| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/PVS/Sturm/ (Beweissystem der NASA Version 6.0.9©) Datei vom 7.10.2014 mit Größe 1 MB |
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Quelle polynomial_pseudo_divide.prfSprache: Lisp |
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(polynomial_pseudo_divide (pseudo_div_TCC1 0 (pseudo_div_TCC1-1 nil 3582634590 ("" (subtype-tcc) nil 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) (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) (/= const-decl "boolean" notequal 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) (rat_max application-judgement "{s: rat | s >= q AND s >= r}" real_defs nil) (int_max application-judgement "{k: int | i <= k AND j <= k}" real_defs nil) (int_plus_int_is_int application-judgement "int" integers nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (posint_plus_nnint_is_posint application-judgement "posint" integers nil) (int_minus_int_is_int application-judgement "int" integers nil)) nil)) (pseudo_div_TCC2 0 (pseudo_div_TCC2-1 nil 3582634590 ("" (subtype-tcc) nil nil) ((bool nonempty-type-eq-decl nil booleans nil) (NOT const-decl "[bool -> bool]" booleans nil) (>= const-decl "bool" reals nil) (nat nonempty-type-eq-decl nil naturalnumbers nil) (/= const-decl "boolean" notequal nil) (listn_0 name-judgement "listn[real](0)" polynomial_division nil) (listn_0 name-judgement "listn[int](0)" polynomial_pseudo_divide nil) (posint_plus_nnint_is_posint application-judgement "posint" integers nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (int_minus_int_is_int application-judgement "int" integers nil) (int nonempty-type-eq-decl nil integers nil) (integer_pred const-decl "[rational -> boolean]" integers nil) (rational nonempty-type-from-decl nil rationals nil) (rational_pred const-decl "[real -> boolean]" rationals nil) (real 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) (array2list const-decl "{l: listn(n) | FORALL (i: below(n)): a(i) = nth(l, i)}" array2list "structures/") (make_divlisttype const-decl "DivListType" polynomial_pseudo_divide nil)) nil)) (pseudo_div_TCC3 0 (pseudo_div_TCC3-1 nil 3582634590 ("" (skeep) (("" (assert) (("" (hide 4) (("" (assert) (("" (grind) nil nil)) nil)) nil)) nil)) nil) ((int_minus_int_is_int application-judgement "int" integers nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (posint_plus_nnint_is_posint application-judgement "posint" integers nil) (listn_0 name-judgement "listn[int](0)" polynomial_pseudo_divide nil) (listn_0 name-judgement "listn[real](0)" polynomial_division nil) (length def-decl "nat" list_props nil) (make_divlisttype const-decl "DivListType" polynomial_pseudo_divide nil) (array2list const-decl "{l: listn(n) | FORALL (i: below(n)): a(i) = nth(l, i)}" array2list "structures/") (array2list_it def-decl "{l: listn(n - i) | FORALL (j: subrange(i, n - 1)): a(j) = nth(l, j - i)}" array2list "structures/") (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)) nil)) (pseudo_div_TCC4 0 (pseudo_div_TCC4-1 nil 3582651046 ("" (skosimp*) (("" (assert) (("" (lift-if) (("" (split -) (("1" (flatten) (("1" (replace -2) (("1" (expand "make_divlisttype" +) (("1" (assert) nil nil)) nil)) nil)) nil) ("2" (flatten) (("2" (typepred "v!1(g!1, n!1)(h!1, m!1)(1 + i!1)") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ((listn_0 name-judgement "listn[int](0)" polynomial_pseudo_divide nil) (listn_0 name-judgement "listn[real](0)" polynomial_division 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) (make_divlisttype const-decl "DivListType" polynomial_pseudo_divide nil) (DivListType type-eq-decl nil polynomial_pseudo_divide