(tmod
(ml3 0
(ml3-1 nil 3406631585
("" (skosimp*)
(("" (expand "abs" )
(("" (lift-if)
(("" (case-replace "floor(i!1 / m!1) = i!1/m!1" )
(("1" (auto-rewrite-theory "real_props" )
(("1" (ground) nil nil )) nil )
("2" (ground)
(("1" (lemma "both_sides_times_pos_lt1" )
(("1" (inst -1 "m!1" "floor(i!1 / m!1)" "i!1 / m!1" )
(("1" (flatten) (("1" (assert ) nil nil )) nil )) nil ))
nil )
("2" (typepred "floor(i!1 / m!1)" )
(("2" (lemma "both_sides_times_pos_lt1" )
(("2"
(inst -1 "m!1" "i!1 / m!1" " 1 + floor(i!1 / m!1)" )
(("2" (flatten)
(("2" (hide -1) (("2" (assert ) nil nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil )
(rat_div_nzrat_is_rat application-judgement "rat" rationals nil )
(number nonempty-type-decl nil numbers nil )
(boolean nonempty-type-decl nil booleans nil )
(= const-decl "[T, T -> boolean]" equalities 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 )
(integer nonempty-type-from-decl nil integers nil )
(bool nonempty-type-eq-decl nil booleans nil )
(AND const-decl "[bool, bool -> bool]" booleans nil )
(<= const-decl "bool" reals nil ) (< const-decl "bool" reals nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil )
(floor const-decl "{i | i <= x & x < i + 1}" floor_ceil nil )
(/= const-decl "boolean" notequal nil )
(nznum nonempty-type-eq-decl nil number_fields nil )
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil )
(int 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 )
(posnat nonempty-type-eq-decl nil integers nil )
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(rat_minus_rat_is_rat application-judgement "rat" rationals nil )
(rat_times_rat_is_rat application-judgement "rat" rationals nil )
(zero_not_lt_zero formula-decl nil real_props nil )
(div_cancel1 formula-decl nil real_props nil )
(NOT const-decl "[bool -> bool]" booleans nil )
(int_plus_int_is_int application-judgement "int" integers nil )
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(both_sides_times_pos_lt1 formula-decl nil real_props nil )
(nonneg_real nonempty-type-eq-decl nil real_types nil )
(posreal nonempty-type-eq-decl nil real_types nil )
(minus_int_is_int application-judgement "int" integers nil )
(int_minus_int_is_int application-judgement "int" integers nil )
(mult_divides2 application-judgement "(divides(m))" divides nil )
(mult_divides1 application-judgement "(divides(n))" divides nil ))
nil ))
(ml4 0
(ml4-1 nil 3406631585
("" (skosimp*)
(("" (expand "abs" )
(("" (lift-if)
(("" (ground)
(("1" (typepred "floor(-i!1 / m!1)" )
(("1" (lemma "both_sides_times_pos_lt1" )
(("1"
(inst -1 "m!1" "-i!1 / m!1" "1 + floor(-i!1 / m!1)" )
(("1" (flatten)
(("1" (hide -1) (("1" (assert ) nil nil )) nil )) nil ))
nil ))
nil ))
nil )
("2" (lemma "both_sides_times_pos_lt1" )
(("2" (inst -1 "m!1" "floor(-i!1 / m!1)" "-i!1 / m!1" )
(("2" (assert )
(("2" (case-replace "floor(-i!1 / m!1) = -i!1/m!1" )
(("1" (assert )
(("1" (auto-rewrite-theory "real_props" )
(("1" (assert ) nil nil )) nil ))
nil )
("2" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil )
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(rat_div_nzrat_is_rat application-judgement "rat" rationals nil )
(both_sides_times_pos_lt1 formula-decl nil real_props nil )
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(nonneg_real nonempty-type-eq-decl nil real_types nil )
(posreal nonempty-type-eq-decl nil real_types 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 )
(integer nonempty-type-from-decl nil integers 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 -> numfield]" number_fields nil )
(floor const-decl "{i | i <= x & x < i + 1}" floor_ceil nil )
(/= const-decl "boolean" notequal nil )
(nznum nonempty-type-eq-decl nil number_fields nil )
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil )
(- const-decl "[numfield -> numfield]" number_fields nil )
(int 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 )
(posnat nonempty-type-eq-decl nil integers nil )
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(int_plus_int_is_int application-judgement "int" integers nil )
(mult_divides2 application-judgement "(divides(m))" divides nil )
(mult_divides1 application-judgement "(divides(n))" divides nil )
(minus_int_is_int application-judgement "int" integers nil ))
nil ))
(mod_TCC1 0
(mod_TCC1-1 nil 3406631585
("" (skosimp*)
(("" (expand "tdiv" )
(("" (case "j!1 >= 0" )
(("1" (lemma "ml3" )
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(("1" (expand "abs" )
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("2" (assert ) nil nil ))
nil ))
nil )
("2" (lemma "ml4" )
(("2" (inst -1 "i!1" "-j!1" )
(("1" (expand "abs" )
(("1" (lift-if) (("1" (ground) nil nil )) nil )) nil )
("2" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((tdiv const-decl "integer" tdiv nil )
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(- const-decl "[numfield -> numfield]" number_fields nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
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(ml3 formula-decl nil tmod nil )
(nzint_abs_is_pos application-judgement "{j: posint | j >= i}"
real_defs nil )
(int_abs_is_nonneg application-judgement "{j: nonneg_int | j >= i}"
real_defs nil )
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil )
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
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(minus_nzint_is_nzint application-judgement "nzint" integers nil )
(nonneg_int nonempty-type-eq-decl nil integers nil )
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(> const-decl "bool" reals nil )
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real_props nil )
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(number nonempty-type-decl nil numbers nil )
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(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 )
(bool nonempty-type-eq-decl nil booleans nil )
(>= const-decl "bool" reals nil )
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(integer_pred const-decl "[rational -> boolean]" integers nil )
(int nonempty-type-eq-decl nil integers nil )
(/= const-decl "boolean" notequal nil )
(nonzero_integer nonempty-type-eq-decl nil integers nil )
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(mult_divides2 application-judgement "(divides(m))" divides nil )
(mult_divides1 application-judgement "(divides(n))" divides nil ))
nil ))
(mod_even 0
(mod_even-1 nil 3406631585
