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real_aux.prf
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3071 lines (3070 loc) · 176 KB
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(real_aux
(e15_16_1 0
(e15_16_1-1 nil 3886974872
("" (skolem-typepred)
(("" (ground)
(("" (lemma "product_rest")
(("" (inst -1 "A!1" "n_f(f!1)") (("" (grind) nil nil)) nil))
nil))
nil))
nil)
((real_times_real_is_real application-judgement "real" reals nil)
(finite_rest application-judgement "finite_set[D]" real_aux nil)
(n_f const-decl "real" real_aux nil)
(finite_remove application-judgement "finite_set[D]" real_aux nil)
(real_minus_real_is_real application-judgement "real" reals nil)
(rest const-decl "set" sets nil)
(/= const-decl "boolean" notequal nil)
(remove const-decl "set" sets nil)
(member const-decl "bool" sets nil)
(empty? const-decl "bool" sets nil)
(injective? const-decl "bool" functions nil)
(nonempty? const-decl "bool" sets nil)
(product_rest formula-decl nil finite_sets_product finite_sets)
(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)
(finite_set type-eq-decl nil finite_sets nil)
(is_finite const-decl "bool" finite_sets nil)
(set type-eq-decl nil sets nil)
(D formal-nonempty-type-decl nil real_aux 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_16_2_TCC1 0
(e15_16_2_TCC1-1 nil 3888781126
("" (lemma "finite_universe")
(("" (skolem-typepred)
(("" (ground)
(("" (skolem-typepred)
(("" (lemma "finite_subset[D]")
(("" (inst -1 "fullset[D]" "x!1")
(("1" (ground)
(("1" (hide -1 -3 2)
(("1" (expand "subset?") (("1" (grind) nil nil))
nil))
nil))
nil)
("2" (hide -1 -3 2) (("2" (grind) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil)
((boolean nonempty-type-decl nil booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(D formal-nonempty-type-decl nil real_aux nil)
(set type-eq-decl nil sets nil)
(is_finite const-decl "bool" finite_sets nil)
(finite_set type-eq-decl nil finite_sets nil)
(fullset const-decl "set" sets nil)
(injective? const-decl "bool" functions nil)
(is_finite_type const-decl "bool" finite_sets nil)
(member const-decl "bool" sets nil)
(subset? const-decl "bool" sets nil)
(subset_is_partial_order name-judgement "(partial_order?[set[D]])"
real_aux nil)
(subset_is_partial_order name-judgement "(partial_order?[set[T]])"
sets_lemmas nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(< const-decl "bool" reals nil)
(below type-eq-decl nil nat_types 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)
(finite_subset formula-decl nil finite_sets nil)
(finite_universe formula-decl nil real_aux nil))
nil
(e15_16_2 subtype
"LAMBDA B: number_fields.*(exponentiation.^(((number_fields.-)(1)), (finite_sets[real_aux.D].card(real_aux.B))), finite_sets_product[real_aux.D, real, 1, restrict[[numfield, numfield], [real, real], numfield].restrict(number_fields.*)].product(real_aux.B, real_aux.f))"
"[set[D] -> real]")))
(e15_16_2 0
(e15_16_2-1 nil 3886975279
("" (skolem-typepred)
(("" (ground)
(("" (lemma "sum_mult")
((""
(inst -1 "powerset(rest(A!1))" "f!1(choose(A!1))"
"LAMBDA B: (-1) ^ (card(B)) * product(B, f!1)")
(("1" (replace -1 * rl t)
(("1" (lemma "sum_eq_funs")
(("1"
(inst -1 "powerset(rest(A!1))" "LAMBDA (t: set[D]):
f!1(choose(A!1)) * ((-1) ^ (card(t)) * product(t, f!1))"
"LAMBDA B: f!1(choose(A!1)) * (-1) ^ (card(B)) * product(B, f!1)")
(("1" (ground) nil nil)
("2" (hide - 2)
(("2" (lemma "finite_universe")
(("2" (skolem-typepred)
(("2" (lemma "finite_subset[D]")
(("2" (inst -1 "fullset[D]" "t!1")
(("1" (ground)
(("1" (expand "subset?")