nil) (max const-decl "{p: real | p >= m AND p >= n}" real_defs nil) (< const-decl "bool" reals nil) (length def-decl "nat" list_props nil) (every adt-def-decl "boolean" list_adt nil) (PRED type-eq-decl nil defined_types nil) (list type-decl nil list_adt nil) (= const-decl "[T, T -> boolean]" equalities nil) (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (numfield nonempty-type-eq-decl nil number_fields nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (/= const-decl "boolean" notequal nil) (nat nonempty-type-eq-decl nil naturalnumbers nil) (>= const-decl "bool" reals nil) (int nonempty-type-eq-decl nil integers nil) (integer_pred const-decl "[rational -> boolean]" integers 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(("2" (assert) (("2" (typepred "v!1(g!1, n!1)(h!1, m!1)(1 + i!1)") (("2" (replaces -2 :dir rl) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((nnint_plus_nnint_is_nnint application-judgement "nonneg_int" integers nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (listn_0 name-judgement "listn[int](0)" polynomial_pseudo_divide nil) (listn_0 name-judgement "listn[real](0)" polynomial_division nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (make_divlisttype const-decl "DivListType" polynomial_pseudo_divide nil) (mult_divides1 application-judgement "(divides(n))" divides nil) (mult_divides2 application-judgement "(divides(m))" divides nil) (boolean nonempty-type-decl nil booleans nil) (bool nonempty-type-eq-decl nil booleans nil) (NOT const-decl "[bool -> bool]" booleans nil) (OR 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) (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) (>= const-decl "bool" reals nil) (nat nonempty-type-eq-decl nil naturalnumbers nil) (/= const-decl "boolean" notequal nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (numfield nonempty-type-eq-decl nil number_fields nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (= const-decl "[T, T -> boolean]" equalities nil) (list type-decl nil list_adt nil) (PRED type-eq-decl nil defined_types nil) (every adt-def-decl "boolean" list_adt nil) (length def-decl "nat" list_props nil) (< const-decl "bool" reals nil) (max const-decl "{p: real | p >= m AND p >= n}" real_defs nil) (DivListType type-eq-decl nil polynomial_pseudo_divide nil) (posint_plus_nnint_is_posint application-judgement "posint" integers nil) (int_minus_int_is_int application-judgement "int" integers nil)) nil)) (pseudo_div_TCC7 0 (pseudo_div_TCC7-1 nil 3582881577 ("" (skosimp*) (("" (assert) (("" (lift-if) (("" (split -) (("1" (flatten) (("1" (replace -2) (("1" (assert) (("1" (expand "make_divlisttype") (("1" (assert) nil nil)) nil)) nil)) nil)) nil) ("2" (flatten) (("2" (typepred "v!1(g!1, n!1)(h!1, m!1)(1 + i!1)") (("2" (replaces -2 :dir rl) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((nnint_plus_nnint_is_nnint application-judgement "nonneg_int" integers nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (listn_0 name-judgement "listn[int](0)" polynomial_pseudo_divide nil) (listn_0 name-judgement "listn[real](0)" polynomial_division nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (make_divlisttype const-decl "DivListType" polynomial_pseudo_divide nil) (DivListType type-eq-decl nil polynomial_pseudo_divide nil) (max const-decl "{p: real | p >= m AND p >= n}" real_defs nil) (< const-decl "bool" reals nil) (length def-decl "nat" list_props nil) (every adt-def-decl "boolean" list_adt nil) (PRED type-eq-decl nil defined_types nil) (list type-decl nil list_adt nil) (= const-decl "[T, T -> boolean]" equalities nil) (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (numfield