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fact
string | imports
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string | __index_level_0__
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Definition divr {R : unitRingType} (x y : R) : R := x / y. | BinInt mathcomp(all_ssreflect all_algebra) CoqEAL(hrel param refinements) CoqEAL(pos binnat rational) tactics rat_of_Z | coq-community-apery/rho_computations | coq-community-apery | 624 |
Definition natr {R : ringType} (x : nat) : R := x%:R. | BinInt mathcomp(all_ssreflect all_algebra) CoqEAL(hrel param refinements) CoqEAL(pos binnat rational) tactics rat_of_Z | coq-community-apery/rho_computations | coq-community-apery | 625 |
Definition alpha := @generic_alpha rat +%R divr *%R (@GRing.exp _) rat_of_Z. | BinInt mathcomp(all_ssreflect all_algebra) CoqEAL(hrel param refinements) CoqEAL(pos binnat rational) tactics rat_of_Z | coq-community-apery/rho_computations | coq-community-apery | 626 |
Definition beta := @generic_beta rat +%R divr (@GRing.exp _) rat_of_Z. | BinInt mathcomp(all_ssreflect all_algebra) CoqEAL(hrel param refinements) CoqEAL(pos binnat rational) tactics rat_of_Z | coq-community-apery/rho_computations | coq-community-apery | 627 |
Definition h := @generic_h rat +%R subr divr *%R (@GRing.exp _) rat_of_Z. | BinInt mathcomp(all_ssreflect all_algebra) CoqEAL(hrel param refinements) CoqEAL(pos binnat rational) tactics rat_of_Z | coq-community-apery/rho_computations | coq-community-apery | 628 |
Definition h_iter := @generic_h_iter rat 0%R +%R subr divr *%R (@GRing.exp _) rat_of_Z natr. | BinInt mathcomp(all_ssreflect all_algebra) CoqEAL(hrel param refinements) CoqEAL(pos binnat rational) tactics rat_of_Z | coq-community-apery/rho_computations | coq-community-apery | 629 |
Parameter rat_of_Z : Z -> rat. | HB(structures) ZArith mathcomp(all_ssreflect all_algebra) tactics | coq-community-apery/rat_of_Z | coq-community-apery | 630 |
Axiom rat_of_ZEdef : rat_of_Z = (fun x : Z => (int_of_Z x)%:Q). | HB(structures) ZArith mathcomp(all_ssreflect all_algebra) tactics | coq-community-apery/rat_of_Z | coq-community-apery | 631 |
Definition rat_of_Z (x : Z) := (int_of_Z x)%:Q. | HB(structures) ZArith mathcomp(all_ssreflect all_algebra) tactics | coq-community-apery/rat_of_Z | coq-community-apery | 632 |
Definition rat_of_ZEdef := erefl rat_of_Z. | HB(structures) ZArith mathcomp(all_ssreflect all_algebra) tactics | coq-community-apery/rat_of_Z | coq-community-apery | 633 |
Definition c_ann := annotated_recs_c.ann c_Sn c_Sk. | mathcomp(all_ssreflect all_algebra) tactics shift binomialz rat_of_Z seq_defs | coq-community-apery/algo_closures | coq-community-apery | 634 |
Definition d_ann := annotated_recs_d.ann d_Sn d_Sk d_Sm. | mathcomp(all_ssreflect all_algebra) tactics shift binomialz rat_of_Z seq_defs | coq-community-apery/algo_closures | coq-community-apery | 635 |
Definition s_ann := annotated_recs_s.ann s_Sn2 s_SnSk s_Sk2. | mathcomp(all_ssreflect all_algebra) tactics shift binomialz rat_of_Z seq_defs | coq-community-apery/algo_closures | coq-community-apery | 636 |
Definition z_ann := annotated_recs_z.ann z_Sn2. | mathcomp(all_ssreflect all_algebra) tactics shift binomialz rat_of_Z seq_defs | coq-community-apery/algo_closures | coq-community-apery | 637 |
Definition u_ann := annotated_recs_s.ann u_Sn2 u_SnSk u_Sk2. | mathcomp(all_ssreflect all_algebra) tactics shift binomialz rat_of_Z seq_defs | coq-community-apery/algo_closures | coq-community-apery | 638 |
