analysis.convex.complex
⟷
Mathlib.Analysis.Convex.Complex
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/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,8 +3,8 @@ Copyright (c) 2019 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Yaël Dillies
-/
-import Mathbin.Analysis.Convex.Basic
-import Mathbin.Data.Complex.Module
+import Analysis.Convex.Basic
+import Data.Complex.Module
#align_import analysis.convex.complex from "leanprover-community/mathlib"@"fe8d0ff42c3c24d789f491dc2622b6cac3d61564"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,15 +2,12 @@
Copyright (c) 2019 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Yaël Dillies
-
-! This file was ported from Lean 3 source module analysis.convex.complex
-! leanprover-community/mathlib commit fe8d0ff42c3c24d789f491dc2622b6cac3d61564
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.Analysis.Convex.Basic
import Mathbin.Data.Complex.Module
+#align_import analysis.convex.complex from "leanprover-community/mathlib"@"fe8d0ff42c3c24d789f491dc2622b6cac3d61564"
+
/-!
# Convexity of half spaces in ℂ
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -22,35 +22,51 @@ are all convex over ℝ.
-/
+#print convex_halfspace_re_lt /-
theorem convex_halfspace_re_lt (r : ℝ) : Convex ℝ {c : ℂ | c.re < r} :=
convex_halfspace_lt (IsLinearMap.mk Complex.add_re Complex.smul_re) _
#align convex_halfspace_re_lt convex_halfspace_re_lt
+-/
+#print convex_halfspace_re_le /-
theorem convex_halfspace_re_le (r : ℝ) : Convex ℝ {c : ℂ | c.re ≤ r} :=
convex_halfspace_le (IsLinearMap.mk Complex.add_re Complex.smul_re) _
#align convex_halfspace_re_le convex_halfspace_re_le
+-/
+#print convex_halfspace_re_gt /-
theorem convex_halfspace_re_gt (r : ℝ) : Convex ℝ {c : ℂ | r < c.re} :=
convex_halfspace_gt (IsLinearMap.mk Complex.add_re Complex.smul_re) _
#align convex_halfspace_re_gt convex_halfspace_re_gt
+-/
+#print convex_halfspace_re_ge /-
theorem convex_halfspace_re_ge (r : ℝ) : Convex ℝ {c : ℂ | r ≤ c.re} :=
convex_halfspace_ge (IsLinearMap.mk Complex.add_re Complex.smul_re) _
#align convex_halfspace_re_ge convex_halfspace_re_ge
+-/
+#print convex_halfspace_im_lt /-
theorem convex_halfspace_im_lt (r : ℝ) : Convex ℝ {c : ℂ | c.im < r} :=
convex_halfspace_lt (IsLinearMap.mk Complex.add_im Complex.smul_im) _
#align convex_halfspace_im_lt convex_halfspace_im_lt
+-/
+#print convex_halfspace_im_le /-
theorem convex_halfspace_im_le (r : ℝ) : Convex ℝ {c : ℂ | c.im ≤ r} :=
convex_halfspace_le (IsLinearMap.mk Complex.add_im Complex.smul_im) _
#align convex_halfspace_im_le convex_halfspace_im_le
+-/
+#print convex_halfspace_im_gt /-
theorem convex_halfspace_im_gt (r : ℝ) : Convex ℝ {c : ℂ | r < c.im} :=
convex_halfspace_gt (IsLinearMap.mk Complex.add_im Complex.smul_im) _
#align convex_halfspace_im_gt convex_halfspace_im_gt
+-/
+#print convex_halfspace_im_ge /-
theorem convex_halfspace_im_ge (r : ℝ) : Convex ℝ {c : ℂ | r ≤ c.im} :=
convex_halfspace_ge (IsLinearMap.mk Complex.add_im Complex.smul_im) _
#align convex_halfspace_im_ge convex_halfspace_im_ge
+-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297
@@ -22,35 +22,35 @@ are all convex over ℝ.
