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5583 lines (5582 loc) · 331 KB
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(p96
(FFS_TCC1 0
(FFS_TCC1-1 nil 3888905413 ("" (assuming-tcc) nil nil)
((restrict const-decl "R" restrict nil)
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nil
(FFS assuming
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"zero_identity: ASSUMPTION operator_defs[finite_sets_sum.R].identity?(finite_sets_sum.+)(finite_sets_sum.zero)")))
(FFS_TCC2 0
(FFS_TCC2-1 nil 3888905413 ("" (assuming-tcc) nil nil)
((real_plus_real_is_real application-judgement "real" reals nil)
(associative? const-decl "bool" operator_defs nil)
(commutative? const-decl "bool" operator_defs nil)
(restrict const-decl "R" restrict nil))
nil
(FFS assuming
"finite_sets@finite_sets_sum[finite_set[finite_set[T]], real, 0, restrict[[numfield, numfield], [real, real], numfield].restrict(number_fields.+)].finite_sets_sum"
"plus_ac: ASSUMPTION booleans.AND(operator_defs[finite_sets_sum.R].associative?(finite_sets_sum.+), operator_defs[finite_sets_sum.R].commutative?(finite_sets_sum.+))")))
(IMP_real_aux_TCC1 0
(IMP_real_aux_TCC1-1 nil 3888598211
("" (lemma "finite_universe[T]")
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(("" (expand "is_finite_type")
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(("1" (prop)
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(("1" (grind) nil nil)
("2" (skolem-typepred)
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nil))
nil))
nil)
("2" (grind) nil nil))
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nil))
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nil))
nil))
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nil)
("2" (hide 2) (("2" (grind) nil nil)) nil))
nil))
nil))
nil))
nil)
((T formal-nonempty-type-decl nil p96 nil)
(powerset_finite2 judgement-tcc nil powerset_aux nil)
(fullset const-decl "set" sets nil)
(is_finite const-decl "bool" finite_sets nil)
(set type-eq-decl nil sets nil)
(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil)
(finite_set type-eq-decl nil finite_sets nil)
(setof type-eq-decl nil defined_types nil)
(IFF const-decl "[bool, bool -> bool]" booleans nil)
(setofsets type-eq-decl nil sets nil)
(powerset const-decl "setofsets" sets nil)
(< const-decl "bool" reals nil)
(below type-eq-decl nil nat_types nil)
(subset? const-decl "bool" sets nil)
(member const-decl "bool" sets nil)
(injective? const-decl "bool" functions nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(nat nonempty-type-eq-decl nil naturalnumbers 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)
(>= const-decl "bool" reals 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)
(number nonempty-type-decl nil numbers nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(is_finite_type const-decl "bool" finite_sets nil)
(finite_universe formula-decl nil p96 nil))
nil
(IMP_real_aux assuming "real_aux[finite_set[T]].real_aux"
"finite_universe: ASSUMPTION finite_sets[real_aux.D].is_finite_type")))
(altcard_TCC1 0
(altcard_TCC1-1 nil 3889242994
("" (skolem-typepred)
(("" (lemma "finite_universe")
(("" (lemma "finite_subset[T]")
((""
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(extend[setof[T], finite_set[T], bool, FALSE](B!1))")
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(("1" (hide 2)
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nil))