Definition v_ann := annotated_recs_v.ann v_Sn2 v_SnSk v_Sk2. | mathcomp(all_ssreflect all_algebra) tactics shift binomialz rat_of_Z seq_defs | coq-community-apery/algo_closures | coq-community-apery | 639 |
Fixpoint b'_rec (n : nat) : rat := match n with | 0 => b 0 | 1 => b 1 | S (S o as o') => let n' := Posz o in - (annotated_recs_c.P_cf0 n' * b'_rec o + annotated_recs_c.P_cf1 n' * b'_rec o') / annotated_recs_c.P_cf2 n' end. | BinInt mathcomp(all_ssreflect all_algebra) tactics binomialz shift rat_of_Z seq_defs | coq-community-apery/reduce_order | coq-community-apery | 640 |
Definition b' (n : int) : rat := match n with | Negz _ => 0 | Posz o => b'_rec o end. | BinInt mathcomp(all_ssreflect all_algebra) tactics binomialz shift rat_of_Z seq_defs | coq-community-apery/reduce_order | coq-community-apery | 641 |
Variables (n i : nat). | ZArith mathcomp(all_ssreflect all_algebra all_field) extra_mathcomp tactics binomialz floor arithmetics posnum rat_of_Z hanson_elem_arith hanson_elem_analysis | coq-community-apery/hanson | coq-community-apery | 642 |
Hypothesis Hain : (a i <= n)%N. | ZArith mathcomp(all_ssreflect all_algebra all_field) extra_mathcomp tactics binomialz floor arithmetics posnum rat_of_Z hanson_elem_arith hanson_elem_analysis | coq-community-apery/hanson | coq-community-apery | 643 |
Definition a' i : algC := exp_quo (a i)%:Q 1%N (a i). | ZArith mathcomp(all_ssreflect all_algebra all_field) extra_mathcomp tactics binomialz floor arithmetics posnum rat_of_Z hanson_elem_arith hanson_elem_analysis | coq-community-apery/hanson | coq-community-apery | 644 |
Definition w_seq k := \prod_(i < k) a' i. | ZArith mathcomp(all_ssreflect all_algebra all_field) extra_mathcomp tactics binomialz floor arithmetics posnum rat_of_Z hanson_elem_arith hanson_elem_analysis | coq-community-apery/hanson | coq-community-apery | 645 |
Definition a'0_ub : rat := rat_of_Z 283 / rat_of_Z 200. | ZArith mathcomp(all_ssreflect all_algebra all_field) extra_mathcomp tactics binomialz floor arithmetics posnum rat_of_Z hanson_elem_arith hanson_elem_analysis | coq-community-apery/hanson | coq-community-apery | 646 |
Definition a'1_ub : rat := rat_of_Z 1443 / rat_of_Z 1000. | ZArith mathcomp(all_ssreflect all_algebra all_field) extra_mathcomp tactics binomialz floor arithmetics posnum rat_of_Z hanson_elem_arith hanson_elem_analysis | coq-community-apery/hanson | coq-community-apery | 647 |
Definition a'2_ub : rat := rat_of_Z 1321 / rat_of_Z 1000. | ZArith mathcomp(all_ssreflect all_algebra all_field) extra_mathcomp tactics binomialz floor arithmetics posnum rat_of_Z hanson_elem_arith hanson_elem_analysis | coq-community-apery/hanson | coq-community-apery | 648 |
Definition a'3_ub : rat := rat_of_Z 273 / rat_of_Z 250. | ZArith mathcomp(all_ssreflect all_algebra all_field) extra_mathcomp tactics binomialz floor arithmetics posnum rat_of_Z hanson_elem_arith hanson_elem_analysis | coq-community-apery/hanson | coq-community-apery | 649 |
Definition a'4_ub : rat := rat_of_Z 201 / rat_of_Z 200. | ZArith mathcomp(all_ssreflect all_algebra all_field) extra_mathcomp tactics binomialz floor arithmetics posnum rat_of_Z hanson_elem_arith hanson_elem_analysis | coq-community-apery/hanson | coq-community-apery | 650 |
Definition w : rat := a'0_ub * a'1_ub * a'2_ub * a'3_ub * a'4_ub ^ 2. | ZArith mathcomp(all_ssreflect all_algebra all_field) extra_mathcomp tactics binomialz floor arithmetics posnum rat_of_Z hanson_elem_arith hanson_elem_analysis | coq-community-apery/hanson | coq-community-apery | 651 |
