| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/PVS/lnexp_fnd/ (Beweissystem der NASA Version 6.0.9©) Datei vom 28.9.2014 mit Größe 5 MB |
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Quelle hyperbolic.prfSprache: Lisp |
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(hyperbolic (noa_abs_lt1 0 (noa_abs_lt1-1 nil 3477842243 ("" (expand "not_one_element?") (("" (skosimp*) (("" (inst-cp + "x!1/2") (("" (inst + "1/2") (("" (assert) nil nil)) nil)) nil)) nil)) nil) ((posrat_div_posrat_is_posrat application-judgement "posrat" rationals nil) (/ const-decl "[numfield, nznum -> numfield]" number_fields nil) (nznum nonempty-type-eq-decl nil number_fields nil) (/= const-decl "boolean" notequal nil) (real_abs_lt1 nonempty-type-eq-decl nil hyperbolic nil) (- const-decl "[numfield -> numfield]" number_fields nil) (numfield nonempty-type-eq-decl nil number_fields nil) (< const-decl "bool" reals nil) (AND const-decl "[bool, bool -> bool]" booleans 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_div_nzreal_is_real application-judgement "real" reals nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (not_one_element? const-decl "bool" deriv_domain_def "analysis/")) shostak)) (noa_posreal_gt1 0 (noa_posreal_gt1-1 nil 3477842295 ("" (expand "not_one_element?") (("" (skosimp*) (("" (inst + "x!1+1") (("" (assert) nil nil)) nil)) nil)) nil) ((+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (numfield nonempty-type-eq-decl nil number_fields nil) (posreal_gt1 nonempty-type-eq-decl nil hyperbolic 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_plus_real_is_real application-judgement "real" reals nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (not_one_element? const-decl "bool" deriv_domain_def "analysis/")) shostak)) (conn_abs_lt1 0 (conn_abs_lt1-1 nil 3477842234 ("" (expand "connected?") (("" (skosimp*) (("" (assert) nil nil)) nil)) nil) ((real_le_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) (minus_odd_is_odd application-judgement "odd_int" integers nil) (connected? const-decl "bool" deriv_domain_def "analysis/")) shostak)) (conn_real 0 (conn_real-1 nil 3477842343 ("" (expand "connected?") (("" (propax) nil nil)) nil) ((connected? const-decl "bool" deriv_domain_def "analysis/")) shostak)) (deriv_domain_abs_lt1 0 (deriv_domain_abs_lt1-1 nil 3477842313 ("" (lemma "deriv_domain_open") (("" (inst - "-1" "1") nil nil)) nil) ((minus_odd_is_odd application-judgement "odd_int" integers 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) (numfield nonempty-type-eq-decl nil number_fields nil) (- const-decl "[numfield -> numfield]" number_fields nil) (deriv_domain_open formula-decl nil deriv_domain "analysis/")) shostak)) (deriv_domain_posreal_gt1 0 (deriv_domain_posreal_gt1-1 nil 3477842351 ("" (expand "deriv_domain?") 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const-decl "bool" real_fun_preds "reals/") (/ const-decl "[numfield, nznum -> numfield]" number_fields nil) (nznum nonempty-type-eq-decl nil number_fields nil) (/= const-decl "boolean" notequal nil) (exp const-decl "{py | x = ln(py)}" ln_exp nil) (ln const-decl "real" ln_exp nil) (= const-decl "[T, T -> boolean]" equalities 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) (exp_diff formula-decl nil ln_exp nil)) shostak)) (tanh_TCC1 0 (tanh_TCC1-1 nil 3253429443 ("" (skosimp*) (("" (typepred "cosh(x!1)") (("" (assert) nil nil)) nil)) nil) ((cosh const-decl "posreal_ge1" hyperbolic nil) (posreal_ge1 nonempty-type-eq-decl nil hyperbolic 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) (bool nonempty-type-eq-decl nil booleans nil) (boolean nonempty-type-decl nil booleans nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil)) shostak)) (tanh_TCC2 0 (tanh_TCC2-1 nil 3253429443 ("" (skosimp*) (("" (lemma "div_mult_pos_lt1") (("" (inst - "cosh(x!1)" "1" "sinh(x!1)") (("1" (lemma "div_mult_pos_lt2") (("1" (inst - "cosh(x!1)" "-1" "sinh(x!1)") (("1" (replace -1) (("1" (replace -2) (("1" (hide (-1 -2)) (("1" (expand "cosh") (("1" (expand "sinh") (("1" (typepred "exp(x!1)") (("1" (typepred "exp(-x!1)") (("1" (name-replace "EPX" "exp(x!1)") (("1" (name-replace "EMX" "exp(-x!1)") (("1" (hide -3 -6) (("1" (grind) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (typepred "cosh(x!1)") (("2" (assert) nil nil)) nil)) nil)) nil) ("2" (typepred "cosh(x!1)") (("2" (assert) nil nil)) nil)) nil)) nil)) nil) ((div_mult_pos_lt1 formula-decl nil real_props nil) (div_mult_pos_lt2 formula-decl nil real_props nil) (real_minus_real_is_real application-judgement "real" reals nil) (minus_real_is_real application-judgement "real" reals nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (posreal_times_posreal_is_posreal application-judgement "posreal" real_types nil) (nzreal_times_nzreal_is_nzreal application-judgement "nzreal" real_types nil) (exp const-decl "{py | x = ln(py)}" ln_exp nil) (ln const-decl "real" ln_exp nil) (= const-decl "[T, T -> boolean]" equalities nil) (NOT const-decl "[bool -> bool]" booleans nil) (posreal_div_posreal_is_posreal application-judgement "posreal" real_types nil) (posreal_plus_nnreal_is_posreal application-judgement "posreal" real_types nil) (numfield nonempty-type-eq-decl nil number_fields nil) (- const-decl "[numfield -> numfield]" number_fields nil) (minus_odd_is_odd application-judgement "odd_int" integers nil) (sinh const-decl "real" hyperbolic 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) (posreal_ge1 nonempty-type-eq-decl nil hyperbolic nil) (cosh const-decl "posreal_ge1" hyperbolic nil) (x!1 skolem-const-decl "real" hyperbolic nil) (> const-decl "bool" reals 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)) shostak)) (csch_TCC1 0 (csch_TCC1-1 nil 3253429443 ("" (skosimp*) (("" (expand "sinh") (("" (lemma "exp_diff" ("x" "0" "y" "n0x!1")) (("" (lemma "exp_0") (("" (replace -1 -2) (("" (replace -2 -3) (("" (rewrite "div_cancel3" -3) (("" (lemma "minus_div1" ("x" "exp(n0x!1)" "y" "1" "n0x" "1" "n0y" "exp(n0x!1)")) (("" (replace -1 -4) (("" (rewrite "div_cancel3" -4) (("" (lemma "exp_strict_increasing") (("" (expand "strict_increasing?") 