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Complex numbers as a field #21
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/- | ||
Copyright (c) 2017 Kevin Buzzard. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Author: Kevin Buzzard | ||
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The complex numbers, modelled as R^2 in the obvious way. | ||
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Natural next step: one could prove that complexes are a topological ring. | ||
-/ | ||
import analysis.real | ||
noncomputable theory | ||
-- because reals are noncomputable | ||
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local attribute [instance] classical.decidable_inhabited classical.prop_decidable | ||
-- because I don't know how to do inverses sensibly otherwise; | ||
-- e.g. I needed to know that if z was non-zero then either its real part | ||
-- was non-zero or its imaginary part was non-zero. | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. This is fine, we're in classical land here, although you probably don't need it since the reals already have a decidable_eq instance. Easy way to find out: remove it and see what breaks. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I removed it and stuff broke so I put it back. |
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structure complex : Type := | ||
(re : ℝ) (im : ℝ) | ||
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notation `ℂ` := complex | ||
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-- definition goes outside namespace, then everything else in it? | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Yes. You don't want the definition being called There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. OK got it. |
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namespace complex | ||
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-- handy checks for equality etc | ||
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theorem eta (z : complex) : complex.mk z.re z.im = z := | ||
cases_on z (λ _ _, rfl) | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. no indentation after There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. OK! |
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-- very useful | ||
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theorem eq_of_re_eq_and_im_eq (z w : complex) : z.re=w.re ∧ z.im=w.im → z=w := | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Don't use the form
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begin | ||
intro H,rw [←eta z,←eta w,H.left,H.right], | ||
end | ||
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-- simp version | ||
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theorem eq_iff_re_eq_and_im_eq (z w : complex) : z=w ↔ z.re=w.re ∧ z.im=w.im := | ||
begin | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I don't think you should use a tactic proof here. I suggest to use a term mode proof here. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Why does anyone care whether I prove a lemma with a tactic proof or a term mode proof? I am serious -- I thought that by proof irrelevance nobody cares what my proofs look like. As long as they compile, we're done. Why am I wrong? There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I guess this is for maintenance reasons. Tactic proofs tend to involve magic (like simp), and magic is fragile. If the list of simp lemma changes (say associativity comes back...) then proofs using simp can break. A term style proof tends to explicitly call out used lemmas. If such a lemma changes, it will probably much easier to understand what went wrong. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. My tactic proof was three lines long and used split, intro, rw, trivial and exact. My term proof uses pretty much the same thing :-) There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. As always, you will get the best results with a combination of tactics and terms. Here's how this is proven in
It is nice but usually surprising to see a successful proof by There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I must admit I didn't look at the proof before writing my comment. But it gives me the opportunity to ask what is this cc tactic. The documentation is not very illuminating to me. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. cc means "congruence closure" but it's one of those tactics that I (and my students) tend to just try occasionally to see if it finishes a (typically proof-by-basic-logic-arguments-only) proof. |
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split, | ||
intro H,rw [H],split;trivial, | ||
exact eq_of_re_eq_and_im_eq _ _, | ||
end | ||
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lemma proj_re (r0 i0 : real) : (complex.mk r0 i0).re = r0 := rfl | ||
lemma proj_im (r0 i0 : real) : (complex.mk r0 i0).im = i0 := rfl | ||
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local attribute [simp] eq_iff_re_eq_and_im_eq proj_re proj_im | ||
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-- Am I right in | ||
-- thinking that the end user should not need to | ||
