| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/PVS/Bernstein/ (Beweissystem der NASA Version 6.0.9©) Datei vom 28.9.2014 mit Größe 751 kB |
|
Quelle multi_bernstein.prfSprache: Lisp |
|||||||||||||||
|
(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") (("1" (skeep) (("1" (assert) (("1" (expand "bsproduct_eval") (("1" (expand "product") (("1" (expand "product") (("1" (expand "multibs_eval_rec") (("1" (rewrite "sigma_scal" :dir rl) (("1" (rewrite "sigma_restrict_eq") (("1" (decompose-equality) (("1" (expand "restrict") (("1" (lift-if) (("1" (ground) (("1" (hide 3) (("1" (expand "multibs_eval_rec") (("1" (expand "product") (("1" (expand "product") (("1" (expand "multibs_eval_mono") (("1" (expand "sigma") (("1" (expand "sigma") (("1" (expand "product") (("1" (expand "product") (("1" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("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") (("1" (assert) (("1" (replace -1 :dir rl) (("1" (hide -1) (("1" (expand "multibs_eval_rec" +) (("1" (rewrite "sigma_restrict_eq") (("1" (decompose-equality) (("1" (expand "restrict") (("1" (lift-if) (("1" (ground) (("1" (hide 3) (("1" (expand "bsproduct_eval") (("1" (inst?) (("1" (inst - "cf") (("1" (mult-by -1 "bspoly(0)(1 + vv)(x!1) * Bern(x!1, bsdegmono(1 + vv))(X(1 + vv))") (("1" (replace -1) (("1" (hide -1) (("1" (case "FORALL (rr:nat,cgm:CoeffMono): rr<=vv IMPLIES multibs_eval_rec(bspoly, bsdegmono, cf, 1 (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)") (("1" (inst?) (("1" (assert) nil nil)) nil) ("2" (hide 3) (("2" (induct "rr") (("1" (skeep) (("1" (expand "multibs_eval_rec") (("1" (hide -1) (("1" (lemma "sigma_scal") (("1" (inst - "(LAMBDA (d: nat): 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") (("1" (replace -1 :dir rl) (("1" (hide -1) (("1" (rewrite "sigma_restrict_eq") (("1" (decompose-equality) (("1" (expand "restrict") (("1" (lift-if) (("1" (ground) (("1" (hide 3) (("1" (expand "multibs_eval_rec") (("1" (expand "multibs_eval_mono") (("1" (expand "sigma") (("1" (expand "sigma") (("1" (expand "product" + 3) (("1" (expand "product" + 3) (("1" (case "NOT product(0, vv, 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)") (("1" (hide 3) (("1" (rewrite "product_restrict_eq") (("1" (hide 2) (("1" (decompose-equality) (("1" (expand "restrict") (("1" (lift-if) (("1" (ground) nil nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (replace -1 :dir rl) (("2" (hide -1) (("2" (case "product(0, vv, 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)))") (("1" (assert) nil nil) ("2" (hide 3) (("2" (rewrite "product_restrict_eq") (("2" (hide 2) (("2" (decompose-equality) (("2" (expand "restrict") (("2" (lift-if) (("2" (ground) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil) ("4" (skosimp*) (("4" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skolem 1 "rr") (("2" (flatten) (("2" (skeep) (("2" (assert) (("2" (expand "multibs_eval_rec" +) (("2" (lemma "sigma_scal") (("2" (inst?) (("2" (inst - "(bspoly(0)(1 + vv)(x!1) * Bern(x!1, bsdegmono(1 + vv))(X(1 + vv)))") (("2" (replace -1 :dir rl) (("2" (hide -1) (("2" (rewrite "sigma_restrict_eq") (("2" (hide 2) (("2" (decompose-equality) (("2" (expand "restrict") (("2" (lift-if) (("2" (ground) (("2" (inst - "cgm WITH [(1 + rr) := x!2]") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil) ("4" (skosimp*) (("4" (assert) nil nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil) ("4" (skosimp*) (("4" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil) ("4" (skosimp*) (("4" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((real_times_real_is_real application-judgement "real" reals nil) (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) (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) (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) (int_minus_int_is_int application-judgement "int" integers nil) (- 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) (even_plus_odd_is_odd application-judgement "odd_int" integers nil) (product def-decl "real" product "reals/") (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) (multibs_eval_mono const-decl "real" multi_bernstein nil) (restrict const-decl "[T -> real]" sigma "reals/") (bsdegmono skolem-const-decl "DegreeMono" multi_bernstein nil) (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) (above nonempty-type-eq-decl nil integers nil) (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< 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" (inst - "X" "bsdegmono" "bspoly" "cf" "nvars" "terms" "0" "terms-1" "terms-1" "terms-1") (("1" (assert) (("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, b 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) (("1" (flatten) (("1" (assert) nil nil)) nil)) nil) ("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, bsdeg 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): IF i <= cfn AND i >= cfn THEN cf(i) ELSE 0 ENDIF, nvars, terms, nvars) (LAMBDA (i: nat): 0)(X)= multibs_eval_rec(bspolynew, bsdegmono, LAMBDA (i: nat): IF i = 0 THEN cf(cfn) ELSE 0 ENDIF, nvars, 1, nvars) (LAMBDA (i: nat): 0)(X)") (("1" (assert) (("1" (flatten) (("1" (assert) nil nil)) nil)) nil) ("2" (hide 2) (("2" (hide -1) (("2" (split) (("1" (expand "multibs_eval") (("1" (expand "sigma" + 1) (("1" (expand "sigma" + 1) (("1" (lemma "sigma_split") (("1" (inst?) (("1" (inst - "cfn") (("1" (assert) (("1" (replace -1) (("1" (hide -1) (("1" (lemma "sigma_split") (("1" (inst - "LAMBDA (i_1: nat): IF i_1 <= cfn AND i_1 >= cfn THEN cf(i_1) ELSE 0 ENDIF * bsproduct_eval(bspoly(i_1), bsdegmono, nvars)(X)" "cfn" "0" "cfn-1") (("1" (assert) (("1" (replace -1) (("1" (hide -1) (("1" (case "sigma(0, cfn - 1, LAMBDA (i_1: nat): IF i_1 <= cfn AND i_1 >= cfn THEN cf(i_1) ELSE 0 ENDIF * bsproduct_eval(bspoly(i_1), bsdegmono, nvars)(X))=0 AND 0= sigma(1 + cfn, terms - 1, LAMBDA (i_1: nat): IF i_1 <= cfn AND i_1 >= cfn THEN cf(i_1) ELSE 0 ENDIF * bsproduct_eval(bspoly(i_1), bsdegmono, nvars)(X))") (("1" (flatten) (("1" (replace -1) (("1" (hide -1) (("1" (replace -1 :dir rl) (("1" (hide -1) (("1" (assert) (("1" (expand "sigma") (("1" (expand "sigma") (("1" (expand "bspolynew") (("1" (propax) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (split) (("1" (rewrite "sigma_restrict_eq_0") nil nil) ("2" (rewrite "sigma_restrict_eq_0") nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (case "FORALL (cfm:CoeffMono,vv:nat): vv<=nvars IMPLIES multibs_eval_rec(bspoly, bsdegmono, LAMBDA (i: nat): IF i <= cfn AND i >= cfn THEN cf(i) ELSE 0 ENDIF, nvars, terms, vv) (cfm)(X) = multibs_eval_rec(bspolynew, bsdegmono, LAMBDA (i: nat): IF i = 0 THEN cf(cfn) ELSE 0 ENDIF, nvars, 1, vv) (cfm)(X)") (("1" (inst?) (("1" (assert) nil nil)) nil) ("2" (hide 2) (("2" (induct "vv") (("1" (skeep) (("1" (expand "multibs_eval_rec") (("1" (case "multibs_eval_mono(bspoly, bsdegmono, LAMBDA (i: nat): IF i <= cfn AND i >= cfn THEN cf(i) ELSE 0 ENDIF, nvars, terms) (cfm) = multibs_eval_mono(bspolynew, bsdegmono, LAMBDA (i: nat): IF i = 0 THEN cf(cfn) ELSE 0 ENDIF, nvars, 1) (cfm)") (("1" (assert) nil nil) ("2" (hide 2) (("2" (expand "multibs_eval_mono") (("2" (expand "sigma" + 2) (("2" (expand "sigma" + 2) (("2" (expand "bspolynew") (("2" (lemma "sigma_split") (("2" (inst?) (("2" (inst - "cfn") (("2" (assert) (("2" (replace -1) (("2" (hide -1) (("2" (lemma "sigma_split") (("2" (inst - "(LAMBDA (i_1: nat): IF i_1 <= cfn AND i_1 >= cfn THEN cf(i_1) ELSE 0 ENDIF * product(0, nvars - 1, LAMBDA (j: nat): bspoly(i_1)(j)(cfm(j))))" "cfn" "0" "cfn-1") (("2" (assert) (("2" (replace -1) (("2" (hide -1) (("2" (case "sigma(0, cfn - 1, (LAMBDA (i_1: nat): IF i_1 <= cfn AND i_1 >= cfn THEN cf(i_1) ELSE 0 ENDIF * product(0, nvars - 1, LAMBDA (j: nat): bspoly(i_1)(j)(cfm(j)))))=0 AND 0= sigma(1 + cfn, terms - 1, LAMBDA (i_1: nat): IF i_1 <= cfn AND i_1 >= cfn THEN cf(i_1) ELSE 0 ENDIF * product(0, nvars - 1, LAMBDA (j: nat): bspoly(i_1)(j)(cfm(j))))") (("1" (flatten) (("1" (replace -1) (("1" (hide -1) (("1" (replace -1 :dir rl) (("1" (hide -1) (("1" (assert) (("1" (expand "sigma") (("1" (expand "sigma") (("1" (propax) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (split) (("1" (rewrite "sigma_restrict_eq_0") nil nil) ("2" (rewrite "sigma_restrict_eq_0") nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skolem 1 "vv") (("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)) nil)) nil)) nil) ("2" (assert) nil nil)) nil)) nil) ("3" (skolem 1 "pz") (("3" (flatten) (("3" (skeep) (("3" (case "lb = 0") (("1" (inst?) (("1" (inst - "cfn+lb") (("1" (assert) nil nil)) nil)) nil) ("2" (label "lbzero" 1) (("2" (hide "lbzero") (("2" (name "cflow" "LAMBDA (i:nat): IF i<=cfn+lb-1 AND i>=cfn THEN cf(i) ELSE 0 ENDIF") (("2" (name "cfhigh" "LAMBDA (i:nat): IF i=cfn+lb THEN cf(i) ELSE 0 ENDIF") (("2" (case "multibs_eval(bspoly, bsdegmono, LAMBDA (i: nat): IF i <= cfn + lb AND i >= cfn THEN cf(i) ELSE 0 ENDIF, nvars, terms) (X) = multibs_eval(bspoly, bsdegmono, cflow, nvars, terms) (X)+multibs_eval(bspoly, bsdegmono, cfhigh, nvars, terms) (X)") (("1" (replace -1) (("1" (case "multibs_eval_rec(bspoly, bsdegmono, LAMBDA (i: nat): IF i <= cfn + lb AND i >= cfn THEN cf(i) ELSE 0 ENDIF, nvars, terms, nvars) (LAMBDA (i: nat): 0)(X) = multibs_eval_rec(bspoly, bsdegmono, cflow, nvars, terms, nvars) (LAMBDA (i: nat): 0)(X)+multibs_eval_rec(bspoly, bsdegmono, cfhigh, nvars, terms, nvars) (LAMBDA (i: nat): 0)(X)") (("1" (replace -1) (("1" (hide -1) (("1" (hide -1) (("1" (reveal "lbzero") (("1" (inst-cp - "X" "bsdegmono" "bspoly" "cf" "nvars" "terms" "cfn" "ct" "lb-1") (("1" (hide "lbzero") (("1" (assert) (("1" (replace -2) (("1" (inst - "X" "bsdegmono" "bspoly" "cf" "nvars" "terms" "cfn+lb" "ct" "0") (("1" (assert) (("1" (case "(LAMBDA (i: nat): IF i <= cfn + lb AND i >= cfn + lb THEN cf(i) ELSE 0 ENDIF) = cfhigh") (("1" (assert) nil nil) ("2" (hide-all-but 1) (("2" (expand "cfhigh") (("2" (decompose-equality) (("2" (lift-if) (("2" (lift-if) (("2" (ground) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (hide -1) (("2" (hide -1) (("2" (hide -1) (("2" (hide -1) (("2" (case "FORALL (v:nat,iggy:CoeffMono): v<=nvars IMPLIES multibs_eval_rec(bspoly, bsdegmono, LAMBDA (i: nat): IF i <= cfn + lb AND i >= cfn THEN cf(i) ELSE 0 ENDIF, nvars, terms, v) (iggy)(X) = multibs_eval_rec(bspoly, bsdegmono, cflow, nvars, terms, v) (iggy)(X) + multibs_eval_rec(bspoly, bsdegmono, cfhigh, nvars, terms, v) (iggy)(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 <= cfn + lb AND i >= cfn THEN cf(i) ELSE 0 ENDIF, nvars, terms) (iggy) = multibs_eval_mono(bspoly, bsdegmono, cfhigh, nvars, terms)(iggy) + multibs_eval_mono(bspoly, bsdegmono, cflow, nvars, terms)(iggy)") (("1" (assert) nil nil) ("2" (hide 2) (("2" (expand "multibs_eval_mono") (("2" (rewrite "sigma_sum") (("2" (hide -1) (("2" (rewrite "sigma_restrict_eq") (("2" (hide 2) (("2" (decompose-equality) (("2" (expand "restrict") (("2" (lift-if) (("2" (ground) (("1" (expand "cfhigh") (("1" (expand "cflow") (("1" (lift-if) (("1" (lift-if) (("1" (ground) nil nil)) nil)) nil)) nil)) nil) ("2" (expand "cfhigh") (("2" (expand "cflow") (("2" (propax) nil nil)) nil)) nil) ("3" (expand "cflow") (("3" (expand "cfhigh") (("3" (propax) nil nil)) nil)) nil)) nil)) nil)) 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_sum" +) (("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)) nil) ("2" (hide 2) (("2" (hide -1) (("2" (hide -1) (("2" (hide -1) (("2" (expand "multibs_eval") (("2" (rewrite "sigma_sum") (("2" (rewrite "sigma_restrict_eq") (("2" (decompose-equality) (("2" (hide 2) (("2" (expand "restrict") (("2" (lift-if) (("2" (ground) (("1" (expand "cfhigh") (("1" (expand "cflow") (("1" (lift-if) (("1" (lift-if) (("1" (ground) nil nil)) nil)) nil)) nil)) nil) ("2" (expand "cfhigh") (("2" (expand "cflow") (("2" (propax) nil nil)) nil)) nil) ("3" (expand "cfhigh") (("3" (expand "cflow") (("3" (propax) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((cfhigh skolem-const-decl "[nat -> real]" multi_bernstein nil) (lb skolem-const-decl "nat" multi_bernstein nil) (sigma_sum formula-decl nil sigma "reals/") (cflow skolem-const-decl "[nat -> real]" multi_bernstein nil) (multibs_eval_1_term formula-decl nil