/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison

! This file was ported from Lean 3 source module topology.unit_interval
! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.Topology.Instances.Real
import Mathbin.Topology.Algebra.Field
import Mathbin.Data.Set.Intervals.ProjIcc
import Mathbin.Data.Set.Intervals.Instances

/-!
# The unit interval, as a topological space

> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.

Use `open_locale unit_interval` to turn on the notation `I := set.Icc (0 : ℝ) (1 : ℝ)`.

We provide basic instances, as well as a custom tactic for discharging
`0 ≤ ↑x`, `0 ≤ 1 - ↑x`, `↑x ≤ 1`, and `1 - ↑x ≤ 1` when `x : I`.

-/

noncomputable section

open Classical Topology Filter

open Set Int Set.Icc

/-! ### The unit interval -/

#print unitInterval /-
/-- The unit interval `[0,1]` in ℝ. -/
abbrev unitInterval : Set ℝ :=
  Set.Icc 0 1
#align unit_interval unitInterval
-/

-- mathport name: unit_interval
scoped[unitInterval] notation "I" => unitInterval

namespace unitInterval

/- warning: unit_interval.zero_mem -> unitInterval.zero_mem is a dubious translation:
lean 3 declaration is
  Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) unitInterval
but is expected to have type
  Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) unitInterval
Case conversion may be inaccurate. Consider using '#align unit_interval.zero_mem unitInterval.zero_memₓ'. -/
theorem zero_mem : (0 : ℝ) ∈ I :=
  ⟨le_rfl, zero_le_one⟩
#align unit_interval.zero_mem unitInterval.zero_mem

/- warning: unit_interval.one_mem -> unitInterval.one_mem is a dubious translation:
lean 3 declaration is
  Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) unitInterval
but is expected to have type
  Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) unitInterval
Case conversion may be inaccurate. Consider using '#align unit_interval.one_mem unitInterval.one_memₓ'. -/
theorem one_mem : (1 : ℝ) ∈ I :=
  ⟨zero_le_one, le_rfl⟩
#align unit_interval.one_mem unitInterval.one_mem

/- warning: unit_interval.mul_mem -> unitInterval.mul_mem is a dubious translation:
lean 3 declaration is
  forall {x : Real} {y : Real}, (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x unitInterval) -> (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) y unitInterval) -> (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.hasMul) x y) unitInterval)
but is expected to have type
  forall {x : Real} {y : Real}, (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x unitInterval) -> (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) y unitInterval) -> (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMulReal) x y) unitInterval)
Case conversion may be inaccurate. Consider using '#align unit_interval.mul_mem unitInterval.mul_memₓ'. -/
theorem mul_mem {x y : ℝ} (hx : x ∈ I) (hy : y ∈ I) : x * y ∈ I :=
  ⟨mul_nonneg hx.1 hy.1, (mul_le_mul hx.2 hy.2 hy.1 zero_le_one).trans_eq <| one_mul 1⟩
#align unit_interval.mul_mem unitInterval.mul_mem

/- warning: unit_interval.div_mem -> unitInterval.div_mem is a dubious translation:
lean 3 declaration is
  forall {x : Real} {y : Real}, (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) x) -> (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) y) -> (LE.le.{0} Real Real.hasLe x y) -> (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) x y) unitInterval)
but is expected to have type
  forall {x : Real} {y : Real}, (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) x) -> (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) y) -> (LE.le.{0} Real Real.instLEReal x y) -> (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) x y) unitInterval)
Case conversion may be inaccurate. Consider using '#align unit_interval.div_mem unitInterval.div_memₓ'. -/
theorem div_mem {x y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hxy : x ≤ y) : x / y ∈ I :=
  ⟨div_nonneg hx hy, div_le_one_of_le hxy hy⟩
#align unit_interval.div_mem unitInterval.div_mem

#print unitInterval.fract_mem /-
theorem fract_mem (x : ℝ) : fract x ∈ I :=
  ⟨fract_nonneg _, (fract_lt_one _).le⟩
#align unit_interval.fract_mem unitInterval.fract_mem
-/

/- warning: unit_interval.mem_iff_one_sub_mem -> unitInterval.mem_iff_one_sub_mem is a dubious translation:
lean 3 declaration is
  forall {t : Real}, Iff (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) t unitInterval) (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) t) unitInterval)
but is expected to have type
  forall {t : Real}, Iff (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) t unitInterval) (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) t) unitInterval)
Case conversion may be inaccurate. Consider using '#align unit_interval.mem_iff_one_sub_mem unitInterval.mem_iff_one_sub_memₓ'. -/
theorem mem_iff_one_sub_mem {t : ℝ} : t ∈ I ↔ 1 - t ∈ I :=
  by
  rw [mem_Icc, mem_Icc]
  constructor <;> intro <;> constructor <;> linarith
#align unit_interval.mem_iff_one_sub_mem unitInterval.mem_iff_one_sub_mem

#print unitInterval.hasZero /-
instance hasZero : Zero I :=
  ⟨⟨0, zero_mem⟩⟩
#align unit_interval.has_zero unitInterval.hasZero
-/

#print unitInterval.hasOne /-
instance hasOne : One I :=
  ⟨⟨1, by constructor <;> norm_num⟩⟩
#align unit_interval.has_one unitInterval.hasOne
-/

/- warning: unit_interval.coe_ne_zero -> unitInterval.coe_ne_zero is a dubious translation:
lean 3 declaration is
  forall {x : coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval}, Iff (Ne.{1} Real ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (HasLiftT.mk.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (CoeTCₓ.coe.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (coeBase.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (coeSubtype.{1} Real (fun (x : Real) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x unitInterval))))) x) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (Ne.{1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) x (OfNat.ofNat.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) 0 (OfNat.mk.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) 0 (Zero.zero.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) unitInterval.hasZero))))
but is expected to have type
  forall {x : Set.Elem.{0} Real unitInterval}, Iff (Ne.{1} Real (Subtype.val.{1} Real (fun (x : Real) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x unitInterval) x) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (Ne.{1} (Set.Elem.{0} Real unitInterval) x (OfNat.ofNat.{0} (Set.Elem.{0} Real unitInterval) 0 (Zero.toOfNat0.{0} (Set.Elem.{0} Real unitInterval) unitInterval.hasZero)))
Case conversion may be inaccurate. Consider using '#align unit_interval.coe_ne_zero unitInterval.coe_ne_zeroₓ'. -/
theorem coe_ne_zero {x : I} : (x : ℝ) ≠ 0 ↔ x ≠ 0 :=
  not_iff_not.mpr coe_eq_zero
#align unit_interval.coe_ne_zero unitInterval.coe_ne_zero

