Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
|---|---|---|---|---|---|---|---|---|---|---|---|
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : β) : List β :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 46 | 48 | theorem length (n m : β) : length (Ico n m) = m - n := by |
dsimp [Ico]
simp [length_range', autoParam]
| 2 | 7.389056 | 1 | 0.9375 | 16 | 794 |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : β) : List β :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 51 | 53 | theorem pairwise_lt (n m : β) : Pairwise (Β· < Β·) (Ico n m) := by |
dsimp [Ico]
simp [pairwise_lt_range', autoParam]
| 2 | 7.389056 | 1 | 0.9375 | 16 | 794 |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : β) : List β :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 56 | 58 | theorem nodup (n m : β) : Nodup (Ico n m) := by |
dsimp [Ico]
simp [nodup_range', autoParam]
| 2 | 7.389056 | 1 | 0.9375 | 16 | 794 |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : β) : List β :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 62 | 69 | theorem mem {n m l : β} : l β Ico n m β n β€ l β§ l < m := by |
suffices n β€ l β§ l < n + (m - n) β n β€ l β§ l < m by simp [Ico, this]
rcases le_total n m with hnm | hmn
Β· rw [Nat.add_sub_cancel' hnm]
Β· rw [Nat.sub_eq_zero_iff_le.mpr hmn, Nat.add_zero]
exact
and_congr_right fun hnl =>
Iff.intro (fun hln => (not_le_of_gt hln hnl).elim) fun hlm => lt_of_lt_of... | 7 | 1,096.633158 | 2 | 0.9375 | 16 | 794 |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : β) : List β :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 72 | 73 | theorem eq_nil_of_le {n m : β} (h : m β€ n) : Ico n m = [] := by |
simp [Ico, Nat.sub_eq_zero_iff_le.mpr h]
| 1 | 2.718282 | 0 | 0.9375 | 16 | 794 |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : β) : List β :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 76 | 77 | theorem map_add (n m k : β) : (Ico n m).map (k + Β·) = Ico (n + k) (m + k) := by |
rw [Ico, Ico, map_add_range', Nat.add_sub_add_right m k, Nat.add_comm n k]
| 1 | 2.718282 | 0 | 0.9375 | 16 | 794 |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : β) : List β :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 80 | 82 | theorem map_sub (n m k : β) (hβ : k β€ n) :
((Ico n m).map fun x => x - k) = Ico (n - k) (m - k) := by |
rw [Ico, Ico, Nat.sub_sub_sub_cancel_right hβ, map_sub_range' _ _ _ hβ]
| 1 | 2.718282 | 0 | 0.9375 | 16 | 794 |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : β) : List β :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 95 | 100 | theorem append_consecutive {n m l : β} (hnm : n β€ m) (hml : m β€ l) :
Ico n m ++ Ico m l = Ico n l := by |
dsimp only [Ico]
convert range'_append n (m-n) (l-m) 1 using 2
Β· rw [Nat.one_mul, Nat.add_sub_cancel' hnm]
Β· rw [Nat.sub_add_sub_cancel hml hnm]
| 4 | 54.59815 | 2 | 0.9375 | 16 | 794 |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : β) : List β :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 104 | 110 | theorem inter_consecutive (n m l : β) : Ico n m β© Ico m l = [] := by |
apply eq_nil_iff_forall_not_mem.2
intro a
simp only [and_imp, not_and, not_lt, List.mem_inter_iff, List.Ico.mem]
intro _ hβ hβ
exfalso
exact not_lt_of_ge hβ hβ
| 6 | 403.428793 | 2 | 0.9375 | 16 | 794 |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : β) : List β :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 120 | 122 | theorem succ_singleton {n : β} : Ico n (n + 1) = [n] := by |
dsimp [Ico]
simp [range', Nat.add_sub_cancel_left]
| 2 | 7.389056 | 1 | 0.9375 | 16 | 794 |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : β) : List β :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 125 | 127 | theorem succ_top {n m : β} (h : n β€ m) : Ico n (m + 1) = Ico n m ++ [m] := by |
rwa [β succ_singleton, append_consecutive]
exact Nat.le_succ _
| 2 | 7.389056 | 1 | 0.9375 | 16 | 794 |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : β) : List β :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 130 | 132 | theorem eq_cons {n m : β} (h : n < m) : Ico n m = n :: Ico (n + 1) m := by |
rw [β append_consecutive (Nat.le_succ n) h, succ_singleton]
rfl
| 2 | 7.389056 | 1 | 0.9375 | 16 | 794 |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : β) : List β :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 136 | 139 | theorem pred_singleton {m : β} (h : 0 < m) : Ico (m - 1) m = [m - 1] := by |
dsimp [Ico]
rw [Nat.sub_sub_self (succ_le_of_lt h)]
simp [β Nat.one_eq_succ_zero]
| 3 | 20.085537 | 1 | 0.9375 | 16 | 794 |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : β) : List β :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 143 | 148 | theorem chain'_succ (n m : β) : Chain' (fun a b => b = succ a) (Ico n m) := by |
by_cases h : n < m
Β· rw [eq_cons h]
exact chain_succ_range' _ _ 1
Β· rw [eq_nil_of_le (le_of_not_gt h)]
trivial
| 5 | 148.413159 | 2 | 0.9375 | 16 | 794 |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : β) : List β :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 153 | 153 | theorem not_mem_top {n m : β} : m β Ico n m := by | simp
