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(continuous_functions_props
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("" (skolem!)
(("" (lemma "connected_domain" ("x" "c!1" "y" "d!1" "z" "u!1"))
(("" (assert) nil))))
nil)
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(AND const-decl "[bool, bool -> bool]" booleans nil)
(<= const-decl "bool" reals nil)
(bool nonempty-type-eq-decl nil booleans nil)
(T formal-subtype-decl nil continuous_functions_props nil)
(T_pred const-decl "[real -> boolean]" continuous_functions_props
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)
(connected_domain formula-decl nil continuous_functions_props nil)
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil))
nil))
(R_TCC1 0
(R_TCC1-1 nil 3473169490
("" (skolem-typepred)
(("" (use "connected_domain" ("z" "u!1")) (("" (assert) nil))))
nil)
((connected_domain formula-decl nil continuous_functions_props nil)
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(J nonempty-type-eq-decl nil continuity_interval nil)
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(<= const-decl "bool" reals nil)
(T formal-subtype-decl nil continuous_functions_props nil)
(T_pred const-decl "[real -> boolean]" continuous_functions_props
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)
(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil))
nil))
(continuous_in_subintervals 0
(continuous_in_subintervals-1 nil 3473169490
("" (skosimp :preds? t)
(("" (auto-rewrite "sub_interval")
(("" (expand "continuous?")
(("" (skolem!)
(("" (inst -3 "x0!1")
(("" (expand "continuous?")
(("" (skosimp*)
(("" (inst -3 "epsilon!1")
(("" (skolem!)
(("" (inst 1 "delta!1")
(("" (skosimp)
(("" (inst -3 "x!1")
(("" (expand "R") (("" (assert) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((real_minus_real_is_real application-judgement "real" reals nil)
(continuous? const-decl "bool" continuous_functions 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)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props 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)
(R const-decl "[J[c, d] -> real]" continuous_functions_props nil)
(J nonempty-type-eq-decl nil continuity_interval nil)
(<= const-decl "bool" reals nil)
(AND const-decl "[bool, bool -> bool]" booleans nil)
(sub_interval formula-decl nil continuous_functions_props 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]" continuous_functions_props
nil)
(T formal-subtype-decl nil continuous_functions_props nil)
(continuous? const-decl "bool" continuous_functions nil))
nil))
(intermediate1 0
(intermediate1-1 nil 3473169490
("" (skosimp)
(("" (assert)
(("" (auto-rewrite "sub_interval")
(("" (lemma "intermediate_value2[a!1, b!1]")
(("" (inst -1 "R(g!1, a!1, b!1)" "x!1")
(("" (rewrite "continuous_in_subintervals")
(("" (expand "R")
(("" (skolem!)
(("" (inst 1 "c!1")
(("" (assert) nil))))))))))))))))))
nil)
((real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(T formal-subtype-decl nil continuous_functions_props nil)
(T_pred const-decl "[real -> boolean]" continuous_functions_props
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)
(intermediate_value2 formula-decl nil continuity_interval nil)
(<= const-decl "bool" reals nil)
(bool nonempty-type-eq-decl nil booleans nil)
(continuous_in_subintervals formula-decl nil
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(continuous? const-decl "bool" continuous_functions nil)
(R const-decl "[J[c, d] -> real]" continuous_functions_props nil)
(J nonempty-type-eq-decl nil continuity_interval nil)
(AND const-decl "[bool, bool -> bool]" booleans nil))
nil))
(intermediate2 0
(intermediate2-1 nil 3473169490
("" (skosimp)
(("" (assert)
(("" (auto-rewrite "sub_interval")
(("" (lemma "intermediate_value4[a!1, b!1]")
(("" (inst -1 "R(g!1, a!1, b!1)" "x!1")
(("" (rewrite "continuous_in_subintervals")
(("" (expand "R")
(("" (skolem!)