-/
-theorem convex_halfspace_re_lt (r : ℝ) : Convex ℝ { c : ℂ | c.re < r } :=
+theorem convex_halfspace_re_lt (r : ℝ) : Convex ℝ {c : ℂ | c.re < r} :=
convex_halfspace_lt (IsLinearMap.mk Complex.add_re Complex.smul_re) _
#align convex_halfspace_re_lt convex_halfspace_re_lt
-theorem convex_halfspace_re_le (r : ℝ) : Convex ℝ { c : ℂ | c.re ≤ r } :=
+theorem convex_halfspace_re_le (r : ℝ) : Convex ℝ {c : ℂ | c.re ≤ r} :=
convex_halfspace_le (IsLinearMap.mk Complex.add_re Complex.smul_re) _
#align convex_halfspace_re_le convex_halfspace_re_le
-theorem convex_halfspace_re_gt (r : ℝ) : Convex ℝ { c : ℂ | r < c.re } :=
+theorem convex_halfspace_re_gt (r : ℝ) : Convex ℝ {c : ℂ | r < c.re} :=
convex_halfspace_gt (IsLinearMap.mk Complex.add_re Complex.smul_re) _
#align convex_halfspace_re_gt convex_halfspace_re_gt
-theorem convex_halfspace_re_ge (r : ℝ) : Convex ℝ { c : ℂ | r ≤ c.re } :=
+theorem convex_halfspace_re_ge (r : ℝ) : Convex ℝ {c : ℂ | r ≤ c.re} :=
convex_halfspace_ge (IsLinearMap.mk Complex.add_re Complex.smul_re) _
#align convex_halfspace_re_ge convex_halfspace_re_ge
-theorem convex_halfspace_im_lt (r : ℝ) : Convex ℝ { c : ℂ | c.im < r } :=
+theorem convex_halfspace_im_lt (r : ℝ) : Convex ℝ {c : ℂ | c.im < r} :=
convex_halfspace_lt (IsLinearMap.mk Complex.add_im Complex.smul_im) _
#align convex_halfspace_im_lt convex_halfspace_im_lt
-theorem convex_halfspace_im_le (r : ℝ) : Convex ℝ { c : ℂ | c.im ≤ r } :=
+theorem convex_halfspace_im_le (r : ℝ) : Convex ℝ {c : ℂ | c.im ≤ r} :=
convex_halfspace_le (IsLinearMap.mk Complex.add_im Complex.smul_im) _
#align convex_halfspace_im_le convex_halfspace_im_le
-theorem convex_halfspace_im_gt (r : ℝ) : Convex ℝ { c : ℂ | r < c.im } :=
+theorem convex_halfspace_im_gt (r : ℝ) : Convex ℝ {c : ℂ | r < c.im} :=
convex_halfspace_gt (IsLinearMap.mk Complex.add_im Complex.smul_im) _
#align convex_halfspace_im_gt convex_halfspace_im_gt
-theorem convex_halfspace_im_ge (r : ℝ) : Convex ℝ { c : ℂ | r ≤ c.im } :=
+theorem convex_halfspace_im_ge (r : ℝ) : Convex ℝ {c : ℂ | r ≤ c.im} :=
convex_halfspace_ge (IsLinearMap.mk Complex.add_im Complex.smul_im) _
#align convex_halfspace_im_ge convex_halfspace_im_ge
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -22,82 +22,34 @@ are all convex over ℝ.
-/
-/- warning: convex_halfspace_re_lt -> convex_halfspace_re_lt is a dubious translation:
-lean 3 declaration is
- forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.hasSmul.{0} Real (Mul.toSMul.{0} Real Real.hasMul)) (setOf.{0} Complex (fun (c : Complex) => LT.lt.{0} Real Real.hasLt (Complex.re c) r))
-but is expected to have type
- forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instSMulComplex.{0} Real (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (Algebra.id.{0} Real Real.instCommSemiringReal))) (setOf.{0} Complex (fun (c : Complex) => LT.lt.{0} Real Real.instLTReal (Complex.re c) r))
-Case conversion may be inaccurate. Consider using '#align convex_halfspace_re_lt convex_halfspace_re_ltₓ'. -/
theorem convex_halfspace_re_lt (r : ℝ) : Convex ℝ { c : ℂ | c.re < r } :=
convex_halfspace_lt (IsLinearMap.mk Complex.add_re Complex.smul_re) _
#align convex_halfspace_re_lt convex_halfspace_re_lt
-/- warning: convex_halfspace_re_le -> convex_halfspace_re_le is a dubious translation:
-lean 3 declaration is
- forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.hasSmul.{0} Real (Mul.toSMul.{0} Real Real.hasMul)) (setOf.{0} Complex (fun (c : Complex) => LE.le.{0} Real Real.hasLe (Complex.re c) r))
