fact stringlengths 8 29k | type stringclasses 12
values | library stringclasses 1
value | imports listlengths 0 132 | filename stringclasses 381
values | symbolic_name stringlengths 1 2.01k | docstring stringclasses 1
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|---|---|---|---|---|---|---|
is_pseudo_lprime (n : Z) (l : list (Z * Z)) := match l with | (p, m) :: l1 => if (n <? m)%Z then true else if ((n mod p) =? 0)%Z then false else is_pseudo_lprime n l1 | _ => true end. | Fixpoint | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | is_pseudo_lprime | |
is_pseudo_lprime_correct n l : 0 <= n -> (forall i j, In (i, j) l -> 1 < i /\ j = i * i) -> is_pseudo_lprime n l = false -> ~ prime n. Proof. intros nP; elim l; simpl; [discriminate| intros [p m] l1 IH Hl]. assert (H : 1 < p /\ m = p * p) by now apply Hl; auto. destruct H as [pP mSp]; subst m. case Z.ltb_spec0; [discri... | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | is_pseudo_lprime_correct | |
check_list l := match l with (i, j) :: l1 => if (1 <? i) then if (j =? (i * i)) then check_list l1 else false else false | [::] => true end. | Fixpoint | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | check_list | |
checklist_ok i j l : check_list l = true -> In (i, j) l -> 1 < i /\ j = i * i. Proof. revert i j; elim l; simpl; [intros ? ? ? []| intros [i1 j1] l1 IH i j]. case Z.ltb_spec0; try discriminate. case Z.eqb_spec; try discriminate. intros H1 H2 H3 [H4|H4]. now inversion H4; subst; auto. now apply IH. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | checklist_ok | |
is_pseudo_prime n := is_pseudo_lprime n primes2. | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | is_pseudo_prime | |
is_pseudo_prime_correct n : 0 <= n -> is_pseudo_prime n = false -> ~ prime n. Proof. intros nP nPP; apply (is_pseudo_lprime_correct n primes2); auto. intros i j; apply checklist_ok; compute; auto. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | is_pseudo_prime_correct | |
lZ_insert (a : Z) (l : list Z) := match l with | b :: l1 => if (a <=? b)%Z then a :: l else b :: lZ_insert a l1 | _ => [::a] end. | Fixpoint | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | lZ_insert | |
lZ_insert_consr a p l : In a l -> In a (lZ_insert p l). Proof. now elim l; [intros H; case H| simpl; intros c l1 IH [Hca|]]; case Z.leb; simpl; auto. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | lZ_insert_consr | |
lZ_insert_consl p l : In p (lZ_insert p l). Proof. now elim l; simpl; [auto | intros c l1 IH]; case Z.leb; simpl; auto. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | lZ_insert_consl | |
lZ_insert_cons a p l : In a (lZ_insert p l) -> a = p \/ In a l. Proof. elim l; [intros H; case H; auto| simpl; intros c l1 IH]. case Z.leb; simpl; auto. - intros [|[|]]; auto. - intros [|]; auto; case (IH H); auto. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | lZ_insert_cons | |
add_ltlist i n l1 l2 := fold_left (fun l z => let v := i * b ^ n + z in if is_pseudo_prime v then lZ_insert v l else l) l1 l2. | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | add_ltlist | |
add_ltlist_subset i n l1 l2 k : In k l2 -> In k (add_ltlist i n l1 l2). Proof. generalize l2; elim l1; simpl; auto. intros a l3 IH l4 Il4. case is_pseudo_prime; apply IH; auto. now apply lZ_insert_consr. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | add_ltlist_subset | |
add_ltlist_correct i n l1 l2 k : 0 <= b -> 0 <= i -> 0 <= k -> In k l1 -> prime(i * b ^ n + k) -> In (i * b ^ n + k) (add_ltlist i n l1 l2). Proof. intros bP iP. generalize l2 k; elim l1; simpl; [intros _ _ _ []| intros z l3 IH l4 k4 k4P Ik4 inkP]. destruct Ik4 as [zE | Ink4l4]. subst z. apply add_ltlist_subset. genera... | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | add_ltlist_correct | |
lnext (n : Z) (l1 : list Z) := let l2 := ldigit in fold_left (fun l i => add_ltlist i n l1 l) l2 [::]. | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | lnext | |