-- have to use this function? | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Yes, they (and you) should use the coercion wherever possible. |
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def of_real : ℝ → ℂ := λ x, { re := x, im := 0 } | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. This should be There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. This I really do not understand -- I mean, I understand that you're asking me to change something, and I understand how to change it, but I do not understand why changing something to something which looks to me to be definitionally equal should be a sensible idea. Of course I do not doubt that it is! But I would really like to understand better where these comments are coming from. What is "the generated simp rule for of_real"? There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. You should try There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. My slow understanding of these comments is currently at the stage where I understand that even though the two approaches give definitionally equal functions, the associated equation lemmas are not the same, however I am unclear about why one prefers Johannes' version, because I am not clear about what is unfolding or whether we want it to unfold. This, as is sometimes the case with me, seems to come back to the fact that I do not actually understand what simp is doing or how it does it. I guess that's my point here -- the two equation lemmas are "different ways of saying exactly the same thing" to me, but for someone who knows better what simp is doing I guess this is not the case. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Let me try the following explanation: Second, it is hard to say why I prefer the |
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-- does one name these instances or not? I've named a random selection | ||
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instance coe_real_complex : has_coe ℝ ℂ := ⟨of_real⟩ | ||
instance : has_zero complex := ⟨of_real 0⟩ | ||
instance : has_one complex := ⟨of_real 1⟩ | ||
instance inhabited_complex : inhabited complex := ⟨0⟩ | ||
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-- dangerously short variable name so I protected it. | ||
-- It's never used in the library (other than in the projection | ||
-- commands) but I think end users will use it. | ||
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protected def i : complex := {re := 0, im := 1} | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Personally, I would want to avoid direct reference to A possibility for There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I've switched to I. |
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def conjugate (z : complex) : complex := {re := z.re, im := -(z.im)} | ||
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-- Are these supposed to be protected too? | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Conventionally, yes we protect There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I've protected add, neg, mul, inv |
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There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I don't think we need explicit constants for There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I see! Thanks. |
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def add : complex → complex → complex := | ||
λ z w, { re :=z.re+w.re, im:=z.im+w.im} | ||
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def neg : complex → complex := | ||
λ z, { re := -z.re, im := -z.im} | ||
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def mul : complex → complex → complex := | ||
λ z w, { re := z.re*w.re - z.im*w.im, | ||
im := z.re*w.im + z.im*w.re} | ||
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def norm_squared : complex → real := | ||
λ z, z.re*z.re+z.im*z.im | ||
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def inv : complex → complex := | ||
λ z, { re := z.re / norm_squared z, | ||
im := -z.im / norm_squared z } | ||
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instance : has_add complex := ⟨complex.add⟩ | ||
instance : has_neg complex := ⟨complex.neg⟩ | ||
instance : has_sub complex := ⟨λx y, x + - y⟩ | ||
instance : has_mul complex := ⟨complex.mul⟩ | ||
instance : has_inv complex := ⟨complex.inv⟩ | ||
instance : has_div complex := ⟨λx y, x * y⁻¹⟩ | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I saw that gitter conversation with Kenny, he diagnosed your problem well. Don't define There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I removed the |
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-- I was initially astounded to find that at some point there was a typo in has_div but | ||
-- this didn't cause any problems at all. I have since understood what is | ||
-- going on: "/" is never used in the field axioms, only ^{-1} . | ||
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-- These are very useful for proofs in the library so I make them local simp lemmas. | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. No harm in having these as global simp lemmas. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I am slightly reluctant to overload simp, especially as I don't really understand how it works. |
