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real_facts.prf
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(real_facts
(bounded?_lem 0
(bounded?_lem-1 nil 3280072578
("" (skosimp*)
(("" (prop)
(("1" (expand "bounded?")
(("1" (flatten)
(("1" (expand "bounded_above?")
(("1" (expand "bounded_below?")
(("1" (skosimp*)
(("1" (expand "lower_bound?")
(("1" (expand "upper_bound?")
(("1" (inst + "max(abs(x!1),abs(x!2))")
(("1" (skosimp*)
(("1" (inst - "x!3")
(("1" (inst - "x!3")
(("1" (expand "max")
(("1" (lift-if)
(("1" (ground)
(("1" (grind) nil nil)
("2" (grind) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (skosimp*)
(("2" (expand "bounded?")
(("2" (prop)
(("1" (expand "bounded_above?")
(("1" (expand "upper_bound?")
(("1" (inst + "B!1")
(("1" (skosimp*) (("1" (inst?) (("1" (grind) nil nil)) nil))
nil))
nil))
nil))
nil)
("2" (expand "bounded_below?")
(("2" (expand "lower_bound?")
(("2" (inst + "-B!1")
(("2" (skosimp*)
(("2" (inst - "s!1") (("2" (grind) nil nil)) nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((bounded_below? const-decl "bool" bounded_real_defs nil)
(lower_bound? const-decl "bool" bounded_real_defs 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)
(AND const-decl "[bool, bool -> bool]" booleans nil)
(>= const-decl "bool" reals nil)
(max const-decl "{p: real | p >= m AND p >= n}" real_defs nil)
(nonneg_real nonempty-type-eq-decl nil real_types nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(- const-decl "[numfield -> numfield]" number_fields nil)
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs nil)
(nonempty? const-decl "bool" sets nil) (set type-eq-decl nil sets nil)
(real_lt_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)
(minus_real_is_real application-judgement "real" reals nil)
(upper_bound? const-decl "bool" bounded_real_defs nil)
(bounded_above? const-decl "bool" bounded_real_defs nil)
(bounded? const-decl "bool" bounded_real_defs nil))
shostak))
(archimedean2 0
(archimedean2-2 "" 3790089399
("" (skolem!)
(("" (lemma "axiom_of_archimedes" ("x" "1/x!1"))
(("" (skolem!)
(("" (assert)
(("" (inst 1 "i!1")
(("" (rewrite "div_mult_pos_lt1")
(("" (rewrite "div_mult_pos_lt1") (("" (assert) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil)
nil shostak)
(archimedean2-1 nil 3258973007
("" (skolem!)
(("" (lemma "axiom_of_archimedes" ("x" "1/x!1"))
(("" (skolem!)
(("" (assert)
(("" (inst 1 "i!1")
(("" (rewrite "div_mult_pos_lt1")
(("" (rewrite "div_mult_pos_lt1")
(("" (assert) 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)
(bool nonempty-type-eq-decl nil booleans nil)
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
(nznum nonempty-type-eq-decl nil number_fields nil)
(/= const-decl "boolean" notequal 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)
(boolean nonempty-type-decl nil booleans nil)
(number nonempty-type-decl nil numbers nil)
(axiom_of_archimedes formula-decl nil real_props nil)
(posreal_div_posreal_is_posreal application-judgement "posreal" real_types
nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(div_mult_pos_lt1 formula-decl nil real_props nil)
(real_times_real_is_real application-judgement "real" reals nil)
(AND const-decl "[bool, bool -> bool]" booleans 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)
(nonneg_int nonempty-type-eq-decl nil integers nil)
(posnat nonempty-type-eq-decl nil integers 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))
nil))
(archimedean3 0
(archimedean3-2 "" 3790089399
("" (skosimp)
(("" (case "x!1 > 0")
(("1" (lemma "archimedean2" ("x" "x!1"))
(("1" (skolem!) (("1" (inst?) (("1" (assert) nil nil)) nil)) nil)
("2" (propax) nil nil))
nil)
("2" (assert) nil nil))
nil))
nil)
nil shostak)
(archimedean3-1 nil 3258973007
("" (skosimp)
(("" (case "x!1 > 0")
(("1" (lemma "archimedean2" ("x" "x!1"))
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("2" (propax) nil)))
("2" (assert) nil))))
nil)
((nonneg_real nonempty-type-eq-decl nil real_types nil)
(>= const-decl "bool" reals 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)
(real_le_is_total_order name-judgement "(total_order?[real])" real_props
nil)
(posrat_div_posrat_is_posrat application-judgement "posrat" rationals nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(posnat nonempty-type-eq-decl nil integers nil)
(nonneg_int nonempty-type-eq-decl nil 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)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(archimedean2 formula-decl nil real_facts nil)
(posreal nonempty-type-eq-decl nil real_types nil))
nil))
(nat_interval 0
(nat_interval-2 "" 3790089399
("" (skolem!)
