| products/sources/formale Sprachen/PVS/numbers/ |
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Quellcode-Bibliothek© Kompilation durch diese Firma[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]Datei: infinite_card.pvs Sprache: PVSUntersuchung PVS© |
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(infinite_primes
(prime_divides 0 (prime_divides-4 nil 3507987320 ("" (skosimp*) (("" (lemma "prime_factors") (("" (inst?) (("" (skosimp*) (("" (expand "product") (("" (lift-if) (("" (split) (("1" (flatten) (("1" (assert) nil nil)) nil) ("2" (flatten) (("2" (expand "product") (("2" (assert) (("2" (inst + "seq(fs!1)(length(fs!1) - 1)") (("2" (expand "ordered_list_of_primes?") (("2" (flatten) (("2" (expand "list_of_primes?") (("2" (inst?) (("2" (assert) (("2" (expand "divides") (("2" (inst?) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((prime_factors formula-decl nil prime_factorization nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (prod_nnr application-judgement "nnreal" product_nat "reals/") (int_minus_int_is_int application-judgement "int" integers nil) (mult_divides2 application-judgement "(divides(m))" divides nil) (mult_divides1 application-judgement "(divides(n))" divides nil) (posint_times_posint_is_posint application-judgement "posint" integers nil) (ordered_list_of_primes? const-decl "bool" prime_factorization nil) (list_of_primes? const-decl "bool" prime_factorization 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/") (divides const-decl "bool" divides nil) (< const-decl "bool" reals nil) (below type-eq-decl nil naturalnumbers nil) (nat nonempty-type-eq-decl nil naturalnumbers nil) (barray type-eq-decl nil fseqs "structures/") (fseq type-eq-decl nil fseqs "structures/") (numfield nonempty-type-eq-decl nil number_fields nil) (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (product def-decl "real" product "reals/") (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (product const-decl "posnat" product_fseq_posnat "reals/") (posnat nonempty-type-eq-decl nil integers nil) (> const-decl "bool" reals nil) (nonneg_int nonempty-type-eq-decl nil integers nil) (>= const-decl "bool" reals nil) (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)) nil) (prime_divides-3 nil 3407858878 ("" (skosimp*) (("" (lemma "prime_factors") (("" (inst?) (("" (skosimp*) (("" (expand "product") (("" (lift-if) (("" (split) (("1" (flatten) (("1" (assert) nil nil)) nil) ("2" (flatten) (("2" (expand "product") (("2" (assert) (("2" (inst + "seq(fs!1)(length(fs!1) - 1)") (("2" (expand "ordered_list_of_primes?") (("2" (flatten) (("2" (expand "list_of_primes?") (("2" (inst?) (("2" (assert) (("2" (expand "divides") (("2" (inst?) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((prime_factors formula-decl nil prime_factorization nil) (prod_posnat application-judgement "posnat" product_nat "reals/") (prod_pr application-judgement "posreal" product_nat "reals/") (prod_nat application-judgement "nat" product_nat "reals/") (prod_nnr application-judgement "nnreal" product_nat "reals/") (ordered_list_of_primes? const-decl "bool" prime_factorization nil) (list_of_primes? const-decl "bool" prime_factorization nil) (T_low type-eq-decl nil product "reals/") (T_high type-eq-decl nil product "reals/") (barray type-eq-decl nil fseqs "structures/") (fseq type-eq-decl nil fseqs "structures/") (product def-decl "real" product "reals/") (product const-decl "posnat" product_fseq_posnat "reals/")) nil) (prime_divides-2 nil 3407849330 ("" (skosimp*) (("" (lemma "prime_factors") (("" (inst?) (("" (skosimp*) (("" (expand "product") (("" (lift-if) (("" (split) (("1" (flatten) (("1" (assert) nil nil)) nil) ("2" (flatten) (("2" (expand "product_rec") (("2" (lift-if) (("2" (ground) (("1" (expand "ordered_list_of_primes?") (("1" (flatten) (("1" (expand "list_of_primes?") (("1" (inst -3 "0") nil nil)) nil)) nil)) nil) ("2" (inst + "seq(fs!1)(length(fs!1) - 1)") (("2" (expand "ordered_list_of_primes?") (("2" (flatten) (("2" (expand "list_of_primes?") (("2" (inst?) (("2" (assert) (("2" (expand "divides?") (("2" (hide -2 -3 -4 1 2 3) (("2" (inst?) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((prime_factors formula-decl nil prime_factorization nil) (ordered_list_of_primes? const-decl "bool" prime_factorization nil) (list_of_primes? const-decl "bool" prime_factorization nil)) nil) (prime_divides-1 nil 3306142515 ("" (skosimp*) (("" (lemma "prime_factors") (("" (inst?) (("" (skosimp*) (("" (expand "product") (("" (lift-if) (("" (split) (("1" (flatten) (("1" (assert) nil nil)) nil) ("2" (flatten) (("2" (expand "product_rec") (("2" (lift-if) (("2" (ground) (("1" (expand "ordered_list_of_primes?") (("1" (flatten) (("1" (expand "list_of_primes?") (("1" (inst -3 "0") nil nil)) nil)) nil)) nil) ("2" (inst + "seq(fs!1)(length(fs!1) - 1)") (("2" (expand "ordered_list_of_primes?") (("2" (flatten) (("2" (expand "list_of_primes?") (("2" (inst?) (("2" (assert) (("2" (expand "divides") (("2" (hide -2 -3 -4 1 2 3) (("2" (inst?) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((prime_factors formula-decl nil prime_factorization nil) (ordered_list_of_primes? const-decl "bool" prime_factorization nil) (list_of_primes? const-decl "bool" prime_factorization nil)) nil)) (divides_product 0 (divides_product-3 nil 3407858913 ("" (skosimp*) (("" (expand "product") (("" (case "(FORALL (n: nat): n < length(fs!1) AND ii!1 <= n IMPLIES divides(seq(fs!1)(ii!1), product(0, n, fs!1`seq)))") (("1" (inst?) (("1" (assert) nil nil)) nil) ("2" (hide 2) (("2" (induct "n" 1 "NAT_induction") (("2" (skosimp*) (("2" (expand "product" +) (("2" (case-replace "ii!1 = j!1") (("1" (hide -2) (("1" (expand "divides") (("1" (inst + "product(0, j!1 - 1, fs!1`seq)") (("1" (flatten) (("1" (assert) nil nil)) nil)) nil)) nil)) nil) ("2" (inst - "j!1-1") (("1" (assert) (("1" (expand "divides") (("1" (skosimp*) (("1" (inst + "x!1 * seq(fs!1)(j!1)") (("1" (assert) nil nil)) nil)) nil)) nil)) nil) ("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((product const-decl "posnat" product_fseq_posnat "reals/") (= const-decl "[T, T -> boolean]" equalities nil) (j!1 skolem-const-decl "nat" infinite_primes 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) (* const-decl "[numfield, numfield -> numfield]" number_fields nil) (NAT_induction formula-decl nil naturalnumbers nil) (pred type-eq-decl nil defined_types nil) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (int_minus_int_is_int application-judgement "int" integers nil) (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (numfield nonempty-type-eq-decl nil number_fields nil) (prod_nnr application-judgement "nnreal" product_nat "reals/") (prod_nat application-judgement "nat" product_nat "reals/") (prod_pr application-judgement "posreal" product_nat "reals/") (prod_posnat application-judgement "posnat" product_nat "reals/") (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (product_nat_nat application-judgement "posnat" product_fseq_posnat "reals/") (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) (IMPLIES const-decl "[bool, bool -> bool]" booleans nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (< const-decl "bool" reals nil) (nonneg_int nonempty-type-eq-decl nil integers nil) (> const-decl "bool" reals nil) (posnat nonempty-type-eq-decl nil integers nil) (barray type-eq-decl nil fseqs "structures/") (fseq type-eq-decl nil fseqs "structures/") (<= const-decl "bool" reals nil) (below type-eq-decl nil naturalnumbers nil) (divides const-decl "bool" divides 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/")) nil) (divides_product-2 nil 3407849284 (";;; Proof divides_product-1 for formula infinite_primes.divides_product" (skosimp*) ((";;; Proof divides_product-1 for formula infinite_primes.divides_product" (expand "product") ((";;; Proof divides_product-1 for formula infinite_primes.divides_product" (case "(FORALL (n: nat): n < length(fs!1) AND ii!1 <= n IMPLIES divides?