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("2" (hide 2)
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nil)
((finite_universe formula-decl nil p96 nil)
(fullset const-decl "set" sets nil)
(setof type-eq-decl nil defined_types nil)
(setofsets type-eq-decl nil sets nil)
(Intersection const-decl "set" sets nil)
(FALSE const-decl "bool" booleans nil)
(extend const-decl "R" extend nil)
(member const-decl "bool" sets nil)
(subset? const-decl "bool" sets nil)
(finite_extend application-judgement "finite_set[T]"
extend_set_props nil)
(Intersection_surjective name-judgement
"(surjective?[setofsets[T], set[T]])" p96 nil)
(Intersection_surjective name-judgement
"(surjective?[setofsets[T], set[T]])" sets_lemmas nil)
(subset_is_partial_order name-judgement "(partial_order?[set[T]])"
p96 nil)
(subset_is_partial_order name-judgement "(partial_order?[set[T]])"
sets_lemmas nil)
(is_finite_type const-decl "bool" finite_sets nil)
(injective? const-decl "bool" functions nil)
(finite_subset formula-decl nil finite_sets nil)
(finite_set type-eq-decl nil finite_sets nil)
(is_finite const-decl "bool" finite_sets nil)
(set type-eq-decl nil sets nil)
(T formal-nonempty-type-decl nil p96 nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil))
nil
(altcard subtype
"sets[p96.T].Intersection(extend[setof[T], finite_set[T], booleans.bool, booleans.FALSE].extend(p96.B))"
"finite_set[T]")))
(e15_22_1_TCC1 0
(e15_22_1_TCC1-1 nil 3889242994
("" (skolem-typepred)
(("" (ground)
(("" (lemma "finite_universe")
(("" (lemma "finite_subset[T]")
((""
(inst -1 "fullset[T]"
"Union[T](extend[setof[T], finite_set[T], bool, FALSE](A!1))")
(("1" (grind) nil nil)
("2" (hide 3)
(("2" (expand "is_finite_type")
(("2" (expand "fullset")
(("2" (hide -2 -3 -4 2)
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nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((Union_surjective name-judgement
"(surjective?[setofsets[T], set[T]])" sets_lemmas nil)
(Union_surjective name-judgement
"(surjective?[setofsets[T], set[T]])" p96 nil)
(finite_extend application-judgement "finite_set[T]"
extend_set_props nil)
(finite_subset formula-decl nil finite_sets nil)
(is_finite_type const-decl "bool" finite_sets nil)
(subset_is_partial_order name-judgement "(partial_order?[set[T]])"
sets_lemmas nil)
(subset_is_partial_order name-judgement "(partial_order?[set[T]])"
p96 nil)
(subset? const-decl "bool" sets nil)
(member const-decl "bool" sets nil)
(setof type-eq-decl nil defined_types nil)
(setofsets type-eq-decl nil sets nil)
(Union const-decl "set" sets nil)
(FALSE const-decl "bool" booleans nil)
(extend const-decl "R" extend nil)
(fullset const-decl "set" sets nil)
(finite_universe formula-decl nil p96 nil)
(non_empty_finite_set type-eq-decl nil finite_sets nil)
(empty? const-decl "bool" sets nil)
(finite_set type-eq-decl nil finite_sets nil)
(is_finite const-decl "bool" finite_sets nil)
(set type-eq-decl nil sets nil)
(T formal-nonempty-type-decl nil p96 nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil))
nil
(e15_22_1 subtype
"sets[p96.T].Union(extend[setof[T], finite_set[T], booleans.bool, booleans.FALSE].extend(p96.A))"
"finite_set[T]")))
(e15_22_1 0
(e15_22_1-1 nil 3887087638
("" (skolem-typepred)
(("" (ground)
(("" (lemma "M_D_sum")
(("" (inst -1 "D!1" "Union(A!1)")
(("" (ground)
(("" (hide 3)
(("" (expand "subset?")