multi_bernstein nil) (even_minus_odd_is_odd application-judgement "odd_int" integers nil) (bspolynew skolem-const-decl "[nat -> Polyproduct]" multi_bernstein nil) (sigma_restrict_eq_0 formula-decl nil sigma "reals/") (real_plus_real_is_real application-judgement "real" reals nil) (sigma_split formula-decl nil sigma "reals/") (sigma def-decl "real" sigma "reals/") (int_plus_int_is_int application-judgement "int" integers nil) (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (le_realorder name-judgement "RealOrder" util nil) (T_low type-eq-decl nil sigma "reals/") (T_high type-eq-decl nil sigma "reals/") (OR const-decl "[bool, bool -> bool]" booleans nil) (bsproduct_eval const-decl "real" multi_bernstein nil) (* const-decl "[numfield, numfield -> numfield]" number_fields nil) (sigma_restrict_eq formula-decl nil sigma "reals/") (real_times_real_is_real application-judgement "real" reals nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (lt_realorder name-judgement "RealOrder" util nil) (restrict const-decl "[T -> real]" sigma "reals/") (nnint_plus_posint_is_posint application-judgement "posint" integers nil) (posint_plus_nnint_is_posint application-judgement "posint" integers nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (gt_realorder name-judgement "RealOrder" util nil) (multibs_eval_mono const-decl "real" multi_bernstein nil) (T_low type-eq-decl nil product "reals/") (T_high type-eq-decl nil product "reals/") (product def-decl "real" product "reals/") (nat_induction formula-decl nil naturalnumbers nil) (pred type-eq-decl nil defined_types nil) (ge_realorder name-judgement "RealOrder" util nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (nnint_plus_nnint_is_nnint application-judgement "nonneg_int" integers nil) (int_minus_int_is_int application-judgement "int" integers nil) (number nonempty-type-decl nil numbers nil) (boolean nonempty-type-decl nil booleans 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) (bool nonempty-type-eq-decl nil booleans nil) (>= const-decl "bool" reals nil) (nat nonempty-type-eq-decl nil naturalnumbers nil) (Vars type-eq-decl nil util nil) (DegreeMono type-eq-decl nil util nil) (Polynomial type-eq-decl nil util nil) (Polyproduct type-eq-decl nil util nil) (MultiBernstein type-eq-decl nil util nil) (Coeff type-eq-decl nil util nil) (nonneg_int nonempty-type-eq-decl nil integers nil) (> const-decl "bool" reals nil) (posnat nonempty-type-eq-decl nil integers nil) (IMPLIES const-decl "[bool, bool -> bool]" booleans 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) (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (= const-decl "[T, T -> boolean]" equalities nil) (multibs_eval const-decl "real" multi_bernstein nil) (CoeffMono type-eq-decl nil util nil) (multibs_eval_rec def-decl "real" multi_bernstein nil) (IF const-decl "[boolean, T, T -> T]" if_def nil)) nil) (multibs_eval_equiv-2 nil 3499083540 ("" (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< 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" (inst - "X" "bsdegmono" "bspoly" "cf" "nvars" "terms" "0" "terms-1" "terms-1" "terms-1") (("1" (assert) (("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, b 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) (("1" (flatten) (("1" (assert) nil))))) ("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))))))))))) ("2" (case "FORALL (v:nat,cfmono:CoeffMono): v<=nvars IMPLIES multibs_eval_rec(bspoly, bsdeg 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))) ("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) ("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))))))))))))))))))) ("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))))))))))))))))))))))))))))))))))))))))) ("2" (hide 2) (("2" (induct "pz") (("1" (assert) 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_equiv2") (("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): IF i <= cfn AND i >= cfn THEN cf(i) ELSE 0 ENDIF, nvars, terms, nvars) (LAMBDA (i: nat): 0)(X)= multibs_eval_rec(bspolynew, bsdegmono, LAMBDA (i: nat): IF i = 0 THEN cf(cfn) ELSE 0 ENDIF, nvars, 1, nvars) (LAMBDA (i: nat): 0)(X)") (("1" (assert) (("1" (flatten) (("1" (assert) nil))))) ("2" (hide 2) (("2" (hide -1) (("2" (split) (("1" (expand "multibs_eval") (("1" (expand "sigma" + 1) (("1" (expand "sigma" + 1) (("1" (lemma "sigma_split") (("1" (inst?) (("1" (inst - "cfn") (("1" (assert) (("1" (replace -1) (("1" (hide -1) (("1" (lemma "sigma_split") (("1" (inst - "LAMBDA (i_1: nat): IF i_1 <= cfn AND i_1 >= cfn THEN cf(i_1) ELSE 0 ENDIF * bsproduct_eval(bspoly(i_1), bsdegmono, nvars)(X)" "cfn" "0" "cfn-1") (("1" (assert) (("1" (replace -1) (("1" (hide -1) (("1" (case "sigma(0, cfn - 1, LAMBDA (i_1: nat): IF i_1 <= cfn AND i_1 >= cfn THEN cf(i_1) ELSE 0 ENDIF * bsproduct_eval(bspoly(i_1), bsdegmono, nvars)(X))=0 AND 0= sigma(1 + cfn, terms - 1, LAMBDA (i_1: nat): IF i_1 <= cfn AND i_1 >= cfn THEN cf(i_1) ELSE 0 ENDIF * bsproduct_eval(bspoly(i_1), bsdegmono, nvars)(X))") (("1" (flatten) (("1" (replace -1) (("1" (hide -1) (("1" (replace -1 :dir rl) (("1" (hide -1) (("1" (assert) (("1" (expand "sigma") (("1" (expand "sigma") (("1" (expand "bspolynew") (("1" (propax) nil))))))))))))))))))) ("2" (hide 2) (("2" (split) (("1" (rewrite "sigma_restrict_eq_0") nil) ("2" (rewrite "sigma_restrict_eq_0") nil))))))))))))))))))))))))))))))))))) ("2" (case "FORALL (cfm:CoeffMono,vv:nat): vv<=nvars IMPLIES multibs_eval_rec(bspoly, bsdegmono, LAMBDA (i: nat): IF i <= cfn AND i >= cfn THEN cf(i) ELSE 0 ENDIF, nvars, terms, vv) (cfm)(X) = multibs_eval_rec(bspolynew, bsdegmono, LAMBDA (i: nat): IF i = 0 THEN cf(cfn) ELSE 0 ENDIF, nvars, 1, vv) (cfm)(X)") (("1" (inst?) (("1" (assert) nil))) ("2" (hide 2) (("2" (induct "vv") (("1" (skeep) (("1" (expand "multibs_eval_rec") (("1" (case "multibs_eval_mono(bspoly, bsdegmono, LAMBDA (i: nat): IF i <= cfn AND i >= cfn THEN cf(i) ELSE 0 ENDIF, nvars, terms) (cfm) = multibs_eval_mono(bspolynew, bsdegmono, LAMBDA (i: nat): IF i = 0 THEN cf(cfn) ELSE 0 ENDIF, nvars, 1) (cfm)") (("1" (assert) nil) ("2" (hide 2) (("2" (expand "multibs_eval_mono") (("2" (expand "sigma" + 2) (("2" (expand "sigma" + 2) (("2" (expand "bspolynew") (("2" (lemma "sigma_split") (("2" (inst?) (("2" (inst - "cfn") (("2" (assert) (("2" (replace -1) (("2" (hide -1) (("2" (lemma "sigma_split") (("2" (inst - "(LAMBDA (i_1: nat): IF i_1 <= cfn AND i_1 >= cfn THEN cf(i_1) ELSE 0 ENDIF * product(0, nvars - 1, LAMBDA (j: nat): bspoly(i_1)(j)(cfm(j))))" "cfn" "0" "cfn-1") (("2" (assert) (("2" (replace -1) (("2" (hide -1) (("2" (case "sigma(0, cfn - 1, (LAMBDA (i_1: nat): IF i_1 <= cfn AND i_1 >= cfn THEN cf(i_1) ELSE 0 ENDIF * product(0, nvars - 1, LAMBDA (j: nat): bspoly(i_1)(j)(cfm(j)))))=0 AND 0= sigma(1 + cfn, terms - 1, LAMBDA (i_1: nat): IF i_1 <= cfn AND i_1 >= cfn THEN cf(i_1) ELSE 0 ENDIF * product(0, nvars - 1, LAMBDA (j: nat): bspoly(i_1)(j)(cfm(j))))") (("1" (flatten) (("1" (replace -1) (("1" (hide -1) (("1" (replace -1 :dir rl) (("1" (hide -1) (("1" (assert) (("1" (expand "sigma") (("1" (expand "sigma") (("1" (propax) nil))))))))))))))))) ("2" (hide 2) (("2" (split) (("1" (rewrite "sigma_restrict_eq_0") nil) ("2" (rewrite "sigma_restrict_eq_0") nil))))))))))))))))))))))))))))))))))))))))))))) ("2" (skolem 1 "vv") (("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))))))))))))))))))))))))))))))))))))))))))))))) ("2" (assert) nil))))) ("3" (skolem 1 "pz") (("3" (flatten) (("3" (skeep) (("3" (case "lb = 0") (("1" (inst?) (("1" (inst - "cfn+lb") (("1" (assert) nil))))) ("2" (label "lbzero" 1) (("2" (hide "lbzero") (("2" (name "cflow" "LAMBDA (i:nat): IF i<=cfn+lb-1 AND i>=cfn THEN cf(i) ELSE 0 ENDIF") (("2" (name "cfhigh" "LAMBDA (i:nat): IF i=cfn+lb THEN cf(i) ELSE 0 ENDIF") (("2" (case "multibs_eval(bspoly, bsdegmono, LAMBDA (i: nat): IF i <= cfn + lb AND i >= cfn THEN cf(i) ELSE 0 ENDIF, nvars, terms) (X) = multibs_eval(bspoly, bsdegmono, cflow, nvars, terms) (X)+multibs_eval(bspoly, bsdegmono, cfhigh, nvars, terms) (X)") (("1" (replace -1) (("1" (case "multibs_eval_rec(bspoly, bsdegmono, LAMBDA (i: nat): IF i <= cfn + lb AND i >= cfn THEN cf(i) ELSE 0 ENDIF, nvars, terms, nvars) (LAMBDA (i: nat): 0)(X) = multibs_eval_rec(bspoly, bsdegmono, cflow, nvars, terms, nvars) (LAMBDA (i: nat): 0)(X)+multibs_eval_rec(bspoly, bsdegmono, cfhigh, nvars, terms, nvars) (LAMBDA (i: nat): 0)(X)") (("1" (replace -1) (("1" (hide -1) (("1" (hide -1) (("1" (reveal "lbzero") (("1" (inst-cp - "X" "bsdegmono" "bspoly" "cf" "nvars" "terms" "cfn" "ct" "lb-1") (("1" (hide "lbzero") (("1" (assert) (("1" (replace -2) (("1" (inst - "X" "bsdegmono" "bspoly" "cf" "nvars" "terms" "cfn+lb" "ct" "0") (("1" (assert) (("1" (case "(LAMBDA (i: nat): IF i <= cfn + lb AND i >= cfn + lb THEN cf(i) ELSE 0 ENDIF) = cfhigh") (("1" (assert) nil) ("2" (hide-all-but 1) (("2" (expand "cfhigh") (("2" (decompose-equality) (("2" (lift-if) (("2" (lift-if) (("2" (ground) nil))))))))))))))))))))))) ("2" (assert) nil))))))))))) ("2" (hide 2) (("2" (hide -1) (("2" (hide -1) (("2" (hide -1) (("2" (hide -1) (("2" (case "FORALL (v:nat,iggy:CoeffMono): v<=nvars IMPLIES multibs_eval_rec(bspoly, bsdegmono, LAMBDA (i: nat): IF i <= cfn + lb AND i >= cfn THEN cf(i) ELSE 0 ENDIF, nvars, terms, v) (iggy)(X) = multibs_eval_rec(bspoly, bsdegmono, cflow, nvars, terms, v) (iggy)(X) + multibs_eval_rec(bspoly, bsdegmono, cfhigh, nvars, terms, v) (iggy)(X)") (("1" (inst?) (("1" (assert) 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 <= cfn + lb AND i >= cfn THEN cf(i) ELSE 0 ENDIF, nvars, terms) (iggy) = multibs_eval_mono(bspoly, bsdegmono, cfhigh, nvars, terms)(iggy) + multibs_eval_mono(bspoly, bsdegmono, cflow, nvars, terms)(iggy)") (("1" (assert) nil) ("2" (hide 2) (("2" (expand "multibs_eval_mono") (("2" (rewrite "sigma_sum") (("2" (hide -1) (("2" (rewrite "sigma_restrict_eq") (("2" (hide 2) (("2" (decompose-equality) (("2" (expand "restrict") (("2" (lift-if) (("2" (ground) (("1" (expand "cfhigh") (("1" (expand "cflow") (("1" (lift-if) (("1" (lift-if) (("1" (lift-if) (("1" (ground) nil))))))))))) ("2" (expand "cfhigh") (("2" (expand "cflow") (("2" (propax) nil))))) ("3" (expand "cflow") (("3" (expand "cfhigh") (("3" (propax) nil))))))))))))))))))))))))))))))) ("2" (skolem 1 "v") (("2" (flatten) (("2" (skeep) (("2" (expand "multibs_eval_rec" +) (("2" (rewrite "sigma_sum" +) (("2" (rewrite "sigma_restrict_eq") (("2" (hide 2) (("2" (decompose-equality) (("2" (expand "restrict") (("2" (lift-if) (("2" (ground) (("2" (inst?) nil))))))))))))))))))))))))))))))))))))))))))) ("2" (hide 2) (("2" (hide -1) (("2" (hide -1) (("2" (hide -1) (("2" (expand "multibs_eval") (("2" (rewrite "sigma_sum") (("2" (rewrite "sigma_restrict_eq") (("2" (decompose-equality) (("2" (hide 2) (("2" (expand "restrict") (("2" (lift-if) (("2" (ground) (("1" (expand "cfhigh") (("1" (expand "cflow") (("1" (lift-if) (("1" (lift-if) (("1" (lift-if) (("1" (ground) nil))))))))))) ("2" (expand "cfhigh") (("2" (expand "cflow") (("2" (propax) nil))))) ("3" (expand "cfhigh") (("3" (expand "cflow") (("3" (propax) nil)))))))))))))))))))))))))))))))))))))))))))))))))))) nil) nil nil) (multibs_eval_equiv-1 nil 3498990121 ("" (case "FORALL (X: Vars, bsdegmono: DegreeMono, bspoly: MultiBernstein, cf: Coeff, nvars, terms: posnat,cfn:nat,ct:nat): cfn<=ct IMPLIES LET cfnew = (LAMBDA (i:nat) 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" (inst - "X" "bsdegmono" "bspoly" "cf" "nvars" "terms" "terms-1" "terms-1") (("1" (assert) (("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, b 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) (("1" (flatten) (("1" (assert) nil nil)) nil)) nil) ("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, bsdeg 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 "ct") (("1" (assert) nil nil) ("2" (case "FORALL (X: Vars, bsdegmono: DegreeMono, bspoly: MultiBernstein, cf: Coeff, nvars, terms: posnat): multibs_eval(bspoly, bsdegmono, cf, nvars, 1) (X) = multibs_eval_rec(bspoly, bsdegmono, cf, nvars, 1, nvars) (LAMBDA (i: nat): 0)(X)") (("1" (skeep) (("1" (case "cfn = 0") (("1" (replace -1) (("1" (hide -3) (("1" (hide -1) (("1" (case "multibs_eval(bspoly, bsdegmono, LAMBDA (i: nat): IF i <= 