/- warning: unit_interval.coe_ne_one -> unitInterval.coe_ne_one is a dubious translation:
lean 3 declaration is
  forall {x : coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval}, Iff (Ne.{1} Real ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (HasLiftT.mk.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (CoeTCₓ.coe.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (coeBase.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (coeSubtype.{1} Real (fun (x : Real) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x unitInterval))))) x) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))) (Ne.{1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) x (OfNat.ofNat.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) 1 (OfNat.mk.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) 1 (One.one.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) unitInterval.hasOne))))
but is expected to have type
  forall {x : Set.Elem.{0} Real unitInterval}, Iff (Ne.{1} Real (Subtype.val.{1} Real (fun (x : Real) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x unitInterval) x) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))) (Ne.{1} (Set.Elem.{0} Real unitInterval) x (OfNat.ofNat.{0} (Set.Elem.{0} Real unitInterval) 1 (One.toOfNat1.{0} (Set.Elem.{0} Real unitInterval) unitInterval.hasOne)))
Case conversion may be inaccurate. Consider using '#align unit_interval.coe_ne_one unitInterval.coe_ne_oneₓ'. -/
theorem coe_ne_one {x : I} : (x : ℝ) ≠ 1 ↔ x ≠ 1 :=
  not_iff_not.mpr coe_eq_one
#align unit_interval.coe_ne_one unitInterval.coe_ne_one

instance : Nonempty I :=
  ⟨0⟩

instance : Mul I :=
  ⟨fun x y => ⟨x * y, mul_mem x.2 y.2⟩⟩

/- warning: unit_interval.mul_le_left -> unitInterval.mul_le_left is a dubious translation:
lean 3 declaration is
  forall {x : coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval} {y : coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval}, LE.le.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) (Subtype.hasLe.{0} Real Real.hasLe (fun (x : Real) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x unitInterval)) (HMul.hMul.{0, 0, 0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) (instHMul.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) unitInterval.hasMul) x y) x
but is expected to have type
  forall {x : Set.Elem.{0} Real unitInterval} {y : Set.Elem.{0} Real unitInterval}, LE.le.{0} (Set.Elem.{0} Real unitInterval) (Subtype.le.{0} Real Real.instLEReal (fun (x : Real) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x unitInterval)) (HMul.hMul.{0, 0, 0} (Set.Elem.{0} Real unitInterval) (Set.Elem.{0} Real unitInterval) (Set.Elem.{0} Real unitInterval) (instHMul.{0} (Set.Elem.{0} Real unitInterval) unitInterval.instMulElemRealUnitInterval) x y) x
Case conversion may be inaccurate. Consider using '#align unit_interval.mul_le_left unitInterval.mul_le_leftₓ'. -/
-- todo: we could set up a `linear_ordered_comm_monoid_with_zero I` instance
theorem mul_le_left {x y : I} : x * y ≤ x :=
  Subtype.coe_le_coe.mp <| (mul_le_mul_of_nonneg_left y.2.2 x.2.1).trans_eq <| mul_one x
#align unit_interval.mul_le_left unitInterval.mul_le_left

/- warning: unit_interval.mul_le_right -> unitInterval.mul_le_right is a dubious translation:
lean 3 declaration is
  forall {x : coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval} {y : coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval}, LE.le.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) (Subtype.hasLe.{0} Real Real.hasLe (fun (x : Real) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x unitInterval)) (HMul.hMul.{0, 0, 0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) (instHMul.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) unitInterval.hasMul) x y) y
but is expected to have type
  forall {x : Set.Elem.{0} Real unitInterval} {y : Set.Elem.{0} Real unitInterval}, LE.le.{0} (Set.Elem.{0} Real unitInterval) (Subtype.le.{0} Real Real.instLEReal (fun (x : Real) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x unitInterval)) (HMul.hMul.{0, 0, 0} (Set.Elem.{0} Real unitInterval) (Set.Elem.{0} Real unitInterval) (Set.Elem.{0} Real unitInterval) (instHMul.{0} (Set.Elem.{0} Real unitInterval) unitInterval.instMulElemRealUnitInterval) x y) y
Case conversion may be inaccurate. Consider using '#align unit_interval.mul_le_right unitInterval.mul_le_rightₓ'. -/
theorem mul_le_right {x y : I} : x * y ≤ y :=
  Subtype.coe_le_coe.mp <| (mul_le_mul_of_nonneg_right x.2.2 y.2.1).trans_eq <| one_mul y
#align unit_interval.mul_le_right unitInterval.mul_le_right

#print unitInterval.symm /-
/-- Unit interval central symmetry. -/
def symm : I → I := fun t => ⟨1 - t, mem_iff_one_sub_mem.mp t.Prop⟩
#align unit_interval.symm unitInterval.symm
-/

-- mathport name: unit_interval.symm
scoped notation "σ" => unitInterval.symm

#print unitInterval.symm_zero /-
@[simp]
theorem symm_zero : σ 0 = 1 :=
  Subtype.ext <| by simp [symm]
#align unit_interval.symm_zero unitInterval.symm_zero
-/

#print unitInterval.symm_one /-
@[simp]
theorem symm_one : σ 1 = 0 :=
  Subtype.ext <| by simp [symm]
#align unit_interval.symm_one unitInterval.symm_one
-/

#print unitInterval.symm_symm /-
@[simp]
theorem symm_symm (x : I) : σ (σ x) = x :=
  Subtype.ext <| by simp [symm]
#align unit_interval.symm_symm unitInterval.symm_symm
-/

/- warning: unit_interval.coe_symm_eq -> unitInterval.coe_symm_eq is a dubious translation:
lean 3 declaration is
  forall (x : coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval), Eq.{1} Real ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (HasLiftT.mk.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (CoeTCₓ.coe.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (coeBase.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (coeSubtype.{1} Real (fun (x : Real) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x unitInterval))))) (unitInterval.symm x)) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (HasLiftT.mk.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (CoeTCₓ.coe.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (coeBase.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (coeSubtype.{1} Real (fun (x : Real) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x unitInterval))))) x))
but is expected to have type
  forall (x : Set.Elem.{0} Real unitInterval), Eq.{1} Real (Subtype.val.{1} Real (fun (x : Real) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x unitInterval) (unitInterval.symm x)) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (Subtype.val.{1} Real (fun (x : Real) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x unitInterval) x))
Case conversion may be inaccurate. Consider using '#align unit_interval.coe_symm_eq unitInterval.coe_symm_eqₓ'. -/
@[simp]
theorem coe_symm_eq (x : I) : (σ x : ℝ) = 1 - x :=
  rfl
#align unit_interval.coe_symm_eq unitInterval.coe_symm_eq

