algebra.char_p.two
⟷
Mathlib.Algebra.CharP.Two
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -160,7 +160,7 @@ variable [Ring R]
theorem neg_one_eq_one_iff [Nontrivial R] : (-1 : R) = 1 ↔ ringChar R = 2 :=
by
refine' ⟨fun h => _, fun h => @CharTwo.neg_eq _ (ringChar.of_eq h) 1⟩
- rw [eq_comm, ← sub_eq_zero, sub_neg_eq_add, ← Nat.cast_one, ← Nat.cast_add] at h
+ rw [eq_comm, ← sub_eq_zero, sub_neg_eq_add, ← Nat.cast_one, ← Nat.cast_add] at h
exact ((Nat.dvd_prime Nat.prime_two).mp (ringChar.dvd h)).resolve_left CharP.ringChar_ne_one
#align neg_one_eq_one_iff neg_one_eq_one_iff
-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,7 +3,7 @@ Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
-import Mathbin.Algebra.CharP.Basic
+import Algebra.CharP.Basic
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"10bf4f825ad729c5653adc039dafa3622e7f93c9"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,14 +2,11 @@
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-
-! This file was ported from Lean 3 source module algebra.char_p.two
-! leanprover-community/mathlib commit 10bf4f825ad729c5653adc039dafa3622e7f93c9
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.Algebra.CharP.Basic
+#align_import algebra.char_p.two from "leanprover-community/mathlib"@"10bf4f825ad729c5653adc039dafa3622e7f93c9"
+
/-!
# Lemmas about rings of characteristic two
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -31,26 +31,38 @@ section Semiring
variable [Semiring R] [CharP R 2]
+#print CharTwo.two_eq_zero /-
theorem two_eq_zero : (2 : R) = 0 := by rw [← Nat.cast_two, CharP.cast_eq_zero]
#align char_two.two_eq_zero CharTwo.two_eq_zero
+-/
+#print CharTwo.add_self_eq_zero /-
@[simp]
theorem add_self_eq_zero (x : R) : x + x = 0 := by rw [← two_smul R x, two_eq_zero, zero_smul]
#align char_two.add_self_eq_zero CharTwo.add_self_eq_zero
+-/
+#print CharTwo.bit0_eq_zero /-
@[simp]
theorem bit0_eq_zero : (bit0 : R → R) = 0 := by funext; exact add_self_eq_zero _
#align char_two.bit0_eq_zero CharTwo.bit0_eq_zero
+-/
+#print CharTwo.bit0_apply_eq_zero /-
theorem bit0_apply_eq_zero (x : R) : (bit0 x : R) = 0 := by simp
#align char_two.bit0_apply_eq_zero CharTwo.bit0_apply_eq_zero
+-/
+#print CharTwo.bit1_eq_one /-
@[simp]
theorem bit1_eq_one : (bit1 : R → R) = 1 := by funext; simp [bit1]
#align char_two.bit1_eq_one CharTwo.bit1_eq_one
+-/
+#print CharTwo.bit1_apply_eq_one /-
theorem bit1_apply_eq_one (x : R) : (bit1 x : R) = 1 := by simp
#align char_two.bit1_apply_eq_one CharTwo.bit1_apply_eq_one
+-/
end Semiring
@@ -58,22 +70,30 @@ section Ring
variable [Ring R] [CharP R 2]
+#print CharTwo.neg_eq /-
@[simp]
theorem neg_eq (x : R) : -x = x := by
rw [neg_eq_iff_add_eq_zero, ← two_smul R x, two_eq_zero, zero_smul]
#align char_two.neg_eq CharTwo.neg_eq
+-/
+#print CharTwo.neg_eq' /-
theorem neg_eq' : Neg.neg = (id : R → R) :=
funext neg_eq
#align char_two.neg_eq' CharTwo.neg_eq'
+-/
+#print CharTwo.sub_eq_add /-
@[simp]
theorem sub_eq_add (x y : R) : x - y = x + y := by rw [sub_eq_add_neg, neg_eq]
#align char_two.sub_eq_add CharTwo.sub_eq_add
+-/
+#print CharTwo.sub_eq_add' /-
theorem sub_eq_add' : Sub.sub = ((· + ·) : R → R → R) :=
funext fun x => funext fun y => sub_eq_add x y
#align char_two.sub_eq_add' CharTwo.sub_eq_add'
+-/
end Ring
@@ -81,23 +101,31 @@ section CommSemiring
variable [CommSemiring R] [CharP R 2]
+#print CharTwo.add_sq /-
theorem add_sq (x y : R) : (x + y) ^ 2 = x ^ 2 + y ^ 2 :=
add_pow_char _ _ _
#align char_two.add_sq CharTwo.add_sq
+-/
+#print CharTwo.add_mul_self /-
theorem add_mul_self (x y : R) : (x + y) * (x + y) = x * x + y * y := by
rw [← pow_two, ← pow_two, ← pow_two, add_sq]
#align char_two.add_mul_self CharTwo.add_mul_self
+-/
open scoped BigOperators
+#print CharTwo.list_sum_sq /-
theorem list_sum_sq (l : List R) : l.Sum ^ 2 = (l.map (· ^ 2)).Sum :=
list_sum_pow_char _ _
#align char_two.list_sum_sq CharTwo.list_sum_sq
+-/
+#print CharTwo.list_sum_mul_self /-
theorem list_sum_mul_self (l : List R) : l.Sum * l.Sum = (List.map (fun x => x * x) l).Sum := by
simp_rw [← pow_two, list_sum_sq]
#align char_two.list_sum_mul_self CharTwo.list_sum_mul_self
+-/
#print CharTwo.multiset_sum_sq /-
theorem multiset_sum_sq (l : Multiset R) : l.Sum ^ 2 = (l.map (· ^ 2)).Sum :=
@@ -105,9 +133,11 @@ theorem multiset_sum_sq (l : Multiset R) : l.Sum ^ 2 = (l.map (· ^ 2)).Sum :=
#align char_two.multiset_sum_sq CharTwo.multiset_sum_sq
-/
+#print CharTwo.multiset_sum_mul_self /-
theorem multiset_sum_mul_self (l : Multiset R) :
l.Sum * l.Sum = (Multiset.map (fun x => x * x) l).Sum := by simp_rw [← pow_two, multiset_sum_sq]
#align char_two.multiset_sum_mul_self CharTwo.multiset_sum_mul_self
+-/
#print CharTwo.sum_sq /-
theorem sum_sq (s : Finset ι) (f : ι → R) : (∑ i in s, f i) ^ 2 = ∑ i in s, f i ^ 2 :=
@@ -115,9 +145,11 @@ theorem sum_sq (s : Finset ι) (f : ι → R) : (∑ i in s, f i) ^ 2 = ∑ i in
#align char_two.sum_sq CharTwo.sum_sq
-/
+#print CharTwo.sum_mul_self /-
theorem sum_mul_self (s : Finset ι) (f : ι → R) :
(∑ i in s, f i) * ∑ i in s, f i = ∑ i in s, f i * f i := by simp_rw [← pow_two, sum_sq]
#align char_two.sum_mul_self CharTwo.sum_mul_self
+-/
end CommSemiring
@@ -127,13 +159,16 @@ section ringChar
variable [Ring R]
+#print neg_one_eq_one_iff /-
theorem neg_one_eq_one_iff [Nontrivial R] : (-1 : R) = 1 ↔ ringChar R = 2 :=
by
refine' ⟨fun h => _, fun h => @CharTwo.neg_eq _ (ringChar.of_eq h) 1⟩
rw [eq_comm, ← sub_eq_zero, sub_neg_eq_add, ← Nat.cast_one, ← Nat.cast_add] at h
exact ((Nat.dvd_prime Nat.prime_two).mp (ringChar.dvd h)).resolve_left CharP.ringChar_ne_one
#align neg_one_eq_one_iff neg_one_eq_one_iff
+-/
+#print orderOf_neg_one /-
@[simp]
theorem orderOf_neg_one [Nontrivial R] : orderOf (-1 : R) = if ringChar R = 2 then 1 else 2 :=
by
@@ -143,6 +178,7 @@ theorem orderOf_neg_one [Nontrivial R] : orderOf (-1 : R) = if ringChar R = 2 th
· simp
simpa [neg_one_eq_one_iff] using h
#align order_of_neg_one orderOf_neg_one
+-/
end ringChar
mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3
@@ -116,7 +116,7 @@ theorem sum_sq (s : Finset ι) (f : ι → R) : (∑ i in s, f i) ^ 2 = ∑ i in
-/
theorem sum_mul_self (s : Finset ι) (f : ι → R) :
- ((∑ i in s, f i) * ∑ i in s, f i) = ∑ i in s, f i * f i := by simp_rw [← pow_two, sum_sq]
+ (∑ i in s, f i) * ∑ i in s, f i = ∑ i in s, f i * f i := by simp_rw [← pow_two, sum_sq]
#align char_two.sum_mul_self CharTwo.sum_mul_self
end CommSemiring
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -130,7 +130,7 @@ variable [Ring R]
theorem neg_one_eq_one_iff [Nontrivial R] : (-1 : R) = 1 ↔ ringChar R = 2 :=
by
refine' ⟨fun h => _, fun h => @CharTwo.neg_eq _ (ringChar.of_eq h) 1⟩
- rw [eq_comm, ← sub_eq_zero, sub_neg_eq_add, ← Nat.cast_one, ← Nat.cast_add] at h
+ rw [eq_comm, ← sub_eq_zero, sub_neg_eq_add, ← Nat.cast_one, ← Nat.cast_add] at h