| 1 | 2.718282 | 0 | 0.9375 | 16 | 794 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section Disjoint
... | Mathlib/GroupTheory/Perm/Support.lean | 50 | 50 | theorem Disjoint.symm : Disjoint f g β Disjoint g f := by | simp only [Disjoint, or_comm, imp_self]
| 1 | 2.718282 | 0 | 0.944444 | 18 | 795 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section Disjoint
... | Mathlib/GroupTheory/Perm/Support.lean | 87 | 90 | theorem disjoint_refl_iff : Disjoint f f β f = 1 := by |
refine β¨fun h => ?_, fun h => h.symm βΈ disjoint_one_left 1β©
ext x
cases' h x with hx hx <;> simp [hx]
| 3 | 20.085537 | 1 | 0.944444 | 18 | 795 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section Disjoint
... | Mathlib/GroupTheory/Perm/Support.lean | 93 | 96 | theorem Disjoint.inv_left (h : Disjoint f g) : Disjoint fβ»ΒΉ g := by |
intro x
rw [inv_eq_iff_eq, eq_comm]
exact h x
| 3 | 20.085537 | 1 | 0.944444 | 18 | 795 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section Disjoint
... | Mathlib/GroupTheory/Perm/Support.lean | 104 | 106 | theorem disjoint_inv_left_iff : Disjoint fβ»ΒΉ g β Disjoint f g := by |
refine β¨fun h => ?_, Disjoint.inv_leftβ©
convert h.inv_left
| 2 | 7.389056 | 1 | 0.944444 | 18 | 795 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section Disjoint
... | Mathlib/GroupTheory/Perm/Support.lean | 110 | 111 | theorem disjoint_inv_right_iff : Disjoint f gβ»ΒΉ β Disjoint f g := by |
rw [disjoint_comm, disjoint_inv_left_iff, disjoint_comm]
| 1 | 2.718282 | 0 | 0.944444 | 18 | 795 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section Disjoint
... | Mathlib/GroupTheory/Perm/Support.lean | 118 | 120 | theorem Disjoint.mul_right (H1 : Disjoint f g) (H2 : Disjoint f h) : Disjoint f (g * h) := by |
rw [disjoint_comm]
exact H1.symm.mul_left H2.symm
| 2 | 7.389056 | 1 | 0.944444 | 18 | 795 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section Disjoint
... | Mathlib/GroupTheory/Perm/Support.lean | 130 | 135 | theorem disjoint_prod_right (l : List (Perm Ξ±)) (h : β g β l, Disjoint f g) :
Disjoint f l.prod := by |
induction' l with g l ih
Β· exact disjoint_one_right _
Β· rw [List.prod_cons]
exact (h _ (List.mem_cons_self _ _)).mul_right (ih fun g hg => h g (List.mem_cons_of_mem _ hg))
| 4 | 54.59815 | 2 | 0.944444 | 18 | 795 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section Disjoint
... | Mathlib/GroupTheory/Perm/Support.lean | 144 | 152 | theorem nodup_of_pairwise_disjoint {l : List (Perm Ξ±)} (h1 : (1 : Perm Ξ±) β l)
(h2 : l.Pairwise Disjoint) : l.Nodup := by |
refine List.Pairwise.imp_of_mem ?_ h2
intro Ο Ο h_mem _ h_disjoint _
subst Ο
suffices (Ο : Perm Ξ±) = 1 by
rw [this] at h_mem
exact h1 h_mem
exact ext fun a => or_self_iff.mp (h_disjoint a)
| 7 | 1,096.633158 | 2 | 0.944444 | 18 | 795 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section IsSwap
va... | Mathlib/GroupTheory/Perm/Support.lean | 248 | 253 | theorem ne_and_ne_of_swap_mul_apply_ne_self {f : Perm Ξ±} {x y : Ξ±} (hy : (swap x (f x) * f) y β y) :
f y β y β§ y β x := by |
simp only [swap_apply_def, mul_apply, f.injective.eq_iff] at *
by_cases h : f y = x
Β· constructor <;> intro <;> simp_all only [if_true, eq_self_iff_true, not_true, Ne]
Β· split_ifs at hy with h h <;> try { simp [*] at * }
| 4 | 54.59815 | 2 | 0.944444 | 18 | 795 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section support
s... | Mathlib/GroupTheory/Perm/Support.lean | 264 | 267 | theorem set_support_inv_eq : { x | pβ»ΒΉ x β x } = { x | p x β x } := by |
ext x
simp only [Set.mem_setOf_eq, Ne]
rw [inv_def, symm_apply_eq, eq_comm]
| 3 | 20.085537 | 1 | 0.944444 | 18 | 795 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section support
s... | Mathlib/GroupTheory/Perm/Support.lean | 270 | 271 | theorem set_support_apply_mem {p : Perm Ξ±} {a : Ξ±} :
p a β { x | p x β x } β a β { x | p x β x } := by | simp
| 1 | 2.718282 | 0 | 0.944444 | 18 | 795 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section support
v... | Mathlib/GroupTheory/Perm/Support.lean | 297 | 298 | theorem mem_support {x : Ξ±} : x β f.support β f x β x := by |
rw [support, mem_filter, and_iff_right (mem_univ x)]
| 1 | 2.718282 | 0 | 0.944444 | 18 | 795 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section support
v... | Mathlib/GroupTheory/Perm/Support.lean | 301 | 301 | theorem not_mem_support {x : Ξ±} : x β f.support β f x = x := by | simp
| 1 | 2.718282 | 0 | 0.944444 | 18 | 795 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section support
v... | Mathlib/GroupTheory/Perm/Support.lean | 304 | 306 | theorem coe_support_eq_set_support (f : Perm Ξ±) : (f.support : Set Ξ±) = { x | f x β x } := by |
ext
simp
| 2 | 7.389056 | 1 | 0.944444 | 18 | 795 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section support
v... | Mathlib/GroupTheory/Perm/Support.lean | 310 | 312 | theorem support_eq_empty_iff {Ο : Perm Ξ±} : Ο.support = β
β Ο = 1 := by |
simp_rw [Finset.ext_iff, mem_support, Finset.not_mem_empty, iff_false_iff, not_not,
Equiv.Perm.ext_iff, one_apply]