(("" (lemma "wf_nat")
(("" (expand "well_founded?")
(("" (inst -1 "lambda (b : nat) : x!1 < b")
(("" (split)
(("1" (skolem!)
(("1" (typepred "y!1")
(("1" (inst -2 "y!1 - 1")
(("1" (assert) nil nil)
("2" (inst 2 "y!1 - 1") (("2" (assert) nil nil)) nil))
nil))
nil))
nil)
("2" (lemma "axiom_of_archimedes" ("x" "x!1"))
(("2" (skolem!) (("2" (inst?) (("2" (assert) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
nil shostak)
(nat_interval-1 nil 3258973007
("" (skolem!)
(("" (lemma "wf_nat")
(("" (expand "well_founded?")
(("" (inst -1 "lambda (b : nat) : x!1 < b")
(("" (beta)
(("" (split)
(("1" (skolem!)
(("1" (typepred "y!1")
(("1" (inst -2 "y!1 - 1")
(("1" (assert) nil)
("2" (inst 2 "y!1 - 1") (("2" (assert) nil)))))))))
("2" (lemma "axiom_of_archimedes" ("x" "x!1"))
(("2" (skolem!)
(("2" (inst?) (("2" (assert) nil))))))))))))))))))
nil)
((wf_nat formula-decl nil naturalnumbers 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)
(pred type-eq-decl nil defined_types nil) (< const-decl "bool" reals nil)
(nonneg_real nonempty-type-eq-decl nil real_types nil)
(axiom_of_archimedes formula-decl nil real_props nil)
(i!1 skolem-const-decl "int" real_facts nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(real_ge_is_total_order name-judgement "(total_order?[real])" real_props
nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(x!1 skolem-const-decl "nonneg_real" real_facts nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(y!1 skolem-const-decl "(LAMBDA (b: nat): x!1 < b)" real_facts nil)
(real_le_is_total_order name-judgement "(total_order?[real])" real_props
nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(well_founded? const-decl "bool" orders nil))
nil))
(int_interval 0
(int_interval-2 "" 3790089400
("" (skolem!)
(("" (case "x!1 >= 0")
(("1" (lemma "nat_interval" ("x" "x!1"))
(("1" (skosimp) (("1" (inst 1 "a!1") (("1" (assert) nil nil)) nil))
nil)
("2" (propax) nil nil))
nil)
("2" (lemma "nat_interval" ("x" "- x!1"))
(("1" (skosimp)
(("1" (case "a!1 = -x!1")
(("1" (inst 2 "- a!1") (("1" (assert) nil nil)) nil)
("2" (inst 3 "-a!1 - 1") (("2" (assert) nil nil)) nil))
nil))
nil)
("2" (assert) nil nil))
nil))
nil))
nil)
nil shostak)
(int_interval-1 nil 3258973007
("" (skolem!)
(("" (case "x!1 >= 0")
(("1" (lemma "nat_interval" ("x" "x!1"))
(("1" (skosimp) (("1" (inst 1 "a!1") (("1" (assert) nil)))))
("2" (propax) nil)))
("2" (lemma "nat_interval" ("x" "- x!1"))
(("1" (skosimp)
(("1" (case "a!1 = -x!1")
(("1" (inst 2 "- a!1") (("1" (assert) nil)))
("2" (inst 3 "-a!1 - 1") (("2" (assert) nil)))))))
("2" (assert) nil))))))
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)
(nnint_plus_posint_is_posint application-judgement "posint" integers nil)
(real_lt_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)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(int nonempty-type-eq-decl nil integers nil)
(integer nonempty-type-from-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_ge_is_total_order name-judgement "(total_order?[real])" real_props
nil)
(nat_interval formula-decl nil real_facts nil)
(nonneg_real nonempty-type-eq-decl nil real_types nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(minus_int_is_int application-judgement "int" integers nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(minus_real_is_real application-judgement "real" reals nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(- const-decl "[numfield -> numfield]" number_fields nil))
nil))
(lub_is_bound 0
(lub_is_bound-1 nil 3258973007
("" (skolem!) (("" (typepred "lub(U!1)") (("" (grind) nil nil)) nil)) nil)
((lub const-decl "{x | least_upper_bound?(x, SA)}" bounded_real_defs nil)
(bounded_above? const-decl "bool" bounded_real_defs nil)
(least_upper_bound? const-decl "bool" bounded_real_defs nil)
(nonempty? const-decl "bool" sets nil) (set type-eq-decl nil 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)
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