|
(polynomial_division
(poly_divide_TCC1 0
(poly_divide_TCC1-1 nil 3582383554 ("" (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_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))
(poly_divide_TCC2 0
(poly_divide_TCC2-1 nil 3582383554 ("" (subtype-tcc) nil nil) nil
nil))
(poly_divide_TCC3 0
(poly_divide_TCC3-1 nil 3582383554 ("" (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))
(poly_divide_TCC4 0
(poly_divide_TCC4-1 nil 3582383554 ("" (subtype-tcc) nil nil)
((make_divtype const-decl "DivType" polynomial_division nil)) nil))
(poly_divide_TCC5 0
(poly_divide_TCC5-1 nil 3582383554
("" (skosimp*) (("" (assert) (("" (postpone) nil nil)) nil)) nil)
((real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(real_div_nzreal_is_real application-judgement "real" reals 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))
nil))
(poly_divide_TCC6 0
(poly_divide_TCC6-1 nil 3582383554 ("" (termination-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)
(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)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(real_div_nzreal_is_real application-judgement "real" reals 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)
(make_divtype const-decl "DivType" polynomial_division nil))
nil))
(poly_divide_TCC7 0
(poly_divide_TCC7-1 nil 3582383609 ("" (termination-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)
(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_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)
(real_div_nzreal_is_real application-judgement "real" reals nil)
(make_divtype const-decl "DivType" polynomial_division nil)
(nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
integers nil))
nil))
(poly_divide_TCC8 0
(poly_divide_TCC8-1 nil 3582544163 ("" (termination-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))
(poly_divide_TCC9 0
(poly_divide_TCC9-1 nil 3582555120 ("" (termination-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_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)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(mult_divides1 application-judgement "(divides(n))" divides nil)
(mult_divides2 application-judgement "(divides(m))" divides nil)
(max const-decl "{p: real | p >= m AND p >= n}" real_defs nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil))
nil))
(poly_divide_def 0
(poly_divide_def-1 nil 3582385231
(""
(case "FORALL (g, h: sequence[real], i, m, n: nat):
h(m) /= 0 AND i<=n-m AND (n-m-i) <= n - m IMPLIES
LET pd = poly_divide(g, n)(h, m)((n-m-i)) IN
FORALL (x: real):
polynomial(g, n)(x) =
polynomial(pd`quot, pd`qdeg)(x) * polynomial(h, m)(x) +
polynomial(pd`rem, pd`rdeg)(x)")
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nil)
("2" (hide 2)
(("2"
(case "NOT FORALL (g, h: sequence[real], i, m, n: nat):
h(m) /= 0 AND i = n - m IMPLIES
LET pd = poly_divide(g, n)(h, m)(i) IN
FORALL (x: real):
polynomial(g, n)(x) =
polynomial(pd`quot, pd`qdeg)(x) * polynomial(h, m)(x) +
polynomial(pd`rem, pd`rdeg)(x)")
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(case "%1=0")
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"x")
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nil
nil))
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nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
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(flatten)
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(label "mz" 1)
(("2"
(hide "mz")
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(skoletin -1)
(("2"
(skoletin -1)
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(assert)
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(assert)
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(case
"NOT polynomial(pd`quot, pd`qdeg)(x) = (g(n)/h(m))*x^(n-m)")
(("1"
(hide 5)
(("1"
(replaces
-1
+
:dir
rl)
(("1"
(assert)
(("1"
(replace
-2
+)
(("1"
(assert)
(("1"
(expand
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(("1"
(lemma
"sigma_eq_one_arg")
(("1"
(inst?)
(("1"
(inst
-
"n-m")
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-1)
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(grind)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(skosimp*)
(("2"
(lift-if)
(("2"
(ground)
nil
nil))
nil))
nil)
("3"
(skosimp*)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(ground)
(("2"
(replaces -1)
(("2"
(lemma
"mul_x_to_n_polynomial")
(("2"
(case
"n = m")
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-1)
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"polynomial"
+
3)
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:dir
rl)
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(hide-all-but
1)
(("2"
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nil
nil))
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5
2
2
1
0
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(-1
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-
"%1"
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"x")
(!
5
2
1
0
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6)
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nil
nil))
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nil
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--> --------------------
--> maximum size reached
--> --------------------
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