(seq(fs!1)(ii!1), product_rec(fs!1, n)))") (("1" (inst?) (("1" (assert) nil))) ("2" (hide 2) (("2" (induct "n" 1 "NAT_induction") (("2" (skosimp*) (("2" (expand "product_rec" +) (("2" (lift-if) (("2" (ground) (("1" (case-replace "ii!1 = 0") (("1" (expand "divides?") (("1" (inst + "1") (("1" (assert) nil))))) ("2" (assert) nil))) ("2" (case-replace "ii!1 = j!1") (("1" (hide -2) (("1" (expand "divides?") (("1" (inst + "product_rec(fs!1, j!1 - 1)") nil))))) ("2" (inst -1 "j!1-1") (("2" (assert) (("2" (expand "divides?") (("2" (skosimp*) (("2" (inst + "k!1 * seq(fs!1)(j!1)") (("2" (assert) (("2" (replace -1) (("2" (hide -1) (("2" (assert) (("2" (name-replace "T1" "seq(fs!1)(j!1)") (("2" (name-replace "T2" "seq(fs!1)(ii!1)") (("2" (assert) nil)))))))))))))))))))))))))))))))))))))))))) ";;; developed with shostak decision procedures") nil nil) (divides_product-1 nil 3306142515 ("" (skosimp*) (("" (expand "product") (("" (case "(FORALL (n: nat): n < length(fs!1) AND ii!1 <= n IMPLIES divides(seq(fs!1)(ii!1), product_rec(fs!1, n)))") (("1" (inst?) (("1" (assert) nil nil)) nil) ("2" (hide 2) (("2" (induct "n" 1 "NAT_induction") (("2" (skosimp*) (("2" (expand "product_rec" +) (("2" (lift-if) (("2" (ground) (("1" (case-replace "ii!1 = 0") (("1" (expand "divides") (("1" (inst + "1") (("1" (assert) nil nil)) nil)) nil) ("2" (assert) nil nil)) nil) ("2" (case-replace "ii!1 = j!1") (("1" (hide -2) (("1" (expand "divides") (("1" (inst + "product_rec(fs!1, j!1 - 1)") nil nil)) nil)) nil) ("2" (inst -1 "j!1-1") (("2" (assert) (("2" (expand "divides") (("2" (skosimp*) (("2" (inst + "k!1 * seq(fs!1)(j!1)") (("2" (assert) (("2" (replace -1) (("2" (hide -1) (("2" (assert) (("2" (name-replace "T1" "seq(fs!1)(j!1)") (("2" (name-replace "T2" "seq(fs!1)(ii!1)") (("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)) (primes_infinite 0 (primes_infinite-5 nil 3514297915 ("" (lemma "bij_exists[nat]") (("" (inst?) (("" (skosimp*) (("" (case "(EXISTS (f: [below(card(Primes)) -> (Primes)]): bijective?(f))") (("1" (skosimp*) (("1" (hide -2) (("1" (name "prime_seq" "(# length := card(Primes), seq := (LAMBDA (ii: nat): IF ii < card(Primes) THEN f!2(ii) ELSE default ENDIF) #)") (("1" (name "M" "product(prime_seq) + 1") (("1" (case "(FORALL (p: (Primes)): NOT divides(p,M))") (("1" (case "prime?(M)") (("1" (expand "bijective?") (("1" (flatten) (("1" (expand "surjective?") (("1" (inst -6 "M") (("1" (skosimp*) (("1" (lemma "product_fseq_ge") (("1" (inst -1 "prime_seq" "x!1") (("1" (replace -5 -1 rl) (("1" (assert) nil nil)) nil) ("2" (replace -4 + rl) (("2" (assert) nil nil)) nil)) nil)) nil)) nil) ("2" (expand "Primes") (("2" (propax) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (lemma "prime_divides") (("2" (inst?) (("2" (assert) (("2" (skosimp*) (("2" (inst?) (("2" (expand "Primes") (("2" (propax) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (lemma "divides_plus_1") (("2" (inst -1 "M-1" "p!1") (("1" (assert) (("1" (split -1) (("1" (hide -1) (("1" (replace -1 * rl) (("1" (assert) (("1" (typepred "p!1") (("1" (expand "bijective?") (("1" (flatten) (("1" (expand "surjective?") (("1" (inst -5 "p!1") (("1" (skosimp*) (("1" (replace -5 + rl) (("1" (case-replace "f!2(x!1) = seq(prime_seq)(x!1)") (("1" (lemma "divides_product") (("1" (ground) (("1" (inst -1 "prime_seq" "x!1") nil nil)) nil)) nil) ("2" (hide -2 2) (("2" (replace -2 :dir rl) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (typepred "p!1") (("2" (expand "Primes") (("2" (lemma "prime_gt_1") (("2" (lemma "prime_gt_1") (("2" (inst?) (("2" (inst?) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (assert) (("2" (skosimp*) (("2" (expand "Primes") (("2" (lemma "prime_gt_1") (("2" (inst?) (("2" (assert) (("2" (typepred "p!1") (("2" (expand "Primes") (("2" (propax) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("3" (prop) (("3" (ground) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (prop) (("1" (skosimp*) (("1" (ground) (("1" (replace -1 * rl) (("1" (assert) (("1" (hide -1) (("1" (typepred "f!2(x1!1)") (("1" (expand "Primes") (("1" (lemma "prime_gt_1") (("1" (inst?) (("1" (assert) (("1" (lemma "bij_inv_is_bij[(Primes),below(card(Primes))]") (("1" (inst?) (("1" (assert) (("1" (hide -1 -2 -3) (("1" (lift-if) (("1" (ground) nil nil)) nil)) nil)) nil) ("2" (ground) (("1" (skosimp*) (("1" (lift-if) (("1" (ground) nil nil)) nil)) nil) ("2" (skosimp*) (("2" (lift-if) (("2" (ground) nil nil)) nil)) nil)) nil)) nil) ("2" (inst + "2") (("2" (expand "Primes") (("2" (rewrite "prime_2") nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (lift-if) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (assert) (("2" (expand "prime_seq") (("2" (propax) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (propax) nil nil)) nil)) nil)) nil) ("2" (lemma "bij_inv_is_bij[(Primes),below(card(Primes))]") (("1" (inst?) (("1" (assert) (("1" (inst?) (("1" (inst + "2") (("1" (expand "Primes") (("1" (assert) (("1" (rewrite "prime_2") nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (inst + "2") (("2" (hide -1 -2 -3) (("2" (expand "Primes") (("2" (rewrite "prime_2") nil nil)) nil)) nil)) nil)) nil) ("3" (expand "Primes") (("3" (skosimp*) nil nil)) nil)) nil)) nil)) nil)) nil) ((Primes const-decl "set[nat]" infinite_primes nil) (is_finite const-decl "bool" finite_sets nil) (set type-eq-decl nil sets nil) (finite_set type-eq-decl nil finite_sets nil) (bijective? const-decl "bool" functions nil) (below type-eq-decl nil naturalnumbers nil) (card const-decl "{n: nat | n = Card(S)}" finite_sets nil) (Card const-decl "nat" finite_sets nil) (= const-decl "[T, T -> boolean]" equalities nil) (< const-decl "bool" reals nil) (posint_plus_nnint_is_posint application-judgement "posint" integers nil) (numfield nonempty-type-eq-decl nil number_fields nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (barray type-eq-decl nil fseqs "structures/") (fseq type-eq-decl nil fseqs "structures/") (product const-decl "posnat" product_fseq_posnat "reals/") (posint nonempty-type-eq-decl nil integers nil) (IMPLIES const-decl "[bool, bool -> bool]" booleans nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (int_minus_int_is_int application-judgement "int" integers nil) (/= const-decl "boolean" notequal nil) (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (p!1 skolem-const-decl "(Primes)" infinite_primes nil) (nzint nonempty-type-eq-decl nil integers nil) (divides_product formula-decl nil infinite_primes nil) (prime_gt_1 formula-decl nil primes "ints/") (divides_plus_1 formula-decl nil divides_lems "ints/") (prime? const-decl "bool" primes "ints/") (M skolem-const-decl "posint" infinite_primes nil) (product_fseq_ge formula-decl nil product_fseq_posnat "reals/") (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (prime_seq skolem-const-decl "[# length: {n: nat | n = Card(Primes)}, seq: [nat -> nat] #]" infinite_primes nil) (x!1 skolem-const-decl "below(card(Primes))" infinite_primes nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (surjective? const-decl "bool" functions nil) (p!1 skolem-const-decl "nat" infinite_primes nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (prime_divides formula-decl nil infinite_primes nil) (divides const-decl "bool" divides nil) (NOT const-decl "[bool -> bool]" booleans nil) (bij_inv_is_bij formula-decl nil function_inverse nil) (TRUE const-decl "bool" booleans nil) (f!2 skolem-const-decl "[below(card(Primes)) -> (Primes)]" infinite_primes nil) (IFF const-decl "[bool, bool -> bool]" booleans nil) (prime_2 formula-decl nil primes "ints/") (IF const-decl "[boolean, T, T -> T]" if_def nil) (nonneg_int nonempty-type-eq-decl nil integers nil) (> const-decl "bool" reals nil) (posnat nonempty-type-eq-decl nil integers nil) (default const-decl "T" fseqs "structures/") (inverse const-decl "D" function_inverse nil) (bij_exists formula-decl nil finite_sets 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)) nil) (primes_infinite-4 nil 3410695243 ("" (lemma "bij_exists[nat]") (("" (inst?) 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