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nil ))
nil ))
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nil ))
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real_props nil )
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nil ))
nil ))
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(number_field_pred const-decl "[number -> boolean]" number_fields
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(boolean nonempty-type-decl nil booleans nil )
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(sigma_lt 0
(sigma_lt-1 nil 3255975799
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(("1" (replace -1 * rl)
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nil ))
nil ))
nil )
("2" (propax) nil nil )
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nil )
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real_props nil )
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(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_minus_real_is_real application-judgement "real" reals nil ))
nil ))
(sigma_with 0
(sigma_with-1 nil 3255975799
("" (skosimp*)
(("" (case "i!1 = low!1" )
(("1" (rewrite "sigma_first" )
(("1" (lemma "sigma_first" )
(("1" (inst -1 "G!1" "high!1" "low!1" )
(("1" (assert )
(("1" (split -1 )
(("1" (assert )
(("1" (replace -1 )
(("1" (replace -5 )
(("1" (hide -1 -5 )
(("1" (assert )
(("1" (replace -1 )
(("1" (assert )
(("1" (rewrite "sigma_eq" ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (assert )
(("2" (expand "sigma" )
(("2" (assert )
(("2" (expand "sigma" ) (("2" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (expand "sigma" )
(("2" (assert )
(("2" (expand "sigma" ) (("2" (assert ) nil nil )) nil )) nil ))
nil))
nil)
("2" (case-replace "i!1 = high!1" )
(("1" (rewrite "sigma_last" )
(("1" (lemma "sigma_last" )
(("1" (inst -1 "G!1" "high!1" "low!1" )
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(("1" (replace -1 )
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(("1" (assert)
(("1" (replace -4 )
(("1" (assert)
(("1" (rewrite "sigma_eq" ) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (lemma "sigma_split" )
(("2" (inst?)
(("2" (inst -1 "i!1" )
(("2" (assert)
(("2" (replace -1 )
(("2" (hide -1 )
(("2" (lemma "sigma_split" )
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(("2" (hide -1 )
(("2"
(lemma "sigma_last" )
(("2"
(inst-cp -1 "F!1" "i!1" "low!1" )
(("2"
(inst -1 "G!1" "i!1" "low!1" )
(("2"
(assert)
(("2"
(replace -1 )
(("2"
(hide -1 )
(("2"
(replace -1 )
(("2"
(hide -1 )
(("2"
(replace -3 )
(("2"
(hide -3 )
(("2"
(ground)
(("2"
(case-replace
"sigma(low!1, i!1 - 1, G!1 WITH [i!1 := a!1]) = sigma(low!1, i!1 - 1, G!1)" )
(("1"
(hide -1 )
(("1"
(assert)
(("1"
(rewrite
"sigma_eq" )
nil
nil))
nil))
nil)
("2"
(hide 4 )
(("2"
(rewrite
"sigma_eq" )
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
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nil))
nil))
nil))
nil))
nil))
nil))
nil)
((T_low type -eq-decl nil old_sigma nil)
(<= const-decl "bool" reals nil)
(OR const-decl "[bool, bool -> bool]" booleans nil)
(bool nonempty-type -eq-decl nil booleans nil)
(T formal-subtype-decl nil old_sigma nil)
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(int nonempty-type -eq-decl nil integers nil)
(integer_pred const-decl "[rational -> boolean]" integers nil)
(rational nonempty-type -from-decl nil rationals nil)
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(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)
(= const-decl "[T, T -> boolean]" equalities nil)
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(minus_odd_is_odd application-judgement "odd_int" integers nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
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("3" (hide 2 )
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(("2" (assert) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("3" (skosimp*) nil nil) ("4" (skosimp*) nil nil)
("5" (skosimp*) nil nil) ("6" (skosimp*) nil nil)
("7" (skosimp*) nil nil) ("8" (skosimp*) nil nil)
("9" (skosimp*) nil nil) ("10" (skosimp*) nil nil)
("11" (skosimp*) nil nil))
nil))
nil)
("3" (hide 2 ) (("3" (skosimp*) nil nil)) nil)
("4" (skosimp*) nil nil) ("5" (skosimp*) nil nil))
nil)
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nil))