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application-judgement "even_int" integers nil) (mult_divides1 application-judgement "(divides(n))" divides nil) (mult_divides2 application-judgement "(divides(m))" divides nil) (/ const-decl "[numfield, nznum -> numfield]" number_fields nil) (nznum nonempty-type-eq-decl nil number_fields nil) (exp const-decl "{py | x = ln(py)}" ln_exp nil) (ln const-decl "real" ln_exp nil) (= const-decl "[T, T -> boolean]" equalities 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, numfield -> numfield]" number_fields nil) (numfield nonempty-type-eq-decl nil number_fields nil) (nonzero_real nonempty-type-eq-decl nil reals nil) (div_cancel3 formula-decl nil real_props nil) (minus_odd_is_odd application-judgement "odd_int" integers nil) (exp_diff formula-decl nil ln_exp 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) (/= const-decl "boolean" notequal nil) (nzreal nonempty-type-eq-decl nil reals nil)) shostak)) (sech_TCC1 0 (sech_TCC1-1 nil 3253429443 ("" (skosimp*) (("" (typepred "cosh(x!1)") (("" (lemma "both_sides_div_pos_ge2" ("py" "cosh(x!1)" "px" "1" "pz" "1")) (("1" (lemma "posreal_div_posreal_is_posreal" ("px" "1" "py" "cosh(x!1)")) (("1" (assert) nil nil)) nil) ("2" (assert) nil nil)) nil)) nil)) nil) ((cosh const-decl "posreal_ge1" hyperbolic nil) (posreal_ge1 nonempty-type-eq-decl nil hyperbolic 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) (bool nonempty-type-eq-decl nil booleans nil) (boolean nonempty-type-decl nil booleans nil) (posreal_div_posreal_is_posreal judgement-tcc nil real_types nil) (nzreal_div_nzreal_is_nzreal application-judgement "nzreal" real_types nil) (posrat_div_posrat_is_posrat application-judgement "posrat" rationals nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (real_gt_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) (both_sides_div_pos_ge2 formula-decl nil real_props 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)) shostak)) (coth_TCC1 0 (coth_TCC1-1 nil 3253429443 ("" (skosimp*) (("" (expand "tanh") (("" (expand "sinh") 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(("" (inst - "0" "x!1") (("" (assert) (("" (rewrite "exp_0") (("" (name-replace "EPX" "exp(x!1)") (("" (name-replace "EPY" "exp(y!1)") (("" (hide (-5 -6 -7 1)) (("" (lemma "posreal_times_posreal_is_posreal" ("px" "EPX*EPY-1" "py" "EPY-EPX")) (("1" (assert) nil nil) ("2" (assert) nil nil) ("3" (lemma "lt_times_lt_pos1") (("3" (inst - "1" "1" "EPY" "EPX") (("3" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((cosh const-decl "posreal_ge1" hyperbolic nil) (exp_0 formula-decl nil ln_exp nil) (minus_odd_is_odd application-judgement "odd_int" integers nil) (div_cancel1 formula-decl nil real_props nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (real_minus_real_is_real application-judgement "real" reals nil) (increasing? const-decl "bool" real_fun_preds "reals/") (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (real_times_real_is_real application-judgement "real" reals nil) (posreal_times_posreal_is_posreal judgement-tcc nil real_types nil) (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (lt_times_lt_pos1 formula-decl nil real_props nil) (exp_increasing formula-decl nil ln_exp nil) (exp_strict_increasing formula-decl nil ln_exp nil) (nonzero_real nonempty-type-eq-decl nil reals nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (/ const-decl "[numfield, nznum -> numfield]" number_fields nil) (nznum nonempty-type-eq-decl nil number_fields nil) (/= const-decl "boolean" notequal nil) (* const-decl "[numfield, numfield -> numfield]" number_fields nil) (both_sides_times_pos_lt1 formula-decl nil real_props nil) (posreal_times_posreal_is_posreal application-judgement "posreal" real_types nil) (posreal_div_posreal_is_posreal application-judgement "posreal" real_types nil) (exp_diff formula-decl nil ln_exp nil) (minus_real_is_real application-judgement "real" reals nil) (posreal_plus_nnreal_is_posreal application-judgement "posreal" real_types nil) (both_sides_div_pos_lt1 formula-decl nil real_props 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) (numfield nonempty-type-eq-decl nil number_fields nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (= const-decl "[T, T -> boolean]" equalities nil) (ln const-decl "real" ln_exp nil) (exp const-decl "{py | x = ln(py)}" ln_exp nil) (- const-decl "[numfield -> numfield]" number_fields nil) (nnreal_cosh const-decl "posreal_ge1" hyperbolic 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) (>= const-decl "bool" reals nil) (nnreal type-eq-decl nil real_types nil) (strict_increasing? const-decl "bool" real_fun_preds "reals/")) shostak)) (tanh_strict_increasing 0 (tanh_strict_increasing-1 nil 3253525103 ("" (lemma "exp_strict_increasing") (("" (expand "strict_increasing?") (("" (expand "tanh") (("" (expand "sinh") (("" (expand "cosh") (("" (skosimp*) (("" (rewrite "div_div1" 1) (("" (rewrite "div_div1" 1) (("" (rewrite "div_cancel1" 1) (("" (rewrite "div_cancel1" 1) (("" (case "exp(-x!1)+exp(x!1) > 0") (("1" (case "exp(-y!1)+exp(y!1) > 0") (("1" (lemma "div_mult_pos_lt1") (("1" (inst - "exp(-x!1) + exp(x!1)" "(exp(y!1) - exp(-y!1)) / (exp(-y!1) + exp(y!1))" "exp(x!1) - exp(-x!1)") (("1" (replace -1 1) (("1" (lemma "div_mult_pos_lt2") (("1" (inst - "exp(-y!1) + exp(y!1)" "exp(x!1) - exp(-x!1)" "(exp(y!1) - exp(-y!1))*(exp(-x!1) + exp(x!1))") (("1" (replace -1 1) (("1" (hide -1 -2) (("1" (inst-cp - "x!1" "y!1") (("1" (inst-cp - "-y!1" "-x!1") (("1" (inst - "x!1-y!1" "y!1-x!1") (("1" (lemma "exp_sum") (("1" (inst-cp - "x!1" "-y!1") (("1" (inst - "y!1" "-x!1") (("1" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (assert) nil nil)) nil) ("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((strict_increasing? const-decl "bool" real_fun_preds "reals/") (real_minus_real_is_real application-judgement "real" reals nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (sinh const-decl "real" hyperbolic nil) (div_mult_pos_lt2 formula-decl nil real_props nil) (nzreal_times_nzreal_is_nzreal application-judgement "nzreal" real_types nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (posreal_times_posreal_is_posreal application-judgement "posreal" real_types nil) (real_plus_real_is_real application-judgement "real" reals nil) (exp_sum formula-decl nil ln_exp nil) (* const-decl "[numfield, numfield -> numfield]" number_fields nil) (div_mult_pos_lt1 formula-decl nil real_props nil) (div_cancel1 formula-decl nil real_props nil) (real_times_real_is_real application-judgement "real" reals nil) (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (/ const-decl "[numfield, nznum -> numfield]" number_fields nil) (nznum nonempty-type-eq-decl nil number_fields nil) (- const-decl "[numfield -> numfield]" number_fields nil) (exp const-decl "{py | x = ln(py)}" ln_exp nil) (ln const-decl "real" ln_exp nil) (= const-decl "[T, T -> boolean]" equalities 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, numfield -> numfield]" number_fields nil) (numfield nonempty-type-eq-decl nil number_fields nil) (nonzero_real nonempty-type-eq-decl nil reals nil) (/= const-decl "boolean" notequal 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) (div_div1 