-but is expected to have type
- forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instSMulComplex.{0} Real (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (Algebra.id.{0} Real Real.instCommSemiringReal))) (setOf.{0} Complex (fun (c : Complex) => LE.le.{0} Real Real.instLEReal (Complex.re c) r))
-Case conversion may be inaccurate. Consider using '#align convex_halfspace_re_le convex_halfspace_re_leₓ'. -/
theorem convex_halfspace_re_le (r : ℝ) : Convex ℝ { c : ℂ | c.re ≤ r } :=
convex_halfspace_le (IsLinearMap.mk Complex.add_re Complex.smul_re) _
#align convex_halfspace_re_le convex_halfspace_re_le
-/- warning: convex_halfspace_re_gt -> convex_halfspace_re_gt is a dubious translation:
-lean 3 declaration is
- forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.hasSmul.{0} Real (Mul.toSMul.{0} Real Real.hasMul)) (setOf.{0} Complex (fun (c : Complex) => LT.lt.{0} Real Real.hasLt r (Complex.re c)))
-but is expected to have type
- forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instSMulComplex.{0} Real (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (Algebra.id.{0} Real Real.instCommSemiringReal))) (setOf.{0} Complex (fun (c : Complex) => LT.lt.{0} Real Real.instLTReal r (Complex.re c)))
-Case conversion may be inaccurate. Consider using '#align convex_halfspace_re_gt convex_halfspace_re_gtₓ'. -/
theorem convex_halfspace_re_gt (r : ℝ) : Convex ℝ { c : ℂ | r < c.re } :=
convex_halfspace_gt (IsLinearMap.mk Complex.add_re Complex.smul_re) _
#align convex_halfspace_re_gt convex_halfspace_re_gt
-/- warning: convex_halfspace_re_ge -> convex_halfspace_re_ge is a dubious translation:
-lean 3 declaration is
- forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.hasSmul.{0} Real (Mul.toSMul.{0} Real Real.hasMul)) (setOf.{0} Complex (fun (c : Complex) => LE.le.{0} Real Real.hasLe r (Complex.re c)))
-but is expected to have type
- forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instSMulComplex.{0} Real (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (Algebra.id.{0} Real Real.instCommSemiringReal))) (setOf.{0} Complex (fun (c : Complex) => LE.le.{0} Real Real.instLEReal r (Complex.re c)))
-Case conversion may be inaccurate. Consider using '#align convex_halfspace_re_ge convex_halfspace_re_geₓ'. -/
theorem convex_halfspace_re_ge (r : ℝ) : Convex ℝ { c : ℂ | r ≤ c.re } :=
convex_halfspace_ge (IsLinearMap.mk Complex.add_re Complex.smul_re) _
#align convex_halfspace_re_ge convex_halfspace_re_ge
-/- warning: convex_halfspace_im_lt -> convex_halfspace_im_lt is a dubious translation:
-lean 3 declaration is
- forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.hasSmul.{0} Real (Mul.toSMul.{0} Real Real.hasMul)) (setOf.{0} Complex (fun (c : Complex) => LT.lt.{0} Real Real.hasLt (Complex.im c) r))
-but is expected to have type
- forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instSMulComplex.{0} Real (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (Algebra.id.{0} Real Real.instCommSemiringReal))) (setOf.{0} Complex (fun (c : Complex) => LT.lt.{0} Real Real.instLTReal (Complex.im c) r))
-Case conversion may be inaccurate. Consider using '#align convex_halfspace_im_lt convex_halfspace_im_ltₓ'. -/
theorem convex_halfspace_im_lt (r : ℝ) : Convex ℝ { c : ℂ | c.im < r } :=
convex_halfspace_lt (IsLinearMap.mk Complex.add_im Complex.smul_im) _
#align convex_halfspace_im_lt convex_halfspace_im_lt
-/- warning: convex_halfspace_im_le -> convex_halfspace_im_le is a dubious translation:
-lean 3 declaration is
- forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.hasSmul.{0} Real (Mul.toSMul.{0} Real Real.hasMul)) (setOf.{0} Complex (fun (c : Complex) => LE.le.{0} Real Real.hasLe (Complex.im c) r))
-but is expected to have type
- forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instSMulComplex.{0} Real (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (Algebra.id.{0} Real Real.instCommSemiringReal))) (setOf.{0} Complex (fun (c : Complex) => LE.le.{0} Real Real.instLEReal (Complex.im c) r))
-Case conversion may be inaccurate. Consider using '#align convex_halfspace_im_le convex_halfspace_im_leₓ'. -/
theorem convex_halfspace_im_le (r : ℝ) : Convex ℝ { c : ℂ | c.im ≤ r } :=
convex_halfspace_le (IsLinearMap.mk Complex.add_im Complex.smul_im) _
#align convex_halfspace_im_le convex_halfspace_im_le
-/- warning: convex_halfspace_im_gt -> convex_halfspace_im_gt is a dubious translation:
-lean 3 declaration is
- forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.hasSmul.{0} Real (Mul.toSMul.{0} Real Real.hasMul)) (setOf.{0} Complex (fun (c : Complex) => LT.lt.{0} Real Real.hasLt r (Complex.im c)))
-but is expected to have type
- forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instSMulComplex.{0} Real (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (Algebra.id.{0} Real Real.instCommSemiringReal))) (setOf.{0} Complex (fun (c : Complex) => LT.lt.{0} Real Real.instLTReal r (Complex.im c)))
-Case conversion may be inaccurate. Consider using '#align convex_halfspace_im_gt convex_halfspace_im_gtₓ'. -/
theorem convex_halfspace_im_gt (r : ℝ) : Convex ℝ { c : ℂ | r < c.im } :=
convex_halfspace_gt (IsLinearMap.mk Complex.add_im Complex.smul_im) _
#align convex_halfspace_im_gt convex_halfspace_im_gt
-/- warning: convex_halfspace_im_ge -> convex_halfspace_im_ge is a dubious translation:
-lean 3 declaration is
- forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.hasSmul.{0} Real (Mul.toSMul.{0} Real Real.hasMul)) (setOf.{0} Complex (fun (c : Complex) => LE.le.{0} Real Real.hasLe r (Complex.im c)))
-but is expected to have type
- forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instSMulComplex.{0} Real (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (Algebra.id.{0} Real Real.instCommSemiringReal))) (setOf.{0} Complex (fun (c : Complex) => LE.le.{0} Real Real.instLEReal r (Complex.im c)))
-Case conversion may be inaccurate. Consider using '#align convex_halfspace_im_ge convex_halfspace_im_geₓ'. -/
theorem convex_halfspace_im_ge (r : ℝ) : Convex ℝ { c : ℂ | r ≤ c.im } :=
convex_halfspace_ge (IsLinearMap.mk Complex.add_im Complex.smul_im) _
#align convex_halfspace_im_ge convex_halfspace_im_ge
mathlib commit https://github.com/leanprover-community/mathlib/commit/738054fa93d43512da144ec45ce799d18fd44248
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Yaël Dillies
! This file was ported from Lean 3 source module analysis.convex.complex
-! leanprover-community/mathlib commit 15730e8d0af237a2ebafeb8cfbbcf71f6160c2e9
+! leanprover-community/mathlib commit fe8d0ff42c3c24d789f491dc2622b6cac3d61564
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -14,6 +14,9 @@ import Mathbin.Data.Complex.Module
/-!
# Convexity of half spaces in ℂ
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
The open and closed half-spaces in ℂ given by an inequality on either the real or imaginary part
are all convex over ℝ.
-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/36b8aa61ea7c05727161f96a0532897bd72aedab
@@ -19,34 +19,82 @@ are all convex over ℝ.
