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intervals_real.prf
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intervals_real.prf
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(interval_reals
(open_ball?_lem 0
(open_ball?_lem-1 nil 3258223423 3292322828
("" (skosimp*)
(("" (prop)
(("1" (expand "open_ball?")
(("1" (skosimp*)
(("1" (expand "open_ball")
(("1" (expand "subset?")
(("1" (expand "member")
(("1" (inst + "delta!1") nil nil)) nil))
nil))
nil))
nil))
nil)
("2" (expand "open_ball?")
(("2" (skosimp*)
(("2" (expand "subset?")
(("2" (expand "member")
(("2" (expand "open_ball") (("2" (inst + "r!1") nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
unchecked
((subset? 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)
(bool nonempty-type-eq-decl nil booleans nil)
(>= const-decl "bool" reals nil)
(nonneg_real nonempty-type-eq-decl nil real_types nil)
(> const-decl "bool" reals nil)
(posreal nonempty-type-eq-decl nil real_types nil)
(member const-decl "bool" sets nil)
(open_ball const-decl "set[T]" open_sets nil)
(open_ball? const-decl "bool" open_sets nil))
30503 110 nil nil))
(open?_lem 0
(open?_lem-1 nil 3258223381 3292322828
("" (skosimp*) (("" (expand "open?") (("" (inst?) nil nil)) nil))
nil)
unchecked
((open? const-decl "bool" open_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)
(T_pred const-decl "[real -> boolean]" open_sets nil)
(T formal-subtype-decl nil open_sets nil)
(bool nonempty-type-eq-decl nil booleans nil)
(set type-eq-decl nil sets nil) nil nil)
20 20 t shostak))
(Open_empty 0
(Open_empty-1 nil 3320064681 3320064682 ("" (skosimp*) nil nil)
proved nil 1331 130 t shostak))
(open_intv_open 0
(open_intv_open-1 nil 3258299476 3292322828
("" (skosimp*)
(("" (expand "open?")
(("" (skosimp*)
(("" (expand "open_ball?")
(("" (inst + "min(x!1 -a!1,b!1-x!1)")
(("1" (skosimp*) (("1" (grind) nil nil)) nil)
("2" (grind) nil nil))
nil))
nil))
nil))
nil))
nil)
unchecked
((open? const-decl "bool" open_sets nil)
(open_ball? const-decl "bool" open_sets nil)
(abs const-decl "{n: nonneg_real | n >= m}" real_defs nil)
(posreal nonempty-type-eq-decl nil real_types nil)
(nonneg_real nonempty-type-eq-decl nil real_types nil)
(boolean nonempty-type-decl nil booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(AND const-decl "[bool, bool -> bool]" booleans 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) (<= const-decl "bool" reals nil)
(min const-decl "{p: real | p <= m AND p <= n}" real_defs nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(T_pred const-decl "[real -> boolean]" open_sets nil)
(T formal-subtype-decl nil open_sets nil)
(< const-decl "bool" reals nil) nil nil nil
(> const-decl "bool" reals nil))
224 220 t shostak))
(above_intv_open 0
(above_intv_open-1 nil 3258299818 3292322829
("" (skosimp*)
(("" (expand "open?")
(("" (skosimp*)
(("" (expand "open_ball?")
(("" (inst + "x!1-a!1")
(("" (skosimp*) (("" (grind) nil nil)) nil)) nil))
nil))
nil))
nil))
nil)
unchecked
((open? const-decl "bool" open_sets nil)
(open_ball? const-decl "bool" open_sets nil)
(abs const-decl "{n: nonneg_real | n >= m}" real_defs nil)
(T formal-subtype-decl nil open_sets nil)
(T_pred const-decl "[real -> boolean]" open_sets nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(posreal nonempty-type-eq-decl nil real_types nil)
(> const-decl "bool" reals nil)
(nonneg_real nonempty-type-eq-decl nil real_types nil)
(>= const-decl "bool" reals nil)
(bool nonempty-type-eq-decl nil booleans 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))
99 100 t shostak))
(below_intv_open 0
(below_intv_open-1 nil 3258299894 3292322829
("" (skosimp*)
(("" (expand "open?")