(("" (skolem-typepred)
(("" (ground)
(("" (expand "Union")
(("" (expand "member")
(("" (skolem-typepred)
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nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((Union_surjective name-judgement
"(surjective?[setofsets[T], set[T]])" sets_lemmas nil)
(Union_surjective name-judgement
"(surjective?[setofsets[T], set[T]])" p96 nil)
(finite_extend application-judgement "finite_set[T]"
extend_set_props nil)
(nil application-judgement "nat" p96 nil)
(setof type-eq-decl nil defined_types nil)
(setofsets type-eq-decl nil sets nil)
(Union const-decl "set" sets nil)
(FALSE const-decl "bool" booleans nil)
(extend const-decl "R" extend nil)
(number nonempty-type-decl nil numbers 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)
(>= 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)
(int nonempty-type-eq-decl nil integers nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(injective? const-decl "bool" functions nil)
(member const-decl "bool" sets nil)
(subset? const-decl "bool" sets nil)
(subset_is_partial_order name-judgement "(partial_order?[set[T]])"
p96 nil)
(subset_is_partial_order name-judgement "(partial_order?[set[T]])"
sets_lemmas nil)
(M_D_sum formula-decl nil M_D_aux nil)
(non_empty_finite_set type-eq-decl nil finite_sets nil)
(empty? const-decl "bool" sets nil)
(finite_set type-eq-decl nil finite_sets nil)
(is_finite const-decl "bool" finite_sets nil)
(set type-eq-decl nil sets nil)
(T formal-nonempty-type-decl nil p96 nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil))
SHOSTAK))
(e15_22_2 0
(e15_22_2-1 nil 3887403016
("" (skolem-typepred)
(("" (ground)
((""
(lemma "sum_f_g"
("S" "D!1" "f" "M_D(Union(A!1))" "g"
"LAMBDA x: 1 - product(A!1, LAMBDA a: neg_M_D(a)(x))"))
(("" (ground)
(("" (hide 3)
(("" (skolem-typepred)
(("" (expand "neg_M_D")
(("" (expand "M_D")
(("" (expand "o ")
(("" (expand "b2n")
(("" (expand "Union")
(("" (lift-if)
(("" (ground)
(("1" (skolem-typepred)
(("1"
(expand "extend")
(("1"
(ground)
(("1"
(lemma
"product_remove[finite_set[T],real,1,*]")
(("1"
(inst
-1
"A!1"
"LAMBDA a: IF (NOT)(a(x!1)) THEN 1 ELSE 0 ENDIF"
"a!1")
(("1"
(ground)
(("1"
(expand "member")
(("1" (grind) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(case "FORALL (a: (A!1)): NOT a(x!1)")
(("1"
(hide 1)
(("1"
(lemma
"product_eq_funs[finite_set[T]]")
(("1"
(inst?)
(("1"
(inst -1 "lambda a: 1")
(("1"
(ground)
(("1"
(replace -1 * lr t)
(("1"
(lemma
"product_const[finite_set[T]]")
(("1"
(inst?)
(("1"
(replace -1 * lr t)
(("1"
(lemma "expt_1i")
(("1"
(inst?)
(("1"
(grind)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(skolem-typepred)
(("2"
(inst? -3)
(("2" (grind) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(skolem-typepred)
(("2"
(inst? 1)
(("2" (grind) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((nil application-judgement "nat" p96 nil)
(Union_surjective name-judgement
"(surjective?[setofsets[T], set[T]])" sets_lemmas nil)
(Union_surjective name-judgement
"(surjective?[setofsets[T], set[T]])" p96 nil)
(finite_extend application-judgement "finite_set[T]"
extend_set_props nil)
(nil application-judgement "int" p96 nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(product_nat application-judgement "nat" p96 nil)
(b2n const-decl "nbit" bit nil)
(product_eq_funs formula-decl nil finite_sets_product_real
finite_sets)
(expt_1i formula-decl nil exponentiation nil)
(posnat_expt application-judgement "posnat" exponentiation nil)