nonempty-type-eq-decl nil number_fields nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (/= const-decl "boolean" notequal nil) (nat nonempty-type-eq-decl nil naturalnumbers nil) (>= const-decl "bool" reals 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) (> 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) (OR const-decl "[bool, bool -> bool]" booleans nil) (NOT const-decl "[bool -> bool]" booleans nil) (bool nonempty-type-eq-decl nil booleans nil) (boolean nonempty-type-decl nil booleans nil) (mult_divides2 application-judgement "(divides(m))" divides nil) (mult_divides1 application-judgement "(divides(n))" divides nil) (posint_plus_nnint_is_posint application-judgement "posint" integers nil) (int_minus_int_is_int application-judgement "int" integers nil)) nil)) (pseudo_div_TCC8 0 (pseudo_div_TCC8-1 nil 3582881577 ("" (subtype-tcc) nil nil) ((posint_plus_nnint_is_posint application-judgement "posint" integers nil) (int_minus_int_is_int application-judgement "int" integers nil) (bool nonempty-type-eq-decl nil booleans nil) (NOT const-decl "[bool -> bool]" booleans nil) (>= const-decl "bool" reals nil) (nat nonempty-type-eq-decl nil naturalnumbers nil) (/= const-decl "boolean" notequal nil) (minus_odd_is_odd application-judgement "odd_int" integers nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (listn_0 name-judgement "listn[real](0)" polynomial_division nil) (listn_0 name-judgement "listn[int](0)" polynomial_pseudo_divide 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) (nnint_plus_nnint_is_nnint application-judgement "nonneg_int" integers nil) (int nonempty-type-eq-decl nil integers nil) (integer_pred const-decl "[rational -> boolean]" integers nil) (rational nonempty-type-from-decl nil rationals nil) (rational_pred const-decl "[real -> boolean]" rationals nil) (real 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) (array2list const-decl "{l: listn(n) | FORALL (i: below(n)): a(i) = nth(l, i)}" array2list "structures/") (make_divlisttype const-decl "DivListType" polynomial_pseudo_divide nil)) nil)) (pseudo_div_TCC9 0 (pseudo_div_TCC9-1 nil 3582881577 ("" (subtype-tcc) nil nil) ((posint_plus_nnint_is_posint application-judgement "posint" integers nil) (int_minus_int_is_int application-judgement "int" integers nil) (bool nonempty-type-eq-decl nil booleans nil) (NOT const-decl "[bool -> 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(array2list const-decl "{l: listn(n) | FORALL (i: below(n)): a(i) = nth(l, i)}" array2list "structures/") (make_divlisttype const-decl "DivListType" polynomial_pseudo_divide nil) (nnint_plus_nnint_is_nnint application-judgement "nonneg_int" integers nil)) nil)) (pseudo_div_TCC10 0 (pseudo_div_TCC10-1 nil 3591348813 ("" (subtype-tcc) nil nil) ((posint_plus_nnint_is_posint application-judgement "posint" integers nil) (int_minus_int_is_int application-judgement "int" integers nil) (/= const-decl "boolean" notequal nil) (nat nonempty-type-eq-decl nil naturalnumbers nil) (>= const-decl "bool" reals nil) (NOT const-decl "[bool -> bool]" booleans nil) (bool nonempty-type-eq-decl nil 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) (listn_0 name-judgement "listn[int](0)" polynomial_pseudo_divide nil) (listn_0 name-judgement "listn[real](0)" polynomial_division 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nil)) nil)) (pseudo_div_TCC11 0 (pseudo_div_TCC11-1 nil 3591348813 ("" (subtype-tcc) nil 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) (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) (/= const-decl "boolean" notequal nil) (minus_odd_is_odd application-judgement "odd_int" integers