| 2 | 7.389056 | 1 | 0.944444 | 18 | 795 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section support
v... | Mathlib/GroupTheory/Perm/Support.lean | 316 | 316 | theorem support_one : (1 : Perm Ξ±).support = β
:= by | rw [support_eq_empty_iff]
| 1 | 2.718282 | 0 | 0.944444 | 18 | 795 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section support
v... | Mathlib/GroupTheory/Perm/Support.lean | 324 | 329 | theorem support_congr (h : f.support β g.support) (h' : β x β g.support, f x = g x) : f = g := by |
ext x
by_cases hx : x β g.support
Β· exact h' x hx
Β· rw [not_mem_support.mp hx, β not_mem_support]
exact fun H => hx (h H)
| 5 | 148.413159 | 2 | 0.944444 | 18 | 795 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section support
v... | Mathlib/GroupTheory/Perm/Support.lean | 339 | 350 | theorem exists_mem_support_of_mem_support_prod {l : List (Perm Ξ±)} {x : Ξ±}
(hx : x β l.prod.support) : β f : Perm Ξ±, f β l β§ x β f.support := by |
contrapose! hx
simp_rw [mem_support, not_not] at hx β’
induction' l with f l ih
Β· rfl
Β· rw [List.prod_cons, mul_apply, ih, hx]
Β· simp only [List.find?, List.mem_cons, true_or]
intros f' hf'
refine hx f' ?_
simp only [List.find?, List.mem_cons]
exact Or.inr hf'
| 10 | 22,026.465795 | 2 | 0.944444 | 18 | 795 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 61 | 61 | theorem ascPochhammer_one : ascPochhammer S 1 = X := by | simp [ascPochhammer]
| 1 | 2.718282 | 0 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 64 | 66 | theorem ascPochhammer_succ_left (n : β) :
ascPochhammer S (n + 1) = X * (ascPochhammer S n).comp (X + 1) := by |
rw [ascPochhammer]
| 1 | 2.718282 | 0 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 69 | 76 | theorem monic_ascPochhammer (n : β) [Nontrivial S] [NoZeroDivisors S] :
Monic <| ascPochhammer S n := by |
induction' n with n hn
Β· simp
Β· have : leadingCoeff (X + 1 : S[X]) = 1 := leadingCoeff_X_add_C 1
rw [ascPochhammer_succ_left, Monic.def, leadingCoeff_mul,
leadingCoeff_comp (ne_zero_of_eq_one <| natDegree_X_add_C 1 : natDegree (X + 1) β 0), hn,
monic_X, one_mul, one_mul, this, one_pow]
| 6 | 403.428793 | 2 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 83 | 87 | theorem ascPochhammer_map (f : S β+* T) (n : β) :
(ascPochhammer S n).map f = ascPochhammer T n := by |
induction' n with n ih
Β· simp
Β· simp [ih, ascPochhammer_succ_left, map_comp]
| 3 | 20.085537 | 1 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 90 | 93 | theorem ascPochhammer_evalβ (f : S β+* T) (n : β) (t : T) :
(ascPochhammer T n).eval t = (ascPochhammer S n).evalβ f t := by |
rw [β ascPochhammer_map f]
exact eval_map f t
| 2 | 7.389056 | 1 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 95 | 99 | theorem ascPochhammer_eval_comp {R : Type*} [CommSemiring R] (n : β) (p : R[X]) [Algebra R S]
(x : S) : ((ascPochhammer S n).comp (p.map (algebraMap R S))).eval x =
(ascPochhammer S n).eval (p.evalβ (algebraMap R S) x) := by |
rw [ascPochhammer_evalβ (algebraMap R S), β evalβ_comp', β ascPochhammer_map (algebraMap R S),
β map_comp, eval_map]
| 2 | 7.389056 | 1 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 104 | 107 | theorem ascPochhammer_eval_cast (n k : β) :
(((ascPochhammer β n).eval k : β) : S) = ((ascPochhammer S n).eval k : S) := by |
rw [β ascPochhammer_map (algebraMap β S), eval_map, β eq_natCast (algebraMap β S),
evalβ_at_natCast,Nat.cast_id]
| 2 | 7.389056 | 1 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 110 | 113 | theorem ascPochhammer_eval_zero {n : β} : (ascPochhammer S n).eval 0 = if n = 0 then 1 else 0 := by |
cases n
Β· simp
Β· simp [X_mul, Nat.succ_ne_zero, ascPochhammer_succ_left]
| 3 | 20.085537 | 1 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 116 | 116 | theorem ascPochhammer_zero_eval_zero : (ascPochhammer S 0).eval 0 = 1 := by | simp
| 1 | 2.718282 | 0 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 120 | 121 | theorem ascPochhammer_ne_zero_eval_zero {n : β} (h : n β 0) : (ascPochhammer S n).eval 0 = 0 := by |
simp [ascPochhammer_eval_zero, h]
| 1 | 2.718282 | 0 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 124 | 134 | theorem ascPochhammer_succ_right (n : β) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X])) := by |
suffices h : ascPochhammer β (n + 1) = ascPochhammer β n * (X + (n : β[X])) by
apply_fun Polynomial.map (algebraMap β S) at h
simpa only [ascPochhammer_map, Polynomial.map_mul, Polynomial.map_add, map_X,
Polynomial.map_natCast] using h
induction' n with n ih
Β· simp
Β· conv_lhs =>
rw [ascPoch... | 9 | 8,103.083928 | 2 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 137 | 140 | theorem ascPochhammer_succ_eval {S : Type*} [Semiring S] (n : β) (k : S) :
(ascPochhammer S (n + 1)).eval k = (ascPochhammer S n).eval k * (k + n) := by |
rw [ascPochhammer_succ_right, mul_add, eval_add, eval_mul_X, β Nat.cast_comm, β C_eq_natCast,
eval_C_mul, Nat.cast_comm, β mul_add]
| 2 | 7.389056 | 1 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 143 | 152 | theorem ascPochhammer_succ_comp_X_add_one (n : β) :