formula-decl nil real_props nil) (minus_real_is_real application-judgement "real" reals nil) (cosh const-decl "posreal_ge1" hyperbolic nil) (posreal_div_posreal_is_posreal application-judgement "posreal" real_types nil) (posreal_plus_nnreal_is_posreal application-judgement "posreal" real_types nil) (tanh const-decl "real_abs_lt1" hyperbolic nil) (exp_strict_increasing formula-decl nil ln_exp nil)) shostak)) (csch_strict_decreasing 0 (csch_strict_decreasing-1 nil 3253526140 ("" (lemma "sinh_strict_increasing") (("" (expand "strict_decreasing?") (("" (skosimp*) (("" (typepred "x!1") (("" (expand "strict_increasing?") (("" (expand "posreal_csch") (("" (expand "csch") (("" (inst-cp - "x!1" "y!1") (("" (inst - "0" "x!1") (("" (name-replace "SX" "sinh(x!1)") (("" (name-replace "SY" "sinh(y!1)") (("" (expand "sinh") (("" (rewrite "exp_0") (("" (assert) (("" (lemma "both_sides_div_pos_lt2" ("pz" "1" "py" "SX" "px" "SY")) (("" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((strict_decreasing? const-decl "bool" real_fun_preds "reals/") (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 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) (posreal_csch const-decl "posreal" hyperbolic nil) (sinh const-decl "real" hyperbolic nil) (= const-decl "[T, T -> boolean]" equalities nil) (real_minus_real_is_real application-judgement "real" reals nil) (nzreal_div_nzreal_is_nzreal application-judgement "nzreal" real_types nil) (odd_minus_odd_is_even application-judgement "even_int" integers nil) (rat_div_nzrat_is_rat application-judgement "rat" rationals nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (real_gt_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) (AND const-decl "[bool, bool -> bool]" booleans nil) (both_sides_div_pos_lt2 formula-decl nil real_props nil) (minus_odd_is_odd application-judgement "odd_int" integers nil) (exp_0 formula-decl nil ln_exp nil) (csch const-decl "real" hyperbolic nil) (strict_increasing? const-decl "bool" real_fun_preds "reals/") (sinh_strict_increasing formula-decl nil hyperbolic nil)) shostak)) (sech_strict_decreasing 0 (sech_strict_decreasing-1 nil 3253526566 ("" (expand "strict_decreasing?") (("" (skosimp*) (("" (typepred "x!1") (("" (expand "nnreal_sech") (("" (expand "sech") (("" (lemma "cosh_strict_increasing") (("" (expand "strict_increasing?") (("" (inst - "x!1" "y!1") (("" (expand "nnreal_cosh") (("" (lemma "both_sides_div_pos_lt2" ("pz" "1" "py" "cosh(x!1)" "px" "cosh(y!1)")) (("1" (assert) nil nil) ("2" (assert) nil nil) ("3" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((nnreal_sech const-decl "posreal_le1" hyperbolic nil) (cosh_strict_increasing formula-decl nil hyperbolic nil) (both_sides_div_pos_lt2 formula-decl nil real_props 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) (posreal_ge1 nonempty-type-eq-decl nil hyperbolic nil) (cosh const-decl "posreal_ge1" hyperbolic 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) (AND const-decl "[bool, bool -> bool]" booleans nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (nzreal_div_nzreal_is_nzreal application-judgement "nzreal" real_types nil) (nnreal_cosh const-decl "posreal_ge1" hyperbolic nil) (strict_increasing? const-decl "bool" real_fun_preds "reals/") (sech const-decl "posreal_le1" hyperbolic 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) (>= const-decl "bool" reals nil) (nnreal type-eq-decl nil real_types nil) (strict_decreasing? const-decl "bool" real_fun_preds "reals/")) shostak)) (coth_strict_decreasing 0 (coth_strict_decreasing-1 nil 3253526884 ("" (expand "strict_decreasing?") (("" (skosimp*) (("" (expand "posreal_coth") (("" (expand "coth") (("" (lemma "tanh_strict_increasing") (("" (expand "strict_increasing?") (("" (inst-cp - "0" "x!1") (("" (inst - "x!1" "y!1") (("" (assert) (("" (name-replace "TX" "tanh(x!1)") (("" (name-replace "TY" "tanh(y!1)") (("" (expand "tanh") (("" (expand "cosh") (("" (expand "sinh") (("" (rewrite "exp_0") (("" (lemma "both_sides_div_pos_lt2" ("pz" "1" "py" "TX" "px" "TY")) (("1" (assert) nil nil) ("2" (assert) nil nil) ("3" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((coth const-decl "real_abs_gt1" hyperbolic nil) (strict_increasing? const-decl "bool" real_fun_preds "reals/") (= const-decl "[T, T -> boolean]" equalities nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (< const-decl "bool" reals nil) (numfield nonempty-type-eq-decl nil number_fields nil) (- const-decl "[numfield -> numfield]" number_fields nil) (real_abs_lt1 nonempty-type-eq-decl nil hyperbolic nil) (tanh const-decl "real_abs_lt1" hyperbolic nil) (sinh const-decl "real" hyperbolic nil) (both_sides_div_pos_lt2 formula-decl nil real_props 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) (rat_div_nzrat_is_rat application-judgement "rat" rationals nil) (odd_minus_odd_is_even application-judgement "even_int" integers nil) (minus_odd_is_odd application-judgement "odd_int" integers nil) (posreal_div_posreal_is_posreal application-judgement "posreal" real_types nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (posreal_plus_nnreal_is_posreal application-judgement "posreal" real_types nil) (real_minus_real_is_real application-judgement "real" reals nil) (exp_0 formula-decl nil ln_exp nil) (cosh const-decl "posreal_ge1" hyperbolic nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (nzreal_div_nzreal_is_nzreal application-judgement "nzreal" real_types 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) (tanh_strict_increasing formula-decl nil hyperbolic nil) (posreal_coth const-decl "posreal_gt1" hyperbolic nil) (strict_decreasing? const-decl "bool" real_fun_preds "reals/")) shostak)) (sinh_0 0 (sinh_0-1 nil 3253386258 ("" (expand "sinh") (("" (rewrite "exp_0") (("" (assert) nil nil)) nil)) nil) ((exp_0 formula-decl nil ln_exp nil) (real_minus_real_is_real application-judgement "real" reals nil) (minus_odd_is_odd application-judgement "odd_int" integers nil) (rat_div_nzrat_is_rat application-judgement "rat" rationals nil) (odd_minus_odd_is_even application-judgement "even_int" integers nil) (sinh const-decl "real" hyperbolic nil)) shostak)) (cosh_0 0 (cosh_0-1 nil 3253386284 ("" (expand "cosh") (("" (rewrite "exp_0") nil nil)) nil) ((posreal_plus_nnreal_is_posreal application-judgement "posreal" real_types nil) (exp_0 formula-decl nil ln_exp nil) (cosh const-decl "posreal_ge1" hyperbolic nil)) shostak)) (tanh_0 0 (tanh_0-1 nil 3253452832 ("" (expand "tanh") (("" (rewrite "sinh_0") (("" (assert) nil nil)) nil)) nil) ((sinh_0 formula-decl nil hyperbolic nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (tanh const-decl "real_abs_lt1" hyperbolic