(("" (skosimp*)
(("" (expand "open_ball?")
(("" (inst + "a!1-x!1") (("" (grind) nil nil)) nil)) nil))
nil))
nil))
nil)
unchecked
((open? const-decl "bool" open_sets nil)
(open_ball? const-decl "bool" open_sets nil)
(abs const-decl "{n: nonneg_real | n >= m}" real_defs nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(< const-decl "bool" reals nil)
(T formal-subtype-decl nil open_sets nil)
(T_pred const-decl "[real -> boolean]" open_sets nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(posreal nonempty-type-eq-decl nil real_types nil)
(> const-decl "bool" reals nil)
(nonneg_real nonempty-type-eq-decl nil real_types nil)
(>= const-decl "bool" reals nil)
(bool nonempty-type-eq-decl nil booleans 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))
103 110 nil nil))
(union_open 0
(union_open-1 nil 3258299981 3292322829
("" (skosimp*)
(("" (expand "open?")
(("" (skosimp*)
(("" (expand "open_ball?")
(("" (typepred "x!1")
(("" (expand "union")
(("" (expand "member")
(("" (split -2)
(("1" (hide -4)
(("1" (inst?)
(("1" (skosimp*)
(("1" (inst + "delta!1")
(("1" (skosimp*)
(("1" (inst - "y!1")
(("1" (assert) nil nil)) nil))
nil))
nil))
nil))
nil))
nil)
("2" (hide -3)
(("2" (inst?)
(("2" (skosimp*)
(("2" (inst + "delta!1")
(("2" (skosimp*)
(("2" (inst - "y!1")
(("2" (assert) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
unchecked
((open? const-decl "bool" open_sets nil)
(open_ball? const-decl "bool" open_sets nil) nil nil nil
(posreal nonempty-type-eq-decl nil real_types nil)
(> const-decl "bool" reals nil)
(nonneg_real nonempty-type-eq-decl nil real_types nil)
(>= const-decl "bool" reals nil)
(member const-decl "bool" sets 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)
(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)
(T_pred const-decl "[real -> boolean]" open_sets nil)
(T formal-subtype-decl nil open_sets nil)
(set type-eq-decl nil sets nil) (union const-decl "set" sets nil))
165 170 t shostak))
(closed_intv_closed 0
(closed_intv_closed-1 nil 3258299631 3292322829
("" (skosimp*)
(("" (expand "closed?")
((""
(case-replace "complement(LAMBDA (x): a!1 <= x AND x <= b!1) =
union(above_intv(b!1),below_intv(a!1))")
(("1" (hide -1)
(("1" (lemma "union_open")
(("1" (inst?)
(("1" (assert)
(("1" (hide 2)
(("1" (lemma "above_intv_open ")
(("1" (inst?)
(("1" (assert)
(("1" (lemma "below_intv_open ")
(("1" (inst?) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (hide 2)
(("2" (apply-extensionality 1 :hide? t)
(("2" (grind) nil nil)) nil))
nil))
nil))
nil))
nil)
unchecked
((closed? const-decl "bool" open_sets nil)
(member const-decl "bool" sets nil)
(below_intv_open formula-decl nil open_sets nil)
(above_intv_open formula-decl nil open_sets nil)
(union_open formula-decl nil open_sets nil)
(< const-decl "bool" reals nil) (> const-decl "bool" reals nil)
(union const-decl "set" sets nil) (<= const-decl "bool" reals nil)
(AND const-decl "[bool, bool -> bool]" booleans nil)
(complement const-decl "set" sets nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(set type-eq-decl nil sets nil)
(bool nonempty-type-eq-decl nil booleans nil)
(T formal-subtype-decl nil open_sets nil)
(T_pred const-decl "[real -> boolean]" open_sets 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))
203 160 t shostak)))