(product_natnil 3351946780 ("" (assuming-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"bool" reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil naturalnumbers nil )nil integers nil )"(total_order?[real])" nil )"(total_order?[real])" nil ))nil ))nil 3352138082"" (skosimp*)"" (inst-cp + 0)"" (inst + "x!1" )"1" (assert ) nil nil ) ("2" (assert ) nil nil )) nil ))nil ))nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil )"(total_order?[real])" nil )"(total_order?[real])" nil ))nil ))nil 3352138082 ("" (judgement-tcc) nil nil ) nil nil ))nil 3403006081"" (skosimp*) (("" (assert ) nil nil )) nil )"int" integers nil )"(total_order?[real])" nil ))nil ))nil 3308675684"" (skosimp*) (("" (rewrite "product_shift_T" ) nil nil )) nil )"nonneg_int" nil )"int" integers nil )nil product nil )OR const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )nil product nil )nil product nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil ))nil ))nil 3308675684"" (skosimp*) (("" (inst + 0) (("" (assert ) nil nil )) nil )) nil )nil nil ))nil 3352138082 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )"bool" reals nil )nil naturalnumbers nil )"int" integers nil )"(total_order?[real])" nil ))nil ))nil 3403006081"" (skosimp*) (("" (assert ) nil nil )) nil )"(strict_total_order?[real])" real_props nil )"int" integers nil )"(total_order?[real])" nil ))nil ))nil 3308675684"" (skosimp*)"" (lemma "product_shift_T2" )"" (inst?)"" (inst - "high!1" "low!1" "0 - m!1" )"" (assert )"" (prop)"1" (replace -1 :hide? t)"1" (rewrite "product_eq" )"1" (skosimp*) (("1" (assert ) nil nil )) nil )) nil ))nil )"2" (case "high!1)"1" (expand "product" ) (("1" (assert ) nil nil )) nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )nil naturalnumbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil product nil )"int" integers nil )OR const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )nil product nil )nil product nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"[bool, bool -> bool]" booleans nil )NOT const-decl "[bool -> bool]" booleans nil )"bool" reals nil )IF const-decl "[boolean, T, T -> T]" if_def nil )nil product nil )"(strict_total_order?[real])" real_props nil )"odd_int" integers nil )"real" product nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"int" integers nil ))nil ))nil 3308675684"" (skosimp*)"" (lemma "product_shift_T2" )"" (inst?)"" (inst - "high!1" "low!1" "0-m!1" )"" (assert )"" (prop)"1" (replace -1 :hide? t)"1" (rewrite "product_eq" )"1" (expand "~" )"1" (assert )"1" (skosimp*)"1" (assert )"1" (lift-if) (("1" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (case "high!1)"1" (expand "product" ) (("1" (assert ) nil nil )) nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )nil naturalnumbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil product nil )"int" integers nil )OR const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )nil product nil )nil product nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"nat" naturalnumbers nil )IF const-decl "[boolean, T, T -> T]" if_def nil )nil product nil )"(strict_total_order?[real])" real_props nil )"odd_int" integers nil )"real" product nil )"(strict_total_order?[real])" real_props nil )"bool" reals nil )"(total_order?[real])" nil )"int" integers nil ))nil ))nil 3411831032 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )"bool" reals nil )nil naturalnumbers nil )"int" integers nil )"(total_order?[real])" nil )"(total_order?[real])" nil ))nil ))nil 3411831032 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )"bool" reals nil )nil naturalnumbers nil )"int" integers nil )"(total_order?[real])" nil )"(total_order?[real])" nil ))nil ))nil 3411831032 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )"bool" reals nil )nil naturalnumbers nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"int" integers nil )"(strict_total_order?[real])" real_props nil ))nil ))nil 3411831064"" (skosimp*)"" (rewrite "product_shift_T2" )"1" (rewrite "product_restrict_eq" )"1" (hide 2)"1" (expand "restrict" ) (("1" (propax) nil nil )) nil )) nil )"2" (skosimp*) (("2" (assert ) nil nil )) nil ))nil )"2" (case "low!1 <= high!1" )"1" (assert ) nil nil ) ("2" (grind) nil nil )) nil ))nil ))nil )nil product nil )OR const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )nil product nil )nil product nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil )"int" integers nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"[T -> real]" product nil )"bool" reals nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )IF const-decl "[boolean, T, T -> T]" if_def nil )nil product nil )NOT const-decl "[bool -> bool]" booleans nil )"[bool, bool -> bool]" booleans nil )"real" product nil )"(strict_total_order?[real])" real_props nil ))nil ))nil 3308675684"" (skosimp*) (("" (rewrite "product_first" ) nil nil )) nil )nil product nil )OR const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )nil product nil )nil product nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil ))nil ))nil 3352162210"" (skosimp*) (("" (inst + 0) (("" (assert ) nil nil )) nil )) nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil )"(total_order?