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lemma proj_zero_re : (0:complex).re=0 := rfl | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Remove the special rules for |
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lemma proj_zero_im : (0:complex).im=0 := rfl | ||
lemma proj_one_re : (1:complex).re=1 := rfl | ||
lemma proj_one_im : (1:complex).im=0 := rfl | ||
lemma proj_i_re : complex.i.re=0 := rfl | ||
lemma proj_i_im : complex.i.im=1 := rfl | ||
lemma proj_conj_re (z : complex) : (conjugate z).re = z.re := rfl | ||
lemma proj_conj_im (z : complex) : (conjugate z).im = -z.im := rfl | ||
lemma proj_add_re (z w: complex) : (z+w).re=z.re+w.re := rfl | ||
lemma proj_add_im (z w: complex) : (z+w).im=z.im+w.im := rfl | ||
lemma proj_neg_re (z: complex) : (-z).re=-z.re := rfl | ||
lemma proj_neg_im (z: complex) : (-z).im=-z.im := rfl | ||
lemma proj_neg_re' (z: complex) : (neg z).re=-z.re := rfl | ||
lemma proj_neg_im' (z: complex) : (neg z).im=-z.im := rfl | ||
lemma proj_sub_re (z w : complex) : (z-w).re=z.re-w.re := rfl | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. same as before. Also move the |
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lemma proj_sub_im (z w : complex) : (z-w).im=z.im-w.im := rfl | ||
lemma proj_mul_re (z w: complex) : (z*w).re=z.re*w.re-z.im*w.im := rfl | ||
lemma proj_mul_im (z w: complex) : (z*w).im=z.re*w.im+z.im*w.re := rfl | ||
lemma proj_of_real_re (r:real) : (of_real r).re = r := rfl | ||
lemma proj_of_real_im (r:real) : (of_real r).im = 0 := rfl | ||
local attribute [simp] proj_zero_re proj_zero_im proj_one_re proj_one_im | ||
local attribute [simp] proj_i_re proj_i_im proj_conj_re proj_conj_im | ||
local attribute [simp] proj_add_re proj_add_im proj_neg_re proj_neg_im | ||
local attribute [simp] proj_neg_re' proj_neg_im' proj_sub_re proj_sub_im | ||
local attribute [simp] proj_mul_re proj_mul_im proj_of_real_re proj_of_real_im | ||
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lemma norm_squared_pos_of_nonzero (z : complex) (H : z ≠ 0) : norm_squared z > 0 := | ||
begin -- far more painful than it should be but I need it for inverses | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. You are right that this is overcomplicated. I'll clean it up after merge |
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suffices : z.re ≠ 0 ∨ z.im ≠ 0, | ||
apply lt_of_le_of_ne, | ||
exact add_nonneg (mul_self_nonneg _) (mul_self_nonneg _), | ||
intro H2, | ||
cases this with Hre Him, | ||
exact Hre (eq_zero_of_mul_self_add_mul_self_eq_zero (eq.symm H2)), | ||
unfold norm_squared at H2,rw [add_comm] at H2, | ||
exact Him (eq_zero_of_mul_self_add_mul_self_eq_zero (eq.symm H2)), | ||
have : ¬ (z.re = 0 ∧ z.im = 0), | ||
intro H2, | ||
exact H (eq_of_re_eq_and_im_eq z 0 H2), | ||
cases classical.em (z.re = 0) with Hre_eq Hre_ne, | ||
right, | ||
intro H2, | ||
apply this, | ||
exact ⟨Hre_eq,H2⟩, | ||
left,assumption, | ||
end | ||
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-- I don't know how to set up | ||
-- real.cast_zero etc | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. It's true that you can define a There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I wrote these for nat and int; I almost wrote one for rat but I needed that the complexes had characteristic zero. I proved this but then for some reason I could not put it all together -- I was getting timeouts. I've left my proof in but commented it out. |
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lemma of_real_injective : function.injective of_real := | ||
begin | ||
intros x₁ x₂ H, | ||
exact congr_arg complex.re H, | ||
end | ||
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lemma of_real_zero : (0:complex) = of_real 0 := rfl | ||
lemma of_real_one : (1:complex) = of_real 1 := rfl | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. This would be a good simp lemma if pointed the other direction. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I don't really understand why but I switched them round. |
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-- amateurish definition of killer tactic but it works! | ||
meta def crunch : tactic unit := do | ||
`[intros], | ||
`[rw [eq_iff_re_eq_and_im_eq]], | ||
`[split;simp[add_mul,mul_add]] | ||
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lemma of_real_neg (r : real) : -of_real r = of_real (-r) := by crunch | ||
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lemma of_real_add (r s: real) : of_real r + of_real s = of_real (r+s) := by crunch | ||
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lemma of_real_sub (r s:real) : of_real r - of_real s = of_real(r-s) := by crunch | ||
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lemma of_real_mul (r s:real) : of_real r * of_real s = of_real (r*s) := by crunch | ||
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lemma of_real_inv (r:real) : (of_real r)⁻¹ = of_real (r⁻¹) := | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Use coercions here There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. OK I changed these to use coercions |
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begin | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more.