(minus_int_is_int application-judgement "int" integers nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(posint_exp application-judgement "posint" exponentiation nil)
(expt def-decl "real" exponentiation nil)
(^ const-decl "real" exponentiation nil)
(injective? const-decl "bool" functions nil)
(subset? const-decl "bool" sets nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(Card const-decl "nat" finite_sets nil)
(card const-decl "{n: nat | n = Card(S)}" finite_sets nil)
(product_const formula-decl nil finite_sets_product_real
finite_sets)
(A!1 skolem-const-decl "finite_set[finite_set[T]]" p96 nil)
(a!1 skolem-const-decl "(A!1)" p96 nil)
(int_times_int_is_int application-judgement "int" integers nil)
(IF const-decl "[boolean, T, T -> T]" if_def nil)
(member const-decl "bool" sets nil)
(finite_remove application-judgement "finite_set[finite_set[T]]"
p96 nil)
(nnint_times_nnint_is_nnint application-judgement "nonneg_int"
integers nil)
(product_remove formula-decl nil finite_sets_product finite_sets)
(O const-decl "T3" function_props nil)
(real_times_real_is_real application-judgement "real" reals nil)
(sum_f_g formula-decl nil finite_sets_sum finite_sets)
(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)
(< const-decl "bool" reals nil) (nbit type-eq-decl nil bit nil)
(M_D const-decl "[T -> nbit]" M_D_aux nil)
(setof type-eq-decl nil defined_types nil)
(setofsets type-eq-decl nil sets nil)
(Union const-decl "set" sets nil)
(FALSE const-decl "bool" booleans nil)
(extend const-decl "R" extend nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(* const-decl "[numfield, numfield -> numfield]" number_fields nil)
(product def-decl "R" finite_sets_product finite_sets)
(neg_M_D const-decl "[T -> nbit]" M_D_aux nil)
(number nonempty-type-decl nil numbers 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)
(numfield nonempty-type-eq-decl nil number_fields nil)
(restrict const-decl "R" restrict nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(non_empty_finite_set type-eq-decl nil finite_sets nil)
(empty? const-decl "bool" sets nil)
(finite_set type-eq-decl nil finite_sets nil)
(is_finite const-decl "bool" finite_sets nil)
(set type-eq-decl nil sets nil)
(T formal-nonempty-type-decl nil p96 nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil))
SHOSTAK))
(e15_22_3 0
(e15_22_3-1 nil 3887402391
("" (skolem-typepred)
(("" (ground)
(("" (lemma "e15_16")
(("" (inst?)
(("" (expand "n_f")
(("" (expand "neg_M_D")
(("" (expand "o ")
(("" (expand "b2n")
(("" (replace -1 * rl t)
(("" (lemma "product_f_g")
(("" (inst?)
(("" (ground)
(("" (skolem-typepred)
(("" (hide 3) (("" (grind) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((product_nat application-judgement "nat" p96 nil)
(nil application-judgement "int" p96 nil)
(powerset_finite application-judgement "finite_set[set[T]]"
finite_sets_of_sets nil)
(powerset_finite3 application-judgement
"non_empty_finite_set[finite_set[finite_set[T]]]" p96 nil)
(nonempty_powerset application-judgement
"(nonempty?[set[finite_set[T]]])" p96 nil)
(int_times_int_is_int application-judgement "int" integers nil)
(int_exp application-judgement "int" exponentiation nil)
(nzreal_exp application-judgement "nzreal" exponentiation nil)
(number nonempty-type-decl nil numbers 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)
(>= const-decl "bool" reals nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(< const-decl "bool" reals nil) (nbit type-eq-decl nil bit nil)
