|
(multi_bernstein
(IMP_sigma_bijection_TCC1 0
(IMP_sigma_bijection_TCC1-1 nil 3618684915
("" (assuming-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)
(integer nonempty-type-from-decl nil 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))
(bsproduct_eval_TCC1 0
(bsproduct_eval_TCC1-1 nil 3498406967 ("" (subtype-tcc) nil nil) nil
nil))
(bsproduct_eval_TCC2 0
(bsproduct_eval_TCC2-1 nil 3498406967 ("" (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_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(ge_realorder name-judgement "RealOrder" util nil))
nil))
(bsproduct_eval_TCC3 0
(bsproduct_eval_TCC3-1 nil 3509275056 ("" (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)
(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)
(gt_realorder name-judgement "RealOrder" util nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(ge_realorder name-judgement "RealOrder" util nil))
nil))
(multibs_eval_rec_TCC1 0
(multibs_eval_rec_TCC1-1 nil 3498990086 ("" (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)
(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)
(gt_realorder name-judgement "RealOrder" util nil)
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(le_realorder name-judgement "RealOrder" util nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(ge_realorder name-judgement "RealOrder" util nil))
nil))
(multibs_eval_rec_TCC2 0
(multibs_eval_rec_TCC2-1 nil 3498990086 ("" (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)
(int_minus_int_is_int application-judgement "int" integers nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(ge_realorder name-judgement "RealOrder" util nil))
nil))
(multibs_eval_rec_TCC3 0
(multibs_eval_rec_TCC3-1 nil 3498990086 ("" (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)
(int_minus_int_is_int application-judgement "int" integers nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(ge_realorder name-judgement "RealOrder" util nil))
nil))
(multibs_eval_rec_TCC4 0
(multibs_eval_rec_TCC4-1 nil 3498990086
("" (termination-tcc) nil nil) nil nil))
(multibs_eval_1_term 0
(multibs_eval_1_term-1 nil 3499083500
("" (expand "multibs_eval")
(("" (expand "sigma")
(("" (expand "sigma")
((""
(case "FORALL (X: Vars, bsdegmono: DegreeMono, bspoly: MultiBernstein,
cf: Coeff, mvars: nat):
cf(0) * bsproduct_eval(bspoly(0), bsdegmono, mvars+1)(X) =
multibs_eval_rec(bspoly, bsdegmono, cf, mvars+1, 1, mvars+1)
(LAMBDA (i: nat): 0)(X)")
(("1" (skeep) (("1" (inst?) (("1" (assert) nil nil)) nil))
nil)
("2" (hide 2)
(("2" (induct "mvars")
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("2" (skolem 1 "vv")
(("2" (flatten)
(("2" (skeep)
(("2" (assert)
(("2" (expand "bsproduct_eval" +)
(("2" (expand "product" + 1)
(("2" (lemma "sigma_scal")
(("2"
(inst - "LAMBDA (j: nat):
IF j > bsdegmono(1 + vv) THEN 0
ELSE bspoly(0)(1 + vv)(j) *
Bern(j, bsdegmono(1 + vv))(X(1 + vv))
ENDIF" "cf(0) *
product(0, vv,
LAMBDA (i: nat):
sigma(0, bsdegmono(i),
LAMBDA (j: nat):
IF j > bsdegmono(i) THEN 0
ELSE bspoly(0)(i)(j) *
Bern(j, bsdegmono(i))(X(i))
ENDIF))"
"bsdegmono(1+vv)" "0")
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-1)
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(cgm)(X)
* (bspoly(0)(1 + vv)(x!1) * Bern(x!1, bsdegmono(1 + vv))(X(1 + vv)))
=
multibs_eval_rec(bspoly, bsdegmono, cf, 2 + vv, 1, 1 + rr)
(cgm WITH [(1 + vv) := x!1])(X)")
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multibs_eval_rec(bspoly, bsdegmono, cf, 1 + vv, 1, 0)
(cgm WITH [(0) := d])(X))"
"(bspoly(0)(1 + vv)(x!1) * Bern(x!1, bsdegmono(1 + vv))(X(1 + vv)))"
"bsdegmono(0)"
"0")
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rl)
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LAMBDA (k: nat):
