-- Mathlib/Topology/Order/IntermediateValue.lean
theorem Continuous.strictMonoOn_of_inj_rigidity {f : α → δ}
    (hf_c : Continuous f) (hf_i : Injective f) {a b : α} (hab : a < b)
    (hf_mono : StrictMonoOn f (Icc a b)) : StrictMono 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
f : α → δ
hf_c : Continuous f
hf_i : Injective f
a b : α
hab : a < b
hf_mono : StrictMonoOn f (Icc a b)
x y : α
hxy : x < y
⊢ f x < f y

Tactic:
let s := min a x

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
f : α → δ
hf_c : Continuous f
hf_i : Injective f
a b : α
hab : a < b
hf_mono : StrictMonoOn f (Icc a b)
x y : α
hxy : x < y
s : α := a ⊓ x
⊢ f x < f y

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
f : α → δ
hf_c : Continuous f
hf_i : Injective f
a b : α
hab : a < b
hf_mono : StrictMonoOn f (Icc a b)
x y : α
hxy : x < y
s : α := a ⊓ x
t : α := b ⊔ y
hsa : s ≤ a
hbt : b ≤ t
this✝ : Fact (s ≤ t)
this : StrictMono ((Icc s t).restrict f) ∨ StrictAnti ((Icc s t).restrict f)
⊢ StrictMonoOn f (Icc s t) ∨ StrictAntiOn f (Icc s t)

Tactic:
exact this.imp strictMono_restrict.mp strictAntiOn_iff_strictAnti.mpr

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
f : α → δ
hf_c : Continuous f
hf_i : Injective f
a b : α
hab : a < b
hf_mono : StrictMonoOn f (Icc a b)
x y : α
hxy : x < y
s : α := a ⊓ x
t : α := b ⊔ y
hsa : s ≤ a
⊢ f x < f y

Tactic:
have hbt : b ≤ t := le_max_left b y

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
f : α → δ
hf_c : Continuous f
hf_i : Injective f
a b : α
hab : a < b
hf_mono : StrictMonoOn f (Icc a b)
x y : α
hxy : x < y
s : α := a ⊓ x
t : α := b ⊔ y
hsa : s ≤ a
hbt : b ≤ t
⊢ f x < f y

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
f : α → δ
hf_c : Continuous f
hf_i : Injective f
a b : α
hab : a < b
hf_mono : StrictMonoOn f (Icc a b)
x y : α
hxy : x < y
s : α := a ⊓ x
t : α := b ⊔ y
⊢ f x < f y

Tactic:
have hsa : s ≤ a := min_le_left a x

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
f : α → δ
hf_c : Continuous f
hf_i : Injective f
a b : α
hab : a < b
hf_mono : StrictMonoOn f (Icc a b)
x y : α
hxy : x < y
s : α := a ⊓ x
t : α := b ⊔ y
hsa : s ≤ a
⊢ f x < f y