Quelle trig_degree.prf Sprache: Lisp

(trig_degree
(rad_deg 0
  (rad_deg-1 nil 3269434094 ("" (grind) nil nil)
   ((Integral const-decl "real" integral_def "analysis/")
    (atan_value const-decl "real" atan nil)
    (pi const-decl "posreal" atan nil)
    (toDeg const-decl "real" trig_degree nil)
    (toRad const-decl "real" trig_degree nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (real_times_real_is_real application-judgement "real" reals nil))
   nil))
(deg_rad 0
  (deg_rad-1 nil 3269434094 ("" (grind) nil nil)
   ((Integral const-decl "real" integral_def "analysis/")
    (atan_value const-decl "real" atan nil)
    (pi const-decl "posreal" atan nil)
    (toRad const-decl "real" trig_degree nil)
    (toDeg const-decl "real" trig_degree nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (real_times_real_is_real application-judgement "real" reals nil))
   nil))
(tand_TCC1 0
  (tand_TCC1-1 nil 3269434094 ("" (subtype-tcc) 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 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)
    (cos_range application-judgement "trig_range" sincos_def nil)
    (Integral const-decl "real" integral_def "analysis/")
    (atan_value const-decl "real" atan nil)
    (pi const-decl "posreal" atan nil)
    (toRad const-decl "real" trig_degree nil)
    (cos const-decl "real" sincos_def nil)
    (Tan? const-decl "bool" sincos_def nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (pi_bound name-judgement "{r: posreal | pi_lb < r AND r < pi_ub}"
     atan_approx nil)
    (posreal_times_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (real_minus_real_is_real application-judgement "real" reals nil))
   nil))
(sq_sin_cos_d_one 0
  (sq_sin_cos_d_one-1 nil 3269434094
   ("" (skolem 1 "a")
    (("" (expand "sind")
      (("" (expand "cosd")
        (("" (lemma "sin2_cos2")
          (("" (inst?) (("" (ground) nil nil)) nil)) nil))
        nil))
      nil))
    nil)
   ((sind const-decl "real" trig_degree nil)
    (sin2_cos2 formula-decl nil trig_basic nil)
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (sin_range application-judgement "trig_range" sincos_def nil)
    (cos_range application-judgement "trig_range" sincos_def nil)
    (toRad const-decl "real" trig_degree 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)
    (cosd const-decl "real" trig_degree nil))
   nil))
(sind_bounds 0
  (sind_bounds-2 nil 3321628718
   ("" (skeep)
    (("" (expand "sind")
      (("" (expand "toRad")
        (("" (case "sin(a * pi / 180) >= sin(a*pi_lb/180)")
          (("1"
            (case "sin(a * pi_lb / 180)>= sin_lb(a * pi_lb / 180,n)")
            (("1" (ground) nil nil)
             ("2" (hide 2)
              (("2" (lemma "sin_bounds")
                (("2" (inst -1 "a * pi_lb / 180" "n")
                  (("2" (ground) nil nil)) nil))
                nil))
              nil))
            nil)
           ("2" (hide 2)
            (("2" (case "a=0")
              (("1" (replace -1 :hide? t) (("1" (ground) nil nil)) nil)
               ("2" (case "a > 0")
                (("1" (hide 1)
                  (("1" (lemma "sin_increasing")
                    (("1" (inst -1 " a * pi / 180" "a * pi_lb / 180")
                      (("1" (ground)
                        (("1" (hide 2 3)
                          (("1" (cross-mult 1)
                            (("1" (cancel-by 1 "a*180") nil nil)) nil))
                          nil)
                         ("2" (cross-mult)
                          (("2" (hide 2)
                            (("2" (cancel-by 1 "2*pi") nil nil)) nil))
                          nil)
                         ("3" (cross-mult)
                          (("3" (hide 2)
                            (("3" (cancel-by 1 "2*pi") nil nil)) nil))
                          nil)
                         ("4" (hide 2)
                          (("4" (cross-mult 1)
                            (("4" (cancel-by 1 "2")
                              (("4" (grind-reals) nil nil)) nil))
                            nil))
                          nil)
                         ("5" (hide 2)
                          (("5" (cross-mult)
                            (("5" (cancel-by 1 "2")
                              (("5"
                                (typepred "pi")
                                (("5" (grind) nil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (assert) nil nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((sind const-decl "real" trig_degree nil)
    (pi_lb const-decl "posreal" atan_approx nil)
    (pi const-decl "posreal" atan 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 "[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)
    (sin const-decl "real" sincos_def 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_div_nzreal_is_real application-judgement "real" reals nil)
    (sin_bounds formula-decl nil trig_approx nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (sin_range application-judgement "trig_range" sincos_def nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (pi_bound name-judgement "{r: posreal | pi_lb < r AND r < pi_ub}"
     atan_approx nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props 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)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (sin_lb const-decl "real" trig_approx nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (nnreal_div_posreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (minus_even_is_even application-judgement "even_int" integers nil)
    (minus_nzint_is_nzint application-judgement "nzint" integers nil)
    (neg_mult formula-decl nil extra_tegies nil)
    (Integral const-decl "real" integral_def "analysis/")
    (atan_value const-decl "real" atan nil)
    (pi_ub const-decl "posreal" atan_approx nil)