exact ((Nat.dvd_prime Nat.prime_two).mp (ringChar.dvd h)).resolve_left CharP.ringChar_ne_one
#align neg_one_eq_one_iff neg_one_eq_one_iff
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -89,7 +89,7 @@ theorem add_mul_self (x y : R) : (x + y) * (x + y) = x * x + y * y := by
rw [← pow_two, ← pow_two, ← pow_two, add_sq]
#align char_two.add_mul_self CharTwo.add_mul_self
-open BigOperators
+open scoped BigOperators
theorem list_sum_sq (l : List R) : l.Sum ^ 2 = (l.map (· ^ 2)).Sum :=
list_sum_pow_char _ _
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -31,60 +31,24 @@ section Semiring
variable [Semiring R] [CharP R 2]
-/- warning: char_two.two_eq_zero -> CharTwo.two_eq_zero is a dubious translation:
-lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))], Eq.{succ u1} R (OfNat.ofNat.{u1} R 2 (OfNat.mk.{u1} R 2 (bit0.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))))) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))))
-but is expected to have type
- forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))], Eq.{succ u1} R (OfNat.ofNat.{u1} R 2 (instOfNat.{u1} R 2 (Semiring.toNatCast.{u1} R _inst_1) (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))
-Case conversion may be inaccurate. Consider using '#align char_two.two_eq_zero CharTwo.two_eq_zeroₓ'. -/
theorem two_eq_zero : (2 : R) = 0 := by rw [← Nat.cast_two, CharP.cast_eq_zero]
#align char_two.two_eq_zero CharTwo.two_eq_zero
-/- warning: char_two.add_self_eq_zero -> CharTwo.add_self_eq_zero is a dubious translation:
-lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (x : R), Eq.{succ u1} R (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) x x) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))))
-but is expected to have type
- forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))] (x : R), Eq.{succ u1} R (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) x x) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))
-Case conversion may be inaccurate. Consider using '#align char_two.add_self_eq_zero CharTwo.add_self_eq_zeroₓ'. -/
@[simp]
theorem add_self_eq_zero (x : R) : x + x = 0 := by rw [← two_smul R x, two_eq_zero, zero_smul]
#align char_two.add_self_eq_zero CharTwo.add_self_eq_zero
-/- warning: char_two.bit0_eq_zero -> CharTwo.bit0_eq_zero is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align char_two.bit0_eq_zero CharTwo.bit0_eq_zeroₓ'. -/
@[simp]
theorem bit0_eq_zero : (bit0 : R → R) = 0 := by funext; exact add_self_eq_zero _
#align char_two.bit0_eq_zero CharTwo.bit0_eq_zero
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-Case conversion may be inaccurate. Consider using '#align char_two.bit0_apply_eq_zero CharTwo.bit0_apply_eq_zeroₓ'. -/
theorem bit0_apply_eq_zero (x : R) : (bit0 x : R) = 0 := by simp
#align char_two.bit0_apply_eq_zero CharTwo.bit0_apply_eq_zero
-/- warning: char_two.bit1_eq_one -> CharTwo.bit1_eq_one is a dubious translation:
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- forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))], Eq.{succ u1} (R -> R) (bit1.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) (OfNat.ofNat.{u1} (R -> R) 1 (OfNat.mk.{u1} (R -> R) 1 (One.one.{u1} (R -> R) (Pi.instOne.{u1, u1} R (fun (a : R) => R) (fun (i : R) => AddMonoidWithOne.toOne.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))))))
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-Case conversion may be inaccurate. Consider using '#align char_two.bit1_eq_one CharTwo.bit1_eq_oneₓ'. -/
@[simp]
theorem bit1_eq_one : (bit1 : R → R) = 1 := by funext; simp [bit1]
#align char_two.bit1_eq_one CharTwo.bit1_eq_one
-/- warning: char_two.bit1_apply_eq_one -> CharTwo.bit1_apply_eq_one is a dubious translation:
-lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (x : R), Eq.{succ u1} R (bit1.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) x) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))))
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-Case conversion may be inaccurate. Consider using '#align char_two.bit1_apply_eq_one CharTwo.bit1_apply_eq_oneₓ'. -/
theorem bit1_apply_eq_one (x : R) : (bit1 x : R) = 1 := by simp
#align char_two.bit1_apply_eq_one CharTwo.bit1_apply_eq_one
@@ -94,43 +58,19 @@ section Ring
variable [Ring R] [CharP R 2]
-/- warning: char_two.neg_eq -> CharTwo.neg_eq is a dubious translation:
-lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (x : R), Eq.{succ u1} R (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))) x) x
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- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (Ring.toAddGroupWithOne.{u1} R _inst_1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))] (x : R), Eq.{succ u1} R (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) x) x
-Case conversion may be inaccurate. Consider using '#align char_two.neg_eq CharTwo.neg_eqₓ'. -/
@[simp]
theorem neg_eq (x : R) : -x = x := by
rw [neg_eq_iff_add_eq_zero, ← two_smul R x, two_eq_zero, zero_smul]
#align char_two.neg_eq CharTwo.neg_eq
-/- warning: char_two.neg_eq' -> CharTwo.neg_eq' is a dubious translation:
-lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))], Eq.{succ u1} (R -> R) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1)))))) (id.{succ u1} R)
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-Case conversion may be inaccurate. Consider using '#align char_two.neg_eq' CharTwo.neg_eq'ₓ'. -/
theorem neg_eq' : Neg.neg = (id : R → R) :=
funext neg_eq
#align char_two.neg_eq' CharTwo.neg_eq'
-/- warning: char_two.sub_eq_add -> CharTwo.sub_eq_add is a dubious translation:
-lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (x : R) (y : R), Eq.{succ u1} R (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (SubNegMonoid.toHasSub.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1)))))) x y) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (Ring.toDistrib.{u1} R _inst_1))) x y)
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- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (Ring.toAddGroupWithOne.{u1} R _inst_1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))] (x : R) (y : R), Eq.{succ u1} R (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (Ring.toSub.{u1} R _inst_1)) x y) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) x y)
-Case conversion may be inaccurate. Consider using '#align char_two.sub_eq_add CharTwo.sub_eq_addₓ'. -/
@[simp]
theorem sub_eq_add (x y : R) : x - y = x + y := by rw [sub_eq_add_neg, neg_eq]
#align char_two.sub_eq_add CharTwo.sub_eq_add
-/- warning: char_two.sub_eq_add' -> CharTwo.sub_eq_add' is a dubious translation:
-lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))], Eq.{succ u1} (R -> R -> R) (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (SubNegMonoid.toHasSub.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (Ring.toDistrib.{u1} R _inst_1))))
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-Case conversion may be inaccurate. Consider using '#align char_two.sub_eq_add' CharTwo.sub_eq_add'ₓ'. -/
theorem sub_eq_add' : Sub.sub = ((· + ·) : R → R → R) :=
funext fun x => funext fun y => sub_eq_add x y
#align char_two.sub_eq_add' CharTwo.sub_eq_add'
@@ -141,44 +81,20 @@ section CommSemiring
variable [CommSemiring R] [CharP R 2]
-/- warning: char_two.add_sq -> CharTwo.add_sq is a dubious translation:
-lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (x : R) (y : R), Eq.{succ u1} R (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x y) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) x (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) y (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))))
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- forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))] (x : R) (y : R), Eq.{succ u1} R (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x y) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) x (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) y (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))))
-Case conversion may be inaccurate. Consider using '#align char_two.add_sq CharTwo.add_sqₓ'. -/
theorem add_sq (x y : R) : (x + y) ^ 2 = x ^ 2 + y ^ 2 :=
add_pow_char _ _ _
#align char_two.add_sq CharTwo.add_sq
-/- warning: char_two.add_mul_self -> CharTwo.add_mul_self is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align char_two.add_mul_self CharTwo.add_mul_selfₓ'. -/
theorem add_mul_self (x y : R) : (x + y) * (x + y) = x * x + y * y := by
rw [← pow_two, ← pow_two, ← pow_two, add_sq]
#align char_two.add_mul_self CharTwo.add_mul_self
open BigOperators
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-Case conversion may be inaccurate. Consider using '#align char_two.list_sum_sq CharTwo.list_sum_sqₓ'. -/
theorem list_sum_sq (l : List R) : l.Sum ^ 2 = (l.map (· ^ 2)).Sum :=
list_sum_pow_char _ _
#align char_two.list_sum_sq CharTwo.list_sum_sq
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-Case conversion may be inaccurate. Consider using '#align char_two.list_sum_mul_self CharTwo.list_sum_mul_selfₓ'. -/
theorem list_sum_mul_self (l : List R) : l.Sum * l.Sum = (List.map (fun x => x * x) l).Sum := by
simp_rw [← pow_two, list_sum_sq]
#align char_two.list_sum_mul_self CharTwo.list_sum_mul_self
@@ -189,12 +105,6 @@ theorem multiset_sum_sq (l : Multiset R) : l.Sum ^ 2 = (l.map (· ^ 2)).Sum :=
#align char_two.multiset_sum_sq CharTwo.multiset_sum_sq
-/
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-Case conversion may be inaccurate. Consider using '#align char_two.multiset_sum_mul_self CharTwo.multiset_sum_mul_selfₓ'. -/
theorem multiset_sum_mul_self (l : Multiset R) :
l.Sum * l.Sum = (Multiset.map (fun x => x * x) l).Sum := by simp_rw [← pow_two, multiset_sum_sq]
#align char_two.multiset_sum_mul_self CharTwo.multiset_sum_mul_self
@@ -205,12 +115,6 @@ theorem sum_sq (s : Finset ι) (f : ι → R) : (∑ i in s, f i) ^ 2 = ∑ i in
#align char_two.sum_sq CharTwo.sum_sq
-/
-/- warning: char_two.sum_mul_self -> CharTwo.sum_mul_self is a dubious translation:
-lean 3 declaration is
- forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (s : Finset.{u2} ι) (f : ι -> R), Eq.{succ u1} R (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) s (fun (i : ι) => f i)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) s (fun (i : ι) => f i))) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) s (fun (i : ι) => HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (f i) (f i)))
-but is expected to have type
- forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))] (s : Finset.{u2} ι) (f : ι -> R), Eq.{succ u1} R (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) s (fun (i : ι) => f i)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) s (fun (i : ι) => f i))) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) s (fun (i : ι) => HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (f i) (f i)))
-Case conversion may be inaccurate. Consider using '#align char_two.sum_mul_self CharTwo.sum_mul_selfₓ'. -/
theorem sum_mul_self (s : Finset ι) (f : ι → R) :
((∑ i in s, f i) * ∑ i in s, f i) = ∑ i in s, f i * f i := by simp_rw [← pow_two, sum_sq]
#align char_two.sum_mul_self CharTwo.sum_mul_self
@@ -223,12 +127,6 @@ section ringChar
variable [Ring R]
-/- warning: neg_one_eq_one_iff -> neg_one_eq_one_iff is a dubious translation:
-lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Iff (Eq.{succ u1} R (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1)))))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1)))))))) (Eq.{1} Nat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)))))
-but is expected to have type
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Iff (Eq.{succ u1} R (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (Eq.{1} Nat (ringChar.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))
-Case conversion may be inaccurate. Consider using '#align neg_one_eq_one_iff neg_one_eq_one_iffₓ'. -/
theorem neg_one_eq_one_iff [Nontrivial R] : (-1 : R) = 1 ↔ ringChar R = 2 :=
by
refine' ⟨fun h => _, fun h => @CharTwo.neg_eq _ (ringChar.of_eq h) 1⟩
@@ -236,12 +134,6 @@ theorem neg_one_eq_one_iff [Nontrivial R] : (-1 : R) = 1 ↔ ringChar R = 2 :=
exact ((Nat.dvd_prime Nat.prime_two).mp (ringChar.dvd h)).resolve_left CharP.ringChar_ne_one
#align neg_one_eq_one_iff neg_one_eq_one_iff
-/- warning: order_of_neg_one -> orderOf_neg_one is a dubious translation:
-lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Eq.{1} Nat (orderOf.{u1} R (Ring.toMonoid.{u1} R _inst_1) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))))))) (ite.{1} Nat (Eq.{1} Nat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (Nat.decidableEq (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)))))
-but is expected to have type
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Eq.{1} Nat (orderOf.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R (Ring.toSemiring.{u1} R _inst_1)))))) (ite.{1} Nat (Eq.{1} Nat (ringChar.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqNat (ringChar.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))
-Case conversion may be inaccurate. Consider using '#align order_of_neg_one orderOf_neg_oneₓ'. -/
@[simp]
theorem orderOf_neg_one [Nontrivial R] : orderOf (-1 : R) = if ringChar R = 2 then 1 else 2 :=
by
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -57,9 +57,7 @@ but is expected to have type
forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))], Eq.{succ u1} (R -> R) (bit0.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) (OfNat.ofNat.{u1} (R -> R) 0 (Zero.toOfNat0.{u1} (R -> R) (Pi.instZero.{u1, u1} R (fun (a : R) => R) (fun (i : R) => MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))))
Case conversion may be inaccurate. Consider using '#align char_two.bit0_eq_zero CharTwo.bit0_eq_zeroₓ'. -/
@[simp]
-theorem bit0_eq_zero : (bit0 : R → R) = 0 := by
- funext
- exact add_self_eq_zero _
+theorem bit0_eq_zero : (bit0 : R → R) = 0 := by funext; exact add_self_eq_zero _
#align char_two.bit0_eq_zero CharTwo.bit0_eq_zero
/- warning: char_two.bit0_apply_eq_zero -> CharTwo.bit0_apply_eq_zero is a dubious translation:
@@ -78,9 +76,7 @@ but is expected to have type
forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))], Eq.{succ u1} (R -> R) (bit1.{u1} R (Semiring.toOne.{u1} R _inst_1) (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) (OfNat.ofNat.{u1} (R -> R) 1 (One.toOfNat1.{u1} (R -> R) (Pi.instOne.{u1, u1} R (fun (a : R) => R) (fun (i : R) => Semiring.toOne.{u1} R _inst_1))))
Case conversion may be inaccurate. Consider using '#align char_two.bit1_eq_one CharTwo.bit1_eq_oneₓ'. -/
@[simp]
-theorem bit1_eq_one : (bit1 : R → R) = 1 := by
- funext
- simp [bit1]
+theorem bit1_eq_one : (bit1 : R → R) = 1 := by funext; simp [bit1]
#align char_two.bit1_eq_one CharTwo.bit1_eq_one
/- warning: char_two.bit1_apply_eq_one -> CharTwo.bit1_apply_eq_one is a dubious translation:
mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8
@@ -133,7 +133,7 @@ theorem sub_eq_add (x y : R) : x - y = x + y := by rw [sub_eq_add_neg, neg_eq]
lean 3 declaration is
forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))], Eq.{succ u1} (R -> R -> R) (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (SubNegMonoid.toHasSub.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (Ring.toDistrib.{u1} R _inst_1))))
but is expected to have type
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (Ring.toAddGroupWithOne.{u1} R _inst_1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))], Eq.{succ u1} (R -> R -> R) (Sub.sub.{u1} R (Ring.toSub.{u1} R _inst_1)) (fun (x._@.Mathlib.Algebra.CharP.Two._hyg.483 : R) (x._@.Mathlib.Algebra.CharP.Two._hyg.485 : R) => HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) x._@.Mathlib.Algebra.CharP.Two._hyg.483 x._@.Mathlib.Algebra.CharP.Two._hyg.485)
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (Ring.toAddGroupWithOne.{u1} R _inst_1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))], Eq.{succ u1} (R -> R -> R) (Sub.sub.{u1} R (Ring.toSub.{u1} R _inst_1)) (fun (x._@.Mathlib.Algebra.CharP.Two._hyg.477 : R) (x._@.Mathlib.Algebra.CharP.Two._hyg.479 : R) => HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) x._@.Mathlib.Algebra.CharP.Two._hyg.477 x._@.Mathlib.Algebra.CharP.Two._hyg.479)
Case conversion may be inaccurate. Consider using '#align char_two.sub_eq_add' CharTwo.sub_eq_add'ₓ'. -/
theorem sub_eq_add' : Sub.sub = ((· + ·) : R → R → R) :=
funext fun x => funext fun y => sub_eq_add x y
mathlib commit https://github.com/leanprover-community/mathlib/commit/08e1d8d4d989df3a6df86f385e9053ec8a372cc1
@@ -231,7 +231,7 @@ variable [Ring R]
lean 3 declaration is
forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Iff (Eq.{succ u1} R (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1)))))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1)))))))) (Eq.{1} Nat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)))))
but is expected to have type
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Iff (Eq.{succ u1} R (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Eq.{1} Nat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Iff (Eq.{succ u1} R (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (Eq.{1} Nat (ringChar.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))
Case conversion may be inaccurate. Consider using '#align neg_one_eq_one_iff neg_one_eq_one_iffₓ'. -/
theorem neg_one_eq_one_iff [Nontrivial R] : (-1 : R) = 1 ↔ ringChar R = 2 :=
by
@@ -244,7 +244,7 @@ theorem neg_one_eq_one_iff [Nontrivial R] : (-1 : R) = 1 ↔ ringChar R = 2 :=
lean 3 declaration is
forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Eq.{1} Nat (orderOf.{u1} R (Ring.toMonoid.{u1} R _inst_1) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))))))) (ite.{1} Nat (Eq.{1} Nat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (Nat.decidableEq (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)))))
but is expected to have type
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Eq.{1} Nat (orderOf.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) (ite.{1} Nat (Eq.{1} Nat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqNat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Eq.{1} Nat (orderOf.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R (Ring.toSemiring.{u1} R _inst_1)))))) (ite.{1} Nat (Eq.{1} Nat (ringChar.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqNat (ringChar.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))
Case conversion may be inaccurate. Consider using '#align order_of_neg_one orderOf_neg_oneₓ'. -/
@[simp]
theorem orderOf_neg_one [Nontrivial R] : orderOf (-1 : R) = if ringChar R = 2 then 1 else 2 :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/9b2b58d6b14b895b2f375108e765cb47de71aebd
@@ -133,7 +133,7 @@ theorem sub_eq_add (x y : R) : x - y = x + y := by rw [sub_eq_add_neg, neg_eq]
lean 3 declaration is
forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))], Eq.{succ u1} (R -> R -> R) (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (SubNegMonoid.toHasSub.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (Ring.toDistrib.{u1} R _inst_1))))
but is expected to have type
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (Ring.toAddGroupWithOne.{u1} R _inst_1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))], Eq.{succ u1} (R -> R -> R) (Sub.sub.{u1} R (Ring.toSub.{u1} R _inst_1)) (fun (x._@.Mathlib.Algebra.CharP.Two._hyg.477 : R) (x._@.Mathlib.Algebra.CharP.Two._hyg.479 : R) => HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) x._@.Mathlib.Algebra.CharP.Two._hyg.477 x._@.Mathlib.Algebra.CharP.Two._hyg.479)
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (Ring.toAddGroupWithOne.{u1} R _inst_1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))], Eq.{succ u1} (R -> R -> R) (Sub.sub.{u1} R (Ring.toSub.{u1} R _inst_1)) (fun (x._@.Mathlib.Algebra.CharP.Two._hyg.483 : R) (x._@.Mathlib.Algebra.CharP.Two._hyg.485 : R) => HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) x._@.Mathlib.Algebra.CharP.Two._hyg.483 x._@.Mathlib.Algebra.CharP.Two._hyg.485)
Case conversion may be inaccurate. Consider using '#align char_two.sub_eq_add' CharTwo.sub_eq_add'ₓ'. -/
theorem sub_eq_add' : Sub.sub = ((· + ·) : R → R → R) :=