nil)) shostak)) (sech_0 0 (sech_0-1 nil 3253452862 ("" (expand "sech") (("" (rewrite "cosh_0") (("" (assert) nil nil)) nil)) nil) ((cosh_0 formula-decl nil hyperbolic nil) (sech const-decl "posreal_le1" hyperbolic nil)) shostak)) (cosh_sinh_one 0 (cosh_sinh_one-1 nil 3253453151 ("" (skosimp*) (("" (expand "cosh") (("" (expand "sinh") (("" (lemma "exp_sum" ("x" "x!1" "y" "-x!1")) (("" (rewrite "exp_0" -1) (("" (name-replace "EPX" "exp(x!1)") (("" (name-replace "EMX" "exp(-x!1)") (("" (grind) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((posreal_plus_nnreal_is_posreal application-judgement "posreal" real_types nil) (cosh const-decl "posreal_ge1" hyperbolic nil) (- const-decl "[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) (boolean nonempty-type-decl nil booleans nil) (number nonempty-type-decl nil numbers nil) (exp_sum formula-decl nil ln_exp nil) (minus_real_is_real application-judgement "real" reals nil) (exp const-decl "{py | x = ln(py)}" ln_exp nil) (ln const-decl "real" ln_exp 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 "[T, T -> boolean]" equalities nil) (minus_odd_is_odd application-judgement "odd_int" integers nil) (sq const-decl "nonneg_real" sq "reals/") (real_times_real_is_real application-judgement "real" reals nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (posreal_div_posreal_is_posreal application-judgement "posreal" real_types nil) (posreal_times_posreal_is_posreal application-judgement "posreal" real_types nil) (sq_nz_pos application-judgement "posreal" sq "reals/") (exp_0 formula-decl nil ln_exp nil) (sinh const-decl "real" hyperbolic nil) (real_minus_real_is_real application-judgement "real" reals nil)) shostak)) (tanh_sech_one 0 (tanh_sech_one-1 nil 3253453367 ("" (skosimp*) (("" (expand "tanh") (("" (expand "sech") (("" (typepred "cosh(x!1)") (("" (lemma "cosh_sinh_one" ("x" "x!1")) (("" (lemma "sq_div" ("d" "cosh(x!1)")) (("" (inst-cp - "1") (("" (inst - "sinh(x!1)") (("" (assert) (("" (replace -1) (("" (replace -2) (("" (lemma "both_sides_div1" ("n0z" "sq(cosh(x!1))" "x" "sq(cosh(x!1)) - sq(sinh(x!1))" "y" "1")) (("" (lemma "minus_div2" ("x" "sq(cosh(x!1))" "y" "sq(sinh(x!1))" "n0x" "sq(cosh(x!1))")) (("" (replace -2 -5 rl) (("" (rewrite "sq_1") (("" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((tanh const-decl "real_abs_lt1" hyperbolic nil) (cosh const-decl "posreal_ge1" hyperbolic nil) (posreal_ge1 nonempty-type-eq-decl nil hyperbolic 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 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(nzreal_div_nzreal_is_nzreal application-judgement "nzreal" real_types nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (cosh_sinh_one formula-decl nil hyperbolic nil) (sech const-decl "posreal_le1" hyperbolic nil)) shostak)) (coth_csch_one 0 (coth_csch_one-1 nil 3253454037 ("" (expand "coth") (("" (expand "csch") (("" (skosimp*) (("" (expand "tanh") (("" (case "sinh(n0x!1) /= 0") (("1" (rewrite "div_div1") (("1" (lemma "sq_div" ("d" "sinh(n0x!1)")) (("1" (replace -2 -1) (("1" (inst-cp - "cosh(n0x!1)") (("1" (inst - "1") (("1" (rewrite "sq_1") (("1" (lemma "cosh_sinh_one" ("x" "n0x!1")) (("1" (lemma "both_sides_div1" ("x" "sq(cosh(n0x!1)) - sq(sinh(n0x!1))" "y" "1" "n0z" "sq(sinh(n0x!1))")) (("1" (lemma "minus_div2" ("n0x" "sq(sinh(n0x!1))" "x" "sq(cosh(n0x!1))" "y" "sq(sinh(n0x!1))")) (("1" (rewrite "div_simp" -1) (("1" (assert) nil nil)) nil)) nil) ("2" (lemma "sq_eq_0" ("a" "sinh(n0x!1)")) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (lemma "sinh_strict_increasing") (("2" (expand "strict_increasing?") (("2" (lemma "sinh_0") (("2" (case "n0x!1>0") (("1" (inst - "0" "n0x!1") (("1" (assert) nil nil)) nil) ("2" (inst - "n0x!1" "0") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((csch const-decl "real" hyperbolic nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (tanh const-decl "real_abs_lt1" hyperbolic nil) (sinh_strict_increasing formula-decl nil hyperbolic nil) (sinh_0 formula-decl nil hyperbolic nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (> const-decl "bool" reals nil) (strict_increasing? const-decl "bool" real_fun_preds "reals/") (div_div1 formula-decl nil real_props nil) (nonzero_real nonempty-type-eq-decl nil reals nil) (bool nonempty-type-eq-decl nil booleans nil) (>= const-decl "bool" reals nil) (posreal_ge1 nonempty-type-eq-decl nil hyperbolic nil) (cosh const-decl 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(real_div_nzreal_is_real application-judgement "real" reals nil) (sinh const-decl "real" hyperbolic nil) (cosh const-decl "posreal_ge1" hyperbolic nil) (posreal_div_posreal_is_posreal application-judgement "posreal" real_types nil) (posreal_plus_nnreal_is_posreal application-judgement "posreal" real_types nil)) shostak)) (sinh_neg 0 (sinh_neg-1 nil 3253452903 ("" (expand "sinh") (("" (propax) nil nil)) nil) ((sinh const-decl "real" hyperbolic nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (real_minus_real_is_real application-judgement "real" reals nil) (minus_real_is_real application-judgement "real" reals nil)) shostak)) (cosh_neg 0 (cosh_neg-1 nil 3253452917 ("" (expand "cosh") (("" (propax) nil nil)) nil) ((cosh const-decl "posreal_ge1" hyperbolic nil) (posreal_plus_nnreal_is_posreal application-judgement "posreal" real_types nil) (minus_real_is_real application-judgement "real" reals nil)) shostak)) (tanh_neg 0 (tanh_neg-1 nil 3253452927 ("" (expand "tanh") 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nonempty-type-decl nil booleans nil) (number nonempty-type-decl nil numbers nil) (cosh_neg formula-decl nil hyperbolic nil)) shostak)) (coth_neg 0 (coth_neg-1 nil 3253453083 ("" (skosimp*) (("" (expand "coth") (("" (rewrite "tanh_neg") (("" (assert) nil nil)) nil)) nil)) nil) ((coth const-decl "real_abs_gt1" hyperbolic nil) (minus_nzreal_is_nzreal application-judgement "nzreal" real_types nil) (minus_real_is_real application-judgement "real" reals nil) (nzreal_div_nzreal_is_nzreal application-judgement "nzreal" real_types nil) (nzreal nonempty-type-eq-decl nil reals nil) (/= const-decl "boolean" notequal 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) (tanh_neg formula-decl nil hyperbolic nil)) shostak)) (sinh_sum 0 (sinh_sum-1 nil 3253474598 ("" (skosimp*) (("" (expand "sinh") (("" (expand "cosh") (("" (lemma "exp_sum" ("x" "x!1" "y" "y!1")) (("" (lemma "exp_diff" ("x" "0")) (("" (rewrite "exp_0") (("" (inst-cp - "x!1+y!1") (("" (inst-cp - "y!1") (("" (inst - "x!1") (("" (replace -1) (("" (replace -2) (("" (replace -3) (("" (lemma "exp_sum" ("x" "x!1" "y" "y!1")) (("" (replace -1 1) (("" (name-replace "EPX" "exp(x!1)") (("" (name-replace "EPY" "exp(y!1)") (("" (hide-all-but 1) (("" (grind) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((real_plus_real_is_real application-judgement "real" reals nil) (real_minus_real_is_real application-judgement "real" reals nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (sinh const-decl "real" hyperbolic 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 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"posreal_ge1" hyperbolic nil) (posreal_div_posreal_is_posreal application-judgement "posreal" real_types nil) (posreal_plus_nnreal_is_posreal application-judgement "posreal" real_types nil)) shostak)) (sinh_diff 0 (sinh_diff-1 nil 3270964764 ("" (skosimp*) (("" (lemma "sinh_sum" ("x" "x!1" "y" "-y!1")) (("" (rewrite "sinh_neg") (("" (rewrite "cosh_neg") (("" (assert) nil nil)) nil)) nil)) nil)) nil) ((- const-decl "[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) (boolean nonempty-type-decl nil booleans nil) (number nonempty-type-decl nil numbers nil) (sinh_sum formula-decl nil hyperbolic nil) (minus_real_is_real application-judgement "real" reals nil) (cosh_neg formula-decl nil hyperbolic nil) (real_plus_real_is_real application-judgement "real" reals nil) (real_minus_real_is_real application-judgement "real" reals nil) (real_times_real_is_real application-judgement "real" reals nil) (sinh_neg formula-decl nil hyperbolic nil)) shostak)) (cosh_sum 0 (cosh_sum-1 nil 3253475454 ("" (skosimp*) (("" (expand "cosh") (("" (expand "sinh") (("" (lemma "exp_diff" ("x" "0")) (("" (inst-cp - "x!1") (("" (inst-cp - "y!1") (("" (inst - "x!1+y!1") (("" (lemma "exp_sum" ("x" "x!1" "y" "y!1")) (("" (replace -1) (("" (rewrite "exp_0") (("" (replace -2 1) (("" (replace -3) (("" (replace -4) (("" (hide-all-but 1) (("" (name-replace "EPX" "exp(x!1)") (("" (name-replace "EPY" "exp(y!1)") (("" (grind) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((real_plus_real_is_real application-judgement "real" reals nil) (posreal_plus_nnreal_is_posreal application-judgement "posreal" real_types nil) (posreal_div_posreal_is_posreal 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application-judgement "posreal" real_types nil) (numfield nonempty-type-eq-decl nil number_fields nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (real_times_real_is_real application-judgement "real" reals nil) (sinh const-decl "real" hyperbolic nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (real_minus_real_is_real application-judgement "real" reals nil)) shostak)) (cosh_diff 0 (cosh_diff-1 nil 3270964814 ("" (skosimp*) (("" (lemma "cosh_sum" ("x" "x!1" "y" "-y!1")) (("" (rewrite "sinh_neg") (("" (rewrite "cosh_neg") (("" (assert) nil nil)) nil)) nil)) nil)) nil) ((- const-decl "[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) (boolean nonempty-type-decl nil booleans nil) (number nonempty-type-decl nil numbers nil) (cosh_sum formula-decl nil hyperbolic nil) (minus_real_is_real application-judgement "real" reals nil) (cosh_neg formula-decl nil hyperbolic nil) (real_plus_real_is_real application-judgement "real" reals nil) (real_minus_real_is_real application-judgement "real" reals nil) (real_times_real_is_real application-judgement "real" reals nil) (sinh_neg formula-decl nil hyperbolic nil)) shostak)) (tanh_sum_TCC1 0 (tanh_sum_TCC1-1 nil 3253471676 ("" (skosimp*) (("" (typepred "tanh(x!1)") (("" (typepred "tanh(y!1)") (("" (name-replace "TX" "tanh(x!1)") (("" (name-replace "TY" "tanh(y!1)") (("" (lemma "abs_mult" ("x" "TX" "y" "TY")) (("" (case "TX=0") (("1" (assert) nil nil) ("2" (lemma "lt_times_lt_pos1") (("2" (inst - "abs(TY)" "abs(TX)" "1" "1") (("1" (split -1) (("1" (grind) nil nil) ("2" (grind) nil nil) ("3" (grind) nil nil)) nil) ("2" (grind) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((tanh const-decl "real_abs_lt1" hyperbolic nil) (real_abs_lt1 nonempty-type-eq-decl nil hyperbolic nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (- const-decl "[numfield -> numfield]" number_fields nil) (numfield nonempty-type-eq-decl nil number_fields 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) (bool nonempty-type-eq-decl nil booleans nil) (boolean nonempty-type-decl nil booleans nil) (= const-decl "[T, T -> boolean]" equalities nil) (real_times_real_is_real application-judgement "real" reals nil) (abs_mult formula-decl nil real_props nil) (lt_times_lt_pos1 formula-decl nil real_props nil) (minus_real_is_real application-judgement "real" reals nil) (posint_times_posint_is_posint application-judgement "posint" integers nil) (odd_times_odd_is_odd application-judgement "odd_int" integers nil) (mult_divides1 application-judgement "(divides(n))" divides nil) (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (posreal nonempty-type-eq-decl nil real_types nil) (TX skolem-const-decl "real_abs_lt1" hyperbolic nil) (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs nil) (nonneg_real nonempty-type-eq-decl nil real_types nil) (>= const-decl "bool" reals nil) (> const-decl "bool" reals nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (abs_nat formula-decl nil abs_lems "reals/") (real_plus_real_is_real application-judgement "real" reals nil) (minus_odd_is_odd application-judgement "odd_int" integers nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (nnreal_times_nnreal_is_nnreal application-judgement "nnreal" 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"[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) (cosh2x formula-decl nil hyperbolic nil) (nnreal_plus_posreal_is_posreal application-judgement "posreal" real_types nil) (nnreal_times_nnreal_is_nnreal application-judgement "nnreal" real_types nil) (real_times_real_is_real application-judgement "real" reals nil) (real_minus_real_is_real application-judgement "real" reals nil) (cosh_sinh_one formula-decl nil hyperbolic nil)) shostak)) (cosh2x_C 0 (cosh2x_C-1 nil 3270916242 ("" (skosimp*) (("" (lemma "cosh2x_B" ("x" "x!1")) (("" (lemma "cosh2x" ("x" "x!1")) (("" (assert) nil nil)) nil)) nil)) 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 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"cosh(x!1)")) (("1" (replace -1 1) (("1" (hide -1) (("1" (expand "sq") (("1" (rewrite "sinh2x") (("1" (rewrite "cosh2x") (("1" (expand "sq") (("1" (lemma "cosh_sinh_one" ("x" "x!1")) (("1" (expand "sq") (("1" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (lemma "sq_nz_pos" ("nz" "cosh(2*x!1)")) (("2" (lemma "sq_nz_pos" ("nz" "cosh(x!1)")) (("2" (lemma "posreal_times_posreal_is_posreal" ("px" "sq(cosh(x!1))" "py" "sq(cosh(2 * x!1))")) (("1" (expand "sq") (("1" (assert) nil nil)) nil) ("2" (propax) nil