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nil ))
nil ))
nil ))
nil )
((mult_divides1 application-judgement "(divides(n))" divides nil )
(mult_divides2 application-judgement "(divides(m))" divides nil )
(mod const-decl "{k | abs(k) < abs(j)}" tmod nil )
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nil )
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(numfield nonempty-type-eq-decl nil number_fields nil )
(/= const-decl "boolean" notequal nil )
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(/ const-decl "[numfield, nznum -> numfield]" number_fields nil )
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(integer? const-decl "bool" integers nil )
(tdiv const-decl "integer" tdiv nil ))
nil ))
(mod_neg 0
(mod_neg-1 nil 3406631585
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(("1" (assert ) nil nil )) nil ))
nil ))
nil ))
nil )
("2" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((rat_div_nzrat_is_rat application-judgement "rat" rationals nil )
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(- const-decl "[numfield -> numfield]" number_fields nil )
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil )
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(number_field nonempty-type-from-decl nil number_fields nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
(number_field_pred const-decl "[number -> boolean]" number_fields
nil )
(= const-decl "[T, T -> boolean]" equalities nil )
(boolean nonempty-type-decl nil booleans nil )
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(closed_neg formula-decl nil integers nil )
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(floor_int formula-decl nil floor_ceil nil )
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(rat_times_rat_is_rat application-judgement "rat" rationals nil )
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(integer? const-decl "bool" integers nil )
(tdiv const-decl "integer" tdiv nil ))
nil ))
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("2" (case "integer_pred(--(i!1/j!1))" )
(("1" (assert ) nil nil )
("2" (assert ) nil nil ))
nil )
("3" (lift-if)
(("3" (ground)
(("3" (hide 1)
(("3"
(lemma "integers.closed_neg" )
(("3"
(inst - "-(i!1/j!1)" )
(("3" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
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(real_pred const-decl "[number_field -> boolean]" reals nil )
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil )
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(number_field nonempty-type-from-decl nil number_fields nil )
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(number_field_pred const-decl "[number -> boolean]" number_fields
nil )
(= const-decl "[T, T -> boolean]" equalities nil )
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(minus_rat_is_rat application-judgement "rat" rationals nil )
(integer nonempty-type-from-decl nil integers nil )
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(rat_minus_rat_is_rat application-judgement "rat" rationals nil )
(rat_times_rat_is_rat application-judgement "rat" rationals nil )
(closed_neg formula-decl nil integers nil )
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(integer? const-decl "bool" integers nil )
(floor_neg formula-decl nil floor_ceil nil )
(tdiv const-decl "integer" tdiv nil ))
nil ))
(mod_eq_arg 0
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nil )
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(int nonempty-type-eq-decl nil integers nil )
(nonzero_integer nonempty-type-eq-decl nil integers nil )
(nzrat_div_nzrat_is_nzrat application-judgement "nzrat" rationals
nil )
(int_minus_int_is_int application-judgement "int" integers nil )
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(mult_divides1 application-judgement "(divides(n))" divides nil )
(mult_divides2 application-judgement "(divides(m))" divides nil ))
nil ))
(mod_lt 0
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nil ))
nil ))
nil )
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(("2" (lift-if)
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nil ))
nil ))
nil )
("3" (rewrite "floor_small" )
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(("3" (ground) (("3" (rewrite "pos_div_ge" ) nil nil ))
nil ))
nil ))
nil )
("4" (rewrite "floor_small" )
(("4" (lift-if)
(("4" (ground) (("4" (rewrite "pos_div_ge" ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil )
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"(strict_total_order?[real])" real_props nil )
(minus_int_is_int application-judgement "int" integers nil )
(minus_nzint_is_nzint application-judgement "nzint" integers nil )
(minus_odd_is_odd application-judgement "odd_int" integers nil )
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real_props nil )
(nonzero_real nonempty-type-eq-decl nil reals nil )
(pos_div_ge formula-decl nil real_props nil )
(nzint_times_nzint_is_nzint application-judgement "nzint" integers
nil )
(int_times_even_is_even application-judgement "even_int" integers
nil )
(int_abs_is_nonneg application-judgement "{j: nonneg_int | j >= i}"
real_defs nil )
(nzint_abs_is_pos application-judgement "{j: posint | j >= i}"
real_defs nil )
(nonzero_integer nonempty-type-eq-decl nil integers nil )
(/= const-decl "boolean" notequal nil )
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nil )
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nil ))
(mod_lt_nat 0
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(("" (rewrite "mod_lt" )
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nil ))
nil )
((mod_lt formula-decl nil tmod nil )
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real_defs nil )
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real_defs nil )
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(sgn const-decl "int" tdiv nil )
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nil ))
nil ))
(mod_sum_pos 0
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(("" (expand "mod" )
(("" (expand "tdiv" )
(("" (lemma "floor_plus_int" )
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nil ))
nil ))
nil ))
nil )
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shostak))
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("2" (rewrite "floor_small" )
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(mod_0_divides 0
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nil )))
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