funext fun x => funext fun y => sub_eq_add x y
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce86f4e05e9a9b8da5e316b22c76ce76440c56a1
@@ -100,7 +100,7 @@ variable [Ring R] [CharP R 2]
/- warning: char_two.neg_eq -> CharTwo.neg_eq is a dubious translation:
lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (x : R), Eq.{succ u1} R (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) x) x
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (x : R), Eq.{succ u1} R (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))) x) x
but is expected to have type
forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (Ring.toAddGroupWithOne.{u1} R _inst_1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))] (x : R), Eq.{succ u1} R (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) x) x
Case conversion may be inaccurate. Consider using '#align char_two.neg_eq CharTwo.neg_eqₓ'. -/
@@ -111,7 +111,7 @@ theorem neg_eq (x : R) : -x = x := by
/- warning: char_two.neg_eq' -> CharTwo.neg_eq' is a dubious translation:
lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))], Eq.{succ u1} (R -> R) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) (id.{succ u1} R)
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))], Eq.{succ u1} (R -> R) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1)))))) (id.{succ u1} R)
but is expected to have type
forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (Ring.toAddGroupWithOne.{u1} R _inst_1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))], Eq.{succ u1} (R -> R) (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1)) (id.{succ u1} R)
Case conversion may be inaccurate. Consider using '#align char_two.neg_eq' CharTwo.neg_eq'ₓ'. -/
@@ -121,7 +121,7 @@ theorem neg_eq' : Neg.neg = (id : R → R) :=
/- warning: char_two.sub_eq_add -> CharTwo.sub_eq_add is a dubious translation:
lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (x : R) (y : R), Eq.{succ u1} R (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (SubNegMonoid.toHasSub.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) x y) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (Ring.toDistrib.{u1} R _inst_1))) x y)
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (x : R) (y : R), Eq.{succ u1} R (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (SubNegMonoid.toHasSub.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1)))))) x y) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (Ring.toDistrib.{u1} R _inst_1))) x y)
but is expected to have type
forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (Ring.toAddGroupWithOne.{u1} R _inst_1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))] (x : R) (y : R), Eq.{succ u1} R (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (Ring.toSub.{u1} R _inst_1)) x y) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) x y)
Case conversion may be inaccurate. Consider using '#align char_two.sub_eq_add CharTwo.sub_eq_addₓ'. -/
@@ -131,7 +131,7 @@ theorem sub_eq_add (x y : R) : x - y = x + y := by rw [sub_eq_add_neg, neg_eq]
/- warning: char_two.sub_eq_add' -> CharTwo.sub_eq_add' is a dubious translation:
lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))], Eq.{succ u1} (R -> R -> R) (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (SubNegMonoid.toHasSub.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (Ring.toDistrib.{u1} R _inst_1))))
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))], Eq.{succ u1} (R -> R -> R) (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (SubNegMonoid.toHasSub.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (Ring.toDistrib.{u1} R _inst_1))))
but is expected to have type
forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (Ring.toAddGroupWithOne.{u1} R _inst_1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))], Eq.{succ u1} (R -> R -> R) (Sub.sub.{u1} R (Ring.toSub.{u1} R _inst_1)) (fun (x._@.Mathlib.Algebra.CharP.Two._hyg.477 : R) (x._@.Mathlib.Algebra.CharP.Two._hyg.479 : R) => HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) x._@.Mathlib.Algebra.CharP.Two._hyg.477 x._@.Mathlib.Algebra.CharP.Two._hyg.479)
Case conversion may be inaccurate. Consider using '#align char_two.sub_eq_add' CharTwo.sub_eq_add'ₓ'. -/
@@ -229,7 +229,7 @@ variable [Ring R]
/- warning: neg_one_eq_one_iff -> neg_one_eq_one_iff is a dubious translation:
lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Iff (Eq.{succ u1} R (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))))) (Eq.{1} Nat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)))))
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Iff (Eq.{succ u1} R (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1)))))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1)))))))) (Eq.{1} Nat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)))))
but is expected to have type
forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Iff (Eq.{succ u1} R (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Eq.{1} Nat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))
Case conversion may be inaccurate. Consider using '#align neg_one_eq_one_iff neg_one_eq_one_iffₓ'. -/
@@ -242,7 +242,7 @@ theorem neg_one_eq_one_iff [Nontrivial R] : (-1 : R) = 1 ↔ ringChar R = 2 :=
/- warning: order_of_neg_one -> orderOf_neg_one is a dubious translation:
lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Eq.{1} Nat (orderOf.{u1} R (Ring.toMonoid.{u1} R _inst_1) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))))) (ite.{1} Nat (Eq.{1} Nat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (Nat.decidableEq (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)))))
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Eq.{1} Nat (orderOf.{u1} R (Ring.toMonoid.{u1} R _inst_1) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))))))) (ite.{1} Nat (Eq.{1} Nat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (Nat.decidableEq (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)))))
but is expected to have type
forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Eq.{1} Nat (orderOf.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) (ite.{1} Nat (Eq.{1} Nat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqNat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))
Case conversion may be inaccurate. Consider using '#align order_of_neg_one orderOf_neg_oneₓ'. -/
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
! This file was ported from Lean 3 source module algebra.char_p.two
-! leanprover-community/mathlib commit 7f1ba1a333d66eed531ecb4092493cd1b6715450
+! leanprover-community/mathlib commit 10bf4f825ad729c5653adc039dafa3622e7f93c9
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -13,6 +13,9 @@ import Mathbin.Algebra.CharP.Basic
/-!