-/
+/- warning: convex_halfspace_re_lt -> convex_halfspace_re_lt is a dubious translation:
+lean 3 declaration is
+ forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.hasSmul.{0} Real (Mul.toSMul.{0} Real Real.hasMul)) (setOf.{0} Complex (fun (c : Complex) => LT.lt.{0} Real Real.hasLt (Complex.re c) r))
+but is expected to have type
+ forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instSMulComplex.{0} Real (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (Algebra.id.{0} Real Real.instCommSemiringReal))) (setOf.{0} Complex (fun (c : Complex) => LT.lt.{0} Real Real.instLTReal (Complex.re c) r))
+Case conversion may be inaccurate. Consider using '#align convex_halfspace_re_lt convex_halfspace_re_ltₓ'. -/
theorem convex_halfspace_re_lt (r : ℝ) : Convex ℝ { c : ℂ | c.re < r } :=
convex_halfspace_lt (IsLinearMap.mk Complex.add_re Complex.smul_re) _
#align convex_halfspace_re_lt convex_halfspace_re_lt
+/- warning: convex_halfspace_re_le -> convex_halfspace_re_le is a dubious translation:
+lean 3 declaration is
+ forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.hasSmul.{0} Real (Mul.toSMul.{0} Real Real.hasMul)) (setOf.{0} Complex (fun (c : Complex) => LE.le.{0} Real Real.hasLe (Complex.re c) r))
+but is expected to have type
+ forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instSMulComplex.{0} Real (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (Algebra.id.{0} Real Real.instCommSemiringReal))) (setOf.{0} Complex (fun (c : Complex) => LE.le.{0} Real Real.instLEReal (Complex.re c) r))
+Case conversion may be inaccurate. Consider using '#align convex_halfspace_re_le convex_halfspace_re_leₓ'. -/
theorem convex_halfspace_re_le (r : ℝ) : Convex ℝ { c : ℂ | c.re ≤ r } :=
convex_halfspace_le (IsLinearMap.mk Complex.add_re Complex.smul_re) _
#align convex_halfspace_re_le convex_halfspace_re_le
+/- warning: convex_halfspace_re_gt -> convex_halfspace_re_gt is a dubious translation:
+lean 3 declaration is
+ forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.hasSmul.{0} Real (Mul.toSMul.{0} Real Real.hasMul)) (setOf.{0} Complex (fun (c : Complex) => LT.lt.{0} Real Real.hasLt r (Complex.re c)))
+but is expected to have type
+ forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instSMulComplex.{0} Real (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (Algebra.id.{0} Real Real.instCommSemiringReal))) (setOf.{0} Complex (fun (c : Complex) => LT.lt.{0} Real Real.instLTReal r (Complex.re c)))
+Case conversion may be inaccurate. Consider using '#align convex_halfspace_re_gt convex_halfspace_re_gtₓ'. -/
theorem convex_halfspace_re_gt (r : ℝ) : Convex ℝ { c : ℂ | r < c.re } :=
convex_halfspace_gt (IsLinearMap.mk Complex.add_re Complex.smul_re) _
#align convex_halfspace_re_gt convex_halfspace_re_gt
+/- warning: convex_halfspace_re_ge -> convex_halfspace_re_ge is a dubious translation:
+lean 3 declaration is
+ forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.hasSmul.{0} Real (Mul.toSMul.{0} Real Real.hasMul)) (setOf.{0} Complex (fun (c : Complex) => LE.le.{0} Real Real.hasLe r (Complex.re c)))
+but is expected to have type
+ forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instSMulComplex.{0} Real (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (Algebra.id.{0} Real Real.instCommSemiringReal))) (setOf.{0} Complex (fun (c : Complex) => LE.le.{0} Real Real.instLEReal r (Complex.re c)))
+Case conversion may be inaccurate. Consider using '#align convex_halfspace_re_ge convex_halfspace_re_geₓ'. -/
theorem convex_halfspace_re_ge (r : ℝ) : Convex ℝ { c : ℂ | r ≤ c.re } :=
convex_halfspace_ge (IsLinearMap.mk Complex.add_re Complex.smul_re) _
#align convex_halfspace_re_ge convex_halfspace_re_ge
+/- warning: convex_halfspace_im_lt -> convex_halfspace_im_lt is a dubious translation:
+lean 3 declaration is
+ forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.hasSmul.{0} Real (Mul.toSMul.{0} Real Real.hasMul)) (setOf.{0} Complex (fun (c : Complex) => LT.lt.{0} Real Real.hasLt (Complex.im c) r))
+but is expected to have type
+ forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instSMulComplex.{0} Real (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (Algebra.id.{0} Real Real.instCommSemiringReal))) (setOf.{0} Complex (fun (c : Complex) => LT.lt.{0} Real Real.instLTReal (Complex.im c) r))