    (< const-decl "bool" reals nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (posint nonempty-type-eq-decl nil integers nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (le_times_le_pos formula-decl nil real_props nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (nzreal_times_nzreal_is_nzreal application-judgement "nzreal"
     real_types nil)
    (div_mult_pos_ge2 formula-decl nil real_props nil)
    (both_sides_times_pos_ge1 formula-decl nil real_props nil)
    (div_mult_pos_le1 formula-decl nil real_props nil)
    (posreal_times_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (div_mult_pos_le2 formula-decl nil real_props nil)
    (both_sides_times_pos_le1 formula-decl nil real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (div_mult_pos_ge1 formula-decl nil real_props nil)
    (zero_times1 formula-decl nil real_props nil)
    (TRUE const-decl "bool" booleans nil)
    (id const-decl "(bijective?[T, T])" identity nil)
    (bijective? const-decl "bool" functions nil)
    (CBD_13 skolem-const-decl "real" trig_degree nil)
    (both_sides_times_pos_le1_imp formula-decl nil extra_real_props
     nil)
    (div_mult_pos_gt2 formula-decl nil extra_real_props nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (times_div2 formula-decl nil real_props nil)
    (div_mult_pos_gt1 formula-decl nil extra_real_props nil)
    (posreal_div_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (minus_nzreal_is_nzreal application-judgement "nzreal" real_types
     nil)
    (nzreal_div_nzreal_is_nzreal application-judgement "nzreal"
     real_types nil)
    (sin_increasing formula-decl nil trig_ineq nil)
    (toRad const-decl "real" trig_degree nil))
   nil))
(cosd_bounds 0
  (cosd_bounds-1 nil 3321628868
   ("" (skeep)
    (("" (expand "cosd")
      (("" (expand "toRad")
        (("" (case "cos(a * pi / 180) <= cos(a*pi_lb/180)")
          (("1"
            (case "cos(a * pi_lb / 180) <= cos_ub(a * pi_lb / 180,n)")
            (("1" (ground) nil nil)
             ("2" (hide 2)
              (("2" (lemma "cos_bounds")
                (("2" (inst -1 "a * pi_lb / 180" "n")
                  (("2" (ground) nil nil)) nil))
                nil))
              nil))
            nil)
           ("2" (hide 2)
            (("2" (lemma "cos_decreasing")
              (("2" (inst -1 "a * pi / 180" "a * pi_lb / 180")
                (("2" (ground)
                  (("1" (hide 2 3)
                    (("1" (cross-mult 1)
                      (("1" (case "a=0")
                        (("1" (replace -1) (("1" (ground) nil nil))
                          nil)
                         ("2" (cancel-by 2 "180*a") nil nil))
                        nil))
                      nil))
                    nil)
                   ("2" (hide 2)
                    (("2" (cross-mult 1)
                      (("2" (cancel-by 1 "pi") nil nil)) nil))
                    nil)
                   ("3" (hide 2)
                    (("3" (cross-mult 1)
                      (("3" (typepred "pi")
                        (("3" (grind-reals) nil nil)) nil))
                      nil))
                    nil)
                   ("4" (hide 2)
                    (("4" (cross-mult)
                      (("4" (typepred "pi")
                        (("4" (swap-rel -3)
                          (("4" (mult-ineq -3 -6)
                            (("4" (assert) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("5" (hide 2)
                    (("5" (cross-mult 1) (("5" (grind) nil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((cosd const-decl "real" trig_degree nil)
    (pi_lb const-decl "posreal" atan_approx nil)
    (pi const-decl "posreal" atan 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)
    (* 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)
    (cos const-decl "real" sincos_def 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_div_nzreal_is_real application-judgement "real" reals nil)
    (cos_bounds formula-decl nil trig_approx nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (cos_range application-judgement "trig_range" sincos_def nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (pi_bound name-judgement "{r: posreal | pi_lb < r AND r < pi_ub}"
     atan_approx 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)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (cos_ub const-decl "real" trig_approx nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (cos_decreasing formula-decl nil trig_ineq 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)
    (div_mult_pos_gt1 formula-decl nil extra_real_props nil)
    (times_div2 formula-decl nil real_props nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (div_mult_pos_gt2 formula-decl nil extra_real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (div_mult_pos_ge1 formula-decl nil real_props nil)
    (TRUE const-decl "bool" booleans nil)
    (id const-decl "(bijective?[T, T])" identity nil)
    (bijective? const-decl "bool" functions nil)
    (CBD_18 skolem-const-decl "real" trig_degree nil)
    (both_sides_times_pos_le1_imp formula-decl nil extra_real_props
     nil)
    (nzreal_div_nzreal_is_nzreal application-judgement "nzreal"
     real_types nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (div_mult_pos_le1 formula-decl nil real_props nil)
    (posreal_times_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (< const-decl "bool" reals nil)
    (pi_ub const-decl "posreal" atan_approx nil)
    (posreal_div_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (both_sides_times_pos_le1 formula-decl nil real_props nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (pos_times_ge formula-decl nil real_props nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (IFF const-decl "[bool, bool -> bool]" booleans nil)
    (gt_times_gt_any1 formula-decl nil extra_real_props nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (toRad const-decl "real" trig_degree nil))
   nil)))

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