# Lemmas about rings of characteristic two
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
This file contains results about `char_p R 2`, in the `char_two` namespace.
The lemmas in this file with a `_sq` suffix are just special cases of the `_pow_char` lemmas
mathlib commit https://github.com/leanprover-community/mathlib/commit/2196ab363eb097c008d4497125e0dde23fb36db2
@@ -28,28 +28,64 @@ section Semiring
variable [Semiring R] [CharP R 2]
+/- warning: char_two.two_eq_zero -> CharTwo.two_eq_zero is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))], Eq.{succ u1} R (OfNat.ofNat.{u1} R 2 (OfNat.mk.{u1} R 2 (bit0.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))))) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))))
+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))], Eq.{succ u1} R (OfNat.ofNat.{u1} R 2 (instOfNat.{u1} R 2 (Semiring.toNatCast.{u1} R _inst_1) (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))
+Case conversion may be inaccurate. Consider using '#align char_two.two_eq_zero CharTwo.two_eq_zeroₓ'. -/
theorem two_eq_zero : (2 : R) = 0 := by rw [← Nat.cast_two, CharP.cast_eq_zero]
#align char_two.two_eq_zero CharTwo.two_eq_zero
+/- warning: char_two.add_self_eq_zero -> CharTwo.add_self_eq_zero is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (x : R), Eq.{succ u1} R (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) x x) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))))
+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))] (x : R), Eq.{succ u1} R (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) x x) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))
+Case conversion may be inaccurate. Consider using '#align char_two.add_self_eq_zero CharTwo.add_self_eq_zeroₓ'. -/
@[simp]
theorem add_self_eq_zero (x : R) : x + x = 0 := by rw [← two_smul R x, two_eq_zero, zero_smul]
#align char_two.add_self_eq_zero CharTwo.add_self_eq_zero
+/- warning: char_two.bit0_eq_zero -> CharTwo.bit0_eq_zero is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))], Eq.{succ u1} (R -> R) (bit0.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) (OfNat.ofNat.{u1} (R -> R) 0 (OfNat.mk.{u1} (R -> R) 0 (Zero.zero.{u1} (R -> R) (Pi.instZero.{u1, u1} R (fun (a : R) => R) (fun (i : R) => MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))))))
+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))], Eq.{succ u1} (R -> R) (bit0.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) (OfNat.ofNat.{u1} (R -> R) 0 (Zero.toOfNat0.{u1} (R -> R) (Pi.instZero.{u1, u1} R (fun (a : R) => R) (fun (i : R) => MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))))
+Case conversion may be inaccurate. Consider using '#align char_two.bit0_eq_zero CharTwo.bit0_eq_zeroₓ'. -/
@[simp]
theorem bit0_eq_zero : (bit0 : R → R) = 0 := by
funext
exact add_self_eq_zero _
#align char_two.bit0_eq_zero CharTwo.bit0_eq_zero
+/- warning: char_two.bit0_apply_eq_zero -> CharTwo.bit0_apply_eq_zero is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (x : R), Eq.{succ u1} R (bit0.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) x) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))))
+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))] (x : R), Eq.{succ u1} R (bit0.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) x) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))
+Case conversion may be inaccurate. Consider using '#align char_two.bit0_apply_eq_zero CharTwo.bit0_apply_eq_zeroₓ'. -/
theorem bit0_apply_eq_zero (x : R) : (bit0 x : R) = 0 := by simp
#align char_two.bit0_apply_eq_zero CharTwo.bit0_apply_eq_zero
+/- warning: char_two.bit1_eq_one -> CharTwo.bit1_eq_one is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))], Eq.{succ u1} (R -> R) (bit1.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) (OfNat.ofNat.{u1} (R -> R) 1 (OfNat.mk.{u1} (R -> R) 1 (One.one.{u1} (R -> R) (Pi.instOne.{u1, u1} R (fun (a : R) => R) (fun (i : R) => AddMonoidWithOne.toOne.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))))))
+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))], Eq.{succ u1} (R -> R) (bit1.{u1} R (Semiring.toOne.{u1} R _inst_1) (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) (OfNat.ofNat.{u1} (R -> R) 1 (One.toOfNat1.{u1} (R -> R) (Pi.instOne.{u1, u1} R (fun (a : R) => R) (fun (i : R) => Semiring.toOne.{u1} R _inst_1))))
+Case conversion may be inaccurate. Consider using '#align char_two.bit1_eq_one CharTwo.bit1_eq_oneₓ'. -/
@[simp]
theorem bit1_eq_one : (bit1 : R → R) = 1 := by
funext
simp [bit1]
#align char_two.bit1_eq_one CharTwo.bit1_eq_one
+/- warning: char_two.bit1_apply_eq_one -> CharTwo.bit1_apply_eq_one is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (x : R), Eq.{succ u1} R (bit1.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) x) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))))
+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))] (x : R), Eq.{succ u1} R (bit1.{u1} R (Semiring.toOne.{u1} R _inst_1) (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) x) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R _inst_1)))
+Case conversion may be inaccurate. Consider using '#align char_two.bit1_apply_eq_one CharTwo.bit1_apply_eq_oneₓ'. -/
theorem bit1_apply_eq_one (x : R) : (bit1 x : R) = 1 := by simp
#align char_two.bit1_apply_eq_one CharTwo.bit1_apply_eq_one
@@ -59,19 +95,43 @@ section Ring
variable [Ring R] [CharP R 2]
+/- warning: char_two.neg_eq -> CharTwo.neg_eq is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (x : R), Eq.{succ u1} R (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) x) x
+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (Ring.toAddGroupWithOne.{u1} R _inst_1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))] (x : R), Eq.{succ u1} R (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) x) x
+Case conversion may be inaccurate. Consider using '#align char_two.neg_eq CharTwo.neg_eqₓ'. -/
@[simp]
theorem neg_eq (x : R) : -x = x := by
rw [neg_eq_iff_add_eq_zero, ← two_smul R x, two_eq_zero, zero_smul]
#align char_two.neg_eq CharTwo.neg_eq
+/- warning: char_two.neg_eq' -> CharTwo.neg_eq' is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))], Eq.{succ u1} (R -> R) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) (id.{succ u1} R)
+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (Ring.toAddGroupWithOne.{u1} R _inst_1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))], Eq.{succ u1} (R -> R) (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1)) (id.{succ u1} R)
+Case conversion may be inaccurate. Consider using '#align char_two.neg_eq' CharTwo.neg_eq'ₓ'. -/
theorem neg_eq' : Neg.neg = (id : R → R) :=
funext neg_eq
#align char_two.neg_eq' CharTwo.neg_eq'
+/- warning: char_two.sub_eq_add -> CharTwo.sub_eq_add is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (x : R) (y : R), Eq.{succ u1} R (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (SubNegMonoid.toHasSub.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) x y) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (Ring.toDistrib.{u1} R _inst_1))) x y)
+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (Ring.toAddGroupWithOne.{u1} R _inst_1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))] (x : R) (y : R), Eq.{succ u1} R (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (Ring.toSub.{u1} R _inst_1)) x y) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) x y)