nil) ("3" (propax) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (lemma "sq_nz_pos" ("nz" "cosh(x!1)")) (("2" (lemma "posreal_times_posreal_is_posreal" ("px" "sq(cosh(x!1))" "py" "cosh(2*x!1)")) (("1" (assert) nil nil) ("2" (propax) nil nil)) nil)) nil)) nil)) nil) ("2" (lemma "sq_nz_pos" ("nz" "cosh(x!1)")) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ((posreal_plus_nnreal_is_posreal application-judgement "posreal" real_types nil) (real_times_real_is_real application-judgement "real" reals nil) (div_cancel4 formula-decl nil 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) (/= const-decl "boolean" notequal nil) (nonzero_real nonempty-type-eq-decl nil reals nil) (numfield nonempty-type-eq-decl nil number_fields nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (bool nonempty-type-eq-decl nil booleans nil) (>= const-decl "bool" reals nil) (nonneg_real nonempty-type-eq-decl nil real_types nil) (sq const-decl "nonneg_real" sq "reals/") (AND const-decl "[bool, bool -> bool]" booleans nil) (< const-decl "bool" reals nil) (- const-decl "[numfield -> numfield]" number_fields nil) (real_abs_lt1 nonempty-type-eq-decl nil hyperbolic nil) (tanh const-decl "real_abs_lt1" hyperbolic nil) (* const-decl "[numfield, numfield -> numfield]" number_fields nil) (cosh const-decl "posreal_ge1" hyperbolic nil) (posreal_ge1 nonempty-type-eq-decl nil hyperbolic nil) (sinh const-decl "real" hyperbolic nil) (sq_div formula-decl nil sq "reals/") (add_div formula-decl nil real_props nil) (real_plus_real_is_real application-judgement "real" reals nil) (cosh2x formula-decl nil hyperbolic nil) (nnreal_times_nnreal_is_nnreal application-judgement "nnreal" real_types nil) (real_minus_real_is_real application-judgement "real" reals nil) (cosh_sinh_one formula-decl nil hyperbolic nil) (sinh2x formula-decl nil hyperbolic nil) (cross_mult formula-decl nil real_props nil) (div_times formula-decl nil real_props nil) (real_div_nzreal_is_real application-judgement "real" reals nil)) shostak)) (sinh3x_TCC1 0 (sinh3x_TCC1-1 nil 3270920091 ("" (skosimp*) (("" (assert) nil nil)) nil) ((real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil)) shostak)) (sinh3x 0 (sinh3x-1 nil 3270918363 ("" (skosimp*) (("" (lemma "sinh_sum" ("x" "2*x!1" "y" "x!1")) (("" (replace -1 1) (("" (hide -1) (("" (rewrite "sinh2x") (("" (rewrite "cosh2x") (("" (lemma "cosh_sinh_one" ("x" "x!1")) (("" (expand "sq") (("" (assert) (("" (expand "^") (("" (expand "expt") (("" (expand "expt") (("" (expand "expt") (("" (expand "expt") (("" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) 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) (boolean nonempty-type-decl nil booleans nil) (number nonempty-type-decl nil numbers nil) (sinh_sum formula-decl nil hyperbolic nil) (real_times_real_is_real application-judgement "real" reals nil) (cosh2x formula-decl nil hyperbolic nil) (nnreal_times_nnreal_is_nnreal application-judgement "nnreal" real_types nil) (real_minus_real_is_real application-judgement "real" reals nil) (sq const-decl "nonneg_real" sq "reals/") (^ const-decl "real" exponentiation nil) (expt def-decl "real" exponentiation nil) (real_plus_real_is_real application-judgement "real" reals nil) (cosh_sinh_one formula-decl nil hyperbolic nil) (sinh2x formula-decl nil hyperbolic nil)) shostak)) (cosh3x 0 (cosh3x-1 nil 3270919010 ("" (skosimp*) (("" (lemma "cosh_sum" ("x" "x!1" "y" "2*x!1")) (("" (replace -1) (("" (rewrite "cosh2x") (("" (rewrite "sinh2x") (("" (hide -1) (("" (expand "^") (("" (expand "expt") (("" (expand "expt") (("" (expand "expt") (("" (expand "expt") (("" (expand "sq") (("" (lemma "cosh_sinh_one" ("x" "x!1")) (("" (expand "sq") (("" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) 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) (boolean nonempty-type-decl nil booleans nil) (number nonempty-type-decl nil numbers nil) (cosh_sum formula-decl nil hyperbolic nil) (real_times_real_is_real application-judgement "real" reals nil) (cosh2x formula-decl nil hyperbolic nil) (nnreal_times_nnreal_is_nnreal application-judgement "nnreal" real_types nil) (real_minus_real_is_real application-judgement "real" reals nil) (expt def-decl "real" exponentiation nil) (sq const-decl "nonneg_real" sq "reals/") (real_plus_real_is_real application-judgement "real" reals nil) (cosh_sinh_one formula-decl nil hyperbolic nil) (^ const-decl "real" exponentiation nil) (sinh2x formula-decl nil hyperbolic nil)) shostak)) (sinh4x 0 (sinh4x-1 nil 3270919340 ("" (skosimp*) (("" (lemma "sinh2x" ("x" "2*x!1")) (("" (rewrite "sinh2x") (("" (rewrite "cosh2x") (("" (expand "sq") (("" (lemma "cosh_sinh_one" ("x" "x!1")) (("" (expand "sq") (("" (assert) (("" (rewrite "sinh2x") (("" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) 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) (boolean nonempty-type-decl nil booleans nil) (number nonempty-type-decl nil numbers nil) (sinh2x formula-decl nil hyperbolic nil) (real_times_real_is_real application-judgement "real" reals nil) (cosh2x formula-decl nil hyperbolic nil) (nnreal_times_nnreal_is_nnreal application-judgement "nnreal" real_types nil) (real_minus_real_is_real application-judgement "real" reals nil) (cosh_sinh_one formula-decl nil hyperbolic nil) (real_plus_real_is_real application-judgement "real" reals nil) (sq const-decl "nonneg_real" sq "reals/") (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal" real_types nil)) shostak)) (cosh4x_TCC1 0 (cosh4x_TCC1-1 nil 3270912606 ("" (skosimp*) (("" (assert) nil nil)) nil) ((real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil)) shostak)) (cosh4x 0 (cosh4x-1 nil 3270919895 ("" (skosimp*) (("" (lemma "cosh2x" ("x" "2*x!1")) (("" (replace -1 1) (("" (hide -1) (("" (rewrite "cosh2x") (("" (expand "sq") (("" (expand "^") (("" (expand "expt") (("" (expand "expt") (("" (expand "expt") (("" (expand "expt") (("" (expand "expt") (("" (lemma "cosh_sinh_one" ("x" "x!1")) (("" (expand "sq") (("" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) 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) (boolean nonempty-type-decl nil booleans nil) (number nonempty-type-decl nil numbers nil) (cosh2x formula-decl nil hyperbolic nil) (real_times_real_is_real application-judgement "real" reals nil) (sq const-decl "nonneg_real" sq "reals/") (expt def-decl "real" exponentiation nil) (cosh_sinh_one formula-decl nil hyperbolic nil) (^ const-decl "real" exponentiation nil) (real_minus_real_is_real application-judgement "real" reals nil) (real_plus_real_is_real application-judgement "real" reals nil) (nnreal_times_nnreal_is_nnreal application-judgement "nnreal" real_types nil)) shostak)) (sinh_times_sinh 