/- ./././Mathport/Syntax/Translate/Tactic/Mathlib/Misc2.lean:301:22: continuitity! not supported at the moment -/
#print unitInterval.continuous_symm /-
@[continuity]
theorem continuous_symm : Continuous σ := by continuity
#align unit_interval.continuous_symm unitInterval.continuous_symm
-/

instance : ConnectedSpace I :=
  Subtype.connectedSpace ⟨nonempty_Icc.mpr zero_le_one, isPreconnected_Icc⟩

/-- Verify there is an instance for `compact_space I`. -/
example : CompactSpace I := by infer_instance

/- warning: unit_interval.nonneg -> unitInterval.nonneg is a dubious translation:
lean 3 declaration is
  forall (x : coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval), LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (HasLiftT.mk.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (CoeTCₓ.coe.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (coeBase.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (coeSubtype.{1} Real (fun (x : Real) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x unitInterval))))) x)
but is expected to have type
  forall (x : Set.Elem.{0} Real unitInterval), LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (Subtype.val.{1} Real (fun (x : Real) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x unitInterval) x)
Case conversion may be inaccurate. Consider using '#align unit_interval.nonneg unitInterval.nonnegₓ'. -/
theorem nonneg (x : I) : 0 ≤ (x : ℝ) :=
  x.2.1
#align unit_interval.nonneg unitInterval.nonneg

/- warning: unit_interval.one_minus_nonneg -> unitInterval.one_minus_nonneg is a dubious translation:
lean 3 declaration is
  forall (x : coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval), LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (HasLiftT.mk.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (CoeTCₓ.coe.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (coeBase.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (coeSubtype.{1} Real (fun (x : Real) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x unitInterval))))) x))
but is expected to have type
  forall (x : Set.Elem.{0} Real unitInterval), LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (Subtype.val.{1} Real (fun (x : Real) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x unitInterval) x))
Case conversion may be inaccurate. Consider using '#align unit_interval.one_minus_nonneg unitInterval.one_minus_nonnegₓ'. -/
theorem one_minus_nonneg (x : I) : 0 ≤ 1 - (x : ℝ) := by simpa using x.2.2
#align unit_interval.one_minus_nonneg unitInterval.one_minus_nonneg

/- warning: unit_interval.le_one -> unitInterval.le_one is a dubious translation:
lean 3 declaration is
  forall (x : coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval), LE.le.{0} Real Real.hasLe ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (HasLiftT.mk.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (CoeTCₓ.coe.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (coeBase.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (coeSubtype.{1} Real (fun (x : Real) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x unitInterval))))) x) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))
but is expected to have type
  forall (x : Set.Elem.{0} Real unitInterval), LE.le.{0} Real Real.instLEReal (Subtype.val.{1} Real (fun (x : Real) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x unitInterval) x) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))
Case conversion may be inaccurate. Consider using '#align unit_interval.le_one unitInterval.le_oneₓ'. -/
theorem le_one (x : I) : (x : ℝ) ≤ 1 :=
  x.2.2
#align unit_interval.le_one unitInterval.le_one

/- warning: unit_interval.one_minus_le_one -> unitInterval.one_minus_le_one is a dubious translation:
lean 3 declaration is
  forall (x : coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval), LE.le.{0} Real Real.hasLe (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (HasLiftT.mk.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (CoeTCₓ.coe.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (coeBase.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (coeSubtype.{1} Real (fun (x : Real) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x unitInterval))))) x)) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))
but is expected to have type
  forall (x : Set.Elem.{0} Real unitInterval), LE.le.{0} Real Real.instLEReal (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (Subtype.val.{1} Real (fun (x : Real) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x unitInterval) x)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))
Case conversion may be inaccurate. Consider using '#align unit_interval.one_minus_le_one unitInterval.one_minus_le_oneₓ'. -/
theorem one_minus_le_one (x : I) : 1 - (x : ℝ) ≤ 1 := by simpa using x.2.1
#align unit_interval.one_minus_le_one unitInterval.one_minus_le_one

/- warning: unit_interval.add_pos -> unitInterval.add_pos is a dubious translation:
lean 3 declaration is
  forall {t : coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval} {x : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) x) -> (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) x ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (HasLiftT.mk.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (CoeTCₓ.coe.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (coeBase.{1, 1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) Real (coeSubtype.{1} Real (fun (x : Real) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x unitInterval))))) t)))
but is expected to have type
  forall {t : Set.Elem.{0} Real unitInterval} {x : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) x) -> (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) x (Subtype.val.{1} Real (fun (x : Real) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x unitInterval) t)))
Case conversion may be inaccurate. Consider using '#align unit_interval.add_pos unitInterval.add_posₓ'. -/
theorem add_pos {t : I} {x : ℝ} (hx : 0 < x) : 0 < (x + t : ℝ) :=
  add_pos_of_pos_of_nonneg hx <| nonneg _
#align unit_interval.add_pos unitInterval.add_pos

/- warning: unit_interval.nonneg' -> unitInterval.nonneg' is a dubious translation:
lean 3 declaration is
  forall {t : coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval}, LE.le.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) (Subtype.hasLe.{0} Real Real.hasLe (fun (x : Real) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x unitInterval)) (OfNat.ofNat.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) 0 (OfNat.mk.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) 0 (Zero.zero.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) unitInterval.hasZero))) t
but is expected to have type
  forall {t : Set.Elem.{0} Real unitInterval}, LE.le.{0} (Set.Elem.{0} Real unitInterval) (Subtype.le.{0} Real Real.instLEReal (fun (x : Real) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x unitInterval)) (OfNat.ofNat.{0} (Set.Elem.{0} Real unitInterval) 0 (Zero.toOfNat0.{0} (Set.Elem.{0} Real unitInterval) unitInterval.hasZero)) t
Case conversion may be inaccurate. Consider using '#align unit_interval.nonneg' unitInterval.nonneg'ₓ'. -/
/-- like `unit_interval.nonneg`, but with the inequality in `I`. -/
theorem nonneg' {t : I} : 0 ≤ t :=
  t.2.1
#align unit_interval.nonneg' unitInterval.nonneg'