+Case conversion may be inaccurate. Consider using '#align char_two.sub_eq_add CharTwo.sub_eq_addₓ'. -/
@[simp]
theorem sub_eq_add (x y : R) : x - y = x + y := by rw [sub_eq_add_neg, neg_eq]
#align char_two.sub_eq_add CharTwo.sub_eq_add
+/- warning: char_two.sub_eq_add' -> CharTwo.sub_eq_add' is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))], Eq.{succ u1} (R -> R -> R) (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (SubNegMonoid.toHasSub.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (Ring.toDistrib.{u1} R _inst_1))))
+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : CharP.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (Ring.toAddGroupWithOne.{u1} R _inst_1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))], Eq.{succ u1} (R -> R -> R) (Sub.sub.{u1} R (Ring.toSub.{u1} R _inst_1)) (fun (x._@.Mathlib.Algebra.CharP.Two._hyg.477 : R) (x._@.Mathlib.Algebra.CharP.Two._hyg.479 : R) => HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) x._@.Mathlib.Algebra.CharP.Two._hyg.477 x._@.Mathlib.Algebra.CharP.Two._hyg.479)
+Case conversion may be inaccurate. Consider using '#align char_two.sub_eq_add' CharTwo.sub_eq_add'ₓ'. -/
theorem sub_eq_add' : Sub.sub = ((· + ·) : R → R → R) :=
funext fun x => funext fun y => sub_eq_add x y
#align char_two.sub_eq_add' CharTwo.sub_eq_add'
@@ -82,36 +142,76 @@ section CommSemiring
variable [CommSemiring R] [CharP R 2]
+/- warning: char_two.add_sq -> CharTwo.add_sq is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (x : R) (y : R), Eq.{succ u1} R (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x y) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) x (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) y (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))))
+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))] (x : R) (y : R), Eq.{succ u1} R (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x y) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) x (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) y (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))))
+Case conversion may be inaccurate. Consider using '#align char_two.add_sq CharTwo.add_sqₓ'. -/
theorem add_sq (x y : R) : (x + y) ^ 2 = x ^ 2 + y ^ 2 :=
add_pow_char _ _ _
#align char_two.add_sq CharTwo.add_sq
+/- warning: char_two.add_mul_self -> CharTwo.add_mul_self is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (x : R) (y : R), Eq.{succ u1} R (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x y) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x y)) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x x) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) y y))
+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))] (x : R) (y : R), Eq.{succ u1} R (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x y) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x y)) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) x x) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) y y))
+Case conversion may be inaccurate. Consider using '#align char_two.add_mul_self CharTwo.add_mul_selfₓ'. -/
theorem add_mul_self (x y : R) : (x + y) * (x + y) = x * x + y * y := by
rw [← pow_two, ← pow_two, ← pow_two, add_sq]
#align char_two.add_mul_self CharTwo.add_mul_self
open BigOperators
+/- warning: char_two.list_sum_sq -> CharTwo.list_sum_sq is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (l : List.{u1} R), Eq.{succ u1} R (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (List.sum.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) l) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (List.sum.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (List.map.{u1, u1} R R (fun (_x : R) => HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) _x (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) l))
+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))] (l : List.{u1} R), Eq.{succ u1} R (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (List.sum.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1)) l) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (List.sum.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1)) (List.map.{u1, u1} R R (fun (_x : R) => HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) _x (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) l))
+Case conversion may be inaccurate. Consider using '#align char_two.list_sum_sq CharTwo.list_sum_sqₓ'. -/
theorem list_sum_sq (l : List R) : l.Sum ^ 2 = (l.map (· ^ 2)).Sum :=
list_sum_pow_char _ _
#align char_two.list_sum_sq CharTwo.list_sum_sq
+/- warning: char_two.list_sum_mul_self -> CharTwo.list_sum_mul_self is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (l : List.{u1} R), Eq.{succ u1} R (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (List.sum.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) l) (List.sum.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) l)) (List.sum.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (List.map.{u1, u1} R R (fun (x : R) => HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x x) l))
+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))] (l : List.{u1} R), Eq.{succ u1} R (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (List.sum.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1)) l) (List.sum.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1)) l)) (List.sum.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1)) (List.map.{u1, u1} R R (fun (x : R) => HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) x x) l))
+Case conversion may be inaccurate. Consider using '#align char_two.list_sum_mul_self CharTwo.list_sum_mul_selfₓ'. -/
theorem list_sum_mul_self (l : List R) : l.Sum * l.Sum = (List.map (fun x => x * x) l).Sum := by
simp_rw [← pow_two, list_sum_sq]
#align char_two.list_sum_mul_self CharTwo.list_sum_mul_self
+#print CharTwo.multiset_sum_sq /-
theorem multiset_sum_sq (l : Multiset R) : l.Sum ^ 2 = (l.map (· ^ 2)).Sum :=
multiset_sum_pow_char _ _
#align char_two.multiset_sum_sq CharTwo.multiset_sum_sq
+-/
+/- warning: char_two.multiset_sum_mul_self -> CharTwo.multiset_sum_mul_self is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (l : Multiset.{u1} R), Eq.{succ u1} R (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) l) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) l)) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Multiset.map.{u1, u1} R R (fun (x : R) => HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x x) l))
+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))] (l : Multiset.{u1} R), Eq.{succ u1} R (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) l) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) l)) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Multiset.map.{u1, u1} R R (fun (x : R) => HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) x x) l))
+Case conversion may be inaccurate. Consider using '#align char_two.multiset_sum_mul_self CharTwo.multiset_sum_mul_selfₓ'. -/
theorem multiset_sum_mul_self (l : Multiset R) :
l.Sum * l.Sum = (Multiset.map (fun x => x * x) l).Sum := by simp_rw [← pow_two, multiset_sum_sq]
#align char_two.multiset_sum_mul_self CharTwo.multiset_sum_mul_self
+#print CharTwo.sum_sq /-
theorem sum_sq (s : Finset ι) (f : ι → R) : (∑ i in s, f i) ^ 2 = ∑ i in s, f i ^ 2 :=