(("1" (grind) nil nil)) nil))
nil)
("2" (hide 2) (("2" (grind) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (hide - 2)
(("2" (skolem-typepred)
(("2" (lemma "finite_universe")
(("2"
(lemma "finite_subset[D]"
("A" "fullset[D]" "s" "x!1"))
(("1" (grind) nil nil)
("2" (hide 2) (("2" (grind) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((real_times_real_is_real application-judgement "real" reals nil)
(powerset_finite application-judgement "finite_set[set[T]]"
finite_sets_of_sets nil)
(nonempty_powerset application-judgement "(nonempty?[set[D]])"
real_aux nil)
(powerset_finite3 application-judgement
"non_empty_finite_set[finite_set[D]]" real_aux nil)
(finite_rest application-judgement "finite_set[D]" real_aux 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)
(setof type-eq-decl nil defined_types nil)
(setofsets type-eq-decl nil sets nil)
(powerset const-decl "setofsets" sets nil)
(rest const-decl "set" sets 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)
(nonempty? const-decl "bool" sets nil)
(choose const-decl "(p)" sets nil)
(numfield nonempty-type-eq-decl nil number_fields 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)
(restrict const-decl "R" restrict nil)
(product def-decl "R" finite_sets_product finite_sets)
(sum_eq_funs formula-decl nil finite_sets_sum_real finite_sets)
(fullset const-decl "set" sets nil)
(subset? const-decl "bool" sets nil)
(member const-decl "bool" sets nil)
(is_finite_type const-decl "bool" finite_sets nil)
(injective? const-decl "bool" functions nil)
(subset_is_partial_order name-judgement "(partial_order?[set[D]])"
real_aux nil)
(subset_is_partial_order name-judgement "(partial_order?[set[T]])"
sets_lemmas nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(< const-decl "bool" reals nil)
(below type-eq-decl nil nat_types nil)
(finite_subset formula-decl nil finite_sets nil)
(finite_universe formula-decl nil real_aux nil)
(sum_mult formula-decl nil finite_sets_sum_real finite_sets)
(finite_set type-eq-decl nil finite_sets nil)
(is_finite const-decl "bool" finite_sets nil)
(set type-eq-decl nil sets nil)
(D formal-nonempty-type-decl nil real_aux 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_16_3 0
(e15_16_3-1 nil 3886975712
("" (skolem-typepred)
(("" (lemma "sum_mult")
(("" (inst?)
(("1" (inst -1 "-1")
(("1" (ground)
(("1" (both-sides "*" "-1" -1)
(("1" (ground)
(("1" (replace -1 * rl)
(("1"
(case "(LAMBDA (t: finite_set[D]): -1 * (f!1(choose(A!1)) * (-1) ^ (card(t)) * product(t, f!1))) = LAMBDA (t: finite_set[D]): (f!1(choose(A!1)) * (-1) ^ (1 + card(t)) * product(t, f!1))")
(("1" (replace -1 * rl t)
(("1" (replace -1 * lr t)
(("1"
(lemma "sum_mult"
("S" "powerset(rest(A!1))" "c" "-1" "f"
"LAMBDA (t: finite_set[D]):
f!1(choose(A!1)) * (-1) ^ (card(t)) * product(t, f!1)"))
(("1" (replace -1 * rl t)
(("1" (beta 1)
(("1"
(ground)
(("1"
(lemma
"sum_eq_funs"
("S"
"powerset(rest(A!1))"
"f"
"LAMBDA (t: finite_set[D]):
-1 * (f!1(choose(A!1)) * (-1) ^ (card(t)) * product(t, f!1))"
"g"
"LAMBDA (t_1: set[D]):
-1 *
(f!1(choose(A!1)) * (-1) ^ (card(t_1)) * product(t_1, f!1))"))
(("1" (ground) nil nil))
nil))
nil))
nil))
nil)
("2" (hide - 2)
(("2" (skolem-typepred)
(("2"
(lemma "finite_universe")
(("2"
(lemma
"finite_subset[D]"
("A" "fullset[D]" "s" "x!1"))
(("1"
(expand "subset?")
(("1"
(prop)
(("1" (grind) nil nil))
nil))
nil)
("2" (grind) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (delete -1 2)
(("2" (apply-extensionality 1 :hide? t)
(("2" (grind) nil nil)) nil))
nil))
nil))
nil))
nil)
("2" (hide - 2)
(("2" (lemma "finite_universe")
(("2" (skolem-typepred)
(("2"
(lemma "finite_subset[D]"
("A" "fullset[D]" "s" "t!1"))
(("1" (prop)
(("1" (expand "subset?")
(("1" (grind) nil nil)) nil))
nil)
("2" (hide 2) (("2" (grind) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (hide - 2)
(("2" (skolem-typepred)
(("2" (lemma "finite_universe")
(("2"
(lemma "finite_subset[D]" ("A" "fullset[D]" "s" "x!1"))
(("1" (expand "subset?")