0 (sinh_times_sinh-1 nil 3270964579 ("" (skosimp*) (("" (rewrite "cosh_sum") (("" (rewrite "cosh_diff") (("" (assert) nil nil)) nil)) nil)) nil) ((cosh_sum formula-decl nil hyperbolic 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) (real_minus_real_is_real application-judgement "real" reals nil) (minus_odd_is_odd application-judgement "odd_int" integers nil) (real_times_real_is_real application-judgement "real" reals nil) (real_plus_real_is_real application-judgement "real" reals nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (cosh_diff formula-decl nil hyperbolic nil)) shostak)) (cosh_times_cosh 0 (cosh_times_cosh-1 nil 3270964883 ("" (skosimp*) (("" (rewrite "cosh_sum") (("" (rewrite "cosh_diff") (("" (assert) nil nil)) nil)) nil)) nil) ((cosh_sum formula-decl nil hyperbolic 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) (minus_odd_is_odd application-judgement "odd_int" integers nil) (real_times_real_is_real application-judgement "real" reals nil) (real_plus_real_is_real application-judgement "real" reals nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (real_minus_real_is_real application-judgement "real" reals nil) (cosh_diff formula-decl nil hyperbolic nil)) shostak)) (sinh_times_cosh 0 (sinh_times_cosh-1 nil 3270964906 ("" (skosimp*) (("" (rewrite "sinh_sum") (("" (rewrite "sinh_diff") (("" (assert) nil nil)) nil)) nil)) nil) ((sinh_sum formula-decl nil hyperbolic 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) (minus_odd_is_odd application-judgement "odd_int" integers nil) (real_times_real_is_real application-judgement "real" reals nil) (real_plus_real_is_real application-judgement "real" reals nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (real_minus_real_is_real application-judgement "real" reals nil) (sinh_diff formula-decl nil hyperbolic nil)) shostak)) (sum_sinh 0 (sum_sinh-1 nil 3270965915 ("" (skosimp*) (("" (lemma "sinh_times_cosh" ("x" "(x!1 + y!1) / 2" "y" "(x!1 - y!1) / 2")) (("" (assert) nil nil)) nil)) nil) ((- const-decl "[numfield, numfield -> numfield]" number_fields nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields 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) (sinh_times_cosh formula-decl nil hyperbolic nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (real_times_real_is_real application-judgement "real" reals nil) (real_plus_real_is_real application-judgement "real" reals nil) (real_minus_real_is_real application-judgement "real" reals nil)) shostak)) (diff_sinh 0 (diff_sinh-1 nil 3270966111 ("" (skosimp*) (("" (lemma "sum_sinh" ("x" "x!1" "y" "-y!1")) (("" (rewrite "sinh_neg") (("" (assert) nil nil)) nil)) nil)) nil) ((- const-decl "[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) (boolean nonempty-type-decl nil booleans nil) (number nonempty-type-decl nil numbers nil) (sum_sinh formula-decl nil hyperbolic nil) (minus_real_is_real application-judgement "real" reals nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (real_times_real_is_real application-judgement "real" reals nil) (real_minus_real_is_real application-judgement "real" reals nil) (real_plus_real_is_real application-judgement "real" reals nil) (sinh_neg formula-decl nil hyperbolic nil)) shostak)) (sum_cosh 0 (sum_cosh-1 nil 3270966232 ("" (skosimp*) (("" (lemma "cosh_times_cosh" ("x" "(x!1+y!1)/2" "y" "(x!1-y!1)/2")) (("" (name-replace "K1" "cosh(x!1)") (("" (name-replace "K2" "cosh(y!1)") (("" (name-replace "K3" "cosh((x!1+y!1)/2)") (("" (name-replace "K4" "cosh((x!1-y!1)/2)") (("" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ((- const-decl "[numfield, numfield -> numfield]" number_fields nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields 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) (cosh_times_cosh formula-decl nil hyperbolic nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (minus_odd_is_odd application-judgement "odd_int" integers nil) (real_plus_real_is_real application-judgement "real" reals nil) (real_minus_real_is_real application-judgement "real" reals nil) (real_times_real_is_real application-judgement "real" reals nil) (= const-decl "[T, T -> boolean]" equalities nil) (bool nonempty-type-eq-decl nil booleans nil) (>= const-decl "bool" reals nil) (posreal_ge1 nonempty-type-eq-decl nil hyperbolic nil) (cosh const-decl "posreal_ge1" hyperbolic nil)) shostak)) (diff_cosh 0 (diff_cosh-1 nil 3270966462 ("" (skosimp*) (("" (lemma "sinh_times_sinh" ("x" "(x!1+y!1)/2" "y" "(x!1-y!1)/2")) (("" (name-replace "K1" "cosh(x!1)") (("" (name-replace "K2" "cosh(y!1)") (("" (name-replace "K3" "sinh((x!1 + y!1) / 2)") (("" (name-replace "K4" "sinh((x!1 - y!1) / 2)") (("" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ((- const-decl "[numfield, numfield -> numfield]" number_fields nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields 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) (sinh_times_sinh formula-decl nil hyperbolic nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (sinh const-decl "real" hyperbolic nil) (real_plus_real_is_real application-judgement "real" reals nil) (real_minus_real_is_real application-judgement "real" reals nil) (minus_odd_is_odd application-judgement "odd_int" integers nil) (real_times_real_is_real application-judgement "real" reals nil) (= const-decl "[T, T -> boolean]" equalities nil) (bool nonempty-type-eq-decl nil booleans nil) (>= const-decl "bool" reals nil) (posreal_ge1 nonempty-type-eq-decl nil hyperbolic nil) (cosh const-decl "posreal_ge1" hyperbolic nil)) shostak)) (sum_tanh_TCC1 0 (sum_tanh_TCC1-1 nil 3270965627 ("" (skosimp*) (("" (lemma "posreal_times_posreal_is_posreal" ("px" "cosh(x!1)" "py" "cosh(y!1)")) (("1" (assert) nil nil) ("2" (assert) nil nil) ("3" (assert) nil nil)) nil)) nil) ((cosh const-decl "posreal_ge1" hyperbolic nil) (posreal_ge1 nonempty-type-eq-decl nil hyperbolic 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) (posreal_times_posreal_is_posreal judgement-tcc nil real_types 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) (AND const-decl "[bool, bool -> bool]" booleans nil) (real_times_real_is_real application-judgement "real" reals nil)) shostak)) (sum_tanh 0 (sum_tanh-1 nil 3270966616 ("" (skosimp*) (("" (expand "tanh") (("" (rewrite "sinh_sum") (("" (lemma "add_div" ("x" "sinh(x!1)" "n0x" "cosh(x!1)" "y" "sinh(y!1)" "n0y" "cosh(y!1)")) (("" (replace -1 1) (("" (propax) nil nil)) nil)) nil)) nil)) nil)) nil) ((tanh const-decl "real_abs_lt1" hyperbolic nil) (sinh const-decl "real" hyperbolic nil) (cosh const-decl "posreal_ge1" hyperbolic nil) (posreal_ge1 nonempty-type-eq-decl nil hyperbolic nil) (>= const-decl "bool" reals nil) (bool nonempty-type-eq-decl nil booleans nil) (nonzero_real nonempty-type-eq-decl nil reals nil) (/= const-decl "boolean" notequal nil) (add_div formula-decl nil real_props nil) (real_plus_real_is_real application-judgement "real" reals nil) (real_times_real_is_real application-judgement "real" reals nil) (real_div_nzreal_is_real application-judgement "real" 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) (boolean nonempty-type-decl nil booleans nil) (number nonempty-type-decl nil numbers nil) (sinh_sum formula-decl nil hyperbolic nil)) shostak)) (sum_coth_TCC1 0 (sum_coth_TCC1-1 nil 3270965675 ("" (case "FORALL (x:nzreal): sinh(x) /=0") (("1" (skosimp*) (("1" (inst-cp - "n0x!1") (("1" (inst - "n0y!1") (("1" (lemma "nzreal_times_nzreal_is_nzreal" ("nzx" "sinh(n0x!1)" "nzy" "sinh(n0y!1)")) 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(posreal_plus_nnreal_is_posreal application-judgement "posreal" real_types nil) (I const-decl "(bijective?