(ascPochhammer S (n + 1)).comp (X + 1) =
ascPochhammer S (n + 1) + (n + 1) β’ (ascPochhammer S n).comp (X + 1) := by |
suffices (ascPochhammer β (n + 1)).comp (X + 1) =
ascPochhammer β (n + 1) + (n + 1) * (ascPochhammer β n).comp (X + 1)
by simpa [map_comp] using congr_arg (Polynomial.map (Nat.castRingHom S)) this
nth_rw 2 [ascPochhammer_succ_left]
rw [β add_mul, ascPochhammer_succ_right β n, mul_comp, mul_comm, add_co... | 7 | 1,096.633158 | 2 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 256 | 256 | theorem descPochhammer_one : descPochhammer R 1 = X := by | simp [descPochhammer]
| 1 | 2.718282 | 0 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 258 | 260 | theorem descPochhammer_succ_left (n : β) :
descPochhammer R (n + 1) = X * (descPochhammer R n).comp (X - 1) := by |
rw [descPochhammer]
| 1 | 2.718282 | 0 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 262 | 269 | theorem monic_descPochhammer (n : β) [Nontrivial R] [NoZeroDivisors R] :
Monic <| descPochhammer R n := by |
induction' n with n hn
Β· simp
Β· have h : leadingCoeff (X - 1 : R[X]) = 1 := leadingCoeff_X_sub_C 1
have : natDegree (X - (1 : R[X])) β 0 := ne_zero_of_eq_one <| natDegree_X_sub_C (1 : R)
rw [descPochhammer_succ_left, Monic.def, leadingCoeff_mul, leadingCoeff_comp this, hn, monic_X,
one_mul, one_m... | 6 | 403.428793 | 2 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 276 | 280 | theorem descPochhammer_map (f : R β+* T) (n : β) :
(descPochhammer R n).map f = descPochhammer T n := by |
induction' n with n ih
Β· simp
Β· simp [ih, descPochhammer_succ_left, map_comp]
| 3 | 20.085537 | 1 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 284 | 287 | theorem descPochhammer_eval_cast (n : β) (k : β€) :
(((descPochhammer β€ n).eval k : β€) : R) = ((descPochhammer R n).eval k : R) := by |
rw [β descPochhammer_map (algebraMap β€ R), eval_map, β eq_intCast (algebraMap β€ R)]
simp only [algebraMap_int_eq, eq_intCast, evalβ_at_intCast, Nat.cast_id, eq_natCast, Int.cast_id]
| 2 | 7.389056 | 1 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 289 | 293 | theorem descPochhammer_eval_zero {n : β} :
(descPochhammer R n).eval 0 = if n = 0 then 1 else 0 := by |
cases n
Β· simp
Β· simp [X_mul, Nat.succ_ne_zero, descPochhammer_succ_left]
| 3 | 20.085537 | 1 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 295 | 295 | theorem descPochhammer_zero_eval_zero : (descPochhammer R 0).eval 0 = 1 := by | simp
| 1 | 2.718282 | 0 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 298 | 299 | theorem descPochhammer_ne_zero_eval_zero {n : β} (h : n β 0) : (descPochhammer R n).eval 0 = 0 := by |
simp [descPochhammer_eval_zero, h]
| 1 | 2.718282 | 0 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 301 | 312 | theorem descPochhammer_succ_right (n : β) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) := by |
suffices h : descPochhammer β€ (n + 1) = descPochhammer β€ n * (X - (n : β€[X])) by
apply_fun Polynomial.map (algebraMap β€ R) at h
simpa [descPochhammer_map, Polynomial.map_mul, Polynomial.map_add, map_X,
Polynomial.map_intCast] using h
induction' n with n ih
Β· simp [descPochhammer]
Β· conv_lhs =>
... | 10 | 22,026.465795 | 2 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 315 | 324 | theorem descPochhammer_natDegree (n : β) [NoZeroDivisors R] [Nontrivial R] :
(descPochhammer R n).natDegree = n := by |
induction' n with n hn
Β· simp
Β· have : natDegree (X - (n : R[X])) = 1 := natDegree_X_sub_C (n : R)
rw [descPochhammer_succ_right,
natDegree_mul _ (ne_zero_of_natDegree_gt <| this.symm βΈ Nat.zero_lt_one), hn, this]
cases n
Β· simp
Β· refine ne_zero_of_natDegree_gt <| hn.symm βΈ Nat.add_one_po... | 8 | 2,980.957987 | 2 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 326 | 329 | theorem descPochhammer_succ_eval {S : Type*} [Ring S] (n : β) (k : S) :
(descPochhammer S (n + 1)).eval k = (descPochhammer S n).eval k * (k - n) := by |
rw [descPochhammer_succ_right, mul_sub, eval_sub, eval_mul_X, β Nat.cast_comm, β C_eq_natCast,
eval_C_mul, Nat.cast_comm, β mul_sub]
| 2 | 7.389056 | 1 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 331 | 339 | theorem descPochhammer_succ_comp_X_sub_one (n : β) :
(descPochhammer R (n + 1)).comp (X - 1) =
descPochhammer R (n + 1) - (n + (1 : R[X])) β’ (descPochhammer R n).comp (X - 1) := by |
suffices (descPochhammer β€ (n + 1)).comp (X - 1) =
descPochhammer β€ (n + 1) - (n + 1) * (descPochhammer β€ n).comp (X - 1)
by simpa [map_comp] using congr_arg (Polynomial.map (Int.castRingHom R)) this
nth_rw 2 [descPochhammer_succ_left]
rw [β sub_mul, descPochhammer_succ_right β€ n, mul_comp, mul_comm, s... | 6 | 403.428793 | 2 | 0.96 | 25 | 796 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace LocalizedModule
universe u v
variable {R : Type u} [CommSemiring R] (S : Submonoid R)
variab... | Mathlib/Algebra/Module/LocalizedModule.lean | 99 | 102 | theorem induction_on {Ξ² : LocalizedModule S M β Prop} (h : β (m : M) (s : S), Ξ² (mk m s)) :
β x : LocalizedModule S M, Ξ² x := by |
rintro β¨β¨m, sβ©β©
exact h m s