/- warning: unit_interval.le_one' -> unitInterval.le_one' is a dubious translation:
lean 3 declaration is
  forall {t : coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval}, LE.le.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) (Subtype.hasLe.{0} Real Real.hasLe (fun (x : Real) => Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) x unitInterval)) t (OfNat.ofNat.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) 1 (OfNat.mk.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) 1 (One.one.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) unitInterval) unitInterval.hasOne)))
but is expected to have type
  forall {t : Set.Elem.{0} Real unitInterval}, LE.le.{0} (Set.Elem.{0} Real unitInterval) (Subtype.le.{0} Real Real.instLEReal (fun (x : Real) => Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) x unitInterval)) t (OfNat.ofNat.{0} (Set.Elem.{0} Real unitInterval) 1 (One.toOfNat1.{0} (Set.Elem.{0} Real unitInterval) unitInterval.hasOne))
Case conversion may be inaccurate. Consider using '#align unit_interval.le_one' unitInterval.le_one'ₓ'. -/
/-- like `unit_interval.le_one`, but with the inequality in `I`. -/
theorem le_one' {t : I} : t ≤ 1 :=
  t.2.2
#align unit_interval.le_one' unitInterval.le_one'

/- warning: unit_interval.mul_pos_mem_iff -> unitInterval.mul_pos_mem_iff is a dubious translation:
lean 3 declaration is
  forall {a : Real} {t : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) a) -> (Iff (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.hasMul) a t) unitInterval) (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) t (Set.Icc.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) a))))
but is expected to have type
  forall {a : Real} {t : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) a) -> (Iff (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMulReal) a t) unitInterval) (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) t (Set.Icc.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) a))))
Case conversion may be inaccurate. Consider using '#align unit_interval.mul_pos_mem_iff unitInterval.mul_pos_mem_iffₓ'. -/
theorem mul_pos_mem_iff {a t : ℝ} (ha : 0 < a) : a * t ∈ I ↔ t ∈ Set.Icc (0 : ℝ) (1 / a) :=
  by
  constructor <;> rintro ⟨h₁, h₂⟩ <;> constructor
  · exact nonneg_of_mul_nonneg_right h₁ ha
  · rwa [le_div_iff ha, mul_comm]
  · exact mul_nonneg ha.le h₁
  · rwa [le_div_iff ha, mul_comm] at h₂
#align unit_interval.mul_pos_mem_iff unitInterval.mul_pos_mem_iff

/- warning: unit_interval.two_mul_sub_one_mem_iff -> unitInterval.two_mul_sub_one_mem_iff is a dubious translation:
lean 3 declaration is
  forall {t : Real}, Iff (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.hasSub) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.hasMul) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne)))) t) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))) unitInterval) (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) t (Set.Icc.{0} Real Real.preorder (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne))))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))))
but is expected to have type
  forall {t : Real}, Iff (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSubReal) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMulReal) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))) t) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))) unitInterval) (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) t (Set.Icc.{0} Real Real.instPreorderReal (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))))
Case conversion may be inaccurate. Consider using '#align unit_interval.two_mul_sub_one_mem_iff unitInterval.two_mul_sub_one_mem_iffₓ'. -/
theorem two_mul_sub_one_mem_iff {t : ℝ} : 2 * t - 1 ∈ I ↔ t ∈ Set.Icc (1 / 2 : ℝ) 1 := by
  constructor <;> rintro ⟨h₁, h₂⟩ <;> constructor <;> linarith
#align unit_interval.two_mul_sub_one_mem_iff unitInterval.two_mul_sub_one_mem_iff

end unitInterval

/- warning: proj_Icc_eq_zero -> projIcc_eq_zero is a dubious translation:
lean 3 declaration is
  forall {x : Real}, Iff (Eq.{1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Icc.{0} Real (PartialOrder.toPreorder.{0} Real (SemilatticeInf.toPartialOrder.{0} Real (Lattice.toSemilatticeInf.{0} Real (LinearOrder.toLattice.{0} Real Real.linearOrder)))) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))))) (Set.projIcc.{0} Real Real.linearOrder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (zero_le_one.{0} Real Real.hasZero Real.hasOne (Preorder.toLE.{0} Real (PartialOrder.toPreorder.{0} Real (SemilatticeInf.toPartialOrder.{0} Real (Lattice.toSemilatticeInf.{0} Real (LinearOrder.toLattice.{0} Real Real.linearOrder))))) (OrderedSemiring.zeroLEOneClass.{0} Real Real.orderedSemiring)) x) (OfNat.ofNat.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Icc.{0} Real (PartialOrder.toPreorder.{0} Real (SemilatticeInf.toPartialOrder.{0} Real (Lattice.toSemilatticeInf.{0} Real (LinearOrder.toLattice.{0} Real Real.linearOrder)))) (Zero.zero.{0} Real Real.hasZero) (One.one.{0} Real Real.hasOne))) 0 (OfNat.mk.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Icc.{0} Real (PartialOrder.toPreorder.{0} Real (SemilatticeInf.toPartialOrder.{0} Real (Lattice.toSemilatticeInf.{0} Real (LinearOrder.toLattice.{0} Real Real.linearOrder)))) (Zero.zero.{0} Real Real.hasZero) (One.one.{0} Real Real.hasOne))) 0 (Zero.zero.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Icc.{0} Real (PartialOrder.toPreorder.{0} Real (SemilatticeInf.toPartialOrder.{0} Real (Lattice.toSemilatticeInf.{0} Real (LinearOrder.toLattice.{0} Real Real.linearOrder)))) (Zero.zero.{0} Real Real.hasZero) (One.one.{0} Real Real.hasOne))) unitInterval.hasZero)))) (LE.le.{0} Real Real.hasLe x (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))
but is expected to have type
  forall {x : Real}, Iff (Eq.{1} (Set.Elem.{0} Real (Set.Icc.{0} Real (PartialOrder.toPreorder.{0} Real (SemilatticeInf.toPartialOrder.{0} Real (Lattice.toSemilatticeInf.{0} Real (DistribLattice.toLattice.{0} Real (instDistribLattice.{0} Real Real.linearOrder))))) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)))) (Set.projIcc.{0} Real Real.linearOrder (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (zero_le_one.{0} Real Real.instZeroReal Real.instOneReal (Preorder.toLE.{0} Real (PartialOrder.toPreorder.{0} Real (SemilatticeInf.toPartialOrder.{0} Real (Lattice.toSemilatticeInf.{0} Real (DistribLattice.toLattice.{0} Real (instDistribLattice.{0} Real Real.linearOrder)))))) (OrderedSemiring.zeroLEOneClass.{0} Real Real.orderedSemiring)) x) (OfNat.ofNat.{0} (Set.Elem.{0} Real (Set.Icc.{0} Real (PartialOrder.toPreorder.{0} Real (SemilatticeInf.toPartialOrder.{0} Real (Lattice.toSemilatticeInf.{0} Real (DistribLattice.toLattice.{0} Real (instDistribLattice.{0} Real Real.linearOrder))))) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)))) 0 (Zero.toOfNat0.{0} (Set.Elem.{0} Real (Set.Icc.{0} Real (PartialOrder.toPreorder.{0} Real (SemilatticeInf.toPartialOrder.{0} Real (Lattice.toSemilatticeInf.{0} Real (DistribLattice.toLattice.{0} Real (instDistribLattice.{0} Real Real.linearOrder))))) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)))) unitInterval.hasZero))) (LE.le.{0} Real Real.instLEReal x (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))
Case conversion may be inaccurate. Consider using '#align proj_Icc_eq_zero projIcc_eq_zeroₓ'. -/
@[simp]
theorem projIcc_eq_zero {x : ℝ} : projIcc (0 : ℝ) 1 zero_le_one x = 0 ↔ x ≤ 0 :=
  projIcc_eq_left zero_lt_one
#align proj_Icc_eq_zero projIcc_eq_zero