0 THEN cf(i) ELSE 0 ENDIF, nvars, terms) (X) = multibs_eval(bspoly, bsdegmono, LAMBDA (i: nat): IF i <= 0 THEN cf(i) ELSE 0 ENDIF, nvars, 1) (X)") (("1" (case " multibs_eval_rec(bspoly, bsdegmono, LAMBDA (i: nat): IF i <= 0 THEN cf(i) ELSE 0 ENDIF, nvars, terms, nvars) (LAMBDA (i: nat): 0)(X) = multibs_eval_rec(bspoly, bsdegmono, LAMBDA (i: nat): IF i <= 0 THEN cf(i) ELSE 0 ENDIF, nvars, 1, nvars) (LAMBDA (i: nat): 0)(X)") (("1" (inst - "X" "bsdegmono" "bspoly" "LAMBDA (i:nat): IF i<=0 THEN cf(i) ELSE 0 ENDIF" "nvars" "1") (("1" (assert) nil nil)) nil) ("2" (hide 2) (("2" (hide -1) (("2" (hide -1) (("2" (case "FORALL (cfmono:CoeffMono,v:nat):v<=nvars IMPLIES multibs_eval_rec(bspoly, bsdegmono, LAMBDA (i: nat): IF i <= 0 THEN cf(i) ELSE 0 ENDIF, nvars, terms, v) (cfmono)(X) = multibs_eval_rec(bspoly, bsdegmono, LAMBDA (i: nat): IF i <= 0 THEN cf(i) ELSE 0 ENDIF, nvars, 1, 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 <= 0 THEN cf(i) ELSE 0 ENDIF, nvars, terms) (cfmono) = multibs_eval_mono(bspoly, bsdegmono, LAMBDA (i: nat): IF i <= 0 THEN cf(i) ELSE 0 ENDIF, nvars, 1) (cfmono)") (("1" (assert) nil nil) ("2" (hide 2) (("2" (expand "multibs_eval_mono") (("2" (lemma "sigma_split") (("2" (inst?) (("2" (inst - "0") (("2" (assert) (("2" (replace -1) (("2" (hide -1) (("2" (assert) (("2" (rewrite "sigma_restrict_eq_0") nil nil)) nil)) 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) ("2" (hide 2) (("2" (hide -1) (("2" (expand "multibs_eval") (("2" (lemma "sigma_split") (("2" (inst?) (("2" (inst - "0") (("2" (assert) (("2" (replace -1) (("2" (hide -1) (("2" (assert) (("2" (rewrite "sigma_restrict_eq_0") nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (assert) nil nil)) nil)) nil) ("2" (hide 2) (("2" (skeep) (("2" (expand "multibs_eval") (("2" (expand "sigma") (("2" (expand "sigma") (("2" (expand "bsproduct_eval") (("2" (case "FORALL (cfmono:CoeffMono,v:nat): v<nvars IMPLIES cf(0) * product(0, v, 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)) = multibs_eval_rec(bspoly, bsdegmono, cf, v+1, 1, v+1) (cfmono)(X)") (("1" (inst?) (("1" (assert) nil nil)) nil) ("2" (hide 2) (("2" (induct "v") (("1" (skeep) (("1" (expand "product") (("1" (expand "product") (("1" (expand "multibs_eval_rec") (("1" (rewrite "sigma_scal" :dir rl) (("1" (rewrite "sigma_restrict_eq") (("1" (decompose-equality) (("1" (expand "restrict") (("1" (lift-if) (("1" (ground) (("1" (hide 3) (("1" (expand "multibs_eval_rec") (("1" (expand "product") (("1" (expand "product") (("1" (expand "multibs_eval_mono") (("1" (expand "sigma") (("1" (expand "product") (("1" (expand "sigma") (("1" (expand "product") (("1" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skolem 1 "v") (("2" (flatten) (("2" (skeep) (("2" (expand "product " +) (("2" (inst?) (("1" (assert) (("1" (case "NOT cf(0) * (sigma(0, bsdegmono(1 + v), LAMBDA (j: nat): IF j > bsdegmono(1 + v) THEN 0 ELSE bspoly(0)(1 + v)(j) * Bern(j, bsdegmono(1 + v))(X(1 + v)) ENDIF) * product(0, v, 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))) = multibs_eval_rec(bspoly, bsdegmono, cf, 1 + v, 1, 1 + v)(cf)(X)*sigma(0, bsdegmon LAMBDA (j: nat): IF j > bsdegmono(1 + v) THEN 0 ELSE bspoly(0)(1 + v)(j) * Bern(j, bsdegmono(1 + v))(X(1 + v)) ENDIF)") (("1" (assert) nil nil) ("2" (replace -1) (("2" (hide -1) (("2" (hide -1) (("2" (expand "multibs_eval_rec" + 2) (("2" (rewrite "sigma_scal" :dir rl) (("2" (rewrite "sigma_restrict_eq") (("1" (decompose-equality) (("1" (expand "restrict") (("1" (lift-if) (("1" (ground) (("1" (hide 3) (("1" (postpone) nil nil)) nil)) nil)) nil)) nil) ("2" (postpone) nil nil) ("3" (postpone) nil nil)) nil) ("2" (postpone) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("3" (postpone) nil nil) ("4" (postpone) nil nil) ("5" (postpone) nil nil)) nil)) nil) ("2" (postpone) nil nil)) nil)) nil)) nil)) nil)) nil) ("3" (postpone) nil nil) ("4" (postpone) nil nil)) nil)) nil) ("3" (postpone) nil nil) ("4" (postpone) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("3" (postpone) nil nil)) nil)) nil)) nil) nil shostak)) (multibs_eval_below_mono 0 (multibs_eval_below_mono-1 nil 3509119235 ("" (skosimp*) (("" (expand "multibs_eval") (("" (rewrite "sigma_eq") (("" (hide 2) (("" (skosimp*) (("" (case "bsproduct_eval(bspoly!1(n!1), bsdegmono!1, nvars!1)(X!1) =bsproduct_eval(bspoly!1(n!1), bsdegmono!1, nvars!1)(Y!1)") (("1" (assert) nil nil) ("2" (hide 2) (("2" (expand "bsproduct_eval") (("2" (rewrite "product_eq") (("1" (hide 2) (("1" (skosimp*) (("1" (rewrite "sigma_eq") (("1" (hide 2) (("1" (skosimp*) (("1" (inst?) (("1" (assert) nil nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((multibs_eval const-decl "real" multi_bernstein nil) (= const-decl "[T, T -> boolean]" equalities nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (subrange type-eq-decl nil integers nil) (ge_realorder name-judgement "RealOrder" util nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (< const-decl "bool" reals nil) (below type-eq-decl nil naturalnumbers nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (gt_realorder name-judgement "RealOrder" util nil) (product_eq formula-decl nil product "reals/") (sigma def-decl "real" sigma "reals/") (IF const-decl "[boolean, T, T -> T]" if_def nil) (above nonempty-type-eq-decl nil integers nil) (Bern const-decl "real" bernstein_polynomials "reals/") (T_high type-eq-decl nil product "reals/") (T_low type-eq-decl nil product "reals/") (NOT const-decl "[bool -> bool]" booleans nil) (IMPLIES const-decl "[bool, bool -> bool]" booleans nil) (le_realorder name-judgement "RealOrder" util nil) (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (lt_realorder name-judgement "RealOrder" util nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props 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) (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_low type-eq-decl nil sigma "reals/") (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (T_high type-eq-decl nil sigma "reals/") (<= const-decl "bool" reals nil) (OR const-decl "[bool, bool -> bool]" booleans nil) (MultiBernstein type-eq-decl nil util nil) (bsproduct_eval const-decl "real" multi_bernstein nil) (Vars type-eq-decl nil util nil) (posnat nonempty-type-eq-decl nil integers nil) (> const-decl "bool" reals nil) (nonneg_int nonempty-type-eq-decl nil integers nil) (DegreeMono type-eq-decl nil util nil) (Polyproduct type-eq-decl nil util nil) (Polynomial type-eq-decl nil util nil) (Coeff type-eq-decl nil util nil) (* const-decl "[numfield, numfield -> numfield]" number_fields nil) (numfield nonempty-type-eq-decl nil number_fields nil) (sigma_eq formula-decl nil sigma "reals/") (int_minus_int_is_int application-judgement "int" integers nil) (real_times_real_is_real application-judgement "real" reals nil)) shostak)) (multibs_eval_id 0 (multibs_eval_id-1 nil 3500206292 ("" (skeep) (("" (expand "multibs_eval") (("" (decompose-equality) (("" (rewrite "sigma_restrict_eq") (("" (hide 2) (("" (decompose-equality) (("" (expand "restrict") (("" (lift-if) (("" (ground) (("" (case "bsproduct_eval(bspoly(x!2), bsdegmono, nvars)(x!1) = bsproduct_eval(bspolz(x!2), bsdegmono, nvars)(x!1)") (("1" (assert) nil nil) ("2" (hide 3) (("2" (expand "bsproduct_eval") (("2" (rewrite "product_restrict_eq") (("1" (hide 2) (("1" (decompose-equality) (("1" (expand "restrict") (("1" (lift-if) (("1" (ground) (("1" (rewrite "sigma_restrict_eq") (("1" (hide 3) (("1" (decompose-equality) (("1" (expand "restrict") (("1" (lift-if) (("1" (ground) (("1" (inst?) (("1" (assert) nil nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((multibs_eval const-decl "real" multi_bernstein nil) (sigma_restrict_eq formula-decl nil sigma "reals/") (restrict const-decl "[T -> real]" sigma "reals/") (= const-decl "[T, T -> boolean]" equalities nil) (< const-decl "bool" reals nil) (below type-eq-decl nil naturalnumbers nil) (upto nonempty-type-eq-decl nil naturalnumbers nil) (x!3 skolem-const-decl "nat" multi_bernstein nil) (restrict const-decl "[T -> real]" product "reals/") (bsdegmono skolem-const-decl "DegreeMono" multi_bernstein nil) (product_restrict_eq formula-decl nil product "reals/") (IF const-decl "[boolean, T, T -> T]" if_def nil) (above nonempty-type-eq-decl nil integers nil) (Bern const-decl "real" bernstein_polynomials "reals/") (T_high type-eq-decl nil product "reals/") (T_low type-eq-decl nil product "reals/") (NOT const-decl "[bool -> bool]" booleans nil) (IMPLIES const-decl "[bool, bool -> bool]" booleans nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (gt_realorder name-judgement "RealOrder" util nil) (number nonempty-type-decl nil numbers nil) (boolean nonempty-type-decl nil booleans 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) (bool nonempty-type-eq-decl nil booleans nil) (>= const-decl "bool" reals nil) (nat nonempty-type-eq-decl nil naturalnumbers nil) (MultiBernstein type-eq-decl nil util nil) (bsproduct_eval const-decl "real" multi_bernstein nil) (DegreeMono type-eq-decl nil util nil) (Polyproduct type-eq-decl nil util nil) (Polynomial type-eq-decl nil util nil) (Coeff type-eq-decl nil util nil) (* const-decl "[numfield, numfield -> numfield]" number_fields 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) (sigma def-decl "real" sigma "reals/") (T_high type-eq-decl nil sigma "reals/") (T_low type-eq-decl nil sigma "reals/") (<= const-decl "bool" reals nil) (OR const-decl "[bool, bool -> bool]" booleans nil) (Vars type-eq-decl nil util nil) (int_minus_int_is_int application-judgement "int" integers nil) (real_times_real_is_real application-judgement "real" reals nil) (ge_realorder name-judgement "RealOrder" util nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (le_realorder name-judgement "RealOrder" util nil) (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (lt_realorder name-judgement "RealOrder" util nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil)) shostak)) (multibscoeff_id 0 (multibscoeff_id-1 nil 3509113819 ("" (skeep) (("" (expand "multibscoeff") (("" (rewrite "sigma_eq") (("" (hide 2) (("" (skosimp*) (("" (case "product(0, nvars - 1, LAMBDA (j: nat): bspoly(n!1)(j)(coeffmono1(j))) = product(0, nvars - 1, LAMBDA (j: nat): bspoly(n!1)(j)(coeffmono2(j)))") (("1" (assert) nil nil) ("2" (hide 2) (("2" (rewrite "product_eq") (("2" (hide 2) (("2" (skosimp*) (("2" (expand "eq_below_mono?") (("2" (inst - "n!2") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((multibscoeff const-decl "real" multi_bernstein nil) (= const-decl "[T, T -> boolean]" equalities nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (subrange type-eq-decl nil integers nil) (product_eq formula-decl nil product "reals/") (below type-eq-decl nil naturalnumbers nil) (< const-decl "bool" reals nil) (eq_below_mono? const-decl "bool" util nil) (le_realorder name-judgement "RealOrder" util nil) (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (lt_realorder name-judgement "RealOrder" util nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props 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) (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_low type-eq-decl nil sigma "reals/") (T_high type-eq-decl nil sigma "reals/") (CoeffMono 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) (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) (product def-decl "real" product "reals/") (T_high type-eq-decl nil product "reals/") (T_low type-eq-decl nil product "reals/") (<= const-decl "bool" reals nil) (OR const-decl "[bool, bool -> bool]" booleans nil) (Coeff type-eq-decl nil util nil) (* const-decl "[numfield, numfield -> numfield]" number_fields nil) (numfield nonempty-type-eq-decl nil number_fields nil) (sigma_eq formula-decl nil sigma "reals/") (real_times_real_is_real application-judgement "real" reals nil) (int_minus_int_is_int application-judgement "int" integers nil) (ge_realorder name-judgement "RealOrder" util nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil)) shostak)) (Bern_coeffs_rel_implic 0 (Bern_coeffs_rel_implic-1 nil 3499007668 ("" (induct "v") (("1" (skeep) (("1" (expand "Bern_coeffs_rel_rec") (("1" (inst?) (("1" (assert) nil nil)) nil)) nil)) nil) ("2" (skolem 1 "v") (("2" (flatten) (("2" (skeep) (("2" (expand "Bern_coeffs_rel_rec" (-3 1)) (("2" (skosimp*) (("2" (inst -3 "d!1") (("2" (inst - "a" "a1" "bsdegmono" "bspoly" "cf" "coeffmono WITH [(v):=d!1]" "nvars" "rel" "terms") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((<= const-decl "bool" reals nil) (upto nonempty-type-eq-decl nil naturalnumbers nil) (nnint_plus_posint_is_posint application-judgement "posint" integers nil) (multibscoeff const-decl "real" multi_bernstein nil) (nat_induction formula-decl nil naturalnumbers nil) (Bern_coeffs_rel_rec def-decl "bool" multi_bernstein nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (IMPLIES const-decl "[bool, bool -> bool]" booleans nil) (posnat nonempty-type-eq-decl nil integers nil) (> const-decl "bool" reals nil) (nonneg_int nonempty-type-eq-decl nil integers nil) (CoeffMono type-eq-decl nil util 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) (pred type-eq-decl nil defined_types 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) (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)) shostak)) (Bern_coeffs_rel_def 0 (Bern_coeffs_rel_def-1 nil 3498927613 ("" (case "FORALL (a: real, bsdegmono: DegreeMono, bspoly: MultiBernstein, cf: Coeff, nvars: posnat, rel: [[real, real] -> bool], terms: posnat,v:nat,iggy,coeffmono:CoeffMono):(FORALL (j:below(nvars)): iggy(j) = coe (coeffmono)=multibscoeff(bspoly, bsdegmono, cf, nvars, terms) (iggy)") (("1" (label "cmlem" -1) (("1" (hide "cmlem") (("1" (case "FORALL (a: real, bsdegmono: DegreeMono, bspoly: MultiBernstein, cf: Coeff, nvars: posnat, rel: [[real, real] -> bool], terms: posnat,v:nat,iggy:CoeffMono):le_below_mono?