快速到达Isabelle的八阶阿贝尔小组的方法

时间:2019-06-29 14:17:57

标签: isabelle

我正在this article在伊莎贝尔市正式成立。在4.1节中,它描述了以下设置:

context 
  fixes c d :: real
  assumes "c ≠ 0" "∃ b. c = b^2" "∃ b'. d = b'^2"
begin
  definition t where "t = sqrt(d/c)"
  definition e' where "e' x y = x^2 + y^2 - 1 - t^2 * x^2 * y^2"

  definition ρ where "ρ x y = (-y,x)"
  definition τ where "τ x y = (1/(t*x),1/(t*y))"

然后将G定义为由ρ和τ生成的八阶阿贝尔群。

有一种简单的方法吗?

  1. 说明ρ和τ产生一个基团。
  2. 既然ρ和τ的阶数为2,并且通勤,我认为其余所有通勤,也许有一个内置定理说,它必须与阶数为8的阿贝尔群相对应?

1 个答案:

答案 0 :(得分:1)

我确实尝试解决了该问题,并提出了一种稍微有力的解决方法:


context 
  fixes c d :: real
  assumes "c ≠ 0" "∃b. c = b^2" "∃b'. d = b'^2"
begin

definition t where "t = sqrt(d/c)"
definition e' where "e' x y = x^2 + y^2 - 1 - t^2 * x^2 * y^2"

context
  assumes nz_t: "t ≠ 0"
begin

definition ρ :: "real × real ⇒ real × real" where 
  "ρ z = (-snd z, fst z)"
definition τ :: "real × real ⇒ real × real" where 
  "τ z = (1/(t*fst z), 1/(t*snd z))"
definition S where
  "S ≡ 
    {
      id,
      (λz. (-snd z, fst z)),
      (λz. (-fst z, -snd z)),
      (λz. (snd z, -fst z)),
      (λz. (1/(t*fst z), 1/(t*snd z))),
      (λz. (-1/(t*snd z), 1/(t*fst z))),
      (λz. (-1/(t*fst z), -1/(t*snd z))),
      (λz. (1/(t*snd z), -1/(t*fst z)))
    }"
definition ρS where
  "ρS ≡ 
    {id, (λz. (-snd z, fst z)), (λz. (-fst z, -snd z)), (λz. (snd z, -fst z))}"
definition τS where
  "τS ≡ {id, (λz. (1/(t*fst z), 1/(t*snd z)))}"

definition BIJ where "BIJ = ⦇carrier = {f. bij f}, mult = comp, one = id⦈"

interpretation bij: group BIJ
  unfolding BIJ_def
  apply unfold_locales
  subgoal by (simp add: bij_comp)
  subgoal by (simp add: comp_assoc)
  subgoal by simp
  subgoal by simp
  subgoal by simp
  subgoal 
    unfolding Units_def
    by clarsimp 
      (metis inj_iff bij_betw_def bij_betw_inv_into inv_o_cancel surj_iff)
  done

(*the proof may take quite a few seconds*)
lemma comp_S: "x ∈ S ⟹ y ∈ S ⟹ x ∘ y ∈ S"
  unfolding comp_apply S_def Set.insert_iff by (elim disjE) fastforce+ 

lemma comm_S: "x ∈ S ⟹ y ∈ S ⟹ x ∘ y = y ∘ x"
  unfolding comp_apply S_def Set.insert_iff by (elim disjE) fastforce+ 

lemma bij_ρ: "bij ρ"
  unfolding bij_def inj_def surj_def ρ_def 
  by clarsimp (metis add.inverse_inverse)

lemma bij_τ: "bij τ"
  unfolding bij_def inj_def surj_def τ_def 
proof(simp add: nz_t, intro allI, intro exI)
  fix a show "a = 1 / (t * (1/(a*t)))" using nz_t by simp
qed

