl nil nat_types nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(posnat nonempty-type-eq-decl nil integers nil)
(> const-decl "bool" reals nil)
(nonneg_int nonempty-type-eq-decl nil integers 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)
(boolean nonempty-type-decl nil booleans nil)
(number nonempty-type-decl nil numbers nil)
(T formal-nonempty-type-decl nil majority_seq nil))
nil)))
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