feat(algebra/module/linear_map): Add linear_map.iterate (#6377)

Co-authored-by: Alex J. Best <alex.j.best@gmail.com>





Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

src/algebra/module/linear_map.lean

@@ -217,6 +217,12 @@ def comp : M →ₗ[R] M₃ := ⟨f ∘ g, by simp, by simp⟩
 @[norm_cast]
 lemma comp_coe : (f : M₂ → M₃) ∘ (g : M → M₂) = f.comp g := rfl
 
+@[simp] theorem comp_id : f.comp id = f :=
+linear_map.ext $ λ x, rfl
+
+@[simp] theorem id_comp : id.comp f = f :=
+linear_map.ext $ λ x, rfl
+
 end
 
 /-- If a function `g` is a left and right inverse of a linear map `f`, then `g` is linear itself. -/

src/linear_algebra/basic.lean

@@ -106,12 +106,6 @@ variables [semimodule R M] [semimodule R M₂] [semimodule R M₃] [semimodule R
 variables (f g : M →ₗ[R] M₂)
 include R
 
-@[simp] theorem comp_id : f.comp id = f :=
-linear_map.ext $ λ x, rfl
-
-@[simp] theorem id_comp : id.comp f = f :=
-linear_map.ext $ λ x, rfl
-
 theorem comp_assoc (g : M₂ →ₗ[R] M₃) (h : M₃ →ₗ[R] M₄) : (h.comp g).comp f = h.comp (g.comp f) :=
 rfl
 
@@ -250,6 +244,32 @@ end
 lemma coe_pow (f : M →ₗ[R] M) (n : ℕ) : ⇑(f^n) = (f^[n]) :=
 by { ext m, apply pow_apply, }
 
+section
+variables {f' : M →ₗ[R] M}
+
+lemma iterate_succ (n : ℕ) : (f' ^ (n + 1)) = comp (f' ^ n) f' :=
+by rw [pow_succ', mul_eq_comp]
+
+lemma iterate_surjective (h : surjective f') : ∀ n : ℕ, surjective ⇑(f' ^ n)
+| 0       := surjective_id
+| (n + 1) := by { rw [iterate_succ], exact surjective.comp (iterate_surjective n) h, }
+
+lemma iterate_injective (h : injective f') : ∀ n : ℕ, injective ⇑(f' ^ n)
+| 0       := injective_id
+| (n + 1) := by { rw [iterate_succ], exact injective.comp (iterate_injective n) h, }
+
+lemma iterate_bijective (h : bijective f') : ∀ n : ℕ, bijective ⇑(f' ^ n)
+| 0       := bijective_id
+| (n + 1) := by { rw [iterate_succ], exact bijective.comp (iterate_bijective n) h, }
+
+lemma injective_of_iterate_injective {n : ℕ} (hn : n ≠ 0) (h : injective ⇑(f' ^ n)) :
+  injective f' :=
+begin
+  rw [← nat.succ_pred_eq_of_pos (pos_iff_ne_zero.mpr hn), iterate_succ, ←comp_coe] at h,
+  exact injective.of_comp h,
+end
+end
+
 section
 open_locale classical