Definition P_horner := annotated_recs_v.P_horner. | coq-community-apery/annotated_recs_b | coq-community-apery | 652 |
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Variable z : int -> rat. | coq-community-apery/ops_for_u | coq-community-apery | 653 |
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Variable s : int -> int -> rat. | coq-community-apery/ops_for_u | coq-community-apery | 654 |
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Variable z_ann : z.Ann z. | coq-community-apery/ops_for_u | coq-community-apery | 655 |
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Variable s_ann : s.Ann s. | coq-community-apery/ops_for_u | coq-community-apery | 656 |
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Definition Sn (n k m : int) := (n >= 0) /\ (m > 0) /\ (n >= m). | coq-community-apery/annotated_recs_d | coq-community-apery | 657 |
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Definition Sk (n k m : int) := true. | coq-community-apery/annotated_recs_d | coq-community-apery | 658 |
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Definition Sm (n k m : int) := (n > 0) /\ (m > 0) /\ (n > m). | coq-community-apery/annotated_recs_d | coq-community-apery | 659 |
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Definition not_D1 (n k m : int) := (0 < m) && (m < n). | coq-community-apery/annotated_recs_d | coq-community-apery | 660 |
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Definition not_D2 (n k m : int) := (0 < m) && (m < n). | coq-community-apery/annotated_recs_d | coq-community-apery | 661 |
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Definition not_D3 (n k m : int) := (0 < m) && (m < n). | coq-community-apery/annotated_recs_d | coq-community-apery | 662 |
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Definition not_D4 (n k m : int) := (0 < m) && (m < n). | coq-community-apery/annotated_recs_d | coq-community-apery | 663 |
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Definition CT1 (d : int -> int -> int -> rat) : Prop := forall (n_ k_ m_ : int), not_D1 n_ k_ m_ -> P1_horner (punk.pfun2 d m_) n_ k_ = Q1_flat d n_ k_ (int.shift 1 m_) - Q1_flat d n_ k_ m_. | coq-community-apery/annotated_recs_d | coq-community-apery | 664 |
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Definition CT2 (d : int -> int -> int -> rat) := forall (n_ k_ m_ : int), not_D2 n_ k_ m_ -> P2_horner (punk.pfun2 d m_) n_ k_ = Q2_flat d n_ k_ (int.shift 1 m_) - Q2_flat d n_ k_ m_. | coq-community-apery/annotated_recs_d | coq-community-apery | 665 |
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Definition CT3 (d : int -> int -> int -> rat) := forall (n_ k_ m_ : int), not_D3 n_ k_ m_ -> P3_horner (punk.pfun2 d m_) n_ k_ = Q3_flat d n_ k_ (int.shift 1 m_) - Q3_flat d n_ k_ m_. | coq-community-apery/annotated_recs_d | coq-community-apery | 666 |
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Definition CT4 (d : int -> int -> int -> rat) := forall (n_ k_ m_ : int), not_D4 n_ k_ m_ -> P4_horner (punk.pfun2 d m_) n_ k_ = Q4_flat d n_ k_ (int.shift 1 m_) - Q4_flat d n_ k_ m_. | coq-community-apery/annotated_recs_d | coq-community-apery | 667 |
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Record Ann d : Type := ann { Sn_ : Sn d; Sk_ : Sk d; Sm_ : Sm d }. | coq-community-apery/annotated_recs_d | coq-community-apery | 668 |
