-- Mathlib/Algebra/Order/BigOperators/Ring/Finset.lean
/-- If `g, h ≤ f` and `g i + h i ≤ f i`, then the product of `f` over `s` is at least the
  sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedAdd`.
-/
lemma prod_add_prod_le' (hi : i ∈ s) (h2i : g i + h i ≤ f i) (hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j)
    (hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j) : ((∏ i ∈ s, g i) + ∏ i ∈ s, h i) ≤ ∏ i ∈ s, f i

Proof state:
ι : Type u_1
R : Type u_2
inst✝² : CommSemiring R
inst✝¹ : PartialOrder R
inst✝ : CanonicallyOrderedAdd R
f g h : ι → R
s : Finset ι
i : ι
hi : i ∈ s
h2i : g i + h i ≤ f i
hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j
hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j
⊢ g i * ∏ i ∈ s \ {i}, g i + h i * ∏ i ∈ s \ {i}, h i ≤ g i * ∏ i ∈ s \ {i}, f i + h i * ∏ i ∈ s \ {i}, f i

Tactic:
gcongr with j hj j hj <;> simp_all

Result:
no goals

Proof state:
ι : Type u_1
R : Type u_2
inst✝² : CommSemiring R
inst✝¹ : PartialOrder R
inst✝ : CanonicallyOrderedAdd R
f g h : ι → R
s : Finset ι
i : ι
hi : i ∈ s
h2i : g i + h i ≤ f i
hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j
hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j
⊢ g i * ∏ i ∈ s \ {i}, g i + h i * ∏ i ∈ s \ {i}, h i ≤ (g i + h i) * ∏ i ∈ s \ {i}, f i

Tactic:
rw [right_distrib]

Result:
ι : Type u_1
R : Type u_2
inst✝² : CommSemiring R
inst✝¹ : PartialOrder R
inst✝ : CanonicallyOrderedAdd R
f g h : ι → R
s : Finset ι
i : ι
hi : i ∈ s
h2i : g i + h i ≤ f i
hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j
hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j
⊢ g i * ∏ i ∈ s \ {i}, g i + h i * ∏ i ∈ s \ {i}, h i ≤ g i * ∏ i ∈ s \ {i}, f i + h i * ∏ i ∈ s \ {i}, f i

Proof state:
ι : Type u_1
R : Type u_2
inst✝² : CommSemiring R
inst✝¹ : PartialOrder R
inst✝ : CanonicallyOrderedAdd R
f g h : ι → R
s : Finset ι
i : ι
hi : i ∈ s
h2i : g i + h i ≤ f i
hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j
hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j
⊢ g i * ∏ i ∈ s \ {i}, g i + h i * ∏ i ∈ s \ {i}, h i ≤ f i * ∏ i ∈ s \ {i}, f i

Tactic:
refine le_trans ?_ (mul_le_mul_right' h2i _)

Result:
ι : Type u_1
R : Type u_2
inst✝² : CommSemiring R
inst✝¹ : PartialOrder R
inst✝ : CanonicallyOrderedAdd R
f g h : ι → R
s : Finset ι
i : ι
hi : i ∈ s
h2i : g i + h i ≤ f i
hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j
hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j
⊢ g i * ∏ i ∈ s \ {i}, g i + h i * ∏ i ∈ s \ {i}, h i ≤ (g i + h i) * ∏ i ∈ s \ {i}, f i

Proof state:
ι : Type u_1
R : Type u_2
inst✝² : CommSemiring R
inst✝¹ : PartialOrder R
inst✝ : CanonicallyOrderedAdd R
f g h : ι → R
s : Finset ι
i : ι
hi : i ∈ s
h2i : g i + h i ≤ f i
hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j
hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j
⊢ ∏ i ∈ s, g i + ∏ i ∈ s, h i ≤ ∏ i ∈ s, f i

Tactic:
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_right' h2i _)
rw [right_distrib]
gcongr with j hj j hj <;> simp_all

Result:
no goals

Proof state:
ι : Type u_1
R : Type u_2
inst✝² : CommSemiring R
inst✝¹ : PartialOrder R
inst✝ : CanonicallyOrderedAdd R
f g h : ι → R
s : Finset ι
i : ι
hi : i ∈ s
h2i : g i + h i ≤ f i
hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j
hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j
⊢ ∏ i ∈ s, g i + ∏ i ∈ s, h i ≤ ∏ i ∈ s, f i

Tactic:
simp_rw [prod_eq_mul_prod_diff_singleton hi]

Result:
ι : Type u_1
R : Type u_2
inst✝² : CommSemiring R
inst✝¹ : PartialOrder R
inst✝ : CanonicallyOrderedAdd R
f g h : ι → R
s : Finset ι
i : ι
hi : i ∈ s
h2i : g i + h i ≤ f i
hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j
hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j
⊢ g i * ∏ i ∈ s \ {i}, g i + h i * ∏ i ∈ s \ {i}, h i ≤ f i * ∏ i ∈ s \ {i}, f i