-- Mathlib/Topology/Order/IntermediateValue.lean
theorem ContinuousOn.strictMonoOn_of_injOn_Icc {a b : α} {f : α → δ}
    (hab : a ≤ b) (hfab : f a ≤ f b)
    (hf_c : ContinuousOn f (Icc a b)) (hf_i : InjOn f (Icc a b)) :
    StrictMonoOn f (Icc a b)

Proof state:
α : Type u
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
this : Fact (a ≤ b)
g : ↑(Icc a b) → δ := (Icc a b).restrict f
⊢ StrictMono g

Tactic:
have hgab : g ⊥ ≤ g ⊤ := by aesop

Result:
α : Type u
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
this : Fact (a ≤ b)
g : ↑(Icc a b) → δ := (Icc a b).restrict f
hgab : g ⊥ ≤ g ⊤
⊢ StrictMono g

Proof state:
α : Type u
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
this : Fact (a ≤ b)
⊢ StrictMonoOn f (Icc a b)

Tactic:
refine StrictMono.of_restrict ?_

Result:
α : Type u
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
this : Fact (a ≤ b)
⊢ StrictMono ((Icc a b).restrict f)

Proof state:
α : Type u
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
⊢ StrictMonoOn f (Icc a b)

Tactic:
have : Fact (a ≤ b) := ⟨hab⟩

Result:
α : Type u
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
this : Fact (a ≤ b)
⊢ StrictMonoOn f (Icc a b)

Proof state:
α : Type u
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
this : Fact (a ≤ b)
g : ↑(Icc a b) → δ := (Icc a b).restrict f
⊢ g ⊥ ≤ g ⊤

Tactic:
aesop

Result:
no goals

Proof state:
α : Type u
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
this : Fact (a ≤ b)
⊢ StrictMono ((Icc a b).restrict f)

Tactic:
set g : Icc a b → δ := Set.restrict (Icc a b) f

Result:
α : Type u
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
this : Fact (a ≤ b)
g : ↑(Icc a b) → δ := (Icc a b).restrict f
⊢ StrictMono g

Proof state:
α : Type u
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hab : a ≤ b
hfab : f a ≤ f b
hf_c : ContinuousOn f (Icc a b)
hf_i : InjOn f (Icc a b)
this : Fact (a ≤ b)
g : ↑(Icc a b) → δ := (Icc a b).restrict f
hgab : g ⊥ ≤ g ⊤
⊢ StrictMono g

Tactic:
exact Continuous.strictMono_of_inj_boundedOrder (f := g) hf_c.restrict hgab hf_i.injective

Result:
no goals