| 2 | 7.389056 | 1 | 1 | 7 | 797 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace LocalizedModule
universe u v
variable {R : Type u} [CommSemiring R] (S : Submonoid R)
variab... | Mathlib/Algebra/Module/LocalizedModule.lean | 106 | 109 | theorem induction_onβ {Ξ² : LocalizedModule S M β LocalizedModule S M β Prop}
(h : β (m m' : M) (s s' : S), Ξ² (mk m s) (mk m' s')) : β x y, Ξ² x y := by |
rintro β¨β¨m, sβ©β© β¨β¨m', s'β©β©
exact h m m' s s'
| 2 | 7.389056 | 1 | 1 | 7 | 797 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace LocalizedModule
universe u v
variable {R : Type u} [CommSemiring R] (S : Submonoid R)
variab... | Mathlib/Algebra/Module/LocalizedModule.lean | 120 | 121 | theorem liftOn_mk {Ξ± : Type*} {f : M Γ S β Ξ±} (wd : β (p p' : M Γ S), p β p' β f p = f p')
(m : M) (s : S) : liftOn (mk m s) f wd = f β¨m, sβ© := by | convert Quotient.liftOn_mk f wd β¨m, sβ©
| 1 | 2.718282 | 0 | 1 | 7 | 797 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace LocalizedModule
universe u v
variable {R : Type u} [CommSemiring R] (S : Submonoid R)
variab... | Mathlib/Algebra/Module/LocalizedModule.lean | 132 | 135 | theorem liftOnβ_mk {Ξ± : Type*} (f : M Γ S β M Γ S β Ξ±)
(wd : β (p q p' q' : M Γ S), p β p' β q β q' β f p q = f p' q') (m m' : M)
(s s' : S) : liftOnβ (mk m s) (mk m' s') f wd = f β¨m, sβ© β¨m', s'β© := by |
convert Quotient.liftOnβ_mk f wd _ _
| 1 | 2.718282 | 0 | 1 | 7 | 797 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace LocalizedModule
universe u v
variable {R : Type u} [CommSemiring R] (S : Submonoid R)
variab... | Mathlib/Algebra/Module/LocalizedModule.lean | 142 | 145 | theorem subsingleton (h : 0 β S) : Subsingleton (LocalizedModule S M) := by |
refine β¨fun a b β¦ ?_β©
induction a,b using LocalizedModule.induction_onβ
exact mk_eq.mpr β¨β¨0, hβ©, by simp only [Submonoid.mk_smul, zero_smul]β©
| 3 | 20.085537 | 1 | 1 | 7 | 797 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
section IsLocalizedModule
universe u v
variable {R : Type*} [CommSemiring R] (S : Submonoid R)
variabl... | Mathlib/Algebra/Module/LocalizedModule.lean | 574 | 588 | theorem IsLocalizedModule.of_linearEquiv (e : M' ββ[R] M'') [hf : IsLocalizedModule S f] :
IsLocalizedModule S (e ββ f : M ββ[R] M'') where
map_units s := by |
rw [show algebraMap R (Module.End R M'') s = e ββ (algebraMap R (Module.End R M') s) ββ e.symm
by ext; simp, Module.End_isUnit_iff, LinearMap.coe_comp, LinearMap.coe_comp,
LinearEquiv.coe_coe, LinearEquiv.coe_coe, EquivLike.comp_bijective, EquivLike.bijective_comp]
exact (Module.End_isUnit_iff _).m... | 12 | 162,754.791419 | 2 | 1 | 7 | 797 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
section IsLocalizedModule
universe u v
variable {R : Type*} [CommSemiring R] (S : Submonoid R)
variabl... | Mathlib/Algebra/Module/LocalizedModule.lean | 599 | 610 | theorem isLocalizedModule_iff_isLocalization {A Aβ} [CommSemiring A] [Algebra R A] [CommSemiring Aβ]
[Algebra A Aβ] [Algebra R Aβ] [IsScalarTower R A Aβ] :
IsLocalizedModule S (IsScalarTower.toAlgHom R A Aβ).toLinearMap β
IsLocalization (Algebra.algebraMapSubmonoid A S) Aβ := by |
rw [isLocalizedModule_iff, isLocalization_iff]
refine and_congr ?_ (and_congr (forall_congr' fun _ β¦ ?_) (forallβ_congr fun _ _ β¦ ?_))
Β· simp_rw [β (Algebra.lmul R Aβ).commutes, Algebra.lmul_isUnit_iff, Subtype.forall,
Algebra.algebraMapSubmonoid, β SetLike.mem_coe, Submonoid.coe_map,
Set.forall_mem_... | 8 | 2,980.957987 | 2 | 1 | 7 | 797 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {Ξ± : Type v} {Ξ² : Type w}
namespace Matrix
def col (w : m β Ξ±) : Matrix m Unit Ξ± :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 61 | 63 | theorem col_add [Add Ξ±] (v w : m β Ξ±) : col (v + w) = col v + col w := by |
ext
rfl
| 2 | 7.389056 | 1 | 1 | 14 | 798 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {Ξ± : Type v} {Ξ² : Type w}
namespace Matrix
def col (w : m β Ξ±) : Matrix m Unit Ξ± :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 67 | 69 | theorem col_smul [SMul R Ξ±] (x : R) (v : m β Ξ±) : col (x β’ v) = x β’ col v := by |
ext
rfl
| 2 | 7.389056 | 1 | 1 | 14 | 798 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {Ξ± : Type v} {Ξ² : Type w}
namespace Matrix
def col (w : m β Ξ±) : Matrix m Unit Ξ± :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 82 | 84 | theorem row_add [Add Ξ±] (v w : m β Ξ±) : row (v + w) = row v + row w := by |
ext
rfl
| 2 | 7.389056 | 1 | 1 | 14 | 798 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {Ξ± : Type v} {Ξ² : Type w}
namespace Matrix
def col (w : m β Ξ±) : Matrix m Unit Ξ± :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 88 | 90 | theorem row_smul [SMul R Ξ±] (x : R) (v : m β Ξ±) : row (x β’ v) = x β’ row v := by |
ext
rfl
| 2 | 7.389056 | 1 | 1 | 14 | 798 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {Ξ± : Type v} {Ξ² : Type w}
namespace Matrix
def col (w : m β Ξ±) : Matrix m Unit Ξ± :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 94 | 96 | theorem transpose_col (v : m β Ξ±) : (Matrix.col v)α΅ = Matrix.row v := by |
ext
rfl
