Begin raising a valuation to a pnat power

Author: PatrickMassot

src/examples/valuations.lean

@@ -0,0 +1,62 @@
+/-
+Stuff in this file should eventually go to the valuation folder,
+but keeping it here is faster at this point.
+-/
+
+import valuation.basic
+
+local attribute [instance, priority 0] classical.decidable_linear_order
+
+section
+@[elab_as_eliminator] protected lemma pnat.induction_on {p : ℕ+ → Prop}
+  (i : ℕ+) (hz : p 1) (hp : ∀j : ℕ+, p j → p (j + 1)) : p i :=
+begin
+  sorry
+end
+
+variables (Γ₀ : Type*)  [linear_ordered_comm_group_with_zero Γ₀]
+open  linear_ordered_structure
+
+lemma linear_ordered_comm_monoid.pow_strict_mono (n : ℕ+) : strict_mono (λ x : Γ₀, x^(n : ℕ)) :=
+begin
+  intros x y h,
+  induction n using pnat.induction_on with n ih,
+  { simpa },
+  { simp only [] at *,
+    rw [pnat.add_coe, pnat.one_coe, pow_succ, pow_succ], -- here we miss some norm_cast attribute
+    by_cases hx : x = 0,
+    { simp [hx] at *,
+
+      sorry },
+    { -- do x * ih (using that x ≠ 0) and then h * y^n (using 0 < x^n < y^n)
+      sorry } }
+end
+
+lemma linear_ordered_comm_monoid.pow_mono (n : ℕ+) : monotone (λ x : Γ₀, x^(n : ℕ)) :=
+(linear_ordered_comm_monoid.pow_strict_mono Γ₀ n).monotone
+
+variables {Γ₀}
+lemma linear_ordered_comm_monoid.pow_le_pow {x y : Γ₀} {n : ℕ+} : x^(n : ℕ) ≤ y^(n : ℕ) ↔ x ≤ y :=
+strict_mono.le_iff_le (linear_ordered_comm_monoid.pow_strict_mono Γ₀ n)
+
+end
+
+variables {R : Type*} [ring R]
+
+variables {Γ₀   : Type*}  [linear_ordered_comm_group_with_zero Γ₀]
+
+instance valuation.pow : has_pow (valuation R Γ₀) ℕ+ :=
+⟨λ v n, { to_fun := λ r, (v r)^n.val,
+  map_one' := by rw [v.map_one, one_pow],
+  map_mul' := λ x y, by rw [v.map_mul, mul_pow],
+  map_zero' := by rw [valuation.map_zero, ← nat.succ_pred_eq_of_pos n.2, pow_succ, zero_mul],
+  map_add' := begin
+    intros x y,
+    wlog vyx : v y ≤ v x using x y,
+    { have : (v y)^n.val ≤ (v x)^n.val, by apply linear_ordered_comm_monoid.pow_mono ; assumption,
+      rw max_eq_left this,
+      apply linear_ordered_comm_monoid.pow_mono,
+      convert v.map_add x y,
+      rw max_eq_left vyx },
+    { rwa [add_comm, max_comm] },
+  end }⟩