IF cgm WITH [(0) := x!2](k) <= bsdegmono(k)
THEN Bern(cgm WITH [(0) := x!2](k), bsdegmono(k))(X(k))
ELSE 1
ENDIF) = product(0, vv,
LAMBDA (k: nat):
IF cgm WITH [(1 + vv) := x!1, (0) := x!2](k) <=
bsdegmono(k)
THEN Bern(cgm WITH [(1 + vv) := x!1, (0) := x!2](k),
bsdegmono(k))
(X(k))
ELSE 1
ENDIF)")
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3)
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"product_restrict_eq")
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2)
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rl)
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(case
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LAMBDA (j: nat): bspoly(0)(j)(cgm WITH [(0) := x!2](j)))=product(0, vv,
LAMBDA (j: nat):
bspoly(0)(j)(cgm WITH [(1 + vv) := x!1, (0) := x!2](j)))")
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nil)
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nil))
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nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
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(sigma def-decl "real" sigma "reals/")
(multibs_eval_rec def-decl "real" multi_bernstein nil)
(CoeffMono type-eq-decl nil util nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(bsproduct_eval const-decl "real" multi_bernstein nil)
(posnat nonempty-type-eq-decl nil integers nil)
(> const-decl "bool" reals nil)
(nonneg_int nonempty-type-eq-decl nil integers nil)
(* const-decl "[numfield, numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(Coeff type-eq-decl nil util nil)
(MultiBernstein type-eq-decl nil util nil)
(Polyproduct type-eq-decl nil util nil)
(Polynomial type-eq-decl nil util nil)
(DegreeMono type-eq-decl nil util nil)
(Vars type-eq-decl nil util nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(>= const-decl "bool" reals 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)
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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)
(boolean nonempty-type-decl nil booleans nil)
(number nonempty-type-decl nil numbers nil)
(nnint_plus_posint_is_posint application-judgement "posint"
integers nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(ge_realorder name-judgement "RealOrder" util nil)
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(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(int_plus_int_is_int application-judgement "int" integers nil)
(pred type-eq-decl nil defined_types nil)
(nat_induction formula-decl nil naturalnumbers nil)
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(sigma_restrict_eq formula-decl nil sigma "reals/")
(gt_realorder name-judgement "RealOrder" util nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
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(restrict const-decl "[T -> real]" sigma "reals/")
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(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(lt_realorder name-judgement "RealOrder" util nil)
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(le_realorder name-judgement "RealOrder" util nil)
(sigma_scal formula-decl nil sigma "reals/")
(IF const-decl "[boolean, T, T -> T]" if_def nil)
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(Bern const-decl "real" bernstein_polynomials "reals/")
(OR const-decl "[bool, bool -> bool]" booleans nil)
(<= const-decl "bool" reals nil)
(T_high type-eq-decl nil sigma "reals/")
(T_low type-eq-decl nil sigma "reals/")
(NOT const-decl "[bool -> bool]" booleans nil)
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil)
(bsdegmono skolem-const-decl "DegreeMono" multi_bernstein nil)
(vv skolem-const-decl "nat" multi_bernstein nil)
(T_high type-eq-decl nil product "reals/")