| 2 | 7.389056 | 1 | 1 | 14 | 798 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {Ξ± : Type v} {Ξ² : Type w}
namespace Matrix
def col (w : m β Ξ±) : Matrix m Unit Ξ± :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 100 | 102 | theorem transpose_row (v : m β Ξ±) : (Matrix.row v)α΅ = Matrix.col v := by |
ext
rfl
| 2 | 7.389056 | 1 | 1 | 14 | 798 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {Ξ± : Type v} {Ξ² : Type w}
namespace Matrix
def col (w : m β Ξ±) : Matrix m Unit Ξ± :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 106 | 108 | theorem conjTranspose_col [Star Ξ±] (v : m β Ξ±) : (col v)α΄΄ = row (star v) := by |
ext
rfl
| 2 | 7.389056 | 1 | 1 | 14 | 798 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {Ξ± : Type v} {Ξ² : Type w}
namespace Matrix
def col (w : m β Ξ±) : Matrix m Unit Ξ± :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 112 | 114 | theorem conjTranspose_row [Star Ξ±] (v : m β Ξ±) : (row v)α΄΄ = col (star v) := by |
ext
rfl
| 2 | 7.389056 | 1 | 1 | 14 | 798 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {Ξ± : Type v} {Ξ² : Type w}
namespace Matrix
def col (w : m β Ξ±) : Matrix m Unit Ξ± :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 117 | 120 | theorem row_vecMul [Fintype m] [NonUnitalNonAssocSemiring Ξ±] (M : Matrix m n Ξ±) (v : m β Ξ±) :
Matrix.row (v α΅₯* M) = Matrix.row v * M := by |
ext
rfl
| 2 | 7.389056 | 1 | 1 | 14 | 798 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {Ξ± : Type v} {Ξ² : Type w}
namespace Matrix
def col (w : m β Ξ±) : Matrix m Unit Ξ± :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 123 | 126 | theorem col_vecMul [Fintype m] [NonUnitalNonAssocSemiring Ξ±] (M : Matrix m n Ξ±) (v : m β Ξ±) :
Matrix.col (v α΅₯* M) = (Matrix.row v * M)α΅ := by |
ext
rfl
| 2 | 7.389056 | 1 | 1 | 14 | 798 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {Ξ± : Type v} {Ξ² : Type w}
namespace Matrix
def col (w : m β Ξ±) : Matrix m Unit Ξ± :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 129 | 132 | theorem col_mulVec [Fintype n] [NonUnitalNonAssocSemiring Ξ±] (M : Matrix m n Ξ±) (v : n β Ξ±) :
Matrix.col (M *α΅₯ v) = M * Matrix.col v := by |
ext
rfl
| 2 | 7.389056 | 1 | 1 | 14 | 798 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {Ξ± : Type v} {Ξ² : Type w}
namespace Matrix
def col (w : m β Ξ±) : Matrix m Unit Ξ± :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 135 | 138 | theorem row_mulVec [Fintype n] [NonUnitalNonAssocSemiring Ξ±] (M : Matrix m n Ξ±) (v : n β Ξ±) :
Matrix.row (M *α΅₯ v) = (M * Matrix.col v)α΅ := by |
ext
rfl
| 2 | 7.389056 | 1 | 1 | 14 | 798 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {Ξ± : Type v} {Ξ² : Type w}
namespace Matrix
def col (w : m β Ξ±) : Matrix m Unit Ξ± :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 148 | 151 | theorem diag_col_mul_row [Mul Ξ±] [AddCommMonoid Ξ±] (a b : n β Ξ±) :
diag (col a * row b) = a * b := by |
ext
simp [Matrix.mul_apply, col, row]
| 2 | 7.389056 | 1 | 1 | 14 | 798 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {Ξ± : Type v} {Ξ² : Type w}
namespace Matrix
def col (w : m β Ξ±) : Matrix m Unit Ξ± :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 154 | 158 | theorem vecMulVec_eq [Mul Ξ±] [AddCommMonoid Ξ±] (w : m β Ξ±) (v : n β Ξ±) :
vecMulVec w v = col w * row v := by |
ext
simp only [vecMulVec, mul_apply, Fintype.univ_punit, Finset.sum_singleton]
rfl
| 3 | 20.085537 | 1 | 1 | 14 | 798 |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.LinearAlgebra.AffineSpace.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.affine_space.affine_map from "leanprover-community/mathlib"@"bd1fc183335ea95a9519a1630bcf901... | Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 135 | 136 | theorem linearMap_vsub (f : P1 βα΅[k] P2) (p1 p2 : P1) : f.linear (p1 -α΅₯ p2) = f p1 -α΅₯ f p2 := by |
conv_rhs => rw [β vsub_vadd p1 p2, map_vadd, vadd_vsub]
| 1 | 2.718282 | 0 | 1 | 2 | 799 |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.LinearAlgebra.AffineSpace.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.affine_space.affine_map from "leanprover-community/mathlib"@"bd1fc183335ea95a9519a1630bcf901... | Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 162 | 169 | theorem ext_linear {f g : P1 βα΅[k] P2} (hβ : f.linear = g.linear) {p : P1} (hβ : f p = g p) :
f = g := by |
ext q
have hgl : g.linear (q -α΅₯ p) = toFun g ((q -α΅₯ p) +α΅₯ q) -α΅₯ toFun g q := by simp
have := f.map_vadd' q (q -α΅₯ p)
rw [hβ, hgl, toFun_eq_coe, map_vadd, linearMap_vsub, hβ] at this
simp at this
exact this
| 6 | 403.428793 | 2 | 1 | 2 | 799 |
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Group.Hom.CompTypeclasses
#align_import algebra.hom.group_action from "leanprover-community/mathlib"@"e7bab9a85e92cf46c02cb4725a7be2f04691e3a7"
assert_not_exists Submonoid
section ... | Mathlib/GroupTheory/GroupAction/Hom.lean | 150 | 154 | theorem _root_.IsScalarTower.smulHomClass [MulOneClass X] [SMul X Y] [IsScalarTower M' X Y]
[MulActionHomClass F X X Y] : MulActionHomClass F M' X Y where
map_smulββ f m x := by |