nil))
nil))
nil)
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(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)
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(number_field nonempty-type -from-decl nil number_fields nil)
(number_field_pred const-decl "[number -> boolean]" number_fields
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(boolean nonempty-type -decl nil booleans nil)
(number nonempty-type -decl nil numbers nil)
(closed_plus formula-decl nil rationals nil)
(closed_plus formula-decl nil integers nil)
(nnint_plus_posint_is_posint application-judgement "posint"
integers nil)
(j!1 skolem-const-decl "nat" old_sigma nil)
(npreal_plus_npreal_is_npreal application-judgement "npreal"
real_types nil)
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil)
(sigma_eq_arg formula-decl nil old_sigma nil)
(nat_induction formula-decl nil naturalnumbers nil)
(pred type -eq-decl nil defined_types nil)
(Fnpi!1 skolem-const-decl "[T -> nonpos_int]" old_sigma nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(high!1 skolem-const-decl "T_high" old_sigma nil)
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(>= const-decl "bool" reals nil)
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(IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
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(AND const-decl "[bool, bool -> bool]" booleans nil)
(sigma def-decl "real" old_sigma nil)
(nonpos_int nonempty-type -eq-decl nil integers nil)
(real_gt_is_strict_total_order name-judgement
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nil)
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("" (skosimp*)
(("" (lemma "sigma_middle" )
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nil))
nil))
nil)
((sigma_middle formula-decl nil old_sigma nil)
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(int_plus_int_is_int application-judgement "int" integers nil)
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real_props nil)
(nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
real_types nil)
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(T_high type -eq-decl nil old_sigma nil)
(<= const-decl "bool" reals nil)
(OR const-decl "[bool, bool -> bool]" booleans nil)
(nonneg_real nonempty-type -eq-decl nil real_types nil)
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(T formal-subtype-decl nil old_sigma nil)
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(int nonempty-type -eq-decl nil integers nil)
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(number_field nonempty-type -from-decl nil number_fields nil)
(number_field_pred const-decl "[number -> boolean]" number_fields
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(boolean nonempty-type -decl nil booleans nil)
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nil))
(sigma_nn_ge_comps 0
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("" (skosimp*)
(("" (lemma "sigma_middle" )
(("" (inst?)
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nil))
nil))
nil)
((sigma_middle formula-decl nil old_sigma nil)
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(int_plus_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)
(nil application-judgement "nonneg_real" old_sigma nil)
(nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
real_types nil)
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(OR const-decl "[bool, bool -> bool]" booleans nil)
(nonneg_real nonempty-type -eq-decl nil real_types nil)
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(bool nonempty-type -eq-decl nil booleans nil)
(T formal-subtype-decl nil old_sigma nil)
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(int nonempty-type -eq-decl nil integers nil)
(integer_pred const-decl "[rational -> boolean]" integers nil)
(rational nonempty-type -from-decl nil rationals nil)
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(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
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(boolean nonempty-type -decl nil booleans nil)
(number nonempty-type -decl nil numbers nil)
(real_plus_real_is_real application-judgement "real" reals nil))
nil)
(sigma_nn_ge_comps-1 nil 3255975877
("" (skosimp*)
(("" (lemma "sigma_middle" )
(("" (inst?)