/- warning: proj_Icc_eq_one -> projIcc_eq_one is a dubious translation:
lean 3 declaration is
  forall {x : Real}, Iff (Eq.{1} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Icc.{0} Real (PartialOrder.toPreorder.{0} Real (SemilatticeInf.toPartialOrder.{0} Real (Lattice.toSemilatticeInf.{0} Real (LinearOrder.toLattice.{0} Real Real.linearOrder)))) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))))) (Set.projIcc.{0} Real Real.linearOrder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (zero_le_one.{0} Real Real.hasZero Real.hasOne (Preorder.toLE.{0} Real (PartialOrder.toPreorder.{0} Real (SemilatticeInf.toPartialOrder.{0} Real (Lattice.toSemilatticeInf.{0} Real (LinearOrder.toLattice.{0} Real Real.linearOrder))))) (OrderedSemiring.zeroLEOneClass.{0} Real Real.orderedSemiring)) x) (OfNat.ofNat.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Icc.{0} Real (PartialOrder.toPreorder.{0} Real (SemilatticeInf.toPartialOrder.{0} Real (Lattice.toSemilatticeInf.{0} Real (LinearOrder.toLattice.{0} Real Real.linearOrder)))) (Zero.zero.{0} Real Real.hasZero) (One.one.{0} Real Real.hasOne))) 1 (OfNat.mk.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Icc.{0} Real (PartialOrder.toPreorder.{0} Real (SemilatticeInf.toPartialOrder.{0} Real (Lattice.toSemilatticeInf.{0} Real (LinearOrder.toLattice.{0} Real Real.linearOrder)))) (Zero.zero.{0} Real Real.hasZero) (One.one.{0} Real Real.hasOne))) 1 (One.one.{0} (coeSort.{1, 2} (Set.{0} Real) Type (Set.hasCoeToSort.{0} Real) (Set.Icc.{0} Real (PartialOrder.toPreorder.{0} Real (SemilatticeInf.toPartialOrder.{0} Real (Lattice.toSemilatticeInf.{0} Real (LinearOrder.toLattice.{0} Real Real.linearOrder)))) (Zero.zero.{0} Real Real.hasZero) (One.one.{0} Real Real.hasOne))) unitInterval.hasOne)))) (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) x)
but is expected to have type
  forall {x : Real}, Iff (Eq.{1} (Set.Elem.{0} Real (Set.Icc.{0} Real (PartialOrder.toPreorder.{0} Real (SemilatticeInf.toPartialOrder.{0} Real (Lattice.toSemilatticeInf.{0} Real (DistribLattice.toLattice.{0} Real (instDistribLattice.{0} Real Real.linearOrder))))) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)))) (Set.projIcc.{0} Real Real.linearOrder (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (zero_le_one.{0} Real Real.instZeroReal Real.instOneReal (Preorder.toLE.{0} Real (PartialOrder.toPreorder.{0} Real (SemilatticeInf.toPartialOrder.{0} Real (Lattice.toSemilatticeInf.{0} Real (DistribLattice.toLattice.{0} Real (instDistribLattice.{0} Real Real.linearOrder)))))) (OrderedSemiring.zeroLEOneClass.{0} Real Real.orderedSemiring)) x) (OfNat.ofNat.{0} (Set.Elem.{0} Real (Set.Icc.{0} Real (PartialOrder.toPreorder.{0} Real (SemilatticeInf.toPartialOrder.{0} Real (Lattice.toSemilatticeInf.{0} Real (DistribLattice.toLattice.{0} Real (instDistribLattice.{0} Real Real.linearOrder))))) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)))) 1 (One.toOfNat1.{0} (Set.Elem.{0} Real (Set.Icc.{0} Real (PartialOrder.toPreorder.{0} Real (SemilatticeInf.toPartialOrder.{0} Real (Lattice.toSemilatticeInf.{0} Real (DistribLattice.toLattice.{0} Real (instDistribLattice.{0} Real Real.linearOrder))))) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)))) unitInterval.hasOne))) (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) x)
Case conversion may be inaccurate. Consider using '#align proj_Icc_eq_one projIcc_eq_oneₓ'. -/
@[simp]
theorem projIcc_eq_one {x : ℝ} : projIcc (0 : ℝ) 1 zero_le_one x = 1 ↔ 1 ≤ x :=
  projIcc_eq_right zero_lt_one
#align proj_Icc_eq_one projIcc_eq_one

namespace Tactic.Interactive

/- warning: tactic.interactive.unit_interval clashes with [anonymous] -> [anonymous]
Case conversion may be inaccurate. Consider using '#align tactic.interactive.unit_interval [anonymous]ₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/
/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/
/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/
/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/
#print [anonymous] /-
/-- A tactic that solves `0 ≤ ↑x`, `0 ≤ 1 - ↑x`, `↑x ≤ 1`, and `1 - ↑x ≤ 1` for `x : I`. -/
unsafe def [anonymous] : tactic Unit :=
  sorry <|> sorry <|> sorry <|> sorry
#align tactic.interactive.unit_interval [anonymous]
-/

end Tactic.Interactive

section

variable {𝕜 : Type _} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [TopologicalRing 𝕜]