lnext_correct n l1 k : 0 <= n -> (forall k, ltprime k -> b ^ n <= k < b ^ (n + 1) -> In k l1) -> ltprime k -> b ^ (n + 1) <= k < b ^ (n + 2) -> In k (lnext (n + 1) l1). Proof. intros nP Hl Hlt Hk. assert (Le : log k = n + 1). apply log_inv; try lia; replace (n + 1 + 1) with (n + 2); lia. pose (k1 := k mod (b ^ (n + 1))... | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | lnext_correct | |
ltprime_tac b := repeat (match goal with |- ltprime _ ?a => let a' := eval compute in a in let v := constr: (log b a') in let v1 := constr: (a' mod (b ^ v)) in let v2 := constr: (a' / (b ^ v)) in let v1' := eval compute in v1 in apply (ltprime_ldecompose b (refl_equal _) v1' v2); [compute; auto| | ] | |- prime ?a => so... | Ltac | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_tac | |
ltprime_list1 := [:: 2; 3; 5; 7]%Z. | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list1 | |
ltprime_list2 := [:: 13; 17; 23; 37; 43; 47; 53; 67; 73; 83; 97]. | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list2 | |
ltprime_list3 := [:: 113; 137; 167; 173; 197; 223; 283; 313; 317; 337; 347; 353; 367; 373; 383; 397; 443; 467; 523; 547; 613; 617; 643; 647; 653; 673; 683; 743; 773; 797; 823; 853; 883; 937; 947; 953; 967; 983; 997]. | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list3 | |
ltprime_list4 := [:: 1223; 1283; 1367; 1373; 1523; 1613; 1823; 1997; 2113; 2137; 2347; 2383; 2467; 2617; 2647; 2683; 2797; 2953; 3137; 3167; 3313; 3347; 3373; 3467; 3547; 3613; 3617; 3643; 3673; 3797; 3823; 3853; 3947; 3967; 4283; 4337; 4373; 4397; 4523; 4547; 4643; 4673; 4937; 4967; 5113; 5167; 5197; 5347; 5443; 5647;... | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list4 | |
ltprime_list5 := [:: 12113; 12347; 12647; 12953; 13313; 13613; 13967; 15443; 15647; 15683; 16547; 16673; 16823; 16883; 18353; 18443; 21283; 21523; 21613; 21997; 23167; 24337; 24373; 24547; 24967; 26113; 26317; 26947; 27283; 27673; 27823; 27883; 29137; 29173; 31223; 32467; 32647; 32797; 33347; 33547; 33613; 33617; 33797... | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list5 | |
ltprime_list6 := [:: 121283; 121523; 121997; 124337; 126317; 132647; 133967; 136373; 139397; 139883; 162683; 163853; 181283; 184523; 184967; 186113; 187547; 192347; 192383; 195443; 196337; 213613; 215443; 231223; 233347; 233617; 234673; 236653; 236947; 237547; 242467; 242797; 243613; 261223; 264283; 266947; 267523; 272... | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list6 | |
ltprime_list7 := [:: 1237547; 1261223; 1279337; 1297523; 1326947; 1327673; 1332467; 1336997; 1338167; 1356197; 1368443; 1384673; 1516883; 1537853; 1549547; 1563467; 1564373; 1629137; 1632647; 1633967; 1636997; 1686197; 1686353; 1812347; 1813613; 1818353; 1833347; 1872953; 1875683; 1891997; 1896353; 1965647; 1966337; 19... | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list7 | |
ltprime_list8 := [:: 12184967; 12336353; 12366173; 12375743; 12463313; 12649613; 12667883; 12723167; 12912953; 12973547; 13272383; 13276883; 13294397; 13564937; 13578167; 13692347; 13834283; 13986113; 15187547; 15367853; 15391823; 15427283; 15439883; 15462467; 15613967; 15675347; 15697673; 15729173; 15979283; 16237547;... | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list8 | |
ltprime_list9 := [:: 124536947; 126934673; 127692647; 132649613; 132723167; 132738317; 133381223; 133924337; 135345953; 135424967; 136266947; 136368443; 136981283; 154981373; 156492467; 157573673; 162195443; 162366173; 163327673; 163381223; 163932647; 163995443; 166516673; 166813613; 166962617; 169516883; 169833347; 16... | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list9 | |
ltprime_list10 := [:: 1219861613; 1231633967; 1266139883; 1273233617; 1296463823; 1324542467; 1329633797; 1336516673; 1339693967; 1354632647; 1363243613; 1365187547; 1383396353; 1393834283; 1398675743; 1399951283; 1518676883; 1539139883; 1543279337; 1546275167; 1564326947; 1564696823; 1566367853; 1593732467; 1632373823... | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list10 | |
ltprime_list11 := [:: 12181833347; 12331891997; 12366421997; 12373924337; 12666391373; 12763327673; 12781332467; 13269915683; 13333924337; 13398675743; 13536676883; 13651356197; 13699537547; 13876537547; 15162366173; 15432729173; 15451813613; 15469326113; 15636631223; 15756373613; 15759192347; 15769833347; 15786373613;... | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list11 | |