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Definition shift1 (z : int) := z + 1. | BinInt ZifyClasses mathcomp(all_ssreflect all_algebra) tactics | coq-community-apery/shift | coq-community-apery | 669 |
Definition shift_ n := iter n shift1. | BinInt ZifyClasses mathcomp(all_ssreflect all_algebra) tactics | coq-community-apery/shift | coq-community-apery | 670 |
Definition shift := nosimpl shift_. | BinInt ZifyClasses mathcomp(all_ssreflect all_algebra) tactics | coq-community-apery/shift | coq-community-apery | 671 |
Definition shift2Z := (zshiftP, shift1E). | BinInt ZifyClasses mathcomp(all_ssreflect all_algebra) tactics | coq-community-apery/shift | coq-community-apery | 672 |
Definition shift2R := (shiftP, shift1P). | BinInt ZifyClasses mathcomp(all_ssreflect all_algebra) tactics | coq-community-apery/shift | coq-community-apery | 673 |
Definition Sn2 (n : int) := (n != - 2%:~R) /\ (n != - 1). | coq-community-apery/annotated_recs_z | coq-community-apery | 674 |
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Record Ann z : Type := ann { Sn2_ : Sn2 z }. | coq-community-apery/annotated_recs_z | coq-community-apery | 675 |
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Definition index_iotaz (mi ni : int) := match mi, ni with | Posz _, Negz _ => [::] | Posz m, Posz n => map Posz (index_iota m n) | Negz m, Negz n => map Negz (rev (index_iota n.+1 m.+1)) | Negz m, Posz n => rev (map Negz (index_iota 0 m.+1)) ++ (map Posz (index_iota 0 n)) end. | mathcomp(all_ssreflect all_algebra) extra_mathcomp | coq-community-apery/bigopz | coq-community-apery | 676 |
Variables (R I : Type) (idx : R) (op : Monoid.com_law idx) (r : seq I). | mathcomp(all_ssreflect all_algebra) extra_mathcomp | coq-community-apery/bigopz | coq-community-apery | 677 |
Variable R : ringType. | mathcomp(all_ssreflect all_algebra) extra_mathcomp | coq-community-apery/bigopz | coq-community-apery | 678 |
Variable R : fieldType. | mathcomp(all_ssreflect all_algebra) extra_mathcomp | coq-community-apery/bigopz | coq-community-apery | 679 |
Structure zifyRing (R : ringType) := ZifyRing { rval : R; zval : int; _ : rval = zval%:~R }. | mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat choice seq) mathcomp(fintype finfun bigop order ssralg ssrnum ssrint) | coq-community-apery/tactics | coq-community-apery | 680 |
Definition zify_zero := ZifyRing 0 0 erefl. | mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat choice seq) mathcomp(fintype finfun bigop order ssralg ssrnum ssrint) | coq-community-apery/tactics | coq-community-apery | 682 |
Definition zify_opp e1 := ZifyRing (- rval e1) (- zval e1) (zify_opp_subproof e1). | mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat choice seq) mathcomp(fintype finfun bigop order ssralg ssrnum ssrint) | coq-community-apery/tactics | coq-community-apery | 683 |
Definition zify_add e1 e2 := ZifyRing (rval e1 + rval e2) (zval e1 + zval e2) (zify_add_subproof e1 e2). | mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat choice seq) mathcomp(fintype finfun bigop order ssralg ssrnum ssrint) | coq-community-apery/tactics | coq-community-apery | 684 |
Definition zify_mulrn e1 n := ZifyRing (rval e1 *+ n) (zval e1 *+ n) (zify_mulrz_subproof e1 n). | mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat choice seq) mathcomp(fintype finfun bigop order ssralg ssrnum ssrint) | coq-community-apery/tactics | coq-community-apery | 685 |