(M_D const-decl "[T -> nbit]" M_D_aux nil)
(neg_M_D const-decl "[T -> nbit]" M_D_aux nil)
(b2n const-decl "nbit" bit nil)
(product_f_g formula-decl nil finite_sets_product finite_sets)
(numfield nonempty-type-eq-decl nil number_fields nil)
(restrict const-decl "R" restrict nil)
(* const-decl "[numfield, numfield -> numfield]" number_fields nil)
(product_int application-judgement "int" p96 nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(subset? const-decl "bool" sets nil)
(member const-decl "bool" sets nil)
(injective? const-decl "bool" functions nil)
(IF const-decl "[boolean, T, T -> T]" if_def nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(O const-decl "T3" function_props nil)
(n_f const-decl "real" real_aux nil)
(minus_odd_is_odd application-judgement "odd_int" integers nil)
(e15_16 formula-decl nil real_aux nil)
(non_empty_finite_set type-eq-decl nil finite_sets nil)
(empty? const-decl "bool" sets nil)
(finite_set type-eq-decl nil finite_sets nil)
(is_finite const-decl "bool" finite_sets nil)
(set type-eq-decl nil sets nil)
(T formal-nonempty-type-decl nil p96 nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil))
SHOSTAK))
(e15_22_3b 0
(e15_22_3b-1 nil 3887416081
("" (skolem-typepred)
(("" (ground)
(("" (lemma "intersection_is_product3")
((""
(lemma "sum_eq_funs[finite_set[finite_set[T]]]"
("S" "powerset(A!1)" "f"
"LAMBDA B: (-1) ^ (card(B)) * product(B, LAMBDA a: M_D(a)(x!1))"
"g" "LAMBDA B:
(-1) ^ (card(B)) *
M_D(Intersection(extend[setof[T], finite_set[T], bool, FALSE](B)))
(x!1)"))
(("" (ground)
(("" (hide 3)
(("" (skolem-typepred)
(("" (inst -2 "t!1" "x!1")
(("" (ground)
(("" (replace -2 * lr t)
(("" (expand "restrict") (("" (propax) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((nil application-judgement "int" p96 nil)
(powerset_finite application-judgement "finite_set[set[T]]"
finite_sets_of_sets nil)
(powerset_finite3 application-judgement
"non_empty_finite_set[finite_set[finite_set[T]]]" p96 nil)
(nonempty_powerset application-judgement
"(nonempty?[set[finite_set[T]]])" p96 nil)
(int_times_int_is_int application-judgement "int" integers nil)
(int_exp application-judgement "int" exponentiation nil)
(nzreal_exp application-judgement "nzreal" exponentiation nil)
(product_nat application-judgement "nat" p96 nil)
(extend const-decl "R" extend nil)
(FALSE const-decl "bool" booleans nil)
(Intersection const-decl "set" sets nil)
(M_D const-decl "[T -> nbit]" M_D_aux nil)
(nbit type-eq-decl nil bit nil) (< const-decl "bool" reals nil)
(product def-decl "R" finite_sets_product finite_sets)
(restrict const-decl "R" restrict nil)
(card const-decl "{n: nat | n = Card(S)}" finite_sets nil)
(Card const-decl "nat" finite_sets nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(- const-decl "[numfield -> numfield]" number_fields nil)
(^ const-decl "real" exponentiation nil)
(>= const-decl "bool" reals nil)
(/= const-decl "boolean" notequal nil)
(OR const-decl "[bool, bool -> bool]" 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)
(* const-decl "[numfield, numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields 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)
(number nonempty-type-decl nil numbers nil)
(powerset const-decl "setofsets" sets nil)
(setofsets type-eq-decl nil sets nil)
(setof type-eq-decl nil defined_types nil)
(sum_eq_funs formula-decl nil finite_sets_sum_real finite_sets)