(("" (inst -1 "i!1" )
(("" (assert) (("" (replace -1 ) (("" (assert) nil nil)) nil))
nil))
nil))
nil))
nil))
nil)
nil shostak))
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("" (skosimp*)
(("" (inst + "low!1" )
(("1" (assert) nil nil)
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nil))
nil)
((low!1 skolem-const-decl "T_low" old_sigma nil)
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(<= const-decl "bool" reals nil)
(T formal-subtype-decl nil old_sigma nil)
(OR const-decl "[bool, bool -> bool]" booleans nil)
(bool nonempty-type -eq-decl nil booleans nil)
(T_pred const-decl "[int -> boolean]" old_sigma 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)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(int_plus_int_is_int application-judgement "int" integers nil)
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(T_high type -eq-decl nil old_sigma nil)
(T_pred_lem formula-decl nil old_sigma nil))
nil))
(sum_it_def_TCC2 0
(sum_it_def_TCC2-1 nil 3255975799
("" (skosimp*) (("" (use "T_pred_lem" ) (("" (assert) nil nil)) nil))
nil)
((T_pred_lem formula-decl nil old_sigma nil)
(T_low type -eq-decl nil old_sigma nil)
(<= const-decl "bool" reals nil)
(T formal-subtype-decl nil old_sigma nil)
(T_pred const-decl "[int -> boolean]" old_sigma nil)
(OR const-decl "[bool, bool -> bool]" booleans 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)
(T_high type -eq-decl nil old_sigma 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))
nil))
(sum_it_def_TCC3 0
(sum_it_def_TCC3-1 nil 3352118808 ("" (subtype-tcc) nil nil)
((int_minus_int_is_int application-judgement "int" integers 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)
(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)
(T_pred const-decl "[int -> boolean]" old_sigma nil)
(T formal-subtype-decl nil old_sigma nil)
(<= const-decl "bool" reals nil)
(T_low type -eq-decl nil old_sigma nil)
(T_high type -eq-decl nil old_sigma nil)
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil)
(int_abs_is_nonneg application-judgement "{j: nonneg_int | j >= i}"
real_defs 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_plus_int_is_int application-judgement "int" integers nil))
nil))
(sum_it_prop_TCC1 0
(sum_it_prop_TCC1-1 nil 3352142246
("" (skosimp* t)
(("" (prop)
(("1" (inst + "low!1" ) (("1" (assert) nil nil)) nil)
("2" (skosimp*)
(("2" (inst + "j!1" ) (("2" (assert) nil nil)) nil)) nil)
("3" (skosimp*)
(("3" (inst + "low!1" ) (("3" (assert) nil nil)) nil)) nil)
("4" (skosimp*)
(("4" (inst + "j!1" ) (("4" (assert) nil nil)) nil)) nil))
nil))
nil)
((int_plus_int_is_int application-judgement "int" integers nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(low!1 skolem-const-decl "T_low" old_sigma 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)
(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)
(T_pred const-decl "[int -> boolean]" old_sigma nil)
(T formal-subtype-decl nil old_sigma nil)
(<= const-decl "bool" reals nil)
(T_high type -eq-decl nil old_sigma nil)
(T_low type -eq-decl nil old_sigma nil))
nil))
(sum_it_prop 0
(sum_it_prop-1 nil 3255975799
("" (skosimp*)
(("" (case "high!1<low!1" )
(("1" (assert)
(("1" (expand "sum_it_def" )
(("1" (expand "sigma" ) (("1" (propax) nil nil)) nil)) nil))
nil)
("2" (lemma "subrange_induction[low!1,high!1]" )
(("1"
(inst -1
"lambda(lh:T| low!1 <= lh AND lh <= high!1): sum_it_def((high!1-lh)+low!1, high!1, F!1, sigma(low!1,(high!1-lh)+low!1-1, F!1)) = sigma(low!1, high!1, F!1)" )
(("1" (expand "extend" )
(("1" (ground)
(("1" (inst -1 "high!1-i!1+low!1-1" )
(("1" (ground) nil nil)
("2" (expand "sum_it_def" )
(("2" (expand "sigma" )
(("2" (assert)
(("2" (lift-if)
(("2" (lift-if) (("2" (ground) nil nil)) nil))
nil))
nil))
nil))
nil))
nil)
("2" (hide -2 -3 -4 3 )
(("2" (expand "sum_it_def" )
(("2" (expand "sum_it_def" )
(("2" (expand "sigma" + 2 )
(("2" (lift-if)
(("2" (ground)
(("2" (expand "sigma" )
(("2" (assert) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil)
("3" (hide -1 -2 3 )
(("3" (use "T_pred_lem" ) (("3" (assert) nil nil)) nil))
nil)
("4" (hide -1 -2 3 )
(("4" (skosimp*)
(("4" (ground)
(("1" (expand "sum_it_def" 1 )
(("1" (rewrite "sigma_rew" -) nil nil)) nil)
("2" (hide-all-but (-3 1 ))
(("2" (lemma "T_pred_lem" )
(("2" (inst?)