+Case conversion may be inaccurate. Consider using '#align convex_halfspace_im_lt convex_halfspace_im_ltₓ'. -/
theorem convex_halfspace_im_lt (r : ℝ) : Convex ℝ { c : ℂ | c.im < r } :=
convex_halfspace_lt (IsLinearMap.mk Complex.add_im Complex.smul_im) _
#align convex_halfspace_im_lt convex_halfspace_im_lt
+/- warning: convex_halfspace_im_le -> convex_halfspace_im_le is a dubious translation:
+lean 3 declaration is
+ forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.hasSmul.{0} Real (Mul.toSMul.{0} Real Real.hasMul)) (setOf.{0} Complex (fun (c : Complex) => LE.le.{0} Real Real.hasLe (Complex.im c) r))
+but is expected to have type
+ forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instSMulComplex.{0} Real (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (Algebra.id.{0} Real Real.instCommSemiringReal))) (setOf.{0} Complex (fun (c : Complex) => LE.le.{0} Real Real.instLEReal (Complex.im c) r))
+Case conversion may be inaccurate. Consider using '#align convex_halfspace_im_le convex_halfspace_im_leₓ'. -/
theorem convex_halfspace_im_le (r : ℝ) : Convex ℝ { c : ℂ | c.im ≤ r } :=
convex_halfspace_le (IsLinearMap.mk Complex.add_im Complex.smul_im) _
#align convex_halfspace_im_le convex_halfspace_im_le
+/- warning: convex_halfspace_im_gt -> convex_halfspace_im_gt is a dubious translation:
+lean 3 declaration is
+ forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.hasSmul.{0} Real (Mul.toSMul.{0} Real Real.hasMul)) (setOf.{0} Complex (fun (c : Complex) => LT.lt.{0} Real Real.hasLt r (Complex.im c)))
+but is expected to have type
+ forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instSMulComplex.{0} Real (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (Algebra.id.{0} Real Real.instCommSemiringReal))) (setOf.{0} Complex (fun (c : Complex) => LT.lt.{0} Real Real.instLTReal r (Complex.im c)))
+Case conversion may be inaccurate. Consider using '#align convex_halfspace_im_gt convex_halfspace_im_gtₓ'. -/
theorem convex_halfspace_im_gt (r : ℝ) : Convex ℝ { c : ℂ | r < c.im } :=
convex_halfspace_gt (IsLinearMap.mk Complex.add_im Complex.smul_im) _
#align convex_halfspace_im_gt convex_halfspace_im_gt
+/- warning: convex_halfspace_im_ge -> convex_halfspace_im_ge is a dubious translation:
+lean 3 declaration is
+ forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.hasSmul.{0} Real (Mul.toSMul.{0} Real Real.hasMul)) (setOf.{0} Complex (fun (c : Complex) => LE.le.{0} Real Real.hasLe r (Complex.im c)))
+but is expected to have type
+ forall (r : Real), Convex.{0, 0} Real Complex Real.orderedSemiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instSMulComplex.{0} Real (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (Algebra.id.{0} Real Real.instCommSemiringReal))) (setOf.{0} Complex (fun (c : Complex) => LE.le.{0} Real Real.instLEReal r (Complex.im c)))
+Case conversion may be inaccurate. Consider using '#align convex_halfspace_im_ge convex_halfspace_im_geₓ'. -/
theorem convex_halfspace_im_ge (r : ℝ) : Convex ℝ { c : ℂ | r ≤ c.im } :=
convex_halfspace_ge (IsLinearMap.mk Complex.add_im Complex.smul_im) _
#align convex_halfspace_im_ge convex_halfspace_im_ge
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
@@ -2,15 +2,12 @@
Copyright (c) 2019 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Yaël Dillies
-
-! This file was ported from Lean 3 source module analysis.convex.complex
-! leanprover-community/mathlib commit 15730e8d0af237a2ebafeb8cfbbcf71f6160c2e9
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Data.Complex.Module
+#align_import analysis.convex.complex from "leanprover-community/mathlib"@"15730e8d0af237a2ebafeb8cfbbcf71f6160c2e9"
+
/-!
# Convexity of half spaces in ℂ
The unported dependencies are
algebra.order.module
init.core
linear_algebra.free_module.finite.rank
algebra.order.monoid.cancel.defs
algebra.abs
algebra.group_power.lemmas
init.data.list.basic
linear_algebra.free_module.rank
algebra.order.monoid.cancel.basic
init.data.list.default
topology.subset_properties
init.logic
The following 1 dependencies have changed in mathlib3 since they were ported, which may complicate porting this file