Theorem: Continuous.strictMono_of_inj
File: Mathlib/Topology/Order/IntermediateValue.lean

tactic: have H {c d : α} (hcd : c < d) : StrictMono f ∨ StrictAnti f :=
  (hf_c.continuousOn.strictMonoOn_of_injOn_Icc' hcd.le hf_i.injOn).imp
    (hf_c.strictMonoOn_of_inj_rigidity hf_i hcd)
    (hf_c.strictMonoOn_of_inj_rigidity (δ := δᵒᵈ) hf_i hcd)
state_before: α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a b : 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
⊢ StrictMono f ∨ StrictAnti f
state_after: α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a b : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
⊢ StrictMono f ∨ StrictAnti f

tactic: by_cases hn : Nonempty α
state_before: α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a b : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
⊢ StrictMono f ∨ StrictAnti f
state_after: case pos
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a b : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
⊢ StrictMono f ∨ StrictAnti f

case neg
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a b : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : ¬Nonempty α
⊢ StrictMono f ∨ StrictAnti f

tactic: let a : α := Classical.choice ‹_›
state_before: case pos
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a b : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
⊢ StrictMono f ∨ StrictAnti f
state_after: case pos
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
⊢ StrictMono f ∨ StrictAnti f

tactic: by_cases h : ∃ b : α, a ≠ b
state_before: case pos
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
⊢ StrictMono f ∨ StrictAnti f
state_after: case pos
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
h : ∃ b, a ≠ b
⊢ StrictMono f ∨ StrictAnti f

case neg
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
h : ¬∃ b, a ≠ b
⊢ StrictMono f ∨ StrictAnti f

tactic: choose b hb using h
state_before: case pos
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
h : ∃ b, a ≠ b
⊢ StrictMono f ∨ StrictAnti f
state_after: case pos
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
b : α
hb : a ≠ b
⊢ StrictMono f ∨ StrictAnti f

tactic: by_cases hab : a < b
state_before: case pos
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
b : α
hb : a ≠ b
⊢ StrictMono f ∨ StrictAnti f
state_after: case pos
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
b : α
hb : a ≠ b
hab : a < b
⊢ StrictMono f ∨ StrictAnti f

case neg
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
b : α
hb : a ≠ b
hab : ¬a < b
⊢ StrictMono f ∨ StrictAnti f

tactic: exact H hab
state_before: case pos
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
b : α
hb : a ≠ b
hab : a < b
⊢ StrictMono f ∨ StrictAnti f
state_after: no goals

tactic: push_neg at hab
state_before: case neg
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
b : α
hb : a ≠ b
hab : ¬a < b
⊢ StrictMono f ∨ StrictAnti f
state_after: case neg
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
b : α
hb : a ≠ b
hab : b ≤ a
⊢ StrictMono f ∨ StrictAnti f

tactic: have : b < a := by exact Ne.lt_of_le (id (Ne.symm hb)) hab
state_before: case neg
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
b : α
hb : a ≠ b
hab : b ≤ a
⊢ StrictMono f ∨ StrictAnti f
state_after: case neg
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
b : α
hb : a ≠ b
hab : b ≤ a
this : b < a
⊢ StrictMono f ∨ StrictAnti f

tactic: exact H this
state_before: case neg
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
b : α
hb : a ≠ b
hab : b ≤ a
this : b < a
⊢ StrictMono f ∨ StrictAnti f
state_after: no goals

tactic: exact Ne.lt_of_le (id (Ne.symm hb)) hab
state_before: α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
b : α
hb : a ≠ b
hab : b ≤ a
⊢ b < a
state_after: no goals

tactic: push_neg at h
state_before: case neg
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
h : ¬∃ b, a ≠ b
⊢ StrictMono f ∨ StrictAnti f
state_after: case neg
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
h : ∀ (b : α), a = b
⊢ StrictMono f ∨ StrictAnti f

tactic: haveI : Subsingleton α := ⟨fun c d => Trans.trans (h c).symm (h d)⟩
state_before: case neg
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
h : ∀ (b : α), a = b
⊢ StrictMono f ∨ StrictAnti f
state_after: case neg
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
h : ∀ (b : α), a = b
this : Subsingleton α
⊢ StrictMono f ∨ StrictAnti f

tactic: exact Or.inl <| Subsingleton.strictMono f
state_before: case neg
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : Nonempty α
a : α := Classical.choice hn
h : ∀ (b : α), a = b
this : Subsingleton α
⊢ StrictMono f ∨ StrictAnti f
state_after: no goals

tactic: aesop
state_before: case neg
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a b : 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
H : ∀ {c d : α}, c < d → StrictMono f ∨ StrictAnti f
hn : ¬Nonempty α
⊢ StrictMono f ∨ StrictAnti f
state_after: no goals