/- warning: affine_homeomorph_image_I -> affineHomeomorph_image_I is a dubious translation:
lean 3 declaration is
  forall {𝕜 : Type.{u1}} [_inst_1 : LinearOrderedField.{u1} 𝕜] [_inst_2 : TopologicalSpace.{u1} 𝕜] [_inst_3 : TopologicalRing.{u1} 𝕜 _inst_2 (NonAssocRing.toNonUnitalNonAssocRing.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))))] (a : 𝕜) (b : 𝕜) (h : LT.lt.{u1} 𝕜 (Preorder.toLT.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (OrderedAddCommGroup.toPartialOrder.{u1} 𝕜 (StrictOrderedRing.toOrderedAddCommGroup.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 0 (OfNat.mk.{u1} 𝕜 0 (Zero.zero.{u1} 𝕜 (MulZeroClass.toHasZero.{u1} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))))))))) a), Eq.{succ u1} (Set.{u1} 𝕜) (Set.image.{u1, u1} 𝕜 𝕜 (coeFn.{succ u1, succ u1} (Homeomorph.{u1, u1} 𝕜 𝕜 _inst_2 _inst_2) (fun (_x : Homeomorph.{u1, u1} 𝕜 𝕜 _inst_2 _inst_2) => 𝕜 -> 𝕜) (Homeomorph.hasCoeToFun.{u1, u1} 𝕜 𝕜 _inst_2 _inst_2) (affineHomeomorph.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1) _inst_2 _inst_3 a b (Ne.symm.{succ u1} 𝕜 (OfNat.ofNat.{u1} 𝕜 0 (OfNat.mk.{u1} 𝕜 0 (Zero.zero.{u1} 𝕜 (MulZeroClass.toHasZero.{u1} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))))))))) a (LT.lt.ne.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (OrderedAddCommGroup.toPartialOrder.{u1} 𝕜 (StrictOrderedRing.toOrderedAddCommGroup.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))) (OfNat.ofNat.{u1} 𝕜 0 (OfNat.mk.{u1} 𝕜 0 (Zero.zero.{u1} 𝕜 (MulZeroClass.toHasZero.{u1} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))))))))) a h)))) (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (OrderedAddCommGroup.toPartialOrder.{u1} 𝕜 (StrictOrderedRing.toOrderedAddCommGroup.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))) (OfNat.ofNat.{u1} 𝕜 0 (OfNat.mk.{u1} 𝕜 0 (Zero.zero.{u1} 𝕜 (MulZeroClass.toHasZero.{u1} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))))))))) (OfNat.ofNat.{u1} 𝕜 1 (OfNat.mk.{u1} 𝕜 1 (One.one.{u1} 𝕜 (AddMonoidWithOne.toOne.{u1} 𝕜 (AddGroupWithOne.toAddMonoidWithOne.{u1} 𝕜 (AddCommGroupWithOne.toAddGroupWithOne.{u1} 𝕜 (Ring.toAddCommGroupWithOne.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))))))))))) (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (OrderedAddCommGroup.toPartialOrder.{u1} 𝕜 (StrictOrderedRing.toOrderedAddCommGroup.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))) b (HAdd.hAdd.{u1, u1, u1} 𝕜 𝕜 𝕜 (instHAdd.{u1} 𝕜 (Distrib.toHasAdd.{u1} 𝕜 (Ring.toDistrib.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))))) a b))
but is expected to have type
  forall {𝕜 : Type.{u1}} [_inst_1 : LinearOrderedField.{u1} 𝕜] [_inst_2 : TopologicalSpace.{u1} 𝕜] [_inst_3 : TopologicalRing.{u1} 𝕜 _inst_2 (NonAssocRing.toNonUnitalNonAssocRing.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1)))))] (a : 𝕜) (b : 𝕜) (h : LT.lt.{u1} 𝕜 (Preorder.toLT.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) a), Eq.{succ u1} (Set.{u1} 𝕜) (Set.image.{u1, u1} 𝕜 𝕜 (FunLike.coe.{succ u1, succ u1, succ u1} (Homeomorph.{u1, u1} 𝕜 𝕜 _inst_2 _inst_2) 𝕜 (fun (_x : 𝕜) => 𝕜) (EmbeddingLike.toFunLike.{succ u1, succ u1, succ u1} (Homeomorph.{u1, u1} 𝕜 𝕜 _inst_2 _inst_2) 𝕜 𝕜 (EquivLike.toEmbeddingLike.{succ u1, succ u1, succ u1} (Homeomorph.{u1, u1} 𝕜 𝕜 _inst_2 _inst_2) 𝕜 𝕜 (Homeomorph.instEquivLikeHomeomorph.{u1, u1} 𝕜 𝕜 _inst_2 _inst_2))) (affineHomeomorph.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1) _inst_2 _inst_3 a b (Ne.symm.{succ u1} 𝕜 (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) a (LT.lt.ne.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) a h)))) (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) b (HAdd.hAdd.{u1, u1, u1} 𝕜 𝕜 𝕜 (instHAdd.{u1} 𝕜 (Distrib.toAdd.{u1} 𝕜 (NonUnitalNonAssocSemiring.toDistrib.{u1} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) a b))
Case conversion may be inaccurate. Consider using '#align affine_homeomorph_image_I affineHomeomorph_image_Iₓ'. -/
-- We only need the ordering on `𝕜` here to avoid talking about flipping the interval over.
-- At the end of the day I only care about `ℝ`, so I'm hesitant to put work into generalizing.
/-- The image of `[0,1]` under the homeomorphism `λ x, a * x + b` is `[b, a+b]`.
-/
theorem affineHomeomorph_image_I (a b : 𝕜) (h : 0 < a) :
    affineHomeomorph a b h.Ne.symm '' Set.Icc 0 1 = Set.Icc b (a + b) := by simp [h]
#align affine_homeomorph_image_I affineHomeomorph_image_I