ltprime_list12 := [:: 121339693967; 123967368443; 124627266947; 127653918443; 129156492467; 129315462467; 133837659467; 133899633797; 151968666173; 153339313613; 154837932647; 154867812347; 154999636997; 159613564937; 163894594397; 165378184523; 165613578167; 166656666173; 183319693967; 186132738317; 195613276883; 1956... | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list12 | |
ltprime_list13 := [:: 1213536676883; 1213876537547; 1218196692347; 1242961965647; 1291566367853; 1333839979337; 1335759192347; 1357564326947; 1518768729173; 1533457816883; 1639627626947; 1651656912953; 1665759192347; 1692373924337; 1837839918353; 1848768729173; 1872493578167; 1899127692647; 1924389973547; 1956645661613... | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list13 | |
ltprime_list14 := [:: 12673876537547; 12967623946997; 13231816543853; 13264242313613; 13986451332467; 15345451813613; 15366127692647; 15421273233617; 15483492961613; 15727653918443; 16215786373613; 16327561813613; 16518427573673; 16579839979337; 16833457816883; 16986451332467; 18195978397283; 18372912366173; 1879935463... | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list14 | |
ltprime_list15 := [:: 121848768729173; 129456645661613; 132195693192347; 136335786373613; 136938367986197; 151518768729173; 156129156492467; 162799354632647; 165487864234673; 165672961965647; 168664392465167; 189463876537547; 212673876537547; 216579839979337; 218799354632647; 231357564326947; 237267627626947; 239793546... | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list15 | |
ltprime_list16 := [:: 1275463876537547; 1399335756373613; 1546215769833347; 1563427653918443; 1635613269915683; 2136938367986197; 2315421273233617; 2429121339693967; 2646216567629137; 2696154867812347; 3315421273233617; 3329121339693967; 3332195693192347; 3394231816543853; 3546216567629137; 3573357564326947; 3726931273... | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list16 | |
ltprime_list17 := [:: 12429121339693967; 12646216567629137; 13573357564326947; 18165672961965647; 18997653319693967; 21546215769833347; 24275463876537547; 24963986391564373; 27669684516387853; 34249872979956113; 36334245663786197; 38997653319693967; 39763986391564373; 39768673651356197; 42429121339693967; 5127546387653... | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list17 | |
ltprime_list18 := [:: 165678739293946997; 198615345451813613; 276812967623946997; 312646216567629137; 351275463876537547; 396334245663786197; 397579333839979337; 484957213536676883; 596334245663786197; 624275463876537547; 624963986391564373; 639763986391564373; 663546216567629137; 675136938367986197; 678493956946986197... | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list18 | |
ltprime_list19 := [:: 1276812967623946997; 3396334245663786197; 3484957213536676883; 4686798799354632647; 5396334245663786197; 6165678739293946997; 6276812967623946997; 6312646216567629137; 6484957213536676883; 6918997653319693967; 7986315421273233617; 8918997653319693967; 8963315421273233617 ]. | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list19 | |
ltprime_list20 := [:: 15396334245663786197; 18918997653319693967; 36484957213536676883; 66276812967623946997; 67986315421273233617; 86312646216567629137 ]. | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list20 | |
ltprime_list21 := [:: 315396334245663786197; 367986315421273233617; 666276812967623946997; 686312646216567629137; 918918997653319693967 ]. | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list21 | |
ltprime_list22 := [:: 5918918997653319693967; 6686312646216567629137; 7686312646216567629137; 9918918997653319693967 ]. | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list22 | |
ltprime_list23 := [:: 57686312646216567629137; 95918918997653319693967; 96686312646216567629137 ]. | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list23 | |
ltprime_list24 := [:: 357686312646216567629137 ]. | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list24 | |
ltprime_list25 : list Z := [:: ]. | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_lprime."