(("1" (prop) (("1" (grind) nil nil)) nil)) nil)
("2" (grind) nil nil))
nil))
nil))
nil))
nil)
("3" (ground) nil nil))
nil))
nil))
nil)
((sum_mult formula-decl nil finite_sets_sum_real finite_sets)
(real_plus_real_is_real application-judgement "real" reals nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(sum def-decl "R" finite_sets_sum finite_sets)
(odd? const-decl "bool" integers nil)
(both_sides_times1 formula-decl nil real_props nil)
(empty? const-decl "bool" sets nil)
(expt def-decl "real" exponentiation nil)
(nzint_times_nzint_is_nzint application-judgement "nzint" integers
nil)
(nzreal_expt application-judgement "nzreal" exponentiation nil)
(int_expt application-judgement "int" exponentiation nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(sum_eq_funs formula-decl nil finite_sets_sum_real finite_sets)
(finite_subset formula-decl nil finite_sets nil)
(fullset const-decl "set" sets nil)
(member const-decl "bool" sets nil)
(is_finite_type const-decl "bool" finite_sets nil)
(injective? const-decl "bool" functions nil)
(subset? const-decl "bool" sets nil)
(below type-eq-decl nil nat_types nil)
(< const-decl "bool" reals nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(finite_universe formula-decl nil real_aux nil)
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil)
(minus_real_is_real application-judgement "real" reals nil)
(nzreal_exp application-judgement "nzreal" exponentiation nil)
(int_exp application-judgement "int" exponentiation nil)
(finite_rest application-judgement "finite_set[D]" real_aux nil)
(product def-decl "R" finite_sets_product finite_sets)
(restrict const-decl "R" restrict nil)
(choose const-decl "(p)" sets 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)
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nil)
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finite_sets_of_sets nil)
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real_aux nil)
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(boolean nonempty-type-decl nil booleans nil))
shostak))
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nil
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"function_props[finite_set[D], finite_set[D], numfield].o((LAMBDA B: number_fields.*(exponentiation.^(((number_fields.-)(1)), (finite_sets[real_aux.D].card(real_aux.B))), finite_sets_product[real_aux.D, real, 1, restrict[[numfield, numfield], [real, real], numfield].restrict(number_fields.*)].product(real_aux.B, real_aux.f))), (LAMBDA B: sets[real_aux.D].add(sets[real_aux.D].choose(real_aux.A), real_aux.B)))"
"[set[D] -> real]")))
(e15_16_4 0
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nil))
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nil))
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integers nil)
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nil)
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(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
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nil)
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(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil))
shostak))
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LAMBDA B: (-1) ^ (card(B)) * product(B, f!1))" 1)
(("1"
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"extend[set[D], (nonempty?[D]), bool, FALSE]
(image(LAMBDA B: add(choose(A!1), B), powerset(rest(A!1))))"
"B"
"extend[setof[D], (nonempty?[D]), bool, FALSE]
(image(LAMBDA (G: set[D]): add(choose(A!1), G),
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"f"
"LAMBDA B: (-1) ^ (card(B)) * product(B, f!1)"))
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1
:hide?
t)
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(FALSE const-decl "bool" booleans nil)
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(setof type-eq-decl nil defined_types nil)
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function_image_aux nil)
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(boolean nonempty-type-decl nil booleans nil))
shostak)
(e15_16_6-1 nil 3887064400
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(("1"
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extend[setof[D], (nonempty?[D]), bool, FALSE]
(image (LAMBDA (G: sets[D].set):
add(choose(A!1), G), sets[D].powerset(sets[D].rest(A!1))))")
(("1" (replace -1 1 lr t)
(("1"
(case "restrict[setof[D], (nonempty?[D]), bool]
(extend[setof[D], (nonempty?[D]), bool, FALSE]
(image(LAMBDA (G: sets[D].set):
add(choose(A!1), G),
sets[D].powerset(sets[D].rest(A!1))))) = image(LAMBDA (G: set[D]): add(choose(A!1), G), powerset(rest(A!1)))")
(("1" (replace -1 1 lr t)
(("1" (expand "extend")
(("1"
(expand "image")
(("1"
(case
"{y: (nonempty?[D]) |
EXISTS (x: (powerset(rest(A!1)))):
y = add(choose(A!1), x)} = LAMBDA (t: setof[D]):
IF (nonempty?[D])(t)
THEN EXISTS (x: (powerset(rest(A!1)))):
t = add(choose(A!1), x)
ELSE FALSE
ENDIF")
(("1"
(replace -1 1 lr t)
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nil))
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