/- warning: Icc_homeo_I -> iccHomeoI is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align Icc_homeo_I iccHomeoIₓ'. -/
/-- The affine homeomorphism from a nontrivial interval `[a,b]` to `[0,1]`.
-/
def iccHomeoI (a b : 𝕜) (h : a < b) : Set.Icc a b ≃ₜ Set.Icc (0 : 𝕜) (1 : 𝕜) :=
  by
  let e := Homeomorph.image (affineHomeomorph (b - a) a (sub_pos.mpr h).Ne.symm) (Set.Icc 0 1)
  refine' (e.trans _).symm
  apply Homeomorph.setCongr
  simp [sub_pos.mpr h]
#align Icc_homeo_I iccHomeoI

/- warning: Icc_homeo_I_apply_coe -> iccHomeoI_apply_coe is a dubious translation:
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a b h) x)) (HDiv.hDiv.{u1, u1, u1} 𝕜 𝕜 𝕜 (instHDiv.{u1} 𝕜 (DivInvMonoid.toHasDiv.{u1} 𝕜 (DivisionRing.toDivInvMonoid.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))) (HSub.hSub.{u1, u1, u1} 𝕜 𝕜 𝕜 (instHSub.{u1} 𝕜 (SubNegMonoid.toHasSub.{u1} 𝕜 (AddGroup.toSubNegMonoid.{u1} 𝕜 (AddGroupWithOne.toAddGroup.{u1} 𝕜 (AddCommGroupWithOne.toAddGroupWithOne.{u1} 𝕜 (Ring.toAddCommGroupWithOne.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))))))) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} 𝕜) Type.{u1} (Set.hasCoeToSort.{u1} 𝕜) (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (OrderedAddCommGroup.toPartialOrder.{u1} 𝕜 (StrictOrderedRing.toOrderedAddCommGroup.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))) a b)) 𝕜 (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} 𝕜) Type.{u1} (Set.hasCoeToSort.{u1} 𝕜) (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (OrderedAddCommGroup.toPartialOrder.{u1} 𝕜 (StrictOrderedRing.toOrderedAddCommGroup.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))) a b)) 𝕜 (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} 𝕜) Type.{u1} (Set.hasCoeToSort.{u1} 𝕜) (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (OrderedAddCommGroup.toPartialOrder.{u1} 𝕜 (StrictOrderedRing.toOrderedAddCommGroup.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))) a b)) 𝕜 (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} 𝕜) Type.{u1} (Set.hasCoeToSort.{u1} 𝕜) (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (OrderedAddCommGroup.toPartialOrder.{u1} 𝕜 (StrictOrderedRing.toOrderedAddCommGroup.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))) a b)) 𝕜 (coeSubtype.{succ u1} 𝕜 (fun (x : 𝕜) => Membership.Mem.{u1, u1} 𝕜 (Set.{u1} 𝕜) (Set.hasMem.{u1} 𝕜) x (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (OrderedAddCommGroup.toPartialOrder.{u1} 𝕜 (StrictOrderedRing.toOrderedAddCommGroup.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))) a b)))))) x) a) (HSub.hSub.{u1, u1, u1} 𝕜 𝕜 𝕜 (instHSub.{u1} 𝕜 (SubNegMonoid.toHasSub.{u1} 𝕜 (AddGroup.toSubNegMonoid.{u1} 𝕜 (AddGroupWithOne.toAddGroup.{u1} 𝕜 (AddCommGroupWithOne.toAddGroupWithOne.{u1} 𝕜 (Ring.toAddCommGroupWithOne.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))))))) b a))
but is expected to have type
  forall {𝕜 : Type.{u1}} [_inst_1 : LinearOrderedField.{u1} 𝕜] [_inst_2 : TopologicalSpace.{u1} 𝕜] [_inst_3 : TopologicalRing.{u1} 𝕜 _inst_2 (NonAssocRing.toNonUnitalNonAssocRing.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1)))))] (a : 𝕜) (b : 𝕜) (h : LT.lt.{u1} 𝕜 (Preorder.toLT.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))) a b) (x : Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)), Eq.{succ u1} 𝕜 (Subtype.val.{succ u1} 𝕜 (fun (x : 𝕜) => Membership.mem.{u1, u1} 𝕜 (Set.{u1} 𝕜) (Set.instMembershipSet.{u1} 𝕜) x (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) (FunLike.coe.{succ u1, succ u1, succ u1} (Homeomorph.{u1, u1} (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) (instTopologicalSpaceSubtype.{u1} 𝕜 (fun (x : 𝕜) => Membership.mem.{u1, u1} 𝕜 (Set.{u1} 𝕜) (Set.instMembershipSet.{u1} 𝕜) x (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) _inst_2) (instTopologicalSpaceSubtype.{u1} 𝕜 (fun (x : 𝕜) => Membership.mem.{u1, u1} 𝕜 (Set.{u1} 𝕜) (Set.instMembershipSet.{u1} 𝕜) x (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) _inst_2)) (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) (fun (_x : Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) => Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) (EmbeddingLike.toFunLike.{succ u1, succ u1, succ u1} (Homeomorph.{u1, u1} (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) (instTopologicalSpaceSubtype.{u1} 𝕜 (fun (x : 𝕜) => Membership.mem.{u1, u1} 𝕜 (Set.{u1} 𝕜) (Set.instMembershipSet.{u1} 𝕜) x (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) _inst_2) 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Case conversion may be inaccurate. Consider using '#align Icc_homeo_I_apply_coe iccHomeoI_apply_coeₓ'. -/
@[simp]
theorem iccHomeoI_apply_coe (a b : 𝕜) (h : a < b) (x : Set.Icc a b) :
    ((iccHomeoI a b h) x : 𝕜) = (x - a) / (b - a) :=
  rfl
#align Icc_homeo_I_apply_coe iccHomeoI_apply_coe