] | src/Coqprime/examples/truncatable/ltprime_init.v | ltprime_list25 | |
rtprime n := no_zero_digit b n /\ forall k, 0 <= k <= log b n -> prime (n / b ^ k). | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime | |
rtprime_small p : prime p -> p < b -> rtprime p. Proof. intros pP pB. assert (H : 0 < p) by now apply GZnZ.p_pos. split. now apply no_zero_digit_small; lia. intros k. replace (log b p) with 0. intros kE; replace k with 0 by lia. now rewrite Z.div_1_r. now apply sym_equal, log_inv; lia. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_small | |
rtprime_prime p : rtprime p -> prime p. Proof. intros [pNZ Hp]. replace p with (p / b ^ 0). now apply Hp; split; [lia | apply log_pos]. now rewrite Z.div_1_r. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_prime | |
rtprime_decompose n m : 0 < m < b -> rtprime n -> prime (n * b + m) -> rtprime (n * b + m). Proof. intros mB [nNZ nM] mnP; split. now apply no_zero_digit_rdecompose; try lia; auto. assert (nPr : prime n) by now apply rtprime_prime. assert (nP : 0 < n) by now apply GZnZ.p_pos. intros k kP. assert (Hl := log_correct b b_... | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_decompose | |
rtprime_div n k : 0 <= k <= log b n -> rtprime n -> rtprime (n / b ^ k). Proof. intros kP [Hn Hnl]; split; [now apply no_zero_digit_div|]. intros k1 k1P. rewrite Zdiv_Zdiv; try lia. rewrite <- Z.pow_add_r; try lia. apply Hnl. now rewrite log_div in k1P; lia. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_div | |
add_rtlist i l l1 := fold_left (fun l z => let v := z * b + i in if is_pseudo_prime v then lZ_insert v l else l) l l1. | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | add_rtlist | |
add_rtlist_subset i l l1 k : In k l1 -> In k (add_rtlist i l l1). Proof. generalize l1; elim l; simpl; auto. intros a l2 IH l3 Il3. case is_pseudo_prime; apply IH; auto. now apply lZ_insert_consr. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | add_rtlist_subset | |
add_rtlist_correct i l l1 k : 0 <= b -> 0 <= i -> 0 <= k -> In k l -> prime(k * b + i) -> In (k * b + i) (add_rtlist i l l1). Proof. intros bP iP. generalize l1 k; elim l; simpl; [intros _ _ _ []| intros z l2 IH l3 k3 k3P Ik3 inkP]. destruct Ik3 as [zE | Ink3l2]. subst z. apply add_rtlist_subset. generalize (is_pseudo_... | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | add_rtlist_correct | |
rnext (l1 : list Z) := let l := ldigit b in fold_left (fun l i => add_rtlist i l1 l) l [::]. | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rnext | |
rnext_correct n l1 k : 0 <= n -> (forall k, rtprime k -> b ^ n <= k < b ^ (n + 1) -> In k l1) -> rtprime k -> b ^ (n + 1) <= k < b ^ (n + 2) -> In k (rnext l1). Proof. intros nP Hl Hlt Hk. assert (Le : log b k = n + 1). apply log_inv; try lia; replace (n + 1 + 1) with (n + 2); lia. pose (k1 := k / b). pose (m := k mod ... | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rnext_correct | |
rtprime_tac b := repeat (match goal with |- rtprime _ ?a => let a' := eval compute in a in let t := constr: (10 <? a) in let t' := eval compute in t in match t' with true => let v1 := constr: (a' mod b) in let v1' := eval compute in v1 in let v2 := constr: (a' / b) in let v2' := eval compute in v2 in apply (rtprime_dec... | Ltac | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_tac | |
rt3prime_list1 := [:: 2; 3; 5; 7]%Z. Compute (1, rt3prime_list1, length rt3prime_list1). | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rt3prime_list1 | |
rt3prime_list2 := Eval compute in rnext 3 rt3prime_list1. Compute (2, rt3prime_list2, length rt3prime_list2). | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rt3prime_list2 | |
rt3prime_list3 := Eval compute in rnext 3 rt3prime_list2. Compute (3, rt3prime_list3, length rt3prime_list3). | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rt3prime_list3 | |
rt3prime_list4 := Eval compute in rnext 3 rt3prime_list3. Compute (4, rt3prime_list4, length rt3prime_list4). | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rt3prime_list4 | |
rt3prime_list5 := Eval compute in rnext 3 rt3prime_list4. *) | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rt3prime_list5 | |