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Also: indentation after |
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rw [eq_iff_re_eq_and_im_eq], | ||
split, | ||
suffices : r/(r*r+0*0) = r⁻¹, | ||
exact this, | ||
cases classical.em (r=0) with Heq Hne, | ||
-- this is taking longer than it should be. | ||
rw [Heq], | ||
simp [inv_zero,div_zero], | ||
rw [mul_zero,add_zero,div_mul_left r Hne,inv_eq_one_div], | ||
suffices : -0/(r*r+0*0) = 0, | ||
exact this, | ||
rw [neg_zero,zero_div], | ||
end | ||
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lemma of_real_abs_squared (r:real) : norm_squared (of_real r) = (abs r)*(abs r) := | ||
begin | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. This is an interesting proof: But it only works if There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Aah! So this is a very good way of explaining in concrete terms the issue that came up earlier. Thanks! |
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rw [abs_mul_abs_self], | ||
suffices : r*r+0*0=r*r, | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. This |
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exact this, | ||
simp, | ||
end | ||
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lemma add_comm : ∀ (a b : ℂ), a + b = b + a := by crunch | ||
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-- I don't think I ever use these actually. | ||
local attribute [simp] of_real_zero of_real_one of_real_neg of_real_add | ||
local attribute [simp] of_real_sub of_real_mul of_real_inv | ||
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instance : discrete_field complex := | ||
{ discrete_field . | ||
zero := 0, | ||
add := (+), | ||
neg := complex.neg, | ||
zero_add := by crunch, | ||
add_zero := by crunch, | ||
add_comm := by crunch, | ||
add_assoc := by crunch, | ||
add_left_neg := by crunch, | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. This is a fairly common pattern. Rather than reproving the new properties the second time around, just omit them entirely, and fill them in with There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I tried this but then got into a real mess with type class inference. Will try again tomorrow. |
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one := 1, | ||
mul := (*), | ||
inv := has_inv.inv, | ||
mul_one := by crunch, | ||
one_mul := by crunch, | ||
mul_comm := by crunch, | ||
mul_assoc := by crunch, | ||
left_distrib := by crunch, | ||
right_distrib := by crunch, | ||
zero_ne_one := begin | ||
intro H, | ||
suffices : (0:complex).re = (1:complex).re, | ||
revert this, | ||
apply zero_ne_one, | ||
rw [←H], | ||
end, | ||
mul_inv_cancel := begin | ||
intros z H, | ||
have H2 : norm_squared z ≠ 0 := ne_of_gt (norm_squared_pos_of_nonzero z H), | ||
apply eq_of_re_eq_and_im_eq, | ||
unfold has_inv.inv inv, | ||
rw [proj_mul_re,proj_mul_im], | ||
split, | ||
suffices : z.re*(z.re/norm_squared z) + -z.im*(-z.im/norm_squared z) = 1, | ||
by simpa, | ||
rw [←mul_div_assoc,←mul_div_assoc,neg_mul_neg,div_add_div_same], | ||
unfold norm_squared at *, | ||
exact div_self H2, | ||
suffices : z.im * (z.re / norm_squared z) + z.re * (-z.im / norm_squared z) = 0, | ||
by simpa, | ||
rw [←mul_div_assoc,←mul_div_assoc,div_add_div_same], | ||
simp [zero_div], | ||
end, | ||
inv_mul_cancel := begin -- let's try cut and pasting mul_inv_cancel proof | ||
intros z H, | ||
have H2 : norm_squared z ≠ 0 := ne_of_gt (norm_squared_pos_of_nonzero z H), | ||
apply eq_of_re_eq_and_im_eq, | ||
unfold has_inv.inv inv, | ||
rw [proj_mul_re,proj_mul_im], | ||
split, | ||
suffices : z.re*(z.re/norm_squared z) + -z.im*(-z.im/norm_squared z) = 1, | ||
by simpa, | ||
rw [←mul_div_assoc,←mul_div_assoc,neg_mul_neg,div_add_div_same], | ||
unfold norm_squared at *, | ||
exact div_self H2, | ||
suffices : z.im * (z.re / norm_squared z) + z.re * (-z.im / norm_squared z) = 0, | ||
by simpa, | ||
rw [←mul_div_assoc,←mul_div_assoc,div_add_div_same], | ||
simp [zero_div], | ||
end, -- it worked without modification! | ||
-- Presumably I could just have proved mul_comm outside the verification that C is a field | ||
-- and then used that too? | ||
inv_zero := begin | ||
unfold has_inv.inv inv add_comm_group.zero, | ||
apply eq_of_re_eq_and_im_eq, | ||
split;simp [zero_div], | ||
end, | ||
has_decidable_eq := by apply_instance } | ||
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-- instance : topological_ring complex := missing | ||
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end complex |
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Obviously, the commentary should be removed. We can continue the discussion here on github.
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I tried to tidy up / remove comments.