/- warning: Icc_homeo_I_symm_apply_coe -> iccHomeoI_symm_apply_coe is a dubious translation:
lean 3 declaration is
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𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))))))))) (OfNat.ofNat.{u1} 𝕜 1 (OfNat.mk.{u1} 𝕜 1 (One.one.{u1} 𝕜 (AddMonoidWithOne.toOne.{u1} 𝕜 (AddGroupWithOne.toAddMonoidWithOne.{u1} 𝕜 (AddCommGroupWithOne.toAddGroupWithOne.{u1} 𝕜 (Ring.toAddCommGroupWithOne.{u1} 𝕜 (StrictOrderedRing.toRing.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))))))))))))))) x)) a)
but is expected to have type
  forall {𝕜 : Type.{u1}} [_inst_1 : LinearOrderedField.{u1} 𝕜] [_inst_2 : TopologicalSpace.{u1} 𝕜] [_inst_3 : TopologicalRing.{u1} 𝕜 _inst_2 (NonAssocRing.toNonUnitalNonAssocRing.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1)))))] (a : 𝕜) (b : 𝕜) (h : LT.lt.{u1} 𝕜 (Preorder.toLT.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1)))))) a b) (x : Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))), Eq.{succ u1} 𝕜 (Subtype.val.{succ u1} 𝕜 (fun (x : 𝕜) => Membership.mem.{u1, u1} 𝕜 (Set.{u1} 𝕜) (Set.instMembershipSet.{u1} 𝕜) x (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) (FunLike.coe.{succ u1, succ u1, succ u1} (Homeomorph.{u1, u1} (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) (instTopologicalSpaceSubtype.{u1} 𝕜 (fun (x : 𝕜) => Membership.mem.{u1, u1} 𝕜 (Set.{u1} 𝕜) (Set.instMembershipSet.{u1} 𝕜) x (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) _inst_2) (instTopologicalSpaceSubtype.{u1} 𝕜 (fun (x : 𝕜) => Membership.mem.{u1, u1} 𝕜 (Set.{u1} 𝕜) (Set.instMembershipSet.{u1} 𝕜) x (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) _inst_2)) (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) (fun (_x : Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) => Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) (EmbeddingLike.toFunLike.{succ u1, succ u1, succ u1} (Homeomorph.{u1, u1} (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) (instTopologicalSpaceSubtype.{u1} 𝕜 (fun (x : 𝕜) => Membership.mem.{u1, u1} 𝕜 (Set.{u1} 𝕜) (Set.instMembershipSet.{u1} 𝕜) x (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) _inst_2) (instTopologicalSpaceSubtype.{u1} 𝕜 (fun (x : 𝕜) => Membership.mem.{u1, u1} 𝕜 (Set.{u1} 𝕜) (Set.instMembershipSet.{u1} 𝕜) x (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) _inst_2)) (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) (EquivLike.toEmbeddingLike.{succ u1, succ u1, succ u1} (Homeomorph.{u1, u1} (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) (instTopologicalSpaceSubtype.{u1} 𝕜 (fun (x : 𝕜) => Membership.mem.{u1, u1} 𝕜 (Set.{u1} 𝕜) (Set.instMembershipSet.{u1} 𝕜) x (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) _inst_2) (instTopologicalSpaceSubtype.{u1} 𝕜 (fun (x : 𝕜) => Membership.mem.{u1, u1} 𝕜 (Set.{u1} 𝕜) (Set.instMembershipSet.{u1} 𝕜) x (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) _inst_2)) (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) (Homeomorph.instEquivLikeHomeomorph.{u1, u1} (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) (instTopologicalSpaceSubtype.{u1} 𝕜 (fun (x : 𝕜) => Membership.mem.{u1, u1} 𝕜 (Set.{u1} 𝕜) (Set.instMembershipSet.{u1} 𝕜) x (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) _inst_2) (instTopologicalSpaceSubtype.{u1} 𝕜 (fun (x : 𝕜) => Membership.mem.{u1, u1} 𝕜 (Set.{u1} 𝕜) (Set.instMembershipSet.{u1} 𝕜) x (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) _inst_2)))) (Homeomorph.symm.{u1, u1} (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) (Set.Elem.{u1} 𝕜 (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) (instTopologicalSpaceSubtype.{u1} 𝕜 (fun (x : 𝕜) => Membership.mem.{u1, u1} 𝕜 (Set.{u1} 𝕜) (Set.instMembershipSet.{u1} 𝕜) x (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) a b)) _inst_2) (instTopologicalSpaceSubtype.{u1} 𝕜 (fun (x : 𝕜) => Membership.mem.{u1, u1} 𝕜 (Set.{u1} 𝕜) (Set.instMembershipSet.{u1} 𝕜) x (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) _inst_2) (iccHomeoI.{u1} 𝕜 _inst_1 _inst_2 _inst_3 a b h)) x)) (HAdd.hAdd.{u1, u1, u1} 𝕜 𝕜 𝕜 (instHAdd.{u1} 𝕜 (Distrib.toAdd.{u1} 𝕜 (NonUnitalNonAssocSemiring.toDistrib.{u1} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) (HMul.hMul.{u1, u1, u1} 𝕜 𝕜 𝕜 (instHMul.{u1} 𝕜 (NonUnitalNonAssocRing.toMul.{u1} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))) (HSub.hSub.{u1, u1, u1} 𝕜 𝕜 𝕜 (instHSub.{u1} 𝕜 (Ring.toSub.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))) b a) (Subtype.val.{succ u1} 𝕜 (fun (x : 𝕜) => Membership.mem.{u1, u1} 𝕜 (Set.{u1} 𝕜) (Set.instMembershipSet.{u1} 𝕜) x (Set.Icc.{u1} 𝕜 (PartialOrder.toPreorder.{u1} 𝕜 (StrictOrderedRing.toPartialOrder.{u1} 𝕜 (LinearOrderedRing.toStrictOrderedRing.{u1} 𝕜 (LinearOrderedCommRing.toLinearOrderedRing.{u1} 𝕜 (LinearOrderedField.toLinearOrderedCommRing.{u1} 𝕜 _inst_1))))) (OfNat.ofNat.{u1} 𝕜 0 (Zero.toOfNat0.{u1} 𝕜 (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (LinearOrderedSemifield.toSemifield.{u1} 𝕜 (LinearOrderedField.toLinearOrderedSemifield.{u1} 𝕜 _inst_1))))))) (OfNat.ofNat.{u1} 𝕜 1 (One.toOfNat1.{u1} 𝕜 (NonAssocRing.toOne.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (DivisionRing.toRing.{u1} 𝕜 (Field.toDivisionRing.{u1} 𝕜 (LinearOrderedField.toField.{u1} 𝕜 _inst_1))))))))) x)) a)
Case conversion may be inaccurate. Consider using '#align Icc_homeo_I_symm_apply_coe iccHomeoI_symm_apply_coeₓ'. -/
@[simp]
theorem iccHomeoI_symm_apply_coe (a b : 𝕜) (h : a < b) (x : Set.Icc (0 : 𝕜) (1 : 𝕜)) :
    ((iccHomeoI a b h).symm x : 𝕜) = (b - a) * x + a :=
  rfl
#align Icc_homeo_I_symm_apply_coe iccHomeoI_symm_apply_coe

end