(("" (inst 1 "c!1")
(("" (assert) nil))))))))))))))))))
nil)
((real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(T formal-subtype-decl nil continuous_functions_props nil)
(T_pred const-decl "[real -> boolean]" continuous_functions_props
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)
(intermediate_value4 formula-decl nil continuity_interval nil)
(<= const-decl "bool" reals nil)
(bool nonempty-type-eq-decl nil booleans nil)
(continuous_in_subintervals formula-decl nil
continuous_functions_props nil)
(continuous? const-decl "bool" continuous_functions nil)
(R const-decl "[J[c, d] -> real]" continuous_functions_props nil)
(J nonempty-type-eq-decl nil continuity_interval nil)
(AND const-decl "[bool, bool -> bool]" booleans nil))
nil))
(max_in_interval 0
(max_in_interval-1 nil 3473169490
("" (skosimp)
(("" (assert)
(("" (auto-rewrite "sub_interval")
(("" (lemma "maximum_exists[a!1, b!1]")
(("" (inst -1 "R(g!1, a!1, b!1)")
(("" (split)
(("1" (grind :if-match nil)
(("1" (inst 1 "c!1")
(("1" (assert)
(("1" (skosimp)
(("1" (inst?) (("1" (assert) nil)))))))))))
("2" (rewrite "continuous_in_subintervals")
nil))))))))))))
nil)
((real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(T formal-subtype-decl nil continuous_functions_props nil)
(T_pred const-decl "[real -> boolean]" continuous_functions_props
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)
(maximum_exists formula-decl nil continuity_interval nil)
(<= const-decl "bool" reals nil)
(bool nonempty-type-eq-decl nil booleans nil)
(sub_interval formula-decl nil continuous_functions_props nil)
(b!1 skolem-const-decl "T" continuous_functions_props nil)
(d!1 skolem-const-decl "T" continuous_functions_props nil)
(a!1 skolem-const-decl "T" continuous_functions_props nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(is_maximum? const-decl "bool" real_fun_preds "reals/")
(continuous_in_subintervals formula-decl nil
continuous_functions_props nil)
(continuous? const-decl "bool" continuous_functions nil)
(R const-decl "[J[c, d] -> real]" continuous_functions_props nil)
(J nonempty-type-eq-decl nil continuity_interval nil)
(AND const-decl "[bool, bool -> bool]" booleans nil))
nil))
(min_in_interval 0
(min_in_interval-1 nil 3473169490
("" (skosimp)
(("" (assert)
(("" (auto-rewrite "sub_interval")
(("" (lemma "minimum_exists[a!1, b!1]")
(("" (inst -1 "R(g!1, a!1, b!1)")
(("" (split)
(("1" (grind :if-match nil)
(("1" (inst 1 "c!1")
(("1" (assert)
(("1" (skosimp)
(("1" (inst?) (("1" (assert) nil)))))))))))
("2" (rewrite "continuous_in_subintervals")
nil))))))))))))
nil)
((real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(T formal-subtype-decl nil continuous_functions_props nil)
(T_pred const-decl "[real -> boolean]" continuous_functions_props
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)
(minimum_exists formula-decl nil continuity_interval nil)
(<= const-decl "bool" reals nil)
(bool nonempty-type-eq-decl nil booleans nil)
(sub_interval formula-decl nil continuous_functions_props nil)
(b!1 skolem-const-decl "T" continuous_functions_props nil)
(d!1 skolem-const-decl "T" continuous_functions_props nil)
(a!1 skolem-const-decl "T" continuous_functions_props nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(is_minimum? const-decl "bool" real_fun_preds "reals/")
(continuous_in_subintervals formula-decl nil
continuous_functions_props nil)
(continuous? const-decl "bool" continuous_functions nil)
(R const-decl "[J[c, d] -> real]" continuous_functions_props nil)
(J nonempty-type-eq-decl nil continuity_interval nil)
(AND const-decl "[bool, bool -> bool]" booleans nil))
nil))
(inj_continuous 0
(inj_continuous-1 nil 3473169490
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(("" (expand "injective?")