[T, T])" identity nil) (identity_derivable_fun formula-decl nil derivatives "analysis/") (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) (sinh const-decl "real" hyperbolic nil) (exp_deriv formula-decl nil ln_exp nil)) nil) (deriv_sinh-1 nil 3270138870 ("" (lemma "exp_deriv") (("" (flatten -1) (("" (expand "sinh") (("" (expand "cosh") (("" (lemma "identity_derivable_fun[real]") (("" (lemma "deriv_id_fun[real]") (("" (expand "I") (("" (expand "const_fun") (("" (lemma "neg_derivable_fun" ("f" "LAMBDA (x: real): x")) (("" (assert) (("" (expand "-") (("" (lemma "deriv_neg_fun" ("ff" "LAMBDA (x: real): x")) (("" 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const-decl "bool" functions nil) (posreal_gt1 nonempty-type-eq-decl nil hyperbolic nil) (both_sides_div_pos_gt2 formula-decl nil real_props nil) (div_cancel4 formula-decl nil real_props nil) (posrat_div_posrat_is_posrat application-judgement "posrat" rationals nil) (div_cancel3 formula-decl nil real_props nil) (x!1 skolem-const-decl "real" hyperbolic nil) (trichotomy formula-decl nil real_axioms nil) (y!1 skolem-const-decl "posreal_gt1" hyperbolic nil) (/ const-decl "[numfield, nznum -> numfield]" number_fields nil) (nznum nonempty-type-eq-decl nil number_fields nil) (real_gt_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) (posreal_div_posreal_is_posreal judgement-tcc nil real_types nil) (strict_increasing? const-decl "bool" real_fun_preds "reals/") (tanh_bij formula-decl nil hyperbolic nil) (posreal_coth const-decl "posreal_gt1" hyperbolic nil)) shostak)) (asinh_alt_def 0 (asinh_alt_def-1 nil 3253394002 ("" (skosimp*) (("" (lemma "sinh_bij") (("" (lemma "bijective_inverse" ("f" "sinh" "y" "x!1" "x" "asinh(x!1)")) (("1" (flatten -1) (("1" (hide -1) (("1" (split -1) (("1" (assert) nil nil) ("2" (hide 2) (("2" (expand "asinh") (("2" (hide -1) (("2" (expand "sinh") (("2" (lemma "sqrt_pos" ("px" "1+sq(x!1)")) (("2" (case "sqrt(1 + sq(x!1)) + x!1 >0") (("1" (rewrite "div_cancel4" 1) (("1" (lemma "exp_diff" ("x" "0" "y" "ln(sqrt(1 + sq(x!1)) + x!1)")) (("1" (rewrite "exp_0") (("1" (replace -1) (("1" (rewrite "exp_ln") (("1" (hide -1) (("1" (lemma "div_cancel3" ("y" "sqrt(1 + sq(x!1))-x!1" "x" "1" "n0z" "sqrt(1 + sq(x!1)) + x!1")) (("1" (flatten -1) (("1" (hide -1) (("1" (split -1) (("1" (assert) nil nil) ("2" (hide 2) (("2" (assert) (("2" (rewrite "sq_rew") (("2" (rewrite "sq_rew") (("2" (rewrite "sq_sqrt") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (assert) nil nil)) nil)) nil) ("2" (hide 2) (("2" (lemma "trichotomy" ("x" "x!1")) (("2" (split -1) (("1" (assert) nil nil) ("2" (assert) nil nil) ("3" (lemma "sq_gt" ("nna" "sqrt(1 + sq(x!1))" "nnb" "-x!1")) (("1" (rewrite "sq_sqrt") (("1" (rewrite "sq_neg") (("1" (assert) nil nil)) nil)) nil) ("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (propax) nil nil)) nil)) nil)) nil) ((sinh_bij formula-decl nil hyperbolic nil) (sqrt_pos application-judgement "posreal" sqrt "reals/") (real_plus_real_is_real application-judgement "real" reals nil) (nnreal type-eq-decl nil real_types nil) (= const-decl "[T, T -> boolean]" equalities nil) (* const-decl "[numfield, numfield -> numfield]" number_fields nil) (sqrt const-decl "{nnz: nnreal | nnz * nnz = nnx}" sqrt "reals/") (exp_diff formula-decl nil ln_exp nil) (nzreal_div_nzreal_is_nzreal application-judgement "nzreal" real_types nil) (posreal_times_posreal_is_posreal application-judgement "posreal" real_types nil) (sq_nz_pos application-judgement "posreal" sq "reals/") (sq_sqrt formula-decl nil sqrt "reals/") (sq_rew formula-decl nil sq "reals/") (div_cancel3 formula-decl nil real_props nil) (posreal_div_posreal_is_posreal application-judgement "posreal" real_types nil) (exp_ln formula-decl nil ln_exp nil) (minus_odd_is_odd application-judgement "odd_int" integers nil) (exp_0 formula-decl nil ln_exp nil) (real_times_real_is_real application-judgement "real" reals nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (minus_real_is_real application-judgement "real" reals nil) (real_minus_real_is_real application-judgement "real" reals nil) (div_cancel4 formula-decl nil real_props nil) (/= const-decl "boolean" notequal nil) (nonzero_real nonempty-type-eq-decl nil reals nil) (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (ln const-decl "real" ln_exp nil) (exp const-decl "{py | x = ln(py)}" ln_exp nil) (- const-decl "[numfield -> numfield]" number_fields nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (trichotomy formula-decl nil real_axioms nil) (sq_gt formula-decl nil sq "reals/") (sq_neg formula-decl nil sq "reals/") (posreal_plus_nnreal_is_posreal application-judgement "posreal" real_types nil) (sqrt_pos judgement-tcc nil sqrt "reals/") (>= 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) (numfield nonempty-type-eq-decl nil number_fields nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (sq const-decl "nonneg_real" sq "reals/") (bijective_inverse formula-decl nil function_inverse nil) (asinh const-decl "real" hyperbolic nil) (bool nonempty-type-eq-decl nil booleans nil) (bijective? const-decl "bool" functions nil) (sinh const-decl "real" hyperbolic 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)) shostak)) (asinh_sinh 0 (asinh_sinh-1 nil 3253392586 ("" (skosimp*) (("" (lemma "asinh_alt_def" ("x" "sinh(x!1)")) (("" (lemma "sinh_bij") (("" (lemma "comp_inverse_left" ("f" "sinh" "x" "x!1")) (("1" (assert) nil nil) ("2" (propax) nil nil)) nil)) nil)) nil)) nil) ((sinh const-decl "real" hyperbolic 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) (asinh_alt_def formula-decl nil hyperbolic nil) (bijective? 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2026-05-26