rtprime_list1 := [:: 2; 3; 5; 7]%Z. Compute (1, rtprime_list1, length rtprime_list1). | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list1 | |
rtprime_list1_rtprime i : In i rtprime_list1 -> rtprime 10 i. Proof. intros H; repeat (inversion_clear H as [|H1]; [now subst; rtprime_tac 10| rename H1 into H]). inversion H. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list1_rtprime | |
rtprime_list1_correct k : rtprime 10 k -> 1 <= k < 10 -> In k rtprime_list1. Proof. intros kLT kB. assert (kPr : prime k) by now apply (rtprime_prime 10). assert (H : k = 1 \/ k = 2 \/ k = 3 \/ k = 4 \/ k = 5 \/ k = 6 \/ k = 7 \/ k = 8 \/ k = 9) by lia. unfold rtprime_list1. destruct H as [H|[H|[H|[H|[H|[H|[H|[H|H]]]]]... | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list1_correct | |
rtprime_list2 := Eval compute in rnext 10 rtprime_list1. Compute (2, rtprime_list2, length rtprime_list2). | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list2 | |
rtprime_list2_rtprime i : In i rtprime_list2 -> rtprime 10 i. Proof. intros H; repeat (inversion_clear H as [|H1]; [now subst; rtprime_tac 10| rename H1 into H]). inversion H. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list2_rtprime | |
rtprime_list2E : rtprime_list2 = rnext 10 rtprime_list1. Proof. now vm_cast_no_check (refl_equal rtprime_list2). Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list2E | |
rtprime_list2_correct k : rtprime 10 k -> 10 <= k < 100 -> In k rtprime_list2. Proof. intros kLT kB. rewrite rtprime_list2E. apply (rnext_correct 10 (refl_equal _) 0); try lia; auto. exact rtprime_list1_correct. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list2_correct | |
rtprime_list3 := Eval compute in rnext 10 rtprime_list2. Compute (3, rtprime_list3, length rtprime_list3). | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list3 | |
rtprime_list3_rtprime i : In i rtprime_list3 -> rtprime 10 i. Proof. intros H; repeat (inversion_clear H as [|H1]; [now subst; rtprime_tac 10| rename H1 into H]). inversion H. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list3_rtprime | |
rtprime_list3E : rtprime_list3 = rnext 10 rtprime_list2. Proof. now vm_cast_no_check (refl_equal rtprime_list3). Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list3E | |
rtprime_list3_correct k : rtprime 10 k -> 10 ^ 2 <= k < 10 ^ 3 -> In k rtprime_list3. Proof. intros kLT kB. rewrite rtprime_list3E. apply (rnext_correct 10 (refl_equal _) 1); try lia; auto. exact rtprime_list2_correct. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list3_correct | |
rtprime_list4 := Eval compute in rnext 10 rtprime_list3. Compute (4, rtprime_list4, length rtprime_list4). | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list4 | |
rtprime_list4_rtprime i : In i rtprime_list4 -> rtprime 10 i. Proof. intros H; repeat (inversion_clear H as [|H1]; [now subst; rtprime_tac 10| rename H1 into H]). inversion H. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list4_rtprime | |
rtprime_list4E : rtprime_list4 = rnext 10 rtprime_list3. Proof. now vm_cast_no_check (refl_equal rtprime_list4). Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list4E | |
rtprime_list4_correct k : rtprime 10 k -> 10 ^ 3 <= k < 10 ^ 4 -> In k rtprime_list4. Proof. intros kLT kB. rewrite rtprime_list4E. apply (rnext_correct 10 (refl_equal _) 2); try lia; auto. exact rtprime_list3_correct. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list4_correct | |
rtprime_list5 := Eval compute in rnext 10 rtprime_list4. Compute (5, rtprime_list5, length rtprime_list5). | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list5 | |
rtprime_list5_rtprime i : In i rtprime_list5 -> rtprime 10 i. Proof. intros H; repeat (inversion_clear H as [|H1]; [now subst; rtprime_tac 10| rename H1 into H]). inversion H. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list5_rtprime | |
rtprime_list5E : rtprime_list5 = rnext 10 rtprime_list4. Proof. now vm_cast_no_check (refl_equal rtprime_list5). Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list5E | |