(("" (assert)
(("" (case "g!1(a!1) <= g!1(c!1)")
(("1" (ground)
(("1"
(lemma "intermediate1"
("g" "g!1" "a" "b!1" "b" "c!1" "x" "g!1(a!1)"))
(("1" (assert)
(("1" (skosimp)
(("1" (inst -5 "a!1" "c!2")
(("1" (assert) nil)))))))))
("2"
(lemma "intermediate1"
("g" "g!1" "a" "a!1" "b" "b!1" "x" "g!1(c!1)"))
(("2" (assert)
(("2" (skosimp)
(("2" (inst -5 "c!1" "c!2")
(("2" (assert) nil)))))))))))
("2" (ground)
(("1"
(lemma "intermediate2"
("g" "g!1" "a" "a!1" "b" "b!1" "x" "g!1(c!1)"))
(("1" (assert)
(("1" (skosimp)
(("1" (inst -4 "c!1" "c!2")
(("1" (assert) nil)))))))))
("2"
(lemma "intermediate2"
("g" "g!1" "a" "b!1" "b" "c!1" "x" "g!1(a!1)"))
(("2" (assert)
(("2" (skosimp)
(("2" (inst -4 "c!2" "a!1")
(("2" (assert) nil))))))))))))))))))
nil)
((injective? const-decl "bool" functions nil)
(continuous? const-decl "bool" continuous_functions nil)
(T formal-subtype-decl nil continuous_functions_props nil)
(T_pred const-decl "[real -> boolean]" continuous_functions_props
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)
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(intermediate2 formula-decl nil continuous_functions_props nil)
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil))
nil))
(inj_monotone 0
(inj_monotone-1 nil 3473169490
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(("" (prop)
(("1" (expand "strict_increasing?")
(("1" (expand "strict_decreasing?")
(("1" (skosimp*)
(("1" (assert)
(("1" (expand "injective?")
(("1" (inst-cp -1 "x!1" "y!1")
(("1" (inst -1 "x!2" "y!2")
(("1" (assert)
(("1" (case "x!1 <= x!2")
(("1"
(lemma "inj_continuous"
("g" "g!1" "a" "x!1" "b" "x!2" "c" "y!2"))
(("1" (assert)
(("1"
(lemma "inj_continuous"
("g"
"g!1"
"a"
"x!1"
"b"
"y!1"
"c"
"y!2"))
(("1"
(lemma
"inj_continuous"
("g"
"g!1"
"a"
"x!1"
"b"
"y!2"
"c"
"y!1"))
(("1" (assert) nil)))))))))
("2"
(lemma "inj_continuous"
("g" "g!1" "a" "x!2" "b" "x!1" "c" "y!1"))
(("2" (assert)
(("2"
(lemma "inj_continuous"
("g"
"g!1"
"a"
"x!2"
"b"
"y!2"
"c"
"y!1"))
(("2"
(lemma
"inj_continuous"
("g"
"g!1"
"a"
"x!2"
"b"
"y!1"
"c"
"y!2"))
(("2"
(assert)
nil)))))))))))))))))))))))))))
("2" (grind :if-match nil)
(("2" (inst-cp -3 "x1!1" "x2!1")
(("2" (inst -3 "x2!1" "x1!1") (("2" (assert) nil)))))))
("3" (grind :if-match nil)
(("3" (inst-cp -3 "x1!1" "x2!1")
(("3" (inst -3 "x2!1" "x1!1") (("3" (assert) nil))))))))))
nil)
((strict_decreasing? const-decl "bool" real_fun_preds "reals/")
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props 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]" continuous_functions_props
nil)
(T formal-subtype-decl nil continuous_functions_props nil)
(inj_continuous formula-decl nil continuous_functions_props nil)
(continuous? const-decl "bool" continuous_functions nil)
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(<= const-decl "bool" reals nil)
(bool nonempty-type-eq-decl nil booleans nil)
(injective? const-decl "bool" functions nil)
(strict_increasing? const-decl "bool" real_fun_preds "reals/")
(NOT const-decl "[bool -> bool]" booleans nil))
nil)))
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