(("2" (inst - "high!1" "low!1" )
(("2" (assert) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (skosimp*)
(("2" (inst + "high!1 -lh!1 +low!1" )
(("1" (assert) nil nil)
("2" (hide-all-but 1 )
(("2" (lemma "T_pred_lem" )
(("2" (inst - "high!1" "low!1" "high!1-lh!1+low!1" )
(("2" (assert) nil nil)) nil))
nil))
nil))
nil))
nil)
("3" (hide-all-but 1 )
(("3" (skosimp*)
(("3" (typepred "high!1" )
(("3" (typepred "low!1" )
(("3" (assert)
(("3" (lemma "T_pred_lem" )
(("3"
(inst - "high!1" "low!1"
"(high!1 - lh!1) + low!1" )
(("3" (assert) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (assert) nil nil))
nil))
nil))
nil)
((T_low type -eq-decl nil old_sigma nil)
(T_high type -eq-decl nil old_sigma nil)
(<= const-decl "bool" reals nil)
(T formal-subtype-decl nil old_sigma nil)
(T_pred const-decl "[int -> boolean]" old_sigma nil)
(OR const-decl "[bool, bool -> bool]" 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)
(< 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)
(sum_it_def def-decl "real" old_sigma nil)
(sigma def-decl "real" old_sigma nil)
(real_lt_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)
(int_minus_int_is_int application-judgement "int" integers nil)
(int_plus_int_is_int application-judgement "int" integers 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)
(high!1 skolem-const-decl "T_high" old_sigma nil)
(low!1 skolem-const-decl "T_low" old_sigma nil)
(AND const-decl "[bool, bool -> bool]" booleans nil)
(subrange type -eq-decl nil integers nil)
(pred type -eq-decl nil defined_types nil)
(FALSE const-decl "bool" booleans nil)
(extend const-decl "R" extend nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(real_plus_real_is_real application-judgement "real" reals nil)
(i!1 skolem-const-decl "T" old_sigma nil)
(T_pred_lem formula-decl nil old_sigma nil)
(sigma_rew formula-decl nil old_sigma nil)
(minus_odd_is_odd application-judgement "odd_int" integers nil)
(lh!1 skolem-const-decl "{lh: T | low!1 <= lh AND lh <= high!1}"
old_sigma nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(>= const-decl "bool" reals nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(subrange_induction formula-decl nil subrange_inductions nil))
nil))
(sum_it_sigma 0
(sum_it_sigma-1 nil 3255975799
("" (skolem 1 ("F" "high" "low" ))
(("" (case "low > high" )
(("1" (expand "sum_it" )
(("1" (expand "sum_it_def" )
(("1" (expand "sigma" ) (("1" (assert) nil nil)) nil)) nil))
nil)
("2" (expand "sum_it" )
(("2" (expand "sum_it_def" )
(("2" (assert)
(("2" (lemma "sum_it_prop" )
(("2" (inst -1 "F" "high" "low" "low" )
(("2" (assert) (("2" (rewrite "sigma_eq_arg" ) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((T_high type -eq-decl nil old_sigma nil)
(T_low type -eq-decl nil old_sigma nil)
(<= const-decl "bool" reals nil)
(T formal-subtype-decl nil old_sigma nil)
(T_pred const-decl "[int -> boolean]" old_sigma nil)
(OR const-decl "[bool, bool -> bool]" 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)
(> 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)
(sum_it_def def-decl "real" old_sigma nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(sigma def-decl "real" old_sigma nil)
(sum_it const-decl "real" old_sigma nil)
(sum_it_prop formula-decl nil old_sigma nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(sigma_eq_arg formula-decl nil old_sigma nil)
(int_plus_int_is_int application-judgement "int" integers nil))
nil)))
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