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Datei: mod_nat.prf Sprache: Lisp

Original von: PVS^©

(mod_nat
(nmod_TCC1 0
  (nmod_TCC1-1 nil 3507028517
   ("" (skosimp*)
    (("" (assert)
      (("" (lemma "div_smaller")
        (("" (inst?)
          (("" (assert) (("" (rewrite "div_rem_small") nil nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_ge_is_total_order name-judgement "(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)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (div_rem_small formula-decl nil div_nat nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (div_smaller formula-decl nil div_nat nil))
   nil))
(nmod_even 0
  (nmod_even-1 nil 3507028517
   ("" (skosimp*)
    (("" (expand "nmod") (("" (rewrite "div_even") nil nil)) nil)) nil)
   ((nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (nmod const-decl "below(m)" mod_nat nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real 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_even formula-decl nil div_nat nil))
   nil))
(nmod_lt_nat 0
  (nmod_lt_nat-1 nil 3507028517
   ("" (skosimp*)
    (("" (expand "nmod")
      (("" (use "div_is_0") (("" (assert) nil nil)) nil)) nil))
    nil)
   ((nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (nmod const-decl "below(m)" mod_nat nil)
    (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)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (div_is_0 formula-decl nil div_nat nil))
   nil))
(nmod_sum 0
  (nmod_sum-1 nil 3507028517
   ("" (skosimp*)
    (("" (expand "nmod")
      (("" (use "div_sum_nat") (("" (assert) nil nil)) nil)) nil))
    nil)
   ((mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (nmod const-decl "below(m)" mod_nat nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (real_ge_is_total_order name-judgement "(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)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (div_sum_nat formula-decl nil div_nat nil))
   nil))
(nmod_sum_nat 0
  (nmod_sum_nat-1 nil 3507028517
   ("" (skosimp*)
    (("" (expand "nmod")
      (("" (lift-if)
        (("" (prop)
          (("1" (lemma "div_is_0")
            (("1" (inst?) (("1" (assert) nil nil)) nil)) nil)
           ("2" (lemma "lt_plus_lt1")
            (("2" (inst?)
              (("2" (inst -1 "m!1" "m!1")
                (("2" (split -1)
                  (("1" (lemma "div_sum_nat")
                    (("1" (inst -1 "-1" "m!1" "n1!1+n2!1")
                      (("1" (assert)
                        (("1" (lemma "div_is_0")
                          (("1" (inst?) (("1" (assert) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (assert) nil nil) ("3" (propax) nil nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (nmod const-decl "below(m)" mod_nat nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_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)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (div_is_0 formula-decl nil div_nat nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nzint_times_nzint_is_nzint application-judgement "nzint" integers
     nil)
    (div_sum_nat formula-decl nil div_nat nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (lt_plus_lt1 formula-decl nil real_props nil))
   nil))
(nmod_it_is 0
  (nmod_it_is-1 nil 3507028517
   ("" (skosimp*)
    (("" (expand "nmod")
      (("" (replace -1)
        (("" (hide -1)
          (("" (lemma "div_sum_nat")
            (("" (inst -1 "c!1" "m!1" "k!1")
              (("" (assert)
                (("" (replace -1)
                  (("" (hide -1)
                    (("" (lemma "div_is_0")
                      (("" (inst?) (("" (assert) nil nil)) nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (nmod const-decl "below(m)" mod_nat nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (div_is_0 formula-decl nil div_nat nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (div_sum_nat formula-decl nil div_nat nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil))
   nil))
(nmod_zero 0
  (nmod_zero-1 nil 3507028517
   ("" (skosimp*) (("" (grind) nil nil)) nil)
   ((div const-decl "upto(n)" div_nat nil)
    (nmod const-decl "below(m)" mod_nat nil)
    (nnrat_div_posrat_is_nnrat application-judgement "nonneg_rat"
     rationals nil)
    (nonneg_floor_is_nat application-judgement "nat" floor_ceil nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil))
   nil))
(nmod_one 0
  (nmod_one-1 nil 3507028517
   ("" (skosimp*)
    (("" (expand "nmod")
      (("" (lift-if)
        (("" (prop)
          (("1" (replace -1)
            (("1" (hide -1)
              (("1" (expand "div") (("1" (assert) nil nil)) nil)) nil))
            nil)
           ("2" (rewrite "div_one") (("2" (assert) nil nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (nmod const-decl "below(m)" mod_nat nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (div const-decl "upto(n)" div_nat nil)
    (nonneg_floor_is_nat application-judgement "nat" floor_ceil nil)
    (int_times_even_is_even application-judgement "even_int" integers
     nil)
    (odd_minus_even_is_odd application-judgement "odd_int" integers
     nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real 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_one formula-decl nil div_nat nil))
   nil))
(nmod_of_nmod 0
  (nmod_of_nmod-1 nil 3507028517
   ("" (skosimp*)
    (("" (rewrite "nmod")
      (("" (lemma "nmod_sum")
        (("" (inst -1 "-div(k!1,m!1)" "m!1" "n!1+k!1")
          (("" (assert)
            (("" (hide 2)
              (("" (lemma "div_smaller")
                (("" (inst?) (("" (assert) nil nil)) nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nmod const-decl "below(m)" mod_nat nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (minus_int_is_int application-judgement "int" integers nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (div const-decl "upto(n)" div_nat nil)
    (upto nonempty-type-eq-decl nil naturalnumbers nil)
    (<= const-decl "bool" reals nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (div_smaller formula-decl nil div_nat nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nmod_sum formula-decl nil mod_nat nil))
   nil))
(nmod_mult 0
  (nmod_mult-1 nil 3507028517
   ("" (skosimp*)
    (("" (rewrite "nmod")
      (("" (lemma "nmod_sum")
        (("" (inst -1 "-div(n!1, mk!1 * m!1) * mk!1" "m!1" "n!1")
          (("" (assert)
            (("" (lemma "div_smaller")
              (("" (inst?) (("" (assert) nil nil)) nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (nmod const-decl "below(m)" mod_nat nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (div const-decl "upto(n)" div_nat nil)
    (upto nonempty-type-eq-decl nil naturalnumbers nil)
    (<= const-decl "bool" reals nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (div_smaller formula-decl nil div_nat nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (minus_int_is_int application-judgement "int" integers nil)
    (nmod_sum formula-decl nil mod_nat nil))
   nil)))

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