(nvars)(iggy,bsdegmono) and v<=nva (Bern_coeffs_rel_rec(bspoly, bsdegmono, cf, nvars, terms,v)(iggy,rel, a) IFF (FORALL (coeffmono): le_below_mono?(nvars)(coeffmono,bsdegmono) AND (FORALL (k:nat): k> rel(multibscoeff(bspoly, bsdegmono, cf, nvars, terms) (coeffmono), a)))") (("1" (skeep) (("1" (ground) (("1" (inst - "a" "bsdegmono" "bspoly" "cf" "nvars" "rel" "terms" "nvars" "LAMBDA (i:nat): 0") (("1" (expand "Bern_coeffs_rel") (("1" (assert) (("1" (skosimp*) (("1" (split -) (("1" (inst - "coeffmono!1") (("1" (assert) (("1" (skosimp*) (("1" (assert) nil nil)) nil)) nil)) nil) ("2" (hide-all-but 1) (("2" (grind) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (inst - "a" "bsdegmono" "bspoly" "cf" "nvars" "rel" "terms" "nvars" "LAMBDA (i:nat): 0") (("2" (ground) (("1" (expand "Bern_coeffs_rel") (("1" (propax) nil nil)) nil) ("2" (skosimp*) (("2" (inst -3 "coeffmono!1") (("2" (assert) nil nil)) nil)) nil) ("3" (hide-all-but 1) (("3" (grind) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (induct "v") (("1" (skeep) (("1" (hide -2) (("1" (ground) (("1" (skosimp*) (("1" (reveal "cmlem") (("1" (inst - "a" "bsdegmono" "bspoly" "cf" "nvars" "rel" "terms" "nvars" "iggy" "coeffmono!1") (("1" (assert) (("1" (expand "Bern_coeffs_rel_rec") (("1" (assert) (("1" (skosimp*) (("1" (inst - "j!1") (("1" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (inst - "iggy") (("2" (assert) (("2" (expand "Bern_coeffs_rel_rec") (("2" (propax) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skolem 1 "vv") (("2" (flatten) (("2" (skeep) (("2" (ground) (("1" (skosimp*) (("1" (expand "Bern_coeffs_rel_rec" -1) (("1" (inst - "coeffmono!1(vv)") (("1" (inst? -4) (("1" (assert) (("1" (split -) (("1" (inst - "coeffmono!1") (("1" (assert) (("1" (skosimp*) (("1" (inst - "k!1") (("1" (assert) (("1" (ground) (("1" (case "k!1 = vv") (("1" (replace -1) (("1" (assert) nil nil)) nil) ("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (expand "le_below_mono?") (("2" (skosimp*) (("2" (case "j!1 = vv") (("1" (inst - "vv") (("1" (assert) nil nil)) nil) ("2" (inst -4 "j!1") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (assert) (("2" (expand "le_below_mono?") (("2" (inst - "vv") nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (assert) (("2" (expand "Bern_coeffs_rel_rec" +) (("2" (skosimp*) (("2" (inst? -2) (("2" (assert) (("2" (split -) (("1" (hide 2) (("1" (skosimp*) (("1" (inst -3 "coeffmono!1") (("1" (assert) (("1" (skosimp*) (("1" (inst -2 "k!1") (("1" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (hide -1) (("2" (expand "le_below_mono?") (("2" (skosimp*) (("2" (inst - "j!1") (("2" (case "j!1 = vv") (("1" (assert) nil nil) ("2" (assert) nil 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" (skeep) (("2" (expand "multibscoeff") (("2" (rewrite "sigma_restrict_eq") (("2" (hide 2) (("2" (decompose-equality) (("2" (expand "restrict") (("2" (lift-if) (("2" (ground) (("2" (case "product(0, nvars - 1, LAMBDA (j: nat): bspoly(x!1)(j)(coeffmono(j))) = product(0, nvars - 1, LAMBDA (j: nat): bspoly(x!1)(j)(iggy(j)))") (("1" (assert) nil nil) ("2" (hide 3) (("2" (rewrite "product_restrict_eq") (("2" (hide 2) (("2" (decompose-equality) (("2" (expand "restrict") (("2" (lift-if) (("2" (ground) (("2" (inst - "x!2") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((gt_realorder name-judgement "RealOrder" util nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (restrict const-decl "[T -> real]" product "reals/") (product_restrict_eq formula-decl nil product "reals/") (restrict const-decl "[T -> real]" sigma "reals/") (T_low type-eq-decl nil sigma "reals/") (T_high type-eq-decl nil sigma "reals/") (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (product def-decl "real" product "reals/") (T_high type-eq-decl nil product "reals/") (T_low type-eq-decl nil product "reals/") (OR const-decl "[bool, bool -> bool]" booleans nil) (* const-decl "[numfield, numfield -> numfield]" number_fields nil) (numfield nonempty-type-eq-decl nil number_fields nil) (sigma_restrict_eq formula-decl nil sigma "reals/") (real_times_real_is_real application-judgement "real" reals nil) (int_minus_int_is_int application-judgement "int" integers nil) (Bern_coeffs_rel_rec def-decl "bool" multi_bernstein nil) (IFF const-decl "[bool, bool -> bool]" booleans nil) (<= const-decl "bool" reals nil) (le_below_mono? const-decl "bool" util nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (Bern_coeffs_rel const-decl "bool" multi_bernstein nil) (NOT const-decl "[bool -> bool]" booleans nil) (ge_realorder name-judgement "RealOrder" util nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (lt_realorder name-judgement "RealOrder" util nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (le_realorder name-judgement "RealOrder" util nil) (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (pred type-eq-decl nil defined_types nil) (nat_induction formula-decl nil naturalnumbers nil) (nnint_plus_posint_is_posint application-judgement "posint" integers nil) (posint_plus_nnint_is_posint application-judgement "posint" integers nil) (upto nonempty-type-eq-decl nil naturalnumbers nil) (bsdegmono skolem-const-decl "DegreeMono" multi_bernstein nil) (vv skolem-const-decl "nat" multi_bernstein nil) (coeffmono!1 skolem-const-decl "CoeffMono" multi_bernstein nil) (number nonempty-type-decl nil numbers nil) (boolean nonempty-type-decl nil booleans 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) (bool nonempty-type-eq-decl nil booleans nil) (>= const-decl "bool" reals nil) (nat nonempty-type-eq-decl nil naturalnumbers nil) (DegreeMono type-eq-decl nil util nil) (Polynomial type-eq-decl nil util nil) (Polyproduct type-eq-decl nil util nil) (MultiBernstein type-eq-decl nil util nil) (Coeff type-eq-decl nil util nil) (nonneg_int nonempty-type-eq-decl nil integers nil) (> const-decl "bool" reals nil) (posnat nonempty-type-eq-decl nil integers nil) (CoeffMono type-eq-decl nil util nil) (IMPLIES const-decl "[bool, bool -> bool]" booleans nil) (< const-decl "bool" reals nil) (below type-eq-decl nil naturalnumbers nil) (= const-decl "[T, T -> boolean]" equalities nil) (multibscoeff const-decl "real" multi_bernstein nil)) shostak)) (multibs_cornerpoint_coeff 0 (multibs_cornerpoint_coeff-2 nil 3509117841 ("" (skosimp*) (("" (assert) (("" (name "Xv" "LAMBDA (i: nat): IF coeffmono!1(i) = 0 THEN 0 ELSE 1 ENDIF") (("" (replace -1) (("" (flatten) (("" (expand "multibscoeff") (("" (expand "multibs_eval") (("" (rewrite "sigma_eq") (("" (hide 2) (("" (skosimp*) (("" (case "bsproduct_eval(bspoly!1(n!1), bsdegmono!1, nvars!1)(Xv) = product(0, nvars!1 - 1, LAMBDA (j: nat): bspoly!1(n!1)(j)(coeffmono!1(j)))") (("1" (assert) nil nil) ("2" (hide 2) (("2" (expand "bsproduct_eval") (("2" (rewrite "product_eq") (("1" (hide 2) (("1" (skosimp*) (("1" (lemma "sigma_split") (("1" (inst?) (("1" (inst - "coeffmono!1(n!2)") (("1" (assert) (("1" (split -) (("1" (replace -1) (("1" (hide -1) (("1" (lemma "sigma_split") (("1" (inst -1 _ _ _ "coeffmono!1(n!2)-1") (("1" (inst?) (("1" (assert) (("1" (replace -1) (("1" (hide -1) (("1" (case "sigma(0, coeffmono!1(n!2) - 1, LAMBDA (j: nat): IF j > bsdegmono!1(n!2) THEN 0 ELSE bspoly!1(n!1)(n!2)(j) * Bern(j, bsdegmono!1(n!2))(Xv(n!2)) ENDIF)=0 AND sigma(1 + coeffmono!1(n!2), bsdegmono!1(n!2), LAMBDA (j: nat): IF j > bsdegmono!1(n!2) THEN 0 ELSE bspoly!1(n!1)(n!2)(j) * Bern(j, bsdegmono!1(n!2))(Xv(n!2)) ENDIF)=0") (("1" (flatten) (("1" (replace -1) (("1" (hide -1) (("1" (replace -1) (("1" (hide -1) (("1" (assert) (("1" (expand "sigma") (("1" (expand "sigma") (("1" (lift-if) (("1" (ground) (("1" (expand "corner_point?") (("1" (inst - "n!2") (("1" (assert) nil nil)) nil)) nil) ("2" (case "Bern(coeffmono!1(n!2), bsdegmono!1(n!2))(Xv(n!2))=1") (("1" (assert) nil nil) ("2" (hide 2) (("2" (hide 2) (("2" (expand "Xv" +) (("2" (lift-if) (("2" (ground) (("1" (expand "Bern") (("1" (rewrite "expt_1i") (("1" (replace -1) (("1" (expand "^") (("1" (expand "expt") (("1" (expand "C") (("1" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (expand "Bern") (("2" (rewrite "expt_1i") (("2" (expand "corner_point?") (("2" (inst - "n!2") (("2" (assert) (("2" (replace -2) (("2" (assert) (("2" (expand "^") (("2" (expand "expt") (("2" (expand "C") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) 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" (split) (("1" (rewrite "sigma_restrict_eq_0") (("1" (hide 2) (("1" (skosimp*) (("1" (lift-if) (("1" (ground) (("1" (expand "Bern") (("1" (expand "Xv" +) (("1" (expand "^") (("1" (expand "expt" + 1) (("1" (lift-if) (("1" (ground) (("1" (expand "corner_point?") (("1" (inst - "n!2") (("1" (ground) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (rewrite "sigma_restrict_eq_0") (("2" (hide 2) (("2" (skosimp*) (("2" (expand "Bern") (("2" (expand "Xv" +) (("2" (lift-if) (("2" (ground) (("1" (expand "^" + 1) (("1" (expand "expt" +) (("1" (propax) nil nil)) nil)) nil) ("2" (expand "^" + 1) (("2" (expand "expt" +) (("2" (lift-if) (("2" (ground) (("2" (expand "corner_point?") (("2" (inst - "n!2") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("3" (hide 2) (("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil) ("4" (skosimp*) (("4" (assert) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (expand "corner_point?") (("2" (inst - "n!2") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((multibscoeff const-decl "real" multi_bernstein nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (ge_realorder name-judgement "RealOrder" util nil) (real_times_real_is_real application-judgement "real" reals nil) (int_minus_int_is_int application-judgement "int" integers nil) (sigma_eq formula-decl nil sigma "reals/") (numfield nonempty-type-eq-decl nil number_fields nil) (* const-decl "[numfield, numfield -> numfield]" number_fields nil) (Coeff type-eq-decl nil util nil) (Polynomial type-eq-decl nil util nil) (Polyproduct type-eq-decl nil util nil) (DegreeMono type-eq-decl nil util nil) (nonneg_int nonempty-type-eq-decl nil integers nil) (> const-decl "bool" reals nil) (posnat nonempty-type-eq-decl nil integers nil) (Vars type-eq-decl nil util nil) (bsproduct_eval const-decl "real" multi_bernstein nil) (MultiBernstein type-eq-decl nil util nil) (OR const-decl "[bool, bool -> bool]" booleans nil) (<= const-decl "bool" reals nil) (T_low type-eq-decl nil product "reals/") (T_high type-eq-decl nil product "reals/") (product def-decl "real" product "reals/") (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (T_high type-eq-decl nil sigma "reals/") (T_low type-eq-decl