-- Mathlib/Analysis/Complex/Hadamard.lean
/-- **Hadamard three-line theorem** on `re ⁻¹' [0, 1]` (Variant in simpler terms): Let `f` be a
bounded function, continuous on the closed strip `re ⁻¹' [0, 1]` and differentiable on open strip
`re ⁻¹' (0, 1)`. If, for all `z.re = 0`, `‖f z‖ ≤ a` for some `a ∈ ℝ` and, similarly, for all
`z.re = 1`, `‖f z‖ ≤ b` for some `b ∈ ℝ` then for all `z` in the closed strip
`re ⁻¹' [0, 1]` the inequality `‖f(z)‖ ≤ a ^ (1 - z.re) * b ^ z.re` holds. -/
lemma norm_le_interp_of_mem_verticalClosedStrip₀₁' (f : ℂ → E) {z : ℂ} {a b : ℝ}
    (hz : z ∈ verticalClosedStrip 0 1) (hd : DiffContOnCl ℂ f (verticalStrip 0 1))
    (hB : BddAbove ((norm ∘ f) '' verticalClosedStrip 0 1))
    (ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a) (hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b) :
    ‖f z‖ ≤ a ^ (1 - z.re) * b ^ z.re

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSupNormIm f 0 ^ (1 - z.re) ≤ a ^ (1 - z.re)

Tactic:
apply Real.rpow_le_rpow (sSupNormIm_nonneg f _) _ (sub_nonneg.mpr hz.2)

Result:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSupNormIm f 0 ≤ a

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSupNormIm f 0 ≤ a

Tactic:
rw [sSupNormIm]

Result:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSup (norm ∘ f '' (re ⁻¹' {0})) ≤ a

Proof state:
case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ 0 ∈ re ⁻¹' {0} ∧ (norm ∘ f) 0 = ‖f 0‖

Tactic:
simp only [mem_preimage, zero_re, mem_singleton_iff, comp_apply,
  and_self]

Result:
no goals

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSup (norm ∘ f '' (re ⁻¹' {0})) ≤ a

Tactic:
apply csSup_le _

Result:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ ∀ b ∈ norm ∘ f '' (re ⁻¹' {0}), b ≤ a

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ (norm ∘ f '' (re ⁻¹' {0})).Nonempty

Proof state:
case neg.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
h0 : 0 ≠ sSupNormIm f 0
h1 : 0 ≠ sSupNormIm f 1
⊢ sSupNormIm f 0 ^ (1 - z).re * sSupNormIm f 1 ^ z.re ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re

Tactic:
simp only [sub_re, one_re, le_refl]

Result:
no goals

Proof state:
case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ 1 ∈ re ⁻¹' {1} ∧ (norm ∘ f) 1 = ‖f 1‖

Tactic:
simp only [mem_preimage, one_re, mem_singleton_iff, comp_apply,
  and_self]

Result:
no goals

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ ∀ b ∈ norm ∘ f '' (re ⁻¹' {0}), b ≤ a

Tactic:
simpa [comp_apply, mem_image, forall_exists_index,
  and_imp, forall_apply_eq_imp_iff₂] using ha

Result:
no goals

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSupNormIm f 1 ^ z.re ≤ b ^ z.re

Tactic:
apply Real.rpow_le_rpow (sSupNormIm_nonneg f _) _ hz.1

Result:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSupNormIm f 1 ≤ b

Proof state:
case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
h : ¬(sSupNormIm f 0 = 0 ∨ sSupNormIm f 1 = 0)
⊢ ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re

Tactic:
push_neg at h

Result:
case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
h : sSupNormIm f 0 ≠ 0 ∧ sSupNormIm f 1 ≠ 0
⊢ ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re