nil)) nil)) (pseudo_div_TCC12 0 (pseudo_div_TCC12-1 nil 3591348813 ("" 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application-judgement "{s: rat | s >= q AND s >= r}" real_defs nil) (int_max application-judgement "{k: int | i <= k AND j <= k}" real_defs nil) (int_plus_int_is_int application-judgement "int" integers 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) (int_minus_int_is_int application-judgement "int" integers nil) (posint_plus_nnint_is_posint application-judgement "posint" integers nil)) nil)) (pseudo_div_lengths 0 (pseudo_div_lengths-1 nil 3582908268 ("" (case "FORALL (g, h: [nat -> int], i, m, n: nat): h(m) /= 0 AND i <= n - m IMPLIES LET psd = pseudo_div(g, n)(h, m)(n-m-i), pd = poly_divide(g, n)(h, m)(n-m-i), qlength = length(psd`quotl), rlength = length(psd`reml) IN psd`qdegl = pd`qdeg AND psd`rdegl = pd`rdeg AND psd`rdegl = max(rlength - 1, 0) and qlength = (IF m = 0 THEN n - m + 1 ELSE i + 1 E (("1" (skeep) (("1" (case "i > n-m") (("1" (hide -2) (("1" (assert) (("1" (expand "pseudo_div") (("1" (expand "make_divlisttype") (("1" (expand "poly_divide") (("1" (expand "make_divtype") (("1" (expand "max") (("1" (lift-if) (("1" (ground) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (inst - "g" "h" "n-m-i" "m" "n") (("1" (assert) nil nil) ("2" (assert) nil nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (induct "i") (("1" (assert) nil nil) ("2" (assert) (("2" (skeep) (("2" (expand "pseudo_div") (("2" (expand "poly_divide") (("2" (lift-if) (("2" (expand "make_divtype") (("2" (expand "make_divlisttype") (("2" (assert) (("2" (ground) (("1" (grind) nil nil) ("2" (expand "max") (("2" (assert) nil nil)) nil) ("3" (expand "length") (("3" (expand "length") (("3" (propax) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("3" (assert) (("3" (expand "max") (("3" (assert) (("3" (skeep) (("3" (skeep) (("3" (inst - "g" "h" "m" 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nil)) nil) ("4" (hide 2) (("4" (skosimp*) (("4" (assert) nil nil)) nil)) nil) ("5" (skosimp*) (("5" (assert) nil nil)) nil)) nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil) ("4" (skosimp*) (("4" (assert) nil nil)) nil)) nil) ((- const-decl "[numfield -> numfield]" number_fields nil) (below type-eq-decl nil nat_types nil) (below type-eq-decl nil naturalnumbers nil) (< const-decl "bool" reals nil) (listn type-eq-decl nil listn "structures/") (nzrat_div_nzrat_is_nzrat application-judgement "nzrat" rationals nil) (minus_odd_is_odd application-judgement "odd_int" integers nil) (NOT const-decl "[bool -> bool]" booleans nil) (nth def-decl "T" list_props nil) (array2list const-decl "{l: listn(n) | FORALL (i: below(n)): a(i) = nth(l, i)}" array2list "structures/") (array2list_it def-decl "{l: listn(n - i) | FORALL (j: subrange(i, n - 1)): a(j) = nth(l, j - i)}" array2list "structures/") (length_singleton formula-decl nil more_list_props "structures/") (max_npreal_0 formula-decl nil min_max "reals/") (length_null formula-decl nil more_list_props "structures/") (real_times_real_is_real application-judgement "real" reals nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (nat_induction formula-decl nil naturalnumbers nil) (pred type-eq-decl nil defined_types nil) (* const-decl "[numfield, numfield -> numfield]" number_fields nil) (i skolem-const-decl "nat" polynomial_pseudo_divide nil) (m skolem-const-decl "nat" polynomial_pseudo_divide nil) (n skolem-const-decl "nat" polynomial_pseudo_divide nil) (posint_plus_nnint_is_posint application-judgement "posint" integers nil) (mult_divides1 application-judgement "(divides(n))" divides nil) (mult_divides2 application-judgement "(divides(m))" divides nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (make_divtype const-decl "DivType" polynomial_division nil) (make_divlisttype const-decl "DivListType" polynomial_pseudo_divide nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (listn_0 name-judgement "listn[real](0)" polynomial_division nil) (listn_0 name-judgement "listn[int](0)" polynomial_pseudo_divide nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (int_minus_int_is_int application-judgement "int" integers nil) (int_max application-judgement "{k: int | i <= k AND j <= k}" real_defs nil) (rat_max application-judgement "{s: rat | s >= q AND s >= r}" real_defs nil) (int_plus_int_is_int application-judgement "int" integers nil) (nnint_plus_posint_is_posint application-judgement "posint" 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) (nat nonempty-type-eq-decl nil naturalnumbers nil) (IMPLIES const-decl "[bool, bool -> bool]" booleans nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (/= const-decl "boolean" notequal nil) (<= const-decl "bool" reals nil) (numfield nonempty-type-eq-decl nil number_fields nil) (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (list type-decl nil list_adt nil) (DivListType type-eq-decl nil polynomial_pseudo_divide nil) (OR const-decl "[bool, bool -> bool]" booleans nil) (> const-decl "bool" reals nil) (= const-decl "[T, T -> boolean]" equalities nil) (PRED type-eq-decl nil defined_types nil) (every adt-def-decl "boolean" list_adt nil) (length def-decl "nat" list_props nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (DivType type-eq-decl nil polynomial_division nil) (max const-decl "{p: real | p >= m AND p >= n}" real_defs nil) (IF const-decl "[boolean, T, T -> T]" if_def nil) (poly_divide def-decl "DivType" polynomial_division nil) (pseudo_div def-decl "{DT: DivListType | (m > n OR i > n - m AND length(DT`reml) = n + 1 AND DT`rdegl = n) OR (m = 0 AND length(DT`reml) = 0 AND DT`rdegl = 0) OR (m > 0 AND length(DT`reml) = m + i AND length(DT`reml) = DT`rdegl + 1)}" polynomial_pseudo_divide nil)) nil)) (HHGGLEM 0 (HHGGLEM-1 nil 3583580204 ("" (grind) nil 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) (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) (nnint_plus_nnint_is_nnint application-judgement "nonneg_int" integers nil) (int_plus_int_is_int application-judgement "int" integers 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) (int_minus_int_is_int application-judgement "int" integers 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) (mult_divides1 application-judgement "(divides(n))" divides nil) (mult_divides2 application-judgement "(divides(m))" divides nil) (GG const-decl "int" polynomial_pseudo_divide nil) (HH const-decl "int" polynomial_pseudo_divide nil) (/= const-decl "boolean" notequal nil) (posint_plus_nnint_is_posint application-judgement "posint" integers nil)) shostak)) (pseudo_div_def_TCC1 0 (pseudo_div_def_TCC1-1 nil 3583595096 ("" (subtype-tcc) nil 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) (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) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (listn_0 name-judgement "listn[int](0)" polynomial_pseudo_divide nil) (listn_0 name-judgement "listn[real](0)" polynomial_division nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (minus_odd_is_odd application-judgement "odd_int" integers nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (DivListType type-eq-decl nil polynomial_pseudo_divide nil) (length def-decl "nat" list_props nil) (every adt-def-decl "boolean" list_adt nil) (PRED type-eq-decl nil defined_types nil) (list