(T_low type-eq-decl nil product "reals/")
(both_sides_times1_imp formula-decl nil extra_real_props nil)
(restrict const-decl "[T -> real]" product "reals/")
(x!1 skolem-const-decl "nat" multi_bernstein nil)
(x!2 skolem-const-decl "nat" multi_bernstein nil)
(cgm skolem-const-decl "CoeffMono" multi_bernstein nil)
(product_restrict_eq formula-decl nil product "reals/")
(multibs_eval const-decl "real" multi_bernstein nil))
nil))
(multibs_eval_equiv 0
(multibs_eval_equiv-3 nil 3499083601
(""
(case "FORALL (X: Vars, bsdegmono: DegreeMono, bspoly: MultiBernstein,
cf: Coeff, nvars, terms: posnat,cfn:nat,ct:nat,lb:nat,pz:nat): lb<=pz AND cfn+lb<=ct AND ct<=terms-1 IMPLIES LET cfnew = (LAMBDA (i:nat): IF i<=cfn+lb AND i>=cfn THEN cf(i) ELSE 0 ENDIF) IN
multibs_eval(bspoly, bsdegmono, cfnew, nvars, terms)(X) =
multibs_eval_rec(bspoly, bsdegmono, cfnew, nvars, terms, nvars)
(LAMBDA (i: nat): 0)(X)")
(("1" (skeep)
(("1"
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(("1"
(case "multibs_eval(bspoly, bsdegmono,
LAMBDA (i: nat):
IF i <= terms - 1 THEN cf(i) ELSE 0 ENDIF,
nvars, terms)
(X) = multibs_eval(bspoly, bsdegmono, cf, nvars, terms)(X) AND multibs_eval_rec(bspoly, bsdegmono,
LAMBDA (i: nat):
IF i <= terms - 1 THEN cf(i) ELSE 0 ENDIF,
nvars, terms, nvars)
(LAMBDA (i: nat): 0)(X) = multibs_eval_rec(bspoly, bsdegmono, cf, nvars, terms, nvars)
(LAMBDA (i: nat): 0)(X)")
(("1" (assert)
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("2" (hide 2)
(("2" (hide -1)
(("2" (split)
(("1" (expand "multibs_eval")
(("1" (rewrite "sigma_restrict_eq")
(("1" (hide 2)
(("1" (decompose-equality)
(("1" (expand "restrict")
(("1" (propax) nil nil)) nil))
nil))
nil))
nil))
nil)
("2"
(case "FORALL (v:nat,cfmono:CoeffMono): v<=nvars IMPLIES multibs_eval_rec(bspoly, bsdegmono,
LAMBDA (i: nat):
IF i <= terms - 1 THEN cf(i) ELSE 0 ENDIF,
nvars, terms, v)
(cfmono)(X)
=
multibs_eval_rec(bspoly, bsdegmono, cf, nvars, terms, v)
(cfmono)(X)")
(("1" (inst?) (("1" (assert) nil nil)) nil)
("2" (hide 2)
(("2" (induct "v")
(("1" (skeep)
(("1" (expand "multibs_eval_rec")
(("1"
(case "multibs_eval_mono(bspoly, bsdegmono,
LAMBDA (i: nat):
IF i <= terms - 1 THEN cf(i) ELSE 0 ENDIF,
nvars, terms)
(cfmono)
=
multibs_eval_mono(bspoly, bsdegmono, cf, nvars, terms)(cfmono)")
(("1" (assert) nil nil)
("2"
(hide 2)
(("2"
(expand "multibs_eval_mono")
(("2"
(rewrite "sigma_restrict_eq")
(("2"
(hide 2)
(("2"
(decompose-equality)
(("2"
(expand "restrict")
(("2" (propax) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (skolem 1 "v")
(("2" (flatten)
(("2" (skeep)
(("2"
(expand "multibs_eval_rec" +)
(("2"
(rewrite "sigma_restrict_eq")
(("2"
(hide 2)
(("2"
(decompose-equality)
(("2"
(expand "restrict")
(("2"
(lift-if)
(("2"
(ground)
(("2" (inst?) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (hide 2)
(("2" (induct "pz")
(("1" (assert) nil nil)
("2" (skeep)
(("2" (case "lb = 0")
(("1" (replace -1)
(("1" (hide -1)
(("1" (hide -1)
(("1"
(name "bspolynew" "LAMBDA (i:nat): bspoly(i+cfn)")
(("1" (lemma "multibs_eval_1_term")
(("1"
(inst - "X" "bsdegmono" "bspolynew"
"LAMBDA (i:nat): IF i=0 THEN cf(cfn) ELSE 0 ENDIF"
"nvars")
(("1"
(case "multibs_eval(bspolynew, bsdegmono,
LAMBDA (i: nat): IF i = 0 THEN cf(cfn) ELSE 0 ENDIF,
nvars, 1)
(X)=
multibs_eval(bspoly, bsdegmono,
LAMBDA (i: nat):
IF i <= cfn AND i >= cfn THEN cf(i) ELSE 0 ENDIF,
nvars, terms)
(X)
and
multibs_eval_rec(bspoly, bsdegmono,
LAMBDA (i: nat):
--> --------------------
--> maximum size reached
--> --------------------
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Lebenszyklus
Die hierunter aufgelisteten Ziele sind für diese Firma wichtig
Ziele
Entwicklung einer Software für die statische Quellcodeanalyse
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