(real_times_real_is_real application-judgement "real" reals nil)
(minus_odd_is_odd application-judgement "odd_int" integers nil)
(finite_extend application-judgement "finite_set[T]"
extend_set_props nil)
(Intersection_surjective name-judgement
"(surjective?[setofsets[T], set[T]])" p96 nil)
(Intersection_surjective name-judgement
"(surjective?[setofsets[T], set[T]])" sets_lemmas nil)
(intersection_is_product3 formula-decl nil M_D_aux nil)
(non_empty_finite_set type-eq-decl nil finite_sets nil)
(empty? const-decl "bool" sets nil)
(finite_set type-eq-decl nil finite_sets nil)
(is_finite const-decl "bool" finite_sets nil)
(set type-eq-decl nil sets nil)
(T formal-nonempty-type-decl nil p96 nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil))
SHOSTAK))
(e15_22_4 0
(e15_22_4-1 nil 3887420576
("" (skolem-typepred)
(("" (ground)
((""
(lemma "sum_distributive2"
("A" "D!1" "f" "lambda x: 1" "g" "LAMBDA x:
FFS.sum(powerset(A!1),
LAMBDA B:
(-1) ^ (card(B)) *
M_D(Intersection(B))(x))"))
(("" (ground)
(("" (replace -1 * rl)
(("" (ground)
(("" (use "sum_const[T]") (("" (ground) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil)
((nil application-judgement "int" p96 nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(int_exp application-judgement "int" exponentiation nil)
(nzreal_exp application-judgement "nzreal" exponentiation nil)
(minus_odd_is_odd application-judgement "odd_int" integers nil)
(nil application-judgement "int" p96 nil)
(nil application-judgement "posnat" p96 nil)
(nnint_times_nnint_is_nnint application-judgement "nonneg_int"
integers nil)
(sum_const formula-decl nil finite_sets_sum_real finite_sets)
(powerset_finite application-judgement "finite_set[set[T]]"
finite_sets_of_sets nil)
(powerset_finite3 application-judgement
"non_empty_finite_set[finite_set[finite_set[T]]]" p96 nil)
(nonempty_powerset application-judgement
"(nonempty?[set[finite_set[T]]])" p96 nil)
(finite_extend application-judgement "finite_set[T]"
extend_set_props nil)
(int_times_int_is_int application-judgement "int" integers nil)
(real_plus_real_is_real application-judgement "real" reals nil)
(sum_distributive2 formula-decl nil finite_sum_aux2 nil)
(number nonempty-type-decl nil numbers 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)
(numfield nonempty-type-eq-decl nil number_fields nil)
(restrict const-decl "R" restrict nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(sum def-decl "R" finite_sets_sum finite_sets)
(setof type-eq-decl nil defined_types nil)
(setofsets type-eq-decl nil sets nil)
(powerset const-decl "setofsets" sets nil)
(* const-decl "[numfield, numfield -> numfield]" number_fields 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)
(OR const-decl "[bool, bool -> bool]" booleans nil)
(/= const-decl "boolean" notequal nil)
(>= const-decl "bool" reals nil)
(^ const-decl "real" exponentiation nil)
(- const-decl "[numfield -> numfield]" number_fields nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(Card const-decl "nat" finite_sets nil)
(card const-decl "{n: nat | n = Card(S)}" finite_sets nil)
(< const-decl "bool" reals nil) (nbit type-eq-decl nil bit nil)
(M_D const-decl "[T -> nbit]" M_D_aux nil)
(Intersection const-decl "set" sets nil)
(FALSE const-decl "bool" booleans nil)
(extend const-decl "R" extend nil)
(non_empty_finite_set type-eq-decl nil finite_sets nil)
(empty? const-decl "bool" sets nil)
(finite_set type-eq-decl nil finite_sets nil)