type-decl nil list_adt 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) (AND const-decl "[bool, bool -> bool]" booleans nil) (> const-decl "bool" reals nil) (OR const-decl "[bool, bool -> bool]" booleans nil) (/= const-decl "boolean" notequal nil) (posint_plus_nnint_is_posint application-judgement "posint" integers nil) (nnint_plus_nnint_is_nnint application-judgement "nonneg_int" integers nil) (int_minus_int_is_int application-judgement "int" integers nil)) nil)) (pseudo_div_def_TCC2 0 (pseudo_div_def_TCC2-1 nil 3583595096 ("" (skeep) (("" (skeep) (("" (skeep) (("" (skeep) (("" (skeep) (("" (skeep) (("" (lemma "pseudo_div_lengths") (("" (inst?) (("" (assert) (("" (flatten) (("" (case "length(pseudo_div(g, n)(h, m)(i)`reml) - 1>=0") (("1" (expand "max") (("1" (assert) nil nil)) nil) ("2" (expand "pseudo_div" 1) (("2" (expand "make_divlisttype") (("2" (assert) (("2" (lift-if) (("2" (assert) (("2" (ground) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) 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) (mult_divides1 application-judgement "(divides(n))" divides nil) (mult_divides2 application-judgement "(divides(m))" divides nil) (rat_max application-judgement "{s: rat | s >= q AND s >= r}" real_defs nil) (int_max application-judgement "{k: int | i <= k AND j <= k}" real_defs nil) (length_null formula-decl nil more_list_props "structures/") (int_plus_int_is_int application-judgement "int" integers nil) (nnint_plus_nnint_is_nnint application-judgement "nonneg_int" integers nil) (make_divlisttype const-decl "DivListType" polynomial_pseudo_divide nil) (max const-decl "{p: real | p >= m AND p >= n}" real_defs nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (numfield nonempty-type-eq-decl nil number_fields nil) (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (list type-decl nil list_adt nil) (PRED type-eq-decl nil defined_types nil) (every adt-def-decl "boolean" list_adt nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (length def-decl "nat" list_props nil) (/= const-decl "boolean" notequal nil) (DivListType type-eq-decl nil polynomial_pseudo_divide nil) (OR const-decl "[bool, bool -> bool]" booleans nil) (> const-decl "bool" reals nil) (= const-decl "[T, T -> boolean]" equalities nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (pseudo_div def-decl "{DT: DivListType | (m > n OR i > n - m AND length(DT`reml) = n + 1 AND DT`rdegl = n) OR (m = 0 AND length(DT`reml) = 0 AND DT`rdegl = 0) OR (m > 0 AND length(DT`reml) = m + i AND length(DT`reml) = DT`rdegl + 1)}" polynomial_pseudo_divide nil) (listn_0 name-judgement "listn[real](0)" polynomial_division nil) (listn_0 name-judgement "listn[int](0)" polynomial_pseudo_divide nil) (rat_exp application-judgement "rat" exponentiation nil) (nzreal_exp application-judgement "nzreal" exponentiation nil) (posint_plus_nnint_is_posint application-judgement "posint" integers 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) (int_minus_int_is_int application-judgement "int" integers nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (pseudo_div_lengths formula-decl nil polynomial_pseudo_divide nil)) nil)) (pseudo_div_def 0 (pseudo_div_def-6 nil 3601033952 ("" (case "FORALL (R, T: nat, g, h: [nat -> int], i, m, n: nat): (h(m) /= 0 AND i <= n - m AND (FORALL (ii: nat): ii > n IMPLIES g(ii) = 0) AND (FORALL (ii: nat): ii > m IMPLIES h(ii) = 0)) IMPLIES LET psd = pseudo_div(g, n)(h, m)(n-m-i), pd = poly_divide(g, n)(h, m)(n-m-i), qlength = length(psd`quotl), rlength = length(psd`reml) IN (pd`quot = (LAMBDA (k: nat): (1 / h(m)) ^ GG(k, n, m, n-m-i, R, T)) * (LAMBDA (k: nat): IF k <= pd`qdeg AND k >= pd`qdeg - qlength + 1 THEN nth(psd`quotl, k - (pd`qdeg - qlength + 1)) ELSE 0 ENDIF)) AND pd`rem = (LAMBDA (k: nat): (1 / h(m)) ^ HH(k, n, m, n-m-i, R, T)) * (LAMBDA (k: nat): IF k <= pd`rdeg AND m > 0 THEN nth(psd`reml, k) ELSE 0 ENDIF)") (("1" (skeep) (("1" (case "i>n-m") (("1" (hide -2) (("1" (assert) (("1" (expand "poly_divide") (("1" (expand "pseudo_div") (("1" (expand "make_divtype") (("1" (expand "make_divlisttype") (("1" (expand "length") (("1" (split) (("1" (decompose-equality) (("1" (expand "*") (("1" (lift-if) (("1" (ground) nil nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil) ("2" (decompose-equality) (("1" (expand "*") (("1" (expand "HH") (("1" (expand "^") (("1" (expand "expt") (("1" (assert) (("1" (lift-if) (("1" (ground) (("1" (typepred "array2list(1 + n)(g)") (("1" (hide -1) (("1" (inst - "x!1") (("1" (assert) nil nil)) nil)) nil)) nil) ("2" (inst - "x!1") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) (("2" (expand "*") (("2" (expand "HH") (("2" (expand "^") (("2" (expand "expt") (("2" (propax) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (inst - "R" "T" "g" "h" "n-m-i" "m" "n") (("1" (assert) (("1" (replace -3) (("1" (replace -2) (("1" (propax) nil nil)) nil)) nil)) nil) ("2" (assert) nil nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (induct "i") (("1" (assert) nil nil) ("2" (assert) (("2" (skeep) (("2" (assert) (("2" (split +) (("1" (assert) (("1" (expand "pseudo_div") (("1" (expand "poly_divide") (("1" (lift-if) (("1" (expand "make_divtype") (("1" (expand "make_divlisttype") (("1" (split +) (("1" (flatten) (("1" (decompose-equality +) (("1" (expand "*") (("1" (assert) (("1" (typepred "array2list(1+n)(g)") (("1" (hide -1) (("1" (replaces -1) (("1" (assert) (("1" (expand "GG") (("1" (lift-if) (("1" (ground) (("1" (inst - "m+x!1") (("1" (replaces -2 :dir rl) (("1" (grind) nil nil)) nil)) nil) ("2" (inst -4 "x!1") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) 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(make_divtype const-decl "DivType" polynomial_division nil) (n skolem-const-decl "nat" polynomial_pseudo_divide nil) (listn type-eq-decl nil listn "structures/") (below type-eq-decl nil naturalnumbers nil) (array2list const-decl "{l: listn(n) | FORALL (i: below(n)): a(i) = nth(l, i)}" array2list "structures/") (g skolem-const-decl "[nat -> int]" polynomial_pseudo_divide nil) (h skolem-const-decl "[nat -> int]" polynomial_pseudo_divide nil) (m skolem-const-decl "nat" polynomial_pseudo_divide nil) (i skolem-const-decl "nat" polynomial_pseudo_divide nil) (R skolem-const-decl "nat" polynomial_pseudo_divide nil) (T skolem-const-decl "nat" polynomial_pseudo_divide nil) (expt def-decl "real" exponentiation nil) (NOT const-decl "[bool -> bool]" booleans nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (listn_0 name-judgement "listn[int](0)" polynomial_pseudo_divide nil) (listn_0 name-judgement "listn[real](0)" polynomial_division nil) (int_times_even_is_even application-judgement "even_int" integers nil) (nnint_times_nnint_is_nnint application-judgement "nonneg_int" integers nil) (null adt-constructor-decl "(null?)" list_adt nil) (null? adt-recognizer-decl "[list -> boolean]" list_adt nil) (length_null formula-decl nil more_list_props "structures/") (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (rat_times_rat_is_rat application-judgement "rat" rationals nil) (make_divlisttype const-decl "DivListType" polynomial_pseudo_divide nil) (mult_divides2 application-judgement "(divides(m))" divides nil) (mult_divides1 application-judgement "(divides(n))" divides nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (nnint_plus_nnint_is_nnint