rtprime_list5_correct k : rtprime 10 k -> 10 ^ 4 <= k < 10 ^ 5 -> In k rtprime_list5. Proof. intros kLT kB. rewrite rtprime_list5E. apply (rnext_correct 10 (refl_equal _) 3); try lia; auto. exact rtprime_list4_correct. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list5_correct | |
rtprime_list6 := Eval compute in rnext 10 rtprime_list5. Compute (6, rtprime_list6, length rtprime_list6). | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list6 | |
rtprime_list6_rtprime i : In i rtprime_list6 -> rtprime 10 i. Proof. intros H; repeat (inversion_clear H as [|H1]; [now subst; rtprime_tac 10| rename H1 into H]). inversion H. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list6_rtprime | |
rtprime_list6E : rtprime_list6 = rnext 10 rtprime_list5. Proof. now vm_cast_no_check (refl_equal rtprime_list6). Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list6E | |
rtprime_list6_correct k : rtprime 10 k -> 10 ^ 5 <= k < 10 ^ 6 -> In k rtprime_list6. Proof. intros kLT kB. rewrite rtprime_list6E. apply (rnext_correct 10 (refl_equal _) 4); try lia; auto. exact rtprime_list5_correct. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list6_correct | |
rtprime_list7 := Eval compute in rnext 10 rtprime_list6. Compute (7, rtprime_list7, length rtprime_list7). | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list7 | |
rtprime_list7_rtprime i : In i rtprime_list7 -> rtprime 10 i. Proof. intros H; repeat (inversion_clear H as [|H1]; [now subst; rtprime_tac 10| rename H1 into H]). inversion H. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list7_rtprime | |
rtprime_list7E : rtprime_list7 = rnext 10 rtprime_list6. Proof. now vm_cast_no_check (refl_equal rtprime_list7). Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list7E | |
rtprime_list7_correct k : rtprime 10 k -> 10 ^ 6 <= k < 10 ^ 7 -> In k rtprime_list7. Proof. intros kLT kB. rewrite rtprime_list7E. apply (rnext_correct 10 (refl_equal _) 5); try lia; auto. exact rtprime_list6_correct. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list7_correct | |
rtprime_list8 := Eval compute in rnext 10 rtprime_list7. Compute (8, rtprime_list8, length rtprime_list8). | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list8 | |
rtprime_list8_rtprime i : In i rtprime_list8 -> rtprime 10 i. Proof. intros H; repeat (inversion_clear H as [|H1]; [now subst; rtprime_tac 10| rename H1 into H]). inversion H. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list8_rtprime | |
rtprime_list8E : rtprime_list8 = rnext 10 rtprime_list7. Proof. now vm_cast_no_check (refl_equal rtprime_list8). Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list8E | |
rtprime_list8_correct k : rtprime 10 k -> 10 ^ 7 <= k < 10 ^ 8 -> In k rtprime_list8. Proof. intros kLT kB. rewrite rtprime_list8E. apply (rnext_correct 10 (refl_equal _) 6); try lia; auto. exact rtprime_list7_correct. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list8_correct | |
rtprime_list9 := Eval compute in rnext 10 rtprime_list8. Compute (9, rtprime_list9, length rtprime_list9). | Definition | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list9 | |
rtprime_list9E : rtprime_list9 = rnext 10 rtprime_list8. Proof. now vm_cast_no_check (refl_equal rtprime_list9). Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list9E | |
rtprime_list9_correct k : 10 ^ 8 <= k < 10 ^ 9 -> ~ rtprime 10 k. Proof. intros kLT kB. assert (H : In k rtprime_list9); [|inversion H]. rewrite rtprime_list9E. apply (rnext_correct 10 (refl_equal _) 7); try lia; auto. exact rtprime_list8_correct. Qed. | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_list9_correct | |
rtprime_main : rtprime 10 73939133 /\ forall k, rtprime 10 k -> k <= 73939133. Proof. split; [now rtprime_tac 10|]. intros k kLT. assert (H : k <= 73939133 \/ 73939133 < k) by lia. destruct H as [|kL]; [lia|]. assert (H : k < 10 ^ 8 \/ 10 ^ 8 <= k) by lia. destruct H as [H1|H1]. assert (In k rtprime_list8). now apply r... | Lemma | src | [
"From Stdlib Require Import ZArith List Lia.",
"From Coqprime Require Import PocklingtonRefl all_rprime ltprime_init."
] | src/Coqprime/examples/truncatable/rtprime.v | rtprime_main |
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