Definition zify_mulrz e1 n := ZifyRing (rval e1 *~ n) (zval e1 *~ n) (zify_mulrz_subproof e1 n). | mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat choice seq) mathcomp(fintype finfun bigop order ssralg ssrnum ssrint) | coq-community-apery/tactics | coq-community-apery | 686 |
Definition zify_one := ZifyRing 1 1 erefl. | mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat choice seq) mathcomp(fintype finfun bigop order ssralg ssrnum ssrint) | coq-community-apery/tactics | coq-community-apery | 687 |
Definition zify_mul e1 e2 := ZifyRing (rval e1 * rval e2) (zval e1 * zval e2) (zify_mul_subproof e1 e2). | mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat choice seq) mathcomp(fintype finfun bigop order ssralg ssrnum ssrint) | coq-community-apery/tactics | coq-community-apery | 688 |
Definition zify_exprn e1 n := ZifyRing (rval e1 ^+ n) (zval e1 ^+ n) (zify_exprn_subproof e1 n). | mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat choice seq) mathcomp(fintype finfun bigop order ssralg ssrnum ssrint) | coq-community-apery/tactics | coq-community-apery | 689 |
Definition exp_quo r p q := q.-root r%:C ^+ p. | mathcomp(all_ssreflect all_algebra all_field) extra_mathcomp posnum hanson_elem_arith | coq-community-apery/hanson_elem_analysis | coq-community-apery | 690 |
Definition floorQ (r : rat) := (numq r %/ denq r)%Z. | mathcomp(all_ssreflect all_algebra) | coq-community-apery/floor | coq-community-apery | 691 |
Definition ghn (m : nat) (n : int) : rat := \sum_(1 <= k < n + 1 :> int) (k %:Q ^ m)^-1. | mathcomp(all_ssreflect all_algebra) tactics shift bigopz | coq-community-apery/harmonic_numbers | coq-community-apery | 692 |
Variable d : int -> int -> int -> rat. | annotated_recs_s | coq-community-apery/ops_for_s | coq-community-apery | 693 |
Variable d_ann : d.Ann d. | annotated_recs_s | coq-community-apery/ops_for_s | coq-community-apery | 694 |
Definition P1_horner := d.P1_horner. | annotated_recs_s | coq-community-apery/ops_for_s | coq-community-apery | 695 |
Definition P2_horner := d.P2_horner. | annotated_recs_s | coq-community-apery/ops_for_s | coq-community-apery | 696 |
Definition P3_horner := d.P3_horner. | annotated_recs_s | coq-community-apery/ops_for_s | coq-community-apery | 697 |
Definition P4_horner := d.P4_horner. | annotated_recs_s | coq-community-apery/ops_for_s | coq-community-apery | 698 |
Definition P1_flat := d.P1_flat. | annotated_recs_s | coq-community-apery/ops_for_s | coq-community-apery | 699 |
Definition P2_flat := d.P2_flat. | annotated_recs_s | coq-community-apery/ops_for_s | coq-community-apery | 700 |
Definition P3_flat := d.P3_flat. | annotated_recs_s | coq-community-apery/ops_for_s | coq-community-apery | 701 |
Definition P4_flat := d.P4_flat. | annotated_recs_s | coq-community-apery/ops_for_s | coq-community-apery | 702 |
Definition multinomial (l : seq nat) : nat := \prod_(0 <= i < size l) (binomial (\sum_(0 <= j < i.+1) l`_j) l`_i). | mathcomp(all_ssreflect all_algebra) extra_mathcomp | coq-community-apery/multinomial | coq-community-apery | 703 |
Example foo : multinomial [::1;2] = 3. | mathcomp(all_ssreflect all_algebra) extra_mathcomp | coq-community-apery/multinomial | coq-community-apery | 704 |
Definition monomial (x : seq R) (e : seq nat) := \prod_(xi <- zip x e) xi.1 ^ xi.2. | mathcomp(all_ssreflect all_algebra) extra_mathcomp | coq-community-apery/multinomial | coq-community-apery | 705 |