[real])" nil )"int" integers nil )"(total_order?[real])" nil ))nil ))nil 3403006081"" (skosimp*) (("" (assert ) nil nil )) nil )"int" integers nil )"(total_order?[real])" nil )"int" integers nil )"(total_order?[real])" nil ))nil ))nil 3308675684"" (skosimp*) (("" (rewrite "product_split" ) nil nil )) nil )nil product nil )OR const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )nil product nil )nil product nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil ))nil ))nil 3620035296 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )"bool" reals nil )nil naturalnumbers nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"int" integers nil )"int" integers nil ))nil ))nil 3620035321"" case "FORALL (n:nat, F:[nat->real], high:int, low:nat): AND n = high-low) IMPLIESLAMBDA (j: nat):IF high + low - j < 0 THEN 0") "1" (skeep)"1" (inst?)"1" (inst -1 "high-low" )"1" (ground) nil nil ) ("2" (ground) nil nil )) nil ))nil ))nil )"2" (hide 2)"2" (induct "n" )"1" (skeep)"1" (both-sides "+" "low" -2)"1" (simplify -2)"1" (replace -2)"1" (lemma "product_eq_arg" )"1" (inst-cp -1 "F" "high" )"1" "LAMBDA (j: nat): IF high + high - j < 0 THEN 0" " high") "1" (hide -1)"1" (replace -1)"1" (replace -2)"1" (lift-if 1)"1" "1" "1" (ground) nil nil ))nil )"2" "2" (ground) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (ground) nil nil )"3" (skeep) (("3" (ground) nil nil )) nil ))nil )"2" (ground) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (skeep)"2" (skeep)"2" (lemma "product_first_ge" )"2" (inst -1 "F" "high" "low" )"2" (split -1)"1" (replace -1)"1" (lemma "product_split_ge" )"1" "LAMBDA (j: nat): IF high + low - j < 0 THEN 0 ELSE F(high + low - j) ENDIF" "high" "high-1" "low" )"1" (split -1)"1" (replace -1)"1" (hide (-1 -2))"1" "1" rewrite "product_eq_arg" )"1" "F(low) = 0" )"1" (ground) nil nil )"2" "F(low)" )"2" "2" "product_shift" )"2" "F" "high-1" "low" "1" )"2" "2" replace -1)"2" "G" "LAMBDA (n: nat): F(1 + n)" )"2" "G" "high-1" "low" )"2" "1" replace -1)"1" "product_eq" )"1" "LAMBDA (j: nat): IF high - 1 + low - j < 0 THEN 0" "LAMBDA (j: nat): IF high - j + low < 0 THEN 0" "high-1" "low" )"1" "1" "1" nil nil )"2" "2" "2" "2" "2" "2" "1" "1" "1" nil nil )"2" "2" nil nil ))nil ))nil ))nil )"2" "2" "1" "1" nil nil ))nil )"2" "2" "2" expand "G" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" nil nil ))nil )"3" "3" nil nil ))nil ))nil ))nil ))nil )"2" nil nil )"3" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" (ground) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (ground) nil nil )"3" (ground) nil nil ))nil )"2" (skeep) (("2" (ground) nil nil )) nil ))nil ))nil ))nil )"2" (ground) nil nil ))nil ))nil ))nil ))nil ))nil )"3" (hide 2)"3" (skeep) (("3" (skeep) (("3" (ground) nil nil )) nil ))nil ))nil )"4" (hide 2) (("4" (skeep) (("4" (assert ) nil nil )) nil ))nil ))nil ))nil )"3" (hide 2)"3" (skeep) (("3" (skeep) (("3" (ground) nil nil )) nil )) nil ))nil )"4" (hide 2) (("4" (skeep) (("4" (grind) nil nil )) nil )) nil ))nil )nil product_nat nil )nil product_nat nil )"posint" nil )"bool" functions nil )"(bijective?[T, T])" identity nil )"bool" booleans nil )"boolean" notequal nil )"real" product_nat nil )"[numfield, nznum -> numfield]" number_fields nil )nil number_fields nil )nil reals nil )nil real_props nil )"nzreal" nil )nil product_nat nil )nil product nil )"[nat -> real]" product_nat nil )"posint" nil )"real" reals nil )"nat" product_nat nil )"int" product_nat nil )nil product nil )"(divides(m))" divides nil )"(divides(n))" divides nil )"even_int" integersnil )"[numfield, numfield -> numfield]" number_fields nil )"(strict_total_order?[real])" real_props nil )"int" product_nat nil )"odd_int" integers nil )"nonneg_int" nil )nil naturalnumbers nil )nil defined_types nil )"(total_order?[real])" nil )"int" product_nat nil )"nat" product_nat nil )"(total_order?[real])" nil )NOT const-decl "[bool -> bool]" booleans nil )"int" integers nil )"int" integers nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil )"[bool, bool -> bool]" booleans nil )AND const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )"[T, T -> boolean]" equalities nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )OR const-decl "[bool, bool -> bool]" booleans nil )nil product nil )nil product nil )"real" product nil )IF const-decl "[boolean, T, T -> T]" if_def nil )"bool" reals nil )"[numfield, numfield -> numfield]" number_fieldsnil ))nil 3403521637"" (skosimp*) (("" (inst + "0" ) (("" (assert ) nil nil )) nil )) nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil )"(total_order?[real])" nil )"(total_order?[real])" nil ))nil ))nil 3403521650"" (skosimp*)"" (expand "product" ) (("" (postpone) nil nil )) nil )) nil )"real" product nil )) shostak)))quality 100%
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