application-judgement "nonneg_int" integers nil) (posint_plus_nnint_is_posint application-judgement "posint" integers nil) (int_minus_int_is_int application-judgement "int" integers nil) (rat_exp application-judgement "rat" exponentiation nil) (nzreal_exp application-judgement "nzreal" exponentiation nil) (nzrat_div_nzrat_is_nzrat application-judgement "nzrat" rationals nil) (int_plus_int_is_int application-judgement "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) (nat nonempty-type-eq-decl nil naturalnumbers nil) (IMPLIES const-decl "[bool, bool -> bool]" booleans nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (/= const-decl "boolean" notequal nil) (<= const-decl "bool" reals nil) (numfield nonempty-type-eq-decl nil number_fields nil) (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (> const-decl "bool" reals nil) (= const-decl "[T, T -> boolean]" equalities nil) (list type-decl nil list_adt nil) (DivListType type-eq-decl nil polynomial_pseudo_divide nil) (OR const-decl "[bool, bool -> bool]" booleans nil) (PRED type-eq-decl nil defined_types nil) (every adt-def-decl "boolean" list_adt nil) (length def-decl "nat" list_props nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (DivType type-eq-decl nil polynomial_division nil) (* const-decl "[T -> real]" real_fun_ops "reals/") (^ const-decl "real" exponentiation nil) (nznum nonempty-type-eq-decl nil number_fields nil) (/ const-decl "[numfield, nznum -> numfield]" number_fields nil) (GG const-decl "int" polynomial_pseudo_divide nil) (IF const-decl "[boolean, T, T -> T]" if_def nil) (< const-decl "bool" reals nil) (below type-eq-decl nil nat_types nil) (nth def-decl "T" list_props nil) (HH const-decl "int" polynomial_pseudo_divide nil) (poly_divide def-decl "DivType" polynomial_division nil) (pseudo_div def-decl "{DT: DivListType | (m > n OR i > n - m AND length(DT`reml) = n + 1 AND DT`rdegl = n) OR (m = 0 AND length(DT`reml) = 0 AND DT`rdegl = 0) OR (m > 0 AND length(DT`reml) = m + i AND length(DT`reml) = DT`rdegl + 1)}" polynomial_pseudo_divide nil)) nil) (pseudo_div_def-5 nil 3593447223 ("" (case "FORALL (R, T: nat, g, h: [nat -> int], i, m, n: nat): (h(m) /= 0 AND i <= n - m AND (FORALL (ii: nat): ii > n IMPLIES g(ii) = 0) AND (FORALL (ii: nat): ii > m IMPLIES h(ii) = 0)) IMPLIES LET psd = pseudo_div(g, n)(h, m)(n-m-i), pd = poly_divide(g, n)(h, m)(n-m-i), qlength = length(psd`quotl), rlength = length(psd`reml) IN (pd`quot = (LAMBDA (k: nat): (1 / h(m)) ^ GG(k, n, m, n-m-i, R, T)) * (LAMBDA (k: nat): IF k <= pd`qdeg AND k >= pd`qdeg - qlength + 1 THEN nth(psd`quotl, k - (pd`qdeg - qlength + 1)) ELSE 0 ENDIF)) AND pd`rem = (LAMBDA (k: nat): (1 / h(m)) ^ HH(k, n, m, n-m-i, R, T)) * (LAMBDA (k: nat): IF k <= pd`rdeg AND m > 0 THEN nth(psd`reml, k) ELSE 0 ENDIF)") (("1" (skeep) (("1" (inst - "R" "T" "g" "h" "n-m-i" "m" "n") (("1" (assert) (("1" (replace -3) (("1" (replace -4) (("1" (propax) nil nil)) nil)) nil)) nil) ("2" (assert) nil nil)) nil)) nil) ("2" (hide 2) (("2" (induct "i") (("1" (assert) nil nil) ("2" (assert) (("2" (skeep) (("2" (assert) (("2" (split +) (("1" (assert) (("1" (expand "pseudo_div") (("1" (expand "poly_divide") (("1" (lift-if) (("1" (expand "make_divtype") (("1" (expand "make_divlisttype") (("1" (split +) (("1" (flatten) (("1" (decompose-equality +) (("1" (expand "*") (("1" (assert) (("1" (typepred "array2list(1+n)(g)") (("1" (hide -1) (("1" (replaces -1) (("1" (assert) (("1" (expand "GG") (("1" (lift-if) (("1" (ground) (("1" (inst - 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--------------------
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