Definition tmap_val {n m} (t : n.-tuple 'I_m) : seq nat := [seq val j | j <- t]. | mathcomp(all_ssreflect all_algebra) extra_mathcomp | coq-community-apery/multinomial | coq-community-apery | 706 |
Variable R : zmodType. | mathcomp(all_ssreflect all_algebra all_field) mathcomp(bigenough) | coq-community-apery/extra_mathcomp | coq-community-apery | 707 |
Variables S T : Type. | mathcomp(all_ssreflect all_algebra all_field) mathcomp(bigenough) | coq-community-apery/extra_mathcomp | coq-community-apery | 709 |
Definition z3seq (n : nat) := ghn3 (Posz n). | mathcomp(all_ssreflect all_algebra) tactics bigopz harmonic_numbers seq_defs | coq-community-apery/z3seq_props | coq-community-apery | 710 |
Fixpoint horner_seqop_rec (cf : seq (int -> R)) (u : int -> R) n n0 := match cf with | [::] => 0 | [:: a] => a n0 * u n | a :: cf' => horner_seqop_rec cf' u (int.shift 1%N n) n0 + a n0 * u n end. | mathcomp(all_ssreflect all_algebra) shift bigopz | coq-community-apery/punk | coq-community-apery | 712 |
Definition horner_seqop cf u n := horner_seqop_rec cf u n n. | mathcomp(all_ssreflect all_algebra) shift bigopz | coq-community-apery/punk | coq-community-apery | 713 |
Variable u : int -> int -> R. | mathcomp(all_ssreflect all_algebra) shift bigopz | coq-community-apery/punk | coq-community-apery | 714 |
Variable Pseq : seq (int -> R). | mathcomp(all_ssreflect all_algebra) shift bigopz | coq-community-apery/punk | coq-community-apery | 715 |
Variable Q : (int -> int -> R) -> (int -> int -> R). | mathcomp(all_ssreflect all_algebra) shift bigopz | coq-community-apery/punk | coq-community-apery | 716 |
Variables (a b : int). | mathcomp(all_ssreflect all_algebra) shift bigopz | coq-community-apery/punk | coq-community-apery | 717 |
Variable not_D : int -> int -> bool. | mathcomp(all_ssreflect all_algebra) shift bigopz | coq-community-apery/punk | coq-community-apery | 718 |
Variables (cf : seq (int -> int -> R)) (u : int -> int -> R). | mathcomp(all_ssreflect all_algebra) shift bigopz | coq-community-apery/punk | coq-community-apery | 720 |
Definition biv_horner_seqop_rec n n0 k := horner_seqop_rec (map (fun a => a ^~ k) cf) (u ^~ k) n n0. | mathcomp(all_ssreflect all_algebra) shift bigopz | coq-community-apery/punk | coq-community-apery | 721 |
Definition biv_horner_seqop n k := biv_horner_seqop_rec n n k. | mathcomp(all_ssreflect all_algebra) shift bigopz | coq-community-apery/punk | coq-community-apery | 722 |
Variables (cf : seq (seq (int -> int -> R))) (u : int -> int -> R). | mathcomp(all_ssreflect all_algebra) shift bigopz | coq-community-apery/punk | coq-community-apery | 723 |
Definition biv_horner_seqop2 : (int -> int -> R) -> int -> int -> R := foldr (fun a r u n k => let u' := fun n_ k_ : int => u n_ (int.shift 1%N k_) in r u' n k + biv_horner_seqop a u n k) (fun u n k => 0) cf. | mathcomp(all_ssreflect all_algebra) shift bigopz | coq-community-apery/punk | coq-community-apery | 724 |
Variable u : int -> int -> int -> R. | mathcomp(all_ssreflect all_algebra) shift bigopz | coq-community-apery/punk | coq-community-apery | 725 |
Variable Pseq : seq (seq (int -> int -> R)). | mathcomp(all_ssreflect all_algebra) shift bigopz | coq-community-apery/punk | coq-community-apery | 726 |
Variable Q : (int -> int -> int -> R) -> (int -> int -> int -> R). | mathcomp(all_ssreflect all_algebra) shift bigopz | coq-community-apery/punk | coq-community-apery | 727 |
Subsets and Splits