(is_finite const-decl "bool" finite_sets nil)
(set type-eq-decl nil sets nil)
(T formal-nonempty-type-decl nil p96 nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil))
SHOSTAK))
(e15_22_5 0
(e15_22_5-6 "grr" 3889249349
("" (skolem-typepred)
(("" (ground)
((""
(lemma "sum_distributive"
("A" "D!1" "f" "LAMBDA (x_1: T):
FFS.sum(singleton(emptyset),
LAMBDA B:
(-1) ^ (card(B)) *
M_D(Intersection(B))(x_1))" "g"
"LAMBDA (x_1: T):
FFS.sum(remove(emptyset, powerset(A!1)),
LAMBDA B:
(-1) ^ (card(B)) *
M_D(Intersection(B))(x_1))"))
(("" (replace -1 * lr t)
(("" (ground)
((""
(lemma "sum_eq_funs"
("S" "D!1" "f"
"LAMBDA x: FFS.sum(powerset(A!1), LAMBDA B: (-1) ^ (card(B)) * M_D(Intersection(B))(x))"
"g"
"LAMBDA (x: T): FFS.sum(singleton(emptyset), LAMBDA B: (-1) ^ (card(B)) * M_D(Intersection(B))(x)) + FFS.sum(remove(emptyset, powerset(A!1)), LAMBDA B: (-1) ^ (card(B)) * M_D(Intersection(B))(x))"))
(("" (prop)
(("" (hide 3)
(("" (skolem-typepred)
((""
(lemma
"sum_remove[finite_set[finite_set[T]],real,0,+]"
("x" "emptyset[finite_set[T]]" "S"
"powerset(A!1)" "f" "LAMBDA B:
(-1) ^ (card(B)) *
M_D(Intersection(extend[setof[T], finite_set[T], bool, FALSE]
(B)))
(t!1)"))
(("" (ground)
(("" (lift-if)
(("" (ground)
(("1" (replace -2 * rl t)
(("1"
(case
"FFS.sum(singleton(emptyset), LAMBDA B: (-1) ^ (card(B)) * M_D(Intersection(extend[setof[T], finite_set[T], bool, FALSE](B)))(t!1)) = (-1) ^ (card(emptyset[finite_set[T]])) *
M_D(Intersection(extend[setof[T], finite_set[T], bool, FALSE](emptyset[finite_set[T]])))(t!1)")
(("1"
(replace -1 * rl t)
(("1"
(ground)
(("1"
(lemma
"sum_eq_doms"
("A"
"{y: finite_set[finite_set[T]] |
emptyset[finite_set[T]] /= y AND
member[finite_set[finite_set[T]]](y, powerset(A!1))}"
"B"
"{y: set[finite_set[T]] |
emptyset[finite_set[T]] /= y AND member(y, powerset(A!1))}"
"f"
"LAMBDA B:
(-1) ^ (card(B)) *
M_D(Intersection(extend[setof[T], finite_set[T], bool, FALSE]
(B)))
(t!1)"))
(("1"
(ground)
(("1"
(expand "remove")
(("1" (propax) nil nil))
nil)
("2"
(hide 2)
(("2"
(apply-extensionality
1
:hide?
t)
(("1" (grind) nil nil)
("2"
(hide - 2)
(("2"
(skolem-typepred)
(("2"
(lemma
"finite_universe")
(("2"
(lemma
"finite_subset[finite_set[T]]")
(("2"
(inst
-1
"powerset(fullset[T])"
"x!1")
(("1"
(ground)
(("1"
(hide 2)
(("1"
(expand
"subset?")
(("1"
(skolem-typepred)
(("1"
(ground)
(("1"
(expand
"member")
(("1"
(expand
"restrict")
(("1"
(grind)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(expand
"restrict")
(("2"
(lemma
"powerset_finite2[T]")
(("2"
(inst
-1
"fullset[T]")
(("1"
(ground)
(("1"
(hide
-1
-3
2)
(("1"
(expand
"is_finite")
(("1"
(skolem-typepred)
(("1"
(inst?)
(("1"
(inst?)
(("1"
(expand
"injective?")
(("1"
(skolem-typepred)
(("1"
(inst?)
nil
nil))
nil))
nil)
("2"
(skolem-typepred)
(("2"
(ground)
(("2"
(lemma
"powerset_finite2[T]")
(("2"
(inst?)
(("1"
(ground)
(("1"
(inst?)
(("1"
(ground)
nil
nil))
nil))
nil)
("2"
(hide
-
2)
(("2"
(lemma
"finite_universe")
(("2"
(expand*
"is_finite_type"
"is_finite")
(("2"
(skolem-typepred)
(("2"
(inst?)
(("2"
(inst?)
(("1"
(expand
"injective?")
(("1"
(skolem-typepred)
(("1"
(inst?)
nil
nil))
nil))
nil)
("2"
(hide
-)
(("2"
(grind)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(hide
2
3)
(("2"
(expand*
"is_finite_type"
"is_finite"
"fullset")
(("2"
(skolem-typepred)
(("2"
(inst?)
(("2"
(inst?)
(("1"
(expand
"injective?")
(("1"
(skolem-typepred)
(("1"
(inst?)
nil
nil))
nil))
nil)
("2"
(grind)
nil
nil))
nil))
nil))
nil))
nil))
nil))