nil sigma "reals/") (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) (IMPLIES const-decl "[bool, bool -> bool]" booleans nil) (NOT const-decl "[bool -> bool]" booleans nil) (Bern const-decl "real" bernstein_polynomials "reals/") (above nonempty-type-eq-decl nil integers nil) (sigma def-decl "real" sigma "reals/") (product_eq formula-decl nil product "reals/") (bsdegmono!1 skolem-const-decl "DegreeMono" multi_bernstein nil) (nvars!1 skolem-const-decl "posnat" multi_bernstein nil) (n!2 skolem-const-decl "subrange(0, nvars!1 - 1)" multi_bernstein nil) (even_minus_odd_is_odd application-judgement "odd_int" integers nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (gt_realorder name-judgement "RealOrder" util nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (below type-eq-decl nil naturalnumbers nil) (< const-decl "bool" reals nil) (corner_point? const-decl "bool" util nil) (Xv skolem-const-decl "[nat -> naturalnumber]" multi_bernstein nil) (expt_1i formula-decl nil exponentiation nil) (nat_exp application-judgement "nat" exponentiation nil) (nnint_times_nnint_is_nnint application-judgement "nonneg_int" integers nil) (mult_divides1 application-judgement "(divides(n))" divides nil) (mult_divides2 application-judgement "(divides(m))" divides nil) (nnrat_exp application-judgement "nnrat" exponentiation nil) (posrat_exp application-judgement "posrat" exponentiation nil) (^ const-decl "real" exponentiation nil) (C const-decl "posnat" binomial "reals/") (posrat_div_posrat_is_posrat application-judgement "posrat" rationals nil) (posint_times_posint_is_posint application-judgement "posint" integers nil) (factorial_0 formula-decl nil factorial "ints/") (expt def-decl "real" exponentiation nil) (posint_exp application-judgement "posint" exponentiation nil) (int_times_even_is_even application-judgement "even_int" integers nil) (posnat_expt application-judgement "posnat" exponentiation nil) (even_times_int_is_even application-judgement "even_int" integers nil) (nat_expt application-judgement "nat" exponentiation nil) (sigma_restrict_eq_0 formula-decl nil sigma "reals/") (nnrat_times_nnrat_is_nnrat application-judgement "nonneg_rat" rationals nil) (odd_minus_even_is_odd application-judgement "odd_int" integers nil) (nzreal_exp application-judgement "nzreal" exponentiation nil) (rat_exp application-judgement "rat" exponentiation nil) (nzrat_times_nzrat_is_nzrat application-judgement "nzrat" rationals nil) (rat_times_rat_is_rat application-judgement "rat" rationals nil) (odd_minus_odd_is_even application-judgement "even_int" integers nil) (int_exp application-judgement "int" exponentiation nil) (real_plus_real_is_real application-judgement "real" reals nil) (posint_plus_nnint_is_posint application-judgement "posint" integers nil) (sigma_split formula-decl nil sigma "reals/") (subrange type-eq-decl nil integers nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (multibs_eval const-decl "real" multi_bernstein nil) (number nonempty-type-decl nil numbers nil) (boolean nonempty-type-decl nil booleans 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) (bool nonempty-type-eq-decl nil booleans nil) (>= const-decl "bool" reals nil) (nat nonempty-type-eq-decl nil naturalnumbers nil) (= const-decl "[T, T -> boolean]" equalities nil) (IF const-decl "[boolean, T, T -> T]" if_def nil) (CoeffMono type-eq-decl nil util nil) (naturalnumber type-eq-decl nil naturalnumbers nil)) nil) (multibs_cornerpoint_coeff-1 nil 3509117746 ("" (skeep) (("" (assert) (("" (name "Xv" "LAMBDA (i: nat): IF coeffmon(i) = 0 THEN 0 ELSE 1 ENDIF") (("" (replace -1) (("" (flatten) (("" (postpone) nil nil)) nil)) nil)) nil)) nil)) nil) nil shostak)) (forall_X_id 0 (forall_X_id-1 nil 3499949735 ("" (case "FORALL (cf: Coeff, bspoly, bspolz, bsdegmono, nvars, terms, rel, a): (FORALL (t: below(terms), v: below(nvars), i: upto(bsdegmono(v))): bspoly(t)(v)(i) = bspolz(t)(v)(i)) IMPLIES forall_X(bspoly, bsdegmono, cf, nvars, terms, rel, a) IMPLIES forall_X(bspolz, bsdegmono, cf, nvars, terms, rel, a)") (("1" (skeep) (("1" (iff) (("1" (split) (("1" (inst?) (("1" (ground) nil nil)) nil) ("2" (inst?) (("2" (ground) (("2" (skeep) (("2" (inst? -2) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (skeep) (("2" (expand "forall_X") (("2" (skeep) (("2" (inst? -2) (("2" (assert) (("2" (lemma "multibs_eval_id") (("2" (inst -1 "bspoly" "bspolz" "bsdegmono" "cf" "nvars" "terms") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((Vars type-eq-decl nil util nil) (multibs_eval_id formula-decl nil multi_bernstein nil) (number nonempty-type-decl nil numbers nil) (boolean nonempty-type-decl nil booleans 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) (bool nonempty-type-eq-decl nil booleans nil) (>= const-decl "bool" reals nil) (nat nonempty-type-eq-decl nil naturalnumbers nil) (Coeff type-eq-decl nil util nil) (Polynomial type-eq-decl nil util nil) (Polyproduct type-eq-decl nil util nil) (MultiBernstein type-eq-decl nil util nil) (DegreeMono type-eq-decl nil util nil) (nonneg_int nonempty-type-eq-decl nil integers nil) (> const-decl "bool" reals nil) (posnat nonempty-type-eq-decl nil integers nil) (IMPLIES const-decl "[bool, bool -> bool]" booleans nil) (< const-decl "bool" reals nil) (below type-eq-decl nil naturalnumbers nil) (<= const-decl "bool" reals nil) (upto nonempty-type-eq-decl nil naturalnumbers nil) (= const-decl "[T, T -> boolean]" equalities nil) (forall_X const-decl "bool" multi_bernstein nil)) shostak)) (forall_X_between_id 0 (forall_X_between_id-1 nil 3507233890 ("" (case "FORALL (cf: Coeff, amin, amax: real) (bspoly, bspolz, bsdegmono, nvars, terms): (FORALL (t: below(terms), v: below(nvars), i: upto(bsdegmono(v))): bspoly(t)(v)(i) = bspolz(t)(v)(i)) IMPLIES forall_X_between(amin, amax)(bspoly, bsdegmono, cf, nvars, terms) IMPLIES forall_X_between(amin, amax)(bspolz, bsdegmono, cf, nvars, terms)") (("1" (skosimp*) (("1" (iff) (("1" (split) (("1" (inst?) (("1" (ground) nil nil)) nil) ("2" (inst -1 "cf!1" "amin!1" "amax!1" "bspolz!1" "bspoly!1" "bsdegmono!1" "nvars!1" "terms!1") (("2" (ground) (("2" (skeep 1) (("2" (inst? -2) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (skosimp*) (("2" (expand "forall_X_between") (("2" (skeep) (("2" (inst? -2) (("2" (assert) (("2" (lemma "multibs_eval_id") (("2" (inst?) (("2" (inst -1 "bspolz!1") (("2" (split -1) (("1" (ground) nil nil) ("2" (propax) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((Vars type-eq-decl nil util nil) (multibs_eval_id formula-decl nil multi_bernstein nil) (le_realorder name-judgement "RealOrder" util nil) (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (number nonempty-type-decl nil numbers nil) (boolean nonempty-type-decl nil booleans 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) (bool nonempty-type-eq-decl nil booleans nil) (>= const-decl "bool" reals nil) (nat nonempty-type-eq-decl nil naturalnumbers nil) (Coeff type-eq-decl nil util nil) (Polynomial type-eq-decl nil util nil) (Polyproduct type-eq-decl nil util nil) (MultiBernstein type-eq-decl nil util nil) (DegreeMono type-eq-decl nil util nil) (nonneg_int nonempty-type-eq-decl nil integers nil) (> const-decl "bool" reals nil) (posnat nonempty-type-eq-decl nil integers nil) (IMPLIES const-decl "[bool, bool -> bool]" booleans nil) (< const-decl "bool" reals nil) (below type-eq-decl nil naturalnumbers nil) (<= const-decl "bool" reals nil) (upto nonempty-type-eq-decl nil naturalnumbers nil) (= const-decl "[T, T -> boolean]" equalities nil) (forall_X_between const-decl "bool" multi_bernstein nil)) nil)) (Bern_coeffs_endpoints_rel_rec_TCC1 0 (Bern_coeffs_endpoints_rel_rec_TCC1-1 nil 3499440937 ("" (skeep) (("" (assert) nil nil)) nil) nil nil)) (Bern_coeffs_endpoints_rel_rec_TCC2 0 (Bern_coeffs_endpoints_rel_rec_TCC2-1 nil 3499440937 ("" (skeep) (("" (assert) nil nil)) nil) nil nil)) (Bern_coeffs_endpoints_rel_rec_TCC3 0 (Bern_coeffs_endpoints_rel_rec_TCC3-1 nil 3503309476 ("" (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)) (Bern_coeffs_endpoints_rel_rec_TCC4 0 (Bern_coeffs_endpoints_rel_rec_TCC4-1 nil 3503309476 ("" (subtype-tcc) nil nil) nil nil)) (Bern_coeffs_endpoints_rel_TCC1 0 (Bern_coeffs_endpoints_rel_TCC1-1 nil 3503309637 ("" (skeep) (("" (assert) nil nil)) nil) nil nil)) (Bern_coeffs_endpoints_rel_def 0 (Bern_coeffs_endpoints_rel_def-1 nil 3499440937 ("" (expand "Bern_coeffs_endpoints_rel") (("" (name "eptest?" "(LAMBDA (bsdegmono:DegreeMono,nvars): (LAMBDA (cfm:CoeffMono): (FORALL (i:below(nvar (("" (skosimp*) (("" (lift-if) (("" (split) (("1" (flatten) (("1" (case "FORALL (a: real, bsdegmono: DegreeMono, bspoly: MultiBernstein, cf: Coeff, nvars: posnat, rel: [[real, real] -> bool], terms: posnat,v:nat,cfm:(eptest?(bsdegmono,nvars))): v<=nvars AND forall_X(bspoly, bsdegmono, cf, nvars, terms, rel, a) IMPLIES null?(Bern_coeffs_endpoints_rel_rec(bspoly, bsdegmono, cf, nvars, terms, v) (cfm, rel, a))") (("1" (inst?) (("1" (assert) nil nil) ("2" (hide 2) (("2" (expand "eptest?") (("2" (propax) nil nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (induct "v") (("1" (skeep) (("1" (expand "Bern_coeffs_endpoints_rel_rec") (("1" (assert) (("1" (lift-if +) (("1" (assert) (("1" (ground) (("1" (hide 2) (("1" (name "Xnew" "LAMBDA (i:nat): IF cfm(i)=0 THEN 0 ELSE 1 ENDIF") (("1" (expand "forall_X") (("1" (inst - "Xnew") (("1" (split -) (("1" (case "multibs_eval(bspoly, bsdegmono, cf, nvars, terms)(Xnew) = multibscoeff(bspoly, bsdegmo (("1" (assert) nil nil) ("2" (hide 2) (("2" (expand "multibscoeff") (("2" (expand "multibs_eval") (("2" (rewrite "sigma_restrict_eq") (("2" (hide 2) (("2" (decompose-equality) (("2" (expand "restrict") (("2" (lift-if) (("2" (ground) (("2" (case "bsproduct_eval(bspoly(x!1), bsdegmono, nvars)(Xnew) = product(0, nvars - 1, LAMBDA (j: nat): bspoly(x!1)(j)(cfm(j)))") (("1" (assert) nil nil) ("2" (hide 3) (("2" (expand "bsproduct_eval" +) (("2" (rewrite "product_restrict_eq") (("1" (hide 2) (("1" (decompose-equality) (("1" (expand "restrict") (("1" (lift-if) (("1" (ground) (("1" (typepred "cfm") (("1" (lemma "sigma_split") (("1" (inst?) (("1" (inst - "cfm(x!2)") (("1" (assert) (("1" (case "cfm(x!2)<=bsdegmono(x!2)") (("1" (assert) (("1" (replace -2) (("1" (hide -2) (("1" (case "sigma(1 + cfm(x!2), bsdegmono(x!2), LAMBDA (j: nat): IF j > bsdegmono(x!2) THEN 0 ELSE bspoly(x!1)(x!2)(j) * Bern(j, bsdegmono(x!2))(Xnew(x!2)) ENDIF) = 0") (("1" (replace -1) (("1" (hide -1) (("1" (assert) (("1" (lemma "sigma_split") (("1" (inst?) (("1" (inst - "cfm(x!2)-1") (("1" (assert) (("1" (replace -1) (("1" (hide -1) (("1" (case "sigma(0, cfm(x!2) - 1, LAMBDA (j: nat): IF j > bsdegmono(x!2) THEN 0 ELSE bspoly(x!1)(x!2)(j) * Bern(j, bsdegmono(x!2))(Xnew(x!2)) ENDIF) = 0") (("1" (replace -1) (("1" (hide -1) (("1" (assert) (("1" (expand "sigma" +) (("1" (expand "sigma" +) (("1" (expand "Xnew" +) (("1" (lift-if) (("1" (ground) (("1" (replace -1) (("1" (expand "Bern" +) (("1" (assert) (("1" (expand "^" +) (("1" (expand "expt" + 1) (("1" (case "FORALL (t:nat): expt(1,t) = 1") (("1" (inst - "bsdegmono(x!2)") (("1" (replace -1) (("1" (assert) (("1" (expand "C" +) (("1" (assert) nil nil)) nil)) nil)) nil)) nil) ("2" (hide-all-but 1) (("2" (induct "t") (("1" (expand "expt") (("1" (propax) nil nil)) nil) ("2" (skeep) (("2" (expand "expt" +) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (typepred "cfm") (("2" (expand "eptest?" -1) (("2" (inst - "x!2") (("2" (assert) (("2" (replace -1) (("2" (expand "Bern" +) (("2" (expand "^" +) (("2" (expand "expt" + 1) (("2" (case "FORALL (t:nat): expt(1,t) = 1") (("1" (inst - "bsdegmono(x!2)") (("1" (replace -1) (("1" (expand "C" +) (("1" (assert) nil nil)) nil)) nil)) nil) ("2" (hide-all-but 1) (("2" (induct "t") (("1" (expand "expt") (("1" (propax) nil nil)) nil) ("2" (skeep) (("2" (expand "expt" +) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 3) (("2" (rewrite "sigma_restrict_eq_0") (("2" (hide 2) (("2" (skosimp*) (("2" (typepred "i!1") (("2" (expand "Xnew" +) (("2" (expand "Bern" +) (("2" (case "FORALL (t:nat): 0^(t+1)=0") (("1" (inst - "bsdegmono(x!2)-i!1-1") (("1" (assert) nil nil)) nil) ("2" (hide-all-but 1) (("2" (induct "t") (("1" (expand "^") (("1" (expand "expt") (("1" (assert) nil nil)) nil)) nil) ("2" (skeep) (("2" (expand "^") (("2" (expand "expt" +) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 3) (("2" (rewrite "sigma_restrict_eq_0") (("2" (hide 2) (("2" (skosimp*) (("2" (expand "Xnew" +) (("2" (lift-if) (("2" (ground) (("1" (expand "Bern" +) (("1" (expand "^") (("1" (expand "expt" + 1) (("1" (propax) nil nil)) nil)) nil)) nil) ("2" (expand "Bern" +) (("2" (typepred "i!1") (("2" (typepred "cfm") (("2" (expand "eptest?" -1) (("2" (inst - "x!2") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide -1) (("2" (expand "eptest?" -1) (("2" (inst - "x!2") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide-all-but 1) (("2" (expand "Xnew") (("2" (grind) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skolem 1 "vv") (("2" (flatten) (("2" (skeep) (("2" (expand "Bern_coeffs_endpoints_rel_rec" +) (("2" (lift-if) (("2" (ground) (("1" (inst?) (("1" (assert) nil nil) ("2" (hide-all-but 1) (("2" (expand "eptest?") (("2" (skosimp*) (("2" (lift-if) (("2" (ground) (("2" (typepred "cfm") (("2" (expand "eptest?") (("2" (inst - "i!1") (("2" (ground) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (assert) (("2" (inst - "a" "bsdegmono" "bspoly" "cf" "nvars" "rel" "terms" "cfm WITH [(vv):=bsdegmono(vv)]") (("1" (assert) nil nil) ("2" (hide-all-but 1) (("2" (expand "eptest?") (("2" (skosimp*) (("2" (lift-if) (("2" (ground) (("2" (typepred "cfm") (("2" (expand "eptest?") (("2" (inst - "i!1") (("2" (ground) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil) ((OR const-decl "[bool, bool -> bool]" booleans nil) (below type-eq-decl nil naturalnumbers nil) (< const-decl "bool" reals nil) (= const-decl "[T, T -> boolean]" equalities nil) (CoeffMono type-eq-decl nil util nil) (posnat nonempty-type-eq-decl nil integers nil) (> const-decl "bool" reals nil) (nonneg_int nonempty-type-eq-decl nil integers nil) (DegreeMono 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) (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) (bsdegmono skolem-const-decl "DegreeMono" multi_bernstein nil) (nvars skolem-const-decl "posnat" multi_bernstein nil) (cfm skolem-const-decl "(eptest?(bsdegmono, nvars))" multi_bernstein nil) (vv skolem-const-decl "nat" multi_bernstein nil) (gt_realorder name-judgement "RealOrder" util nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (NOT const-decl "[bool -> bool]" booleans nil) (Bern const-decl "real" bernstein_polynomials "reals/") (above nonempty-type-eq-decl nil integers nil) (sigma def-decl "real" sigma "reals/") (product_restrict_eq formula-decl nil product "reals/") (bsdegmono skolem-const-decl "DegreeMono" multi_bernstein nil) (restrict const-decl "[T -> real]" product "reals/") (x!2 skolem-const-decl "nat" multi_bernstein nil) (even_minus_odd_is_odd application-judgement "odd_int" integers nil) (real_plus_real_is_real application-judgement "real" reals nil) (subrange type-eq-decl nil integers nil) (nat_expt application-judgement "nat" exponentiation nil) (even_times_int_is_even application-judgement "even_int" integers nil) (int_plus_int_is_int application-judgement "int" integers nil) (nnrat_times_nnrat_is_nnrat application-judgement "nonneg_rat" rationals nil) (nnrat_exp application-judgement "nnrat" exponentiation nil) (/= const-decl "boolean" notequal nil) (nnint_plus_posint_is_posint application-judgement "posint" integers nil) (sigma_restrict_eq_0 formula-decl nil sigma "reals/") (posint_exp application-judgement "posint" exponentiation nil) (nat_exp application-judgement "nat" exponentiation nil) (mult_divides2 application-judgement "(divides(m))" divides nil) (mult_divides1 application-judgement "(divides(n))" divides nil) (nnint_times_nnint_is_nnint application-judgement "nonneg_int" integers nil) (expt def-decl "real" exponentiation nil) (factorial_0 formula-decl nil factorial "ints/") (posrat_div_posrat_is_posrat application-judgement "posrat" rationals nil) (C const-decl "posnat" binomial "reals/") (posint_times_posint_is_posint application-judgement "posint" integers nil) (posnat_expt application-judgement "posnat" exponentiation nil) (^ const-decl "real" exponentiation nil) (Xnew skolem-const-decl "[nat -> naturalnumber]" multi_bernstein nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (posint_plus_nnint_is_posint application-judgement "posint" integers nil) (sigma_split formula-decl nil sigma "reals/") (restrict const-decl "[T -> real]" sigma "reals/") (lt_realorder name-judgement "RealOrder" util nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (real_times_real_is_real application-judgement "real" reals nil) (int_minus_int_is_int application-judgement "int" integers nil) (sigma_restrict_eq formula-decl nil sigma "reals/") (numfield nonempty-type-eq-decl nil number_fields nil) (* const-decl "[numfield, numfield -> numfield]" number_fields nil) (bsproduct_eval const-decl "real" multi_bernstein nil) (T_low type-eq-decl nil product "reals/") (T_high type-eq-decl nil product "reals/") (product def-decl "real" product "reals/") (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (T_high type-eq-decl nil sigma "reals/") (T_low type-eq-decl nil sigma "reals/") (ge_realorder name-judgement "RealOrder" util nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (multibscoeff const-decl "real" multi_bernstein nil) (multibs_eval const-decl "real" multi_bernstein nil) (naturalnumber type-eq-decl nil naturalnumbers nil) (IF const-decl "[boolean, T, T -> T]" if_def nil) (nat_induction formula-decl nil naturalnumbers nil) (pred type-eq-decl nil defined_types nil) (eptest? skolem-const-decl "[[DegreeMono, posnat] -> [CoeffMono -> boolean]]" multi_bernstein nil) (bsdegmono!1 skolem-const-decl "DegreeMono" multi_bernstein nil) (nvars!1 skolem-const-decl "posnat" multi_bernstein nil) (le_realorder name-judgement "RealOrder" util nil) (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (Polynomial type-eq-decl nil util nil) (Polyproduct type-eq-decl nil util nil) (MultiBernstein type-eq-decl nil util nil) (Coeff type-eq-decl nil util nil) (IMPLIES const-decl "[bool, bool -> bool]" booleans nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (<= const-decl "bool" reals nil) (forall_X const-decl "bool" multi_bernstein nil) (list type-decl nil list_adt nil) (PRED type-eq-decl nil defined_types nil) (every adt-def-decl "boolean" list_adt nil) (null? adt-recognizer-decl "[list -> boolean]" list_adt nil) (cons? adt-recognizer-decl "[list -> boolean]" list_adt nil) (Vars type-eq-decl nil util nil) (unitbox? const-decl "bool" util nil) (list2array def-decl "T" array2list "structures/") (length def-decl "nat" list_props nil) (Bern_coeffs_endpoints_rel_rec def-decl "{l: list[real] | cons?(l) IMPLIES unitbox?(nvars)(list2array(0)(l)) AND length(l) = nvars}" multi_bernstein nil) (Bern_coeffs_endpoints_rel const-decl "{l: list[real] | cons?(l) IMPLIES unitbox?(nvars)(list2array(0)(l)) AND length(l) = nvars}" multi_bernstein nil)) shostak)) (Bern_coeffs_counterexample 0 (Bern_coeffs_counterexample-1 nil 3499800124 ("" (expand "Bern_coeffs_endpoints_rel") (("" (name "eptest?" "(LAMBDA (bsdegmono:DegreeMono,nvars): (LAMBDA (cfm:CoeffMono): (FORALL (i:below(nvar (("" (case "NOT FORALL (a: real, bsdegmono: DegreeMono, bspoly: MultiBernstein, cf: Coeff, nvars: posnat, rel: [[real, real] -> bool], terms: posnat,v:nat,cfm:(eptest?(bsdegmono,nvars))): (NOT null?(Bern_coeffs_endpoints_rel_rec(bspoly, bsdegmono, cf, nvars, terms, v) (cfm, rel, a))) IMPLIES length(Bern_coeffs_endpoints_rel_rec(bspoly, bsdegmono, cf, nvars, terms, v) (cfm, rel, a)) = nvars") (("1" (hide-all-but (-1 1)) (("1" (skeep) (("1" (typepred "Bern_coeffs_endpoints_rel_rec(bspoly, bsdegmono, cf, nvars, terms, v) (cfm, rel, a)") (("1" (assert) nil nil)) nil)) nil)) nil) ("2" (case "FORALL (a: real, bsdegmono: DegreeMono, bspoly: MultiBernstein, cf: Coeff, nvars: posnat, rel: [[real, real] -> bool], terms: posnat,v:nat,cfm:(eptest?(bsdegmono,nvars))): (NOT null?(Bern_coeffs_endpoints_rel_rec(bspoly, bsdegmono, cf, nvars, terms, v) (cfm, rel, a))) IMPLIES NOT rel(multibs_eval(bspoly, bsdegmono, cf, nvars, terms) (list2array(0) (Bern_coeffs_endpoints_rel_rec (bspoly, bsdegmono, cf, nvars, terms, v) (cfm, rel, a))),a)") (("1" (skeep) (("1" (case "FORALL (j: below(nvars)): intendpts(j)`1 AND intendpts(j)`2") (("1" (replace -1) (("1" (inst? -3) (("1" (assert) (("1" (inst? -2) (("1" (assert) nil nil) ("2" (hide-all-but 1) (("2" (expand "eptest?") (("2" (propax) nil nil)) nil)) nil)) nil)) nil) ("2" (hide-all-but 1) (("2" (expand "eptest?") (("2" (propax) nil nil)) nil)) nil)) nil)) nil) ("2" (replace 1) (("2" (assert) nil nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (induct "v") (("1" (skeep) (("1" (inst?) (("1" (assert) (("1" (expand "Bern_coeffs_endpoints_rel_rec") (("1" (lift-if) (("1" (ground) (("1" (hide 2 3) (("1" (name "Xnew" "LAMBDA (i:nat): IF i<nvars THEN (IF cfm(i)=0 THEN 0 ELSE 1 ENDIF) ELSE 0 ENDIF") (("1" (case "list2array(0)(corner_to_point(cfm, nvars)) = Xnew") (("1" (replace -1) (("1" (hide -1) (("1" (case "multibs_eval(bspoly, bsdegmono, cf, nvars, terms)(Xnew) = multibscoeff(bspoly, bsdegmo (("1" (assert) nil nil) ("2" (hide 2) (("2" (hide 2) (("2" (expand "multibscoeff") (("2" (expand "multibs_eval" +) (("2" (rewrite "sigma_restrict_eq") (("2" (hide 2) (("2" (decompose-equality) (("2" (expand "restrict") (("2" (lift-if) (("2" (ground) (("2" (case "bsproduct_eval(bspoly(x!1), bsdegmono, nvars)(Xnew) = product(0, nvars - 1, LAMBDA (j: nat): bspoly(x!1)(j)(cfm(j)))") (("1" (assert) nil nil) ("2" (hide 3) (("2" (expand "bsproduct_eval" +) (("2" (rewrite "product_restrict_eq") (("1" (hide 2) (("1" (decompose-equality) (("1" (expand "restrict") (("1" (lift-if) (("1" (ground) (("1" (typepred "cfm") (("1" (lemma "sigma_split") (("1" (inst?) (("1" (inst - "cfm(x!2)") (("1" (assert) (("1" (case "cfm(x!2)<=bsdegmono(x!2)") (("1" (assert) (("1" (replace -2) (("1" (hide -2) (("1" (case "sigma(1 + cfm(x!2), bsdegmono(x!2), LAMBDA (j: nat): IF j > bsdegmono(x!2) THEN 0 ELSE bspoly(x!1)(x!2)(j) * Bern(j, bsdegmono(x!2))(Xnew(x!2)) ENDIF) = 0") (("1" (replace -1) (("1" (hide -1) (("1" (assert) (("1" (lemma "sigma_split") (("1" (inst?) (("1" (inst - "cfm(x!2)-1") (("1" (assert) (("1" (replace -1) (("1" (hide -1) (("1" (case "sigma(0, cfm(x!2) - 1, LAMBDA (j: nat): IF j > bsdegmono(x!2) THEN 0 ELSE bspoly(x!1)(x!2)(j) * Bern(j, bsdegmono(x!2))(Xnew(x!2)) ENDIF) = 0") (("1" (replace -1) (("1" (hide -1) (("1" (assert) (("1" (expand "sigma" +) (("1" (expand "sigma" +) (("1" (expand "Xnew" +) (("1" (lift-if) (("1" (ground) (("1" (replace -1) (("1" (expand "Bern" +) (("1" (assert) (("1" (expand "^" +) (("1" (expand "expt" + 1) (("1" (lemma "expt_1i") (("1" (inst - "bsdegmono(x!2)") (("1" (assert) (("1" (expand "C" +) (("1" (assert) (("1" (expand "^" -1) (("1" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (typepred "cfm") (("2" (expand "eptest?" -1) (("2" (inst - "x!2") (("2" (assert) (("2" (replace -1) (("2" (expand "Bern" +) (("2" (expand "^" +) (("2" (expand "expt" + 1) (("2" (lemma "expt_1i") (("2" (inst - "bsdegmono(x!2)") (("2" (expand "^" -1) (("2" (replace -1) (("2" (expand "C" +) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 3) (("2" (rewrite "sigma_restrict_eq_0") (("2" (hide 2) (("2" (skosimp*) (("2" (typepred "i!1") (("2" (expand "Xnew" +) (("2" (expand "Bern" +) (("2" (expand "^" +) (("2" (expand "expt" +) (("2" (propax) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 3) (("2" (rewrite "sigma_restrict_eq_0") (("2" (hide 2) (("2" (skosimp*) (("2" (expand "Xnew" +) (("2" (lift-if) (("2" (ground) (("1" (expand "Bern" +) (("1" (expand "^") (("1" (expand "expt" + 1) (("1" (propax) nil nil)) nil)) nil)) nil) ("2" (expand "Bern" +) (("2" (typepred "i!1") (("2" (typepred "cfm") (("2" (expand "eptest?" -1) (("2" (inst - "x!2") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide -1) (("2" (expand "eptest?" -1) (("2" (inst - "x!2") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (replaces -1 :dir rl) (("2" (decompose-equality 1) (("2" (expand "corner_to_point") (("2" (case "x!1 < nvars") (("1" (assert) (("1" (rewrite "array2list_inv[real]") nil nil)) nil) ("2" (assert) (("2" (rewrite "list2array_sound") nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skolem 1 "vv") (("2" (flatten) (("2" (skeep) (("2" (expand "Bern_coeffs_endpoints_rel_rec" (-2 1)) (("2" (lift-if) (("2" (ground) (("1" (inst? -4) (("1" (assert) nil nil) ("2" (hide-all-but 1) (("2" (typepred "cfm") (("2" (expand "eptest?") (("2" (skosimp*) (("2" (inst - "i!1") (("2" (lift-if) (("2" (ground) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (inst? -2) (("1" (assert) nil nil) ("2" (hide-all-but 1) (("2" (typepred "cfm") (("2" (expand "eptest?") (("2" (skosimp*) (("2" (inst - "i!1") (("2" (lift-if) (("2" (ground) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((OR const-decl "[bool, bool -> bool]" booleans nil) (below type-eq-decl nil naturalnumbers nil) (< const-decl "bool" reals nil) (= const-decl "[T, T -> boolean]" equalities nil) (CoeffMono type-eq-decl nil util nil) (posnat nonempty-type-eq-decl nil integers nil) (> const-decl "bool" reals nil) (nonneg_int nonempty-type-eq-decl nil integers nil) (DegreeMono 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) (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) (multibs_eval const-decl "real" multi_bernstein nil) (IntervalEndpoints type-eq-decl nil util nil) (eptest? skolem-const-decl "[[DegreeMono, posnat] -> [CoeffMono -> boolean]]" multi_bernstein nil) (bsdegmono skolem-const-decl "DegreeMono" multi_bernstein nil) (nvars skolem-const-decl "posnat" multi_bernstein nil) (pred type-eq-decl nil defined_types nil) (nat_induction formula-decl nil naturalnumbers nil) (length_null formula-decl nil more_list_props "structures/") (IF const-decl "[boolean, T, T -> T]" if_def nil) (naturalnumber type-eq-decl nil naturalnumbers nil) (list2array_sound formula-decl nil array2list "structures/") (below type-eq-decl nil nat_types nil) (nth def-decl "T" list_props nil) (array2list const-decl "{l: listn(n) | FORALL (i: below(n)): a(i) = nth(l, i)}" array2list "structures/") (array2list_inv formula-decl nil array2list "structures/") (multibscoeff const-decl "real" multi_bernstein nil) (gt_realorder name-judgement "RealOrder" util nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (Bern const-decl "real" bernstein_polynomials "reals/") (above nonempty-type-eq-decl nil integers nil) (sigma def-decl "real" sigma "reals/") (product_restrict_eq formula-decl nil product "reals/") (bsdegmono skolem-const-decl "DegreeMono" multi_bernstein nil) (restrict const-decl "[T -> real]" product "reals/") (x!2 skolem-const-decl "nat" multi_bernstein nil) (even_minus_odd_is_odd application-judgement "odd_int" integers nil) (real_plus_real_is_real application-judgement "real" reals nil) (subrange type-eq-decl nil integers nil) (sigma_restrict_eq_0 formula-decl nil sigma "reals/") (posint_exp application-judgement "posint" exponentiation nil) (nat_exp application-judgement "nat" exponentiation nil) (mult_divides2 application-judgement "(divides(m))" divides nil) (mult_divides1 application-judgement "(divides(n))" divides nil) (nnint_times_nnint_is_nnint application-judgement "nonneg_int" integers nil) (expt def-decl "real" exponentiation nil) (C const-decl "posnat" binomial "reals/") (factorial_0 formula-decl nil factorial "ints/") (posrat_times_posrat_is_posrat application-judgement "posrat" rationals nil) (posrat_div_posrat_is_posrat application-judgement "posrat" rationals nil) (posint_times_posint_is_posint application-judgement "posint" integers nil) (posnat_expt application-judgement "posnat" exponentiation nil) (expt_1i formula-decl nil exponentiation nil) (^ const-decl "real" exponentiation nil) (Xnew skolem-const-decl "[nat -> naturalnumber]" multi_bernstein nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (posint_plus_nnint_is_posint application-judgement "posint" integers nil) (sigma_split formula-decl nil sigma "reals/") (restrict const-decl "[T -> real]" sigma "reals/") (le_realorder name-judgement "RealOrder" util nil) (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (lt_realorder name-judgement "RealOrder" util nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (real_times_real_is_real application-judgement "real" reals nil) (int_minus_int_is_int application-judgement "int" integers nil) (sigma_restrict_eq formula-decl nil sigma "reals/") (numfield nonempty-type-eq-decl nil number_fields nil) (* const-decl "[numfield, numfield -> numfield]" number_fields nil) (bsproduct_eval const-decl "real" multi_bernstein nil) (<= const-decl "bool" reals nil) (T_low type-eq-decl nil product "reals/") (T_high type-eq-decl nil product "reals/") (product def-decl "real" product "reals/") (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (T_high type-eq-decl nil sigma "reals/") (T_low type-eq-decl nil sigma "reals/") (ge_realorder name-judgement "RealOrder" util nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (corner_to_point const-decl "{l: listn[real](nvars) | unitbox?(nvars)(list2array(0)(l))}" util nil) (listn type-eq-decl nil listn "structures/") (vv skolem-const-decl "nat" multi_bernstein nil) (cfm skolem-const-decl "(eptest?(bsdegmono, nvars))" multi_bernstein nil) (nvars skolem-const-decl "posnat" multi_bernstein nil) (bsdegmono skolem-const-decl "DegreeMono" multi_bernstein nil) (NOT const-decl "[bool -> bool]" booleans nil) (Polynomial type-eq-decl nil util nil) (Polyproduct type-eq-decl nil util nil) (MultiBernstein type-eq-decl nil util nil) (Coeff type-eq-decl nil util nil) (IMPLIES const-decl "[bool, bool -> bool]" booleans nil) (list type-decl nil list_adt nil) (PRED type-eq-decl nil defined_types nil) (every adt-def-decl "boolean" list_adt nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (null? adt-recognizer-decl "[list -> boolean]" list_adt nil) (cons? adt-recognizer-decl "[list -> boolean]" list_adt nil) (Vars type-eq-decl nil util nil) (unitbox? const-decl "bool" util nil) (list2array def-decl "T" array2list "structures/") (length def-decl "nat" list_props nil) (Bern_coeffs_endpoints_rel_rec def-decl "{l: list[real] | cons?(l) IMPLIES unitbox?(nvars)(list2array(0)(l)) AND length(l) = nvars}" multi_bernstein nil) (Bern_coeffs_endpoints_rel const-decl "{l: list[real] | cons?(l) IMPLIES unitbox?(nvars)(list2array(0)(l)) AND length(l) = nvars}" multi_bernstein nil)) nil)) (Bern_le 0 (Bern_le-1 nil 3499875743 ("" (skeep) (("" (rewrite "Bern_coeffs_rel_def") (("" (expand "forall_X") (("" (skosimp*) (("" (rewrite "multibs_eval_equiv") (("" (name "cfnew" "LAMBDA (i:nat): a/(terms)") (("" (name "bspolynew" "LAMBDA(i:nat)(j:nat)(k:nat): 1") (("" (case "multibs_eval_rec(bspoly, bsdegmono, cf, nvars, terms, nvars) (LAMBDA (i: nat): 0)(X!1) <= multibs_eval_rec(bspolynew, bsdegmono, cfnew, nvars, terms, nv (LAMBDA (i: nat): 0)(X!1) AND multibs_eval_rec(bspolynew, bsdegmono, cfnew, nvars, terms, (LAMBDA (i: nat): 0)(X!1) =a") (("1" (flatten) (("1" (assert) nil nil)) nil) ("2" (hide 2) (("2" (split) (("1" (case "FORALL (cmo:CoeffMono,v:nat): v<=nvars AND le_below_mono?(nvars)(cmo,bsdegmono) (cmo)(X!1) <= multibs_eval_rec(bspolynew, bsdegmono, cfnew, nvars, terms, v) (cmo)(X!1)") (("1" (inst?) (("1" (assert) (("1" (hide-all-but 1) (("1" (grind) nil nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (induct "v") (("1" (skeep) (("1" (expand "multibs_eval_rec" +) (("1" (case "product(0, nvars - 1, LAMBDA (k: nat): IF cmo(k) <= bsdegmono(k) THEN Bern(cmo(k), bsdegmono(k))(X!1(k)) ELSE 1 ENDIF) >= 0") (("1" (case "multibs_eval_mono(bspoly, bsdegmono, cf, nvars, terms)(cmo) <= multibs_eval_mono(bspolynew, bsdegmono, cfnew, nvars, terms)(cmo)") (("1" (mult-by -1 "product(0, nvars - 1, LAMBDA (k: nat): IF cmo(k) <= bsdegmono(k) THEN Bern(cmo(k), bsdegmono(k))(X!1(k)) ELSE 1 ENDIF)") (("1" (hide 2) (("1" (skosimp*) (("1" (assert) nil nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (hide -1) (("2" (expand "multibs_eval_mono") (("2" (expand "bspolynew" +) (("2" (expand "cfnew" +) (("2" (expand "multibscoeff") (("2" (inst - "cmo") (("2" (assert) (("2" (case "FORALL (j:nat): j<terms IMPLIES sigma(0, j, LAMBDA (i: nat): a / (terms) * product(0, nvars - 1, LAMBDA (j: nat): 1)) = a*((j+1)/(terms))") (("1" (inst - "terms-1") (("1" (assert) nil nil)) nil) ("2" (hide 2) (("2" (case "FORALL (kr:nat): product(0,kr,LAMBDA (j:nat): 1) = 1") (("1" (induct "j") (("1" (assert) (("1" (expand "sigma" +) (("1" (expand "sigma" +) (("1" (inst - "nvars-1") (("1" (assert) nil nil)) nil)) nil)) nil)) nil) ("2" (skolem 1 "j") (("2" (flatten) (("2" (assert) (("2" (expand "sigma" +) (("2" (replace -1) (("2" (inst - "nvars-1") (("2" (replace -3) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (induct "kr") (("1" (expand "product") (("1" (expand "product") (("1" (propax) nil nil)) nil)) nil) ("2" (skolem 1 "kr") (("2" (flatten) (("2" (expand "product" +) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (case "FORALL (kr:nat): kr<=nvars-1 IMPLIES product(0, kr, LAMBDA (k: nat): IF cmo(k) <= bsdegmono(k) THEN Bern(cmo(k), bsdegmono(k))(X!1(k)) ELSE 1 ENDIF) >= 0") (("1" (inst - "nvars-1") (("1" (assert) nil nil)) nil) ("2" (hide 2) (("2" (induct "kr") (("1" (flatten) (("1" (expand "product") (("1" (expand "product") (("1" (lift-if) (("1" (ground) (("1" (lemma "Bern_nonnegative") (("1" (inst - "X!1(0)") (("1" (split -) (("1" (inst?) nil nil) ("2" (expand "unitbox?") (("2" (ground) (("2" (inst -8 "0") (("2" (ground) nil nil)) nil)) nil)) nil) ("3" (ground) (("3" (expand "unitbox?") (("3" (inst -8 "0") (("3" (ground) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skeep) (("2" (expand "product" +) (("2" (lift-if) (("2" (ground) (("2" (mult-by -2 "Bern(cmo(1 + j), bsdegmono(1 + j))(X!1(1 + j))") (("1" (assert) nil nil) ("2" (hide 2) (("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil) ("3" (hide 2) (("3" (lemma "Bern_nonnegative") (("3" (inst - "X!1(1+j)") (("3" (expand "unitbox?") (("3" (inst -10 "1+j") (("3" (split -) (("1" (inst?) nil nil) ("2" (ground) nil nil) ("3" (ground) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil)) nil) ("2" (skeep) (("2" (skeep) (("2" (expand "multibs_eval_rec" +) (("2" (rewrite "sigma_le") (("2" (hide 2) (("2" (skosimp*) (("2" (inst?) (("2" (assert) (("2" (hide 1) (("2" (expand "le_below_mono?") (("2" (skosimp*) (("2" (case "j!1 = j") (("1" (assert) nil nil) ("2" (inst -2 "j!1") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (rewrite "multibs_eval_equiv" + :dir rl) (("2" (expand "multibs_eval") (("2" (case "(LAMBDA (i: nat): cfnew(i) * bsproduct_eval(bspolynew(i), bsdegmono, nvars)(X!1)) = (LAMBDA (i:nat): a/terms)") (("1" (replace -1) (("1" (hide -1) (("1" (case "FORALL (j:nat): j<terms IMPLIES sigma(0,j,(LAMBDA (i:nat): a/terms)) = (j+1)*a/terms") (("1" (inst - "terms-1") (("1" (assert) nil nil)) nil) ("2" (hide 2) (("2" (induct "j") (("1" (assert) (("1" (expand "sigma") (("1" (expand "sigma") (("1" (assert) nil nil)) nil)) nil)) nil) ("2" (skolem 1 "j") (("2" (flatten) (("2" (assert) (("2" (expand "sigma" +) (("2" (replace -1) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (decompose-equality) (("2" (expand "cfnew") (("2" (expand "bsproduct_eval") (("2" (expand "bspolynew") (("2" (case "(LAMBDA (i: nat): sigma(0, bsdegmono(i), LAMBDA (j: nat): IF j > bsdegmono(i) THEN 0 ELSE 1 * Bern(j, bsdegmono(i))(X!1(i)) ENDIF)) = (LAMBDA (i:nat): 1)") (("1" (replace -1) (("1" (hide -1) (("1" (case "FORALL (kr:nat): product(0,kr,LAMBDA (j:nat): 1) = 1") (("1" (inst - "nvars-1") (("1" (replace -1) (("1" (assert) nil nil)) nil)) nil) ("2" (hide 2) (("2" (induct "kr") (("1" (expand "product") (("1" (expand "product") (("1" (propax) nil nil)) nil)) nil) ("2" (skolem 1 "jj") (("2" (flatten) (("2" (expand "product" +) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (decompose-equality) (("1" (lemma "Bernstein_partition_of_unity") (("1" (inst - "bsdegmono(x!2)" "X!1(x!2)") (("1" (assert) (("1" (replace -1) (("1" (propax) nil nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) nil nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil) ("4" (skosimp*) (("4" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((Bern_coeffs_rel_def formula-decl nil multi_bernstein nil) (number nonempty-type-decl nil numbers nil) (boolean nonempty-type-decl nil booleans 