rw [β mul_one (m β’ x), β smul_eq_mul, map_smul, smul_assoc, β map_smul,
smul_eq_mul, mul_one, id_eq]
| 2 | 7.389056 | 1 | 1 | 1 | 800 |
import Mathlib.MeasureTheory.Group.Arithmetic
#align_import measure_theory.group.pointwise from "leanprover-community/mathlib"@"66f7114a1d5cba41c47d417a034bbb2e96cf564a"
open Pointwise
open Set
@[to_additive]
| Mathlib/MeasureTheory/Group/Pointwise.lean | 24 | 28 | theorem MeasurableSet.const_smul {G Ξ± : Type*} [Group G] [MulAction G Ξ±] [MeasurableSpace G]
[MeasurableSpace Ξ±] [MeasurableSMul G Ξ±] {s : Set Ξ±} (hs : MeasurableSet s) (a : G) :
MeasurableSet (a β’ s) := by |
rw [β preimage_smul_inv]
exact measurable_const_smul _ hs
| 2 | 7.389056 | 1 | 1 | 3 | 801 |
import Mathlib.MeasureTheory.Group.Arithmetic
#align_import measure_theory.group.pointwise from "leanprover-community/mathlib"@"66f7114a1d5cba41c47d417a034bbb2e96cf564a"
open Pointwise
open Set
@[to_additive]
theorem MeasurableSet.const_smul {G Ξ± : Type*} [Group G] [MulAction G Ξ±] [MeasurableSpace G]
[Measu... | Mathlib/MeasureTheory/Group/Pointwise.lean | 32 | 36 | theorem MeasurableSet.const_smul_of_ne_zero {Gβ Ξ± : Type*} [GroupWithZero Gβ] [MulAction Gβ Ξ±]
[MeasurableSpace Gβ] [MeasurableSpace Ξ±] [MeasurableSMul Gβ Ξ±] {s : Set Ξ±}
(hs : MeasurableSet s) {a : Gβ} (ha : a β 0) : MeasurableSet (a β’ s) := by |
rw [β preimage_smul_invβ ha]
exact measurable_const_smul _ hs
| 2 | 7.389056 | 1 | 1 | 3 | 801 |
import Mathlib.MeasureTheory.Group.Arithmetic
#align_import measure_theory.group.pointwise from "leanprover-community/mathlib"@"66f7114a1d5cba41c47d417a034bbb2e96cf564a"
open Pointwise
open Set
@[to_additive]
theorem MeasurableSet.const_smul {G Ξ± : Type*} [Group G] [MulAction G Ξ±] [MeasurableSpace G]
[Measu... | Mathlib/MeasureTheory/Group/Pointwise.lean | 39 | 44 | theorem MeasurableSet.const_smulβ {Gβ Ξ± : Type*} [GroupWithZero Gβ] [Zero Ξ±]
[MulActionWithZero Gβ Ξ±] [MeasurableSpace Gβ] [MeasurableSpace Ξ±] [MeasurableSMul Gβ Ξ±]
[MeasurableSingletonClass Ξ±] {s : Set Ξ±} (hs : MeasurableSet s) (a : Gβ) :
MeasurableSet (a β’ s) := by |
rcases eq_or_ne a 0 with (rfl | ha)
exacts [(subsingleton_zero_smul_set s).measurableSet, hs.const_smul_of_ne_zero ha]
| 2 | 7.389056 | 1 | 1 | 3 | 801 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.Topology.Algebra.Module.Basic
open Function
structure ContinuousAffineEquiv (k Pβ Pβ : Type*) {Vβ Vβ : Type*} [Ring k]
[AddCommGroup Vβ] [Module k Vβ] [AddTorsor Vβ Pβ] [TopologicalSpace Pβ]
[AddCommGroup Vβ] [Module k Vβ] [AddTorsor Vβ P... | Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean | 65 | 67 | theorem toAffineEquiv_injective : Injective (toAffineEquiv : (Pβ βα΅L[k] Pβ) β Pβ βα΅[k] Pβ) := by |
rintro β¨e, econt, einv_contβ© β¨e', e'cont, e'inv_contβ© H
congr
| 2 | 7.389056 | 1 | 1 | 2 | 802 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.Topology.Algebra.Module.Basic
open Function
structure ContinuousAffineEquiv (k Pβ Pβ : Type*) {Vβ Vβ : Type*} [Ring k]
[AddCommGroup Vβ] [Module k Vβ] [AddTorsor Vβ Pβ] [TopologicalSpace Pβ]
[AddCommGroup Vβ] [Module k Vβ] [AddTorsor Vβ P... | Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean | 84 | 87 | theorem coe_injective : Function.Injective ((β) : (Pβ βα΅L[k] Pβ) β Pβ βα΅[k] Pβ) := by |
intro e e' H
cases e
congr
| 3 | 20.085537 | 1 | 1 | 2 | 802 |
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.ContinuousFunction.Ordered
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.homotopy.basic from "leanprover-community/mathlib"@"11c53f174270aa43140c0b26dabce5fc4a253e80"
noncomputable section
universe u v ... | Mathlib/Topology/Homotopy/Basic.lean | 166 | 169 | theorem extend_apply_of_le_zero (F : Homotopy fβ fβ) {t : β} (ht : t β€ 0) (x : X) :
F.extend t x = fβ x := by |
rw [β F.apply_zero]
exact ContinuousMap.congr_fun (Set.IccExtend_of_le_left (zero_le_one' β) F.curry ht) x
| 2 | 7.389056 | 1 | 1 | 2 | 803 |
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.ContinuousFunction.Ordered
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.homotopy.basic from "leanprover-community/mathlib"@"11c53f174270aa43140c0b26dabce5fc4a253e80"
noncomputable section
universe u v ... | Mathlib/Topology/Homotopy/Basic.lean | 172 | 175 | theorem extend_apply_of_one_le (F : Homotopy fβ fβ) {t : β} (ht : 1 β€ t) (x : X) :
F.extend t x = fβ x := by |
rw [β F.apply_one]
exact ContinuousMap.congr_fun (Set.IccExtend_of_right_le (zero_le_one' β) F.curry ht) x
| 2 | 7.389056 | 1 | 1 | 2 | 803 |
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.path_connected from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter unitInterval Set Fun... | Mathlib/Topology/Connected/PathConnected.lean | 165 | 165 | theorem refl_range {a : X} : range (Path.refl a) = {a} := by | simp [Path.refl, CoeFun.coe]
| 1 | 2.718282 | 0 | 1 | 4 | 804 |
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.path_connected from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter unitInterval Set Fun... | Mathlib/Topology/Connected/PathConnected.lean | 178 | 181 | theorem symm_symm (Ξ³ : Path x y) : Ξ³.symm.symm = Ξ³ := by |