sum_pow_char _ _ _
#align char_two.sum_sq CharTwo.sum_sq
+-/
+/- warning: char_two.sum_mul_self -> CharTwo.sum_mul_self is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))] (s : Finset.{u2} ι) (f : ι -> R), Eq.{succ u1} R (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) s (fun (i : ι) => f i)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) s (fun (i : ι) => f i))) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) s (fun (i : ι) => HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (f i) (f i)))
+but is expected to have type
+ forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CharP.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))] (s : Finset.{u2} ι) (f : ι -> R), Eq.{succ u1} R (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) s (fun (i : ι) => f i)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) s (fun (i : ι) => f i))) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) s (fun (i : ι) => HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (f i) (f i)))
+Case conversion may be inaccurate. Consider using '#align char_two.sum_mul_self CharTwo.sum_mul_selfₓ'. -/
theorem sum_mul_self (s : Finset ι) (f : ι → R) :
((∑ i in s, f i) * ∑ i in s, f i) = ∑ i in s, f i * f i := by simp_rw [← pow_two, sum_sq]
#align char_two.sum_mul_self CharTwo.sum_mul_self
@@ -124,6 +224,12 @@ section ringChar
variable [Ring R]
+/- warning: neg_one_eq_one_iff -> neg_one_eq_one_iff is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Iff (Eq.{succ u1} R (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))))) (Eq.{1} Nat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)))))
+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Iff (Eq.{succ u1} R (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Eq.{1} Nat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))
+Case conversion may be inaccurate. Consider using '#align neg_one_eq_one_iff neg_one_eq_one_iffₓ'. -/
theorem neg_one_eq_one_iff [Nontrivial R] : (-1 : R) = 1 ↔ ringChar R = 2 :=
by
refine' ⟨fun h => _, fun h => @CharTwo.neg_eq _ (ringChar.of_eq h) 1⟩
@@ -131,6 +237,12 @@ theorem neg_one_eq_one_iff [Nontrivial R] : (-1 : R) = 1 ↔ ringChar R = 2 :=
exact ((Nat.dvd_prime Nat.prime_two).mp (ringChar.dvd h)).resolve_left CharP.ringChar_ne_one
#align neg_one_eq_one_iff neg_one_eq_one_iff
+/- warning: order_of_neg_one -> orderOf_neg_one is a dubious translation:
+lean 3 declaration is
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Eq.{1} Nat (orderOf.{u1} R (Ring.toMonoid.{u1} R _inst_1) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))))) (ite.{1} Nat (Eq.{1} Nat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (Nat.decidableEq (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)))))
+but is expected to have type
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_2 : Nontrivial.{u1} R], Eq.{1} Nat (orderOf.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) (ite.{1} Nat (Eq.{1} Nat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqNat (ringChar.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))
+Case conversion may be inaccurate. Consider using '#align order_of_neg_one orderOf_neg_oneₓ'. -/
@[simp]
theorem orderOf_neg_one [Nontrivial R] : orderOf (-1 : R) = if ringChar R = 2 then 1 else 2 :=
by
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
CharP+Prime
to ExpChar
in frobenius
(#10016)
Consequently, the part about frobenius
in Algebra/CharP/Basic is moved to CharP/ExpChar, and imports are adjusted as necessary.
Add instances from CharP+Fact(Nat.Prime)
and CharZero
to ExpChar
, to allow lemmas generalized to ExpChar still apply to CharP.
Remove lemmas in Algebra/CharP/ExpChar from [#9799](https://github.com/leanprover-community/mathlib4/commit/1e74fcfff8d5ffe5a3a9881864cf10fa39f619e6) because they coincide with the generalized lemmas, and golf the other lemmas (in Algebra/CharP/Basic).
Define the RingHom iterateFrobenius
and the semilinear map LinearMap.(iterate)Frobenius
for an algebra. When the characteristic is zero (ExpChar is 1), these are all equal to the identity map (· ^ 1). Also define iterateFrobeniusEquiv
for perfect rings.
Fix and/or generalize other files.
Co-authored-by: Junyan Xu <junyanxu.math@gmail.com> Co-authored-by: acmepjz <acme_pjz@hotmail.com>
@@ -3,7 +3,7 @@ Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
-import Mathlib.Algebra.CharP.Basic
+import Mathlib.Algebra.CharP.ExpChar
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
Some of these are likely porting errors. Statements should always be about the heterogenous versions because these are the ones with notation.
For places where we are abusing defeq, this debuts the trick of using (by exact a : B) = (by exact a1) + (by exact b2)
to ensure the =
and +
are typed as B
instead of A
.
@@ -73,7 +73,7 @@ theorem neg_eq' : Neg.neg = (id : R → R) :=
theorem sub_eq_add (x y : R) : x - y = x + y := by rw [sub_eq_add_neg, neg_eq]
#align char_two.sub_eq_add CharTwo.sub_eq_add
-theorem sub_eq_add' : Sub.sub = ((· + ·) : R → R → R) :=
+theorem sub_eq_add' : HSub.hSub = ((· + ·) : R → R → R) :=
funext fun x => funext fun y => sub_eq_add x y
#align char_two.sub_eq_add' CharTwo.sub_eq_add'
Type _
and Sort _
(#6499)
We remove all possible occurences of Type _
and Sort _
in favor of Type*
and Sort*
.
This has nice performance benefits.
@@ -17,7 +17,7 @@ elsewhere, with a shorter name for ease of discovery, and no need for a `[Fact (
-/
-variable {R ι : Type _}
+variable {R ι : Type*}
namespace CharTwo
@@ -2,14 +2,11 @@
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-
-! This file was ported from Lean 3 source module algebra.char_p.two
-! leanprover-community/mathlib commit 7f1ba1a333d66eed531ecb4092493cd1b6715450
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.Algebra.CharP.Basic
+#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
+
/-!
# Lemmas about rings of characteristic two
fix-comments.py
on all files.@@ -16,7 +16,7 @@ import Mathlib.Algebra.CharP.Basic
This file contains results about `CharP R 2`, in the `CharTwo` namespace.
The lemmas in this file with a `_sq` suffix are just special cases of the `_pow_char` lemmas
-elsewhere, with a shorter name for ease of discovery, and no need for a `[Fact (prime 2)]` argument.
+elsewhere, with a shorter name for ease of discovery, and no need for a `[Fact (Prime 2)]` argument.
-/
Now that leanprover/lean4#2210 has been merged, this PR:
set_option synthInstance.etaExperiment true
commands (and some etaExperiment%
term elaborators)set_option maxHeartbeats
commandsCo-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Matthew Ballard <matt@mrb.email>
@@ -63,7 +63,6 @@ section Ring
variable [Ring R] [CharP R 2]
-set_option synthInstance.etaExperiment true in
@[simp]
theorem neg_eq (x : R) : -x = x := by
rw [neg_eq_iff_add_eq_zero, ← two_smul R x, two_eq_zero, zero_smul]
This is to fix timeouts in https://github.com/leanprover-community/mathlib4/pull/3552.
See discussion at https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/!4.233552.20.28LinearAlgebra.2EMatrix.2EToLin.29.
Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
@@ -63,6 +63,7 @@ section Ring
variable [Ring R] [CharP R 2]
+set_option synthInstance.etaExperiment true in
@[simp]
theorem neg_eq (x : R) : -x = x := by
rw [neg_eq_iff_add_eq_zero, ← two_smul R x, two_eq_zero, zero_smul]
The unported dependencies are