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) (bool nonempty-type-eq-decl nil booleans nil) (>= const-decl "bool" reals nil) (nat nonempty-type-eq-decl nil naturalnumbers nil) (DegreeMono type-eq-decl nil util nil) (Polynomial type-eq-decl nil util nil) (Polyproduct type-eq-decl nil util nil) (MultiBernstein type-eq-decl nil util nil) (Coeff type-eq-decl nil util nil) (nonneg_int nonempty-type-eq-decl nil integers nil) (> const-decl "bool" reals nil) (posnat nonempty-type-eq-decl nil integers nil) (<= const-decl "bool" reals nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (numfield nonempty-type-eq-decl nil number_fields nil) (= const-decl "[T, T -> boolean]" equalities nil) (/= const-decl "boolean" notequal nil) (nznum nonempty-type-eq-decl nil number_fields nil) (/ const-decl "[numfield, nznum -> numfield]" number_fields nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (CoeffMono type-eq-decl nil util nil) (multibs_eval_rec def-decl "real" multi_bernstein nil) (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (le_realorder name-judgement "RealOrder" util nil) (subrange type-eq-decl nil integers nil) (sigma_le formula-decl nil sigma "reals/") (int_minus_int_is_int application-judgement "int" integers nil) (OR const-decl "[bool, bool -> bool]" booleans nil) (T_low type-eq-decl nil product "reals/") (T_high type-eq-decl nil product "reals/") (product def-decl "real" product "reals/") (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (IF const-decl "[boolean, T, T -> T]" if_def nil) (above nonempty-type-eq-decl nil integers nil) (Bern const-decl "real" bernstein_polynomials "reals/") (cfnew skolem-const-decl "[nat -> real]" multi_bernstein nil) (T_low type-eq-decl nil sigma "reals/") (T_high type-eq-decl nil sigma "reals/") (sigma def-decl "real" sigma "reals/") (* const-decl "[numfield, numfield -> numfield]" number_fields nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (posrat_div_posrat_is_posrat application-judgement "posrat" rationals nil) (prod_nat application-judgement "nat" product_nat "reals/") (prod_pr application-judgement "posreal" product_nat "reals/") (prod_posnat application-judgement "posnat" product_nat "reals/") (posint_plus_nnint_is_posint application-judgement "posint" integers nil) (nnint_plus_posint_is_posint application-judgement "posint" integers nil) (real_plus_real_is_real application-judgement "real" reals nil) (mult_divides2 application-judgement "(divides(m))" divides nil) (mult_divides1 application-judgement "(divides(n))" divides nil) (posint_times_posint_is_posint application-judgement "posint" integers nil) (prod_nnr application-judgement "nnreal" product_nat "reals/") (multibscoeff const-decl "real" multi_bernstein nil) (bspolynew skolem-const-decl "[nat -> [nat -> [nat -> posint]]]" multi_bernstein nil) (both_sides_times_pos_le1_imp formula-decl nil extra_real_props nil) (nonneg_real nonempty-type-eq-decl nil real_types nil) (X!1 skolem-const-decl "Vars" multi_bernstein nil) (bsdegmono skolem-const-decl "DegreeMono" multi_bernstein nil) (cmo skolem-const-decl "CoeffMono" multi_bernstein nil) (nvars skolem-const-decl "posnat" multi_bernstein nil) (ge_realorder name-judgement "RealOrder" util nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (real_times_real_is_real application-judgement "real" reals nil) (gt_realorder name-judgement "RealOrder" util nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (multibs_eval_mono const-decl "real" multi_bernstein nil) (Bern_nonnegative formula-decl nil bernstein_polynomials "reals/") (unitbox? const-decl "bool" util nil) (j skolem-const-decl "nat" multi_bernstein nil) (both_sides_times_pos_ge1_imp formula-decl nil extra_real_props nil) (nat_induction formula-decl nil naturalnumbers nil) (pred type-eq-decl nil defined_types nil) (below type-eq-decl nil naturalnumbers nil) (< const-decl "bool" reals nil) (NOT const-decl "[bool -> bool]" booleans nil) (lt_realorder name-judgement "RealOrder" util nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (le_below_mono? const-decl "bool" util nil) (IMPLIES const-decl "[bool, bool -> bool]" booleans nil) (multibs_eval const-decl "real" multi_bernstein nil) (Bernstein_partition_of_unity formula-decl nil bernstein_polynomials "reals/") (bsproduct_eval const-decl "real" multi_bernstein nil) (posint nonempty-type-eq-decl nil integers nil) (multibs_eval_equiv formula-decl nil multi_bernstein nil) (Vars type-eq-decl nil util nil) (forall_X const-decl "bool" multi_bernstein nil)) nil)) (Bern_lt 0 (Bern_lt-1 nil 3499875835 ("" (expand "Bern_coeffs_rel") (("" (case "FORALL (a: real, bsdegmono: DegreeMono, bspoly: MultiBernstein, cf: Coeff, nvars, terms: posnat,v:nat,cmo:CoeffMono):v<=nvars AND Bern_coeffs_rel_rec(bspoly, bsdegmono, cf, nvars, terms, v) (cmo, <, a) IMPLIES (EXISTS (anew:real): anew<a AND Bern_coeffs_rel_rec(bspoly, bsdegmono, c (cmo, <=, anew))") (("1" (skeep) (("1" (inst?) (("1" (assert) (("1" (skeep -1) (("1" (lemma "Bern_le") (("1" (inst - "anew" "bsdegmono" "bspoly" "cf" "nvars" "terms") (("1" (expand "Bern_coeffs_rel") (("1" (assert) (("1" (expand "forall_X") (("1" (skosimp*) (("1" (inst - "X!1") (("1" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (induct "v") (("1" (skeep) (("1" (expand "Bern_coeffs_rel_rec") (("1" (inst + "(multibscoeff(bspoly, bsdegmono, cf, nvars, terms)(cmo) + a)/2") (("1" (assert) nil nil)) nil)) nil)) nil) ("2" (skolem 1 "v") (("2" (flatten) (("2" (skeep) (("2" (expand "Bern_coeffs_rel_rec" (-3 1)) (("2" (case "FORALL (d:nat): d<=bsdegmono(v) IMPLIES (EXISTS (anew: real):FORALL (gg:nat):gg<=d I anew < a AND Bern_coeffs_rel_rec(bspoly, bsdegmono, cf, nvars, terms, v) (cmo WITH [(v):=gg], <=, anew))") (("1" (inst - "bsdegmono(v)") (("1" (assert) (("1" (skeep -1) (("1" (inst + "anew") (("1" (assert) (("1" (case "anew < a") (("1" (assert) (("1" (skosimp*) (("1" (inst - "d!1") (("1" (assert) nil nil)) nil)) nil)) nil) ("2" (inst - "0") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (induct "d") (("1" (assert) (("1" (flatten) (("1" (inst - "a" "bsdegmono" "bspoly" "cf" "nvars" "terms" "cmo WITH [(v):=0]") (("1" (assert) (("1" (inst -4 "0") (("1" (assert) (("1" (skosimp*) (("1" (inst + "anew!1") (("1" (skosimp*) (("1" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skolem 1 "dd") (("2" (flatten) (("2" (assert) (("2" (skosimp*) (("2" (inst - "a" "bsdegmono" "bspoly" "cf" "nvars" "terms" "cmo WITH [(v):=dd+1]") (("2" (assert) (("2" (inst -5 "dd+1") (("2" (assert) (("2" (skosimp*) (("2" (inst + "max(anew!1,anew!2)") (("2" (skosimp*) (("2" (split +) (("1" (inst - "0") (("1" (assert) (("1" (flatten) (("1" (expand "max") (("1" (lift-if) (("1" (ground) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (case "gg!1 = 1+dd") (("1" (assert) (("1" (replace -1) (("1" (lemma "Bern_coeffs_rel_implic") (("1" (inst?) (("1" (inst - "anew!2") (("1" (assert) (("1" (hide-all-but 1) (("1" (grind) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (inst - "gg!1") (("2" (assert) (("2" (flatten) (("2" (lemma "Bern_coeffs_rel_implic") (("2" (inst?) (("2" (inst - "anew!1") (("2" (assert) (("2" (hide-all-but 1) (("2" (grind) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((< const-decl "bool" reals nil) (Bern_coeffs_rel_rec def-decl "bool" multi_bernstein nil) (<= const-decl "bool" reals nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (IMPLIES const-decl "[bool, bool -> bool]" booleans nil) (CoeffMono type-eq-decl nil util nil) (posnat nonempty-type-eq-decl nil integers nil) (> const-decl "bool" reals nil) (nonneg_int nonempty-type-eq-decl nil integers 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) (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) (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) (Vars type-eq-decl nil util nil) (forall_X const-decl "bool" multi_bernstein nil) (Bern_le formula-decl nil multi_bernstein nil) (lt_realorder name-judgement "RealOrder" util nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (le_realorder name-judgement "RealOrder" util nil) (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (pred type-eq-decl nil defined_types nil) (nat_induction formula-decl nil naturalnumbers nil) (real_plus_real_is_real application-judgement "real" reals nil) (numfield nonempty-type-eq-decl nil number_fields 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 -> numfield]" number_fields nil) (multibscoeff const-decl "real" multi_bernstein nil) (real_div_nzreal_is_real application-judgement "real" reals nil) (nnint_plus_posint_is_posint application-judgement "posint" integers nil) (= const-decl "[T, T -> boolean]" equalities nil) (NOT const-decl "[bool -> bool]" booleans nil) (Bern_coeffs_rel_implic formula-decl nil multi_bernstein nil) (max const-decl "{p: real | p >= m AND p >= n}" real_defs nil) (posint_plus_nnint_is_posint application-judgement "posint" integers nil) (upto nonempty-type-eq-decl nil naturalnumbers nil) (Bern_coeffs_rel const-decl "bool" multi_bernstein nil)) nil)) (Bern_ge 0 (Bern_ge-1 nil 3499875940 ("" (skeep) (("" (rewrite "Bern_coeffs_rel_def") (("" (expand "forall_X") (("" (skosimp*) (("" (rewrite "multibs_eval_equiv") (("" (name "cfnew" "LAMBDA (i:nat): a/(terms)") (("" (name "bspolynew" "LAMBDA(i:nat)(j:nat)(k:nat): 1") (("" (case "multibs_eval_rec(bspoly, bsdegmono, cf, nvars, terms, nvars) (LAMBDA (i: nat): 0)(X!1) >= multibs_eval_rec(bspolynew, bsdegmono, cfnew, nvars, terms, nv (LAMBDA (i: nat): 0)(X!1) AND multibs_eval_rec(bspolynew, bsdegmono, cfnew, nvars, terms, (LAMBDA (i: nat): 0)(X!1) =a") (("1" (flatten) (("1" (assert) nil nil)) nil) ("2" (hide 2) (("2" (split) (("1" (case "FORALL (cmo:CoeffMono,v:nat): v<=nvars AND le_below_mono?(nvars)(cmo,bsdegmono) (cmo)(X!1) >= multibs_eval_rec(bspolynew, bsdegmono, cfnew, nvars, terms, v) (cmo)(X!1)") (("1" (inst?) (("1" (assert) (("1" (hide-all-but 1) (("1" (grind) nil nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (induct "v") (("1" (skeep) (("1" (expand "multibs_eval_rec" +) (("1" (case "product(0, nvars - 1, LAMBDA (k: nat): IF cmo(k) <= bsdegmono(k) THEN Bern(cmo(k), bsdegmono(k))(X!1(k)) ELSE 1 ENDIF) >= 0") (("1" (case "multibs_eval_mono(bspoly, bsdegmono, cf, nvars, terms)(cmo) >= multibs_eval_mono(bspolynew, bsdegmono, cfnew, nvars, terms)(cmo)") (("1" (mult-by -1 "product(0, nvars - 1, LAMBDA (k: nat): IF cmo(k) <= bsdegmono(k) THEN Bern(cmo(k), bsdegmono(k))(X!1(k)) ELSE 1 ENDIF)") (("1" (hide 2) (("1" (skosimp*) (("1" (assert) nil nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (hide -1) (("2" (expand "multibs_eval_mono") (("2" (expand "bspolynew" +) (("2" (expand "cfnew" +) (("2" (expand "multibscoeff") (("2" (inst - "cmo") (("2" (assert) (("2" (case "FORALL (j:nat): j<terms IMPLIES sigma(0, j, LAMBDA (i: nat): a / (terms) * product(0, nvars - 1, LAMBDA (j: nat): 1)) = a*((j+1)/(terms))") (("1" (inst - "terms-1") (("1" (assert) nil nil)) nil) ("2" (hide 2) (("2" (case "FORALL (kr:nat): product(0,kr,LAMBDA (j:nat): 1) = 1") (("1" (induct "j") (("1" (assert) (("1" (expand "sigma" +) (("1" (expand "sigma" +) (("1" (inst - "nvars-1") (("1" (assert) nil nil)) nil)) nil)) nil)) nil) ("2" (skolem 1 "j") (("2" (flatten) (("2" (assert) (("2" (expand "sigma" +) (("2" (replace -1) (("2" (inst - "nvars-1") (("2" (replace -3) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (induct "kr") (("1" (expand "product") (("1" (expand "product") (("1" (propax) nil nil)) nil)) nil) ("2" (skolem 1 "kr") (("2" (flatten) (("2" (expand "product" +) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (case "FORALL (kr:nat): kr<=nvars-1 IMPLIES product(0, kr, LAMBDA (k: nat): IF cmo(k) <= bsdegmono(k) THEN Bern(cmo(k), bsdegmono(k))(X!1(k)) ELSE 1 ENDIF) >= 0") (("1" (inst - "nvars-1") (("1" (assert) nil nil)) nil) ("2" (hide 2) (("2" (induct "kr") (("1" (flatten) (("1" (expand "product") (("1" (expand "product") (("1" (lift-if) (("1" (ground) (("1" (lemma "Bern_nonnegative") (("1" (inst - "X!1(0)") (("1" (split -) (("1" --> -------------------- --> maximum size reached --> --------------------
¤ Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.0.907Bemerkung: (vorverarbeitet am 2026-04-28) ¤*Bot Zugriff |
|
||||||||||||||
2026-05-26