ext t
show Ξ³ (Ο (Ο t)) = Ξ³ t
rw [unitInterval.symm_symm]
| 3 | 20.085537 | 1 | 1 | 4 | 804 |
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.path_connected from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter unitInterval Set Fun... | Mathlib/Topology/Connected/PathConnected.lean | 188 | 190 | theorem refl_symm {a : X} : (Path.refl a).symm = Path.refl a := by |
ext
rfl
| 2 | 7.389056 | 1 | 1 | 4 | 804 |
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.path_connected from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter unitInterval Set Fun... | Mathlib/Topology/Connected/PathConnected.lean | 194 | 200 | theorem symm_range {a b : X} (Ξ³ : Path a b) : range Ξ³.symm = range Ξ³ := by |
ext x
simp only [mem_range, Path.symm, DFunLike.coe, unitInterval.symm, SetCoe.exists, comp_apply,
Subtype.coe_mk]
constructor <;> rintro β¨y, hy, hxyβ© <;> refine β¨1 - y, mem_iff_one_sub_mem.mp hy, ?_β© <;>
convert hxy
simp
| 6 | 403.428793 | 2 | 1 | 4 | 804 |
import Mathlib.Data.Sum.Order
import Mathlib.Order.InitialSeg
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
#align_import set_theory.ordinal.basic from "leanprover-community/mathlib"@"8ea5598db6caeddde6cb734aa179cc2408dbd345"
assert_not_exists Module
assert_not_exists Field
noncomputabl... | Mathlib/SetTheory/Ordinal/Basic.lean | 137 | 139 | theorem eta (o : WellOrder) : mk o.Ξ± o.r o.wo = o := by |
cases o
rfl
| 2 | 7.389056 | 1 | 1 | 1 | 805 |
import Mathlib.Logic.Function.Iterate
import Mathlib.Init.Data.Int.Order
import Mathlib.Order.Compare
import Mathlib.Order.Max
import Mathlib.Order.RelClasses
import Mathlib.Tactic.Choose
#align_import order.monotone.basic from "leanprover-community/mathlib"@"554bb38de8ded0dafe93b7f18f0bfee6ef77dc5d"
open Functio... | Mathlib/Order/Monotone/Basic.lean | 1,014 | 1,018 | theorem Nat.rel_of_forall_rel_succ_of_le_of_lt (r : Ξ² β Ξ² β Prop) [IsTrans Ξ² r] {f : β β Ξ²} {a : β}
(h : β n, a β€ n β r (f n) (f (n + 1))) β¦b c : ββ¦ (hab : a β€ b) (hbc : b < c) :
r (f b) (f c) := by |
induction' hbc with k b_lt_k r_b_k
exacts [h _ hab, _root_.trans r_b_k (h _ (hab.trans_lt b_lt_k).le)]
| 2 | 7.389056 | 1 | 1 | 1 | 806 |
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Smooth.Basic
import Mathlib.RingTheory.Unramified.Basic
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
-- Porting note: added to make the syntax work below.
open scoped TensorProdu... | Mathlib/RingTheory/Etale/Basic.lean | 66 | 69 | theorem iff_unramified_and_smooth :
FormallyEtale R A β FormallyUnramified R A β§ FormallySmooth R A := by |
rw [formallyUnramified_iff, formallySmooth_iff, formallyEtale_iff]
simp_rw [β forall_and, Function.Bijective]
| 2 | 7.389056 | 1 | 1 | 1 | 807 |
import Mathlib.Logic.UnivLE
import Mathlib.SetTheory.Ordinal.Basic
set_option autoImplicit true
noncomputable section
open Cardinal
| Mathlib/SetTheory/Cardinal/UnivLE.lean | 19 | 27 | theorem univLE_iff_cardinal_le : UnivLE.{u, v} β univ.{u, v+1} β€ univ.{v, u+1} := by |
rw [β not_iff_not, UnivLE]; simp_rw [small_iff_lift_mk_lt_univ]; push_neg
-- strange: simp_rw [univ_umax.{v,u}] doesn't work
refine β¨fun β¨Ξ±, leβ© β¦ ?_, fun h β¦ ?_β©
Β· rw [univ_umax.{v,u}, β lift_le.{u+1}, lift_univ, lift_lift] at le
exact le.trans_lt (lift_lt_univ'.{u,v+1} #Ξ±)
Β· obtain β¨β¨Ξ±β©, hβ© := lt_univ'... | 8 | 2,980.957987 | 2 | 1 | 2 | 808 |
import Mathlib.Logic.UnivLE
import Mathlib.SetTheory.Ordinal.Basic
set_option autoImplicit true
noncomputable section
open Cardinal
theorem univLE_iff_cardinal_le : UnivLE.{u, v} β univ.{u, v+1} β€ univ.{v, u+1} := by
rw [β not_iff_not, UnivLE]; simp_rw [small_iff_lift_mk_lt_univ]; push_neg
-- strange: simp_r... | Mathlib/SetTheory/Cardinal/UnivLE.lean | 30 | 31 | theorem univLE_total : UnivLE.{u, v} β¨ UnivLE.{v, u} := by |
simp_rw [univLE_iff_cardinal_le]; apply le_total
| 1 | 2.718282 | 0 | 1 | 2 | 808 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Ring.Regular
import Mathlib.Tactic.Common
#align_import algebra.gcd_monoid.basic from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
variable {Ξ± : Type*}
-- Porting note: mathlib3 had a `@[protect_proj]` here, but adding `protect... | Mathlib/Algebra/GCDMonoid/Basic.lean | 148 | 148 | theorem normalize_coe_units (u : Ξ±Λ£) : normalize (u : Ξ±) = 1 := by | simp
| 1 | 2.718282 | 0 | 1 | 4 | 809 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Ring.Regular
import Mathlib.Tactic.Common
#align_import algebra.gcd_monoid.basic from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
variable {Ξ± : Type*}
-- Porting note: mathlib3 had a `@[protect_proj]` here, but adding `protect... | Mathlib/Algebra/GCDMonoid/Basic.lean | 162 | 166 | theorem normUnit_mul_normUnit (a : Ξ±) : normUnit (a * normUnit a) = 1 := by |
nontriviality Ξ± using Subsingleton.elim a 0
obtain rfl | h := eq_or_ne a 0
Β· rw [normUnit_zero, zero_mul, normUnit_zero]
Β· rw [normUnit_mul h (Units.ne_zero _), normUnit_coe_units, mul_inv_eq_one]
| 4 | 54.59815 | 2 | 1 | 4 | 809 |
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