lemma generate_ρτ: "generate BIJ {ρ, τ} = S"
proof(standard; intro subsetI)
  have inv_τ: "inv⇘BIJ⇙ τ = τ"
    unfolding m_inv_def
  proof(standard)
    show "τ ∈ carrier BIJ ∧ τ ⊗⇘BIJ⇙ τ = ?⇘BIJ⇙ ∧ τ ⊗⇘BIJ⇙ τ = ?⇘BIJ⇙"
      unfolding BIJ_def apply(intro conjI)
      subgoal using bij_τ by simp
      subgoal unfolding τ_def using nz_t by auto 
      subgoal unfolding τ_def using nz_t by auto 
      done
    then show 
      "y ∈ carrier BIJ ∧ τ ⊗⇘BIJ⇙ y = ?⇘BIJ⇙ ∧ y ⊗⇘BIJ⇙ τ = ?⇘BIJ⇙ ⟹ y = τ" 
      for y
      unfolding BIJ_def by (auto intro: left_right_inverse_eq)
  qed
  define ρ' :: "real × real ⇒ real × real" where "ρ' = (λz. (snd z, -fst z))"
  have bij_ρ': "bij ρ'"
    unfolding bij_def inj_def surj_def ρ'_def
    by simp (metis add.inverse_inverse)
  have inv_ρ: "inv⇘BIJ⇙ ρ = ρ'"
    unfolding m_inv_def
  proof(standard)
    show "ρ' ∈ carrier BIJ ∧ ρ ⊗⇘BIJ⇙ ρ' = ?⇘BIJ⇙ ∧ ρ' ⊗⇘BIJ⇙ ρ = ?⇘BIJ⇙"
      unfolding BIJ_def apply(intro conjI)
      subgoal using bij_ρ' by auto
      subgoal unfolding ρ_def ρ'_def by auto 
      subgoal unfolding ρ_def ρ'_def by auto 
      done
    then show 
      "y ∈ carrier BIJ ∧ ρ ⊗⇘BIJ⇙ y = ?⇘BIJ⇙ ∧ y ⊗⇘BIJ⇙ ρ = ?⇘BIJ⇙ ⟹ y = ρ'" 
      for y
      unfolding BIJ_def by (auto intro: left_right_inverse_eq)
  qed
  have ττ: "τ ⊗⇘BIJ⇙ τ = ?⇘BIJ⇙" 
    unfolding BIJ_def τ_def comp_def by (auto simp: nz_t)
  show "x ∈ generate BIJ {ρ, τ} ⟹ x ∈ S" for x
    apply(induction rule: generate.induct)
    subgoal unfolding BIJ_def S_def by auto
    subgoal unfolding BIJ_def S_def ρ_def τ_def by auto
    subgoal 
      unfolding Set.insert_iff apply(elim disjE)
      subgoal using inv_ρ unfolding BIJ_def S_def ρ_def ρ'_def by simp    
      subgoal using inv_τ unfolding BIJ_def S_def τ_def by simp
      subgoal by simp
      done
    subgoal unfolding BIJ_def by (metis monoid.select_convs(1) comp_S)
    done
  show "x ∈ S ⟹ x ∈ generate BIJ {ρ, τ}" for x
    unfolding S_def Set.insert_iff
  proof(elim disjE; clarsimp)
    show "id ∈ generate BIJ {ρ, τ}"
      unfolding BIJ_def using generate.simps by fastforce
    show ρ_gen: "(λz. (- snd z, fst z)) ∈ generate BIJ {ρ, τ}"
      by (fold ρ_def, rule generate.simps[THEN iffD2]) simp
    show τ_gen: "(λz. (1 / (t * fst z), 1 / (t * snd z))) ∈ generate BIJ {ρ, τ}"
      by (fold τ_def) (simp add: generate.incl)
    from inv_ρ show inv_ρ_gen: "(λz. (snd z, - fst z)) ∈ generate BIJ {ρ, τ}"
      by (fold ρ'_def) (auto simp: generate.inv insertI1)
    show ρρ_gen: "(λz. (- fst z, - snd z)) ∈ generate BIJ {ρ, τ}"
    proof-
      have ρρ: "(λz. (- fst z, - snd z)) = ρ ⊗⇘BIJ⇙ ρ"
        unfolding ρ_def BIJ_def by auto
      show ?thesis 
        apply(rule generate.simps[THEN iffD2])
        using ρρ ρ_gen[folded ρ_def] by auto
    qed
    show "(λz. (- (1 / (t * snd z)), 1 / (t * fst z))) ∈ generate BIJ {ρ, τ}"
    proof-
      have ρτ: "(λz. (- (1 / (t * snd z)), 1 / (t * fst z))) = ρ ⊗⇘BIJ⇙ τ"
        unfolding ρ_def τ_def BIJ_def by auto
      show ?thesis 
        apply(rule generate.simps[THEN iffD2])
        using ρτ ρ_gen[folded ρ_def] τ_gen[folded τ_def] by auto
    qed
    show 
      "(λz. (- (1 / (t * fst z)), - (1 / (t * snd z)))) ∈ generate BIJ {ρ, τ}"
    proof-
      have ρρτ: 
        "(λz. (- (1 / (t * fst z)), - (1 / (t * snd z)))) = 
        (λz. (- fst z, - snd z)) ⊗⇘BIJ⇙ τ"
        unfolding τ_def BIJ_def by auto
      show ?thesis 
        apply(rule generate.simps[THEN iffD2])
        using ρρτ ρρ_gen τ_gen[folded τ_def] by auto
    qed
    show "(λz. (1 / (t * snd z), - (1 / (t * fst z)))) ∈ generate BIJ {ρ, τ}"
    proof-
      have inv_ρ_τ: 
        "(λz. (1 / (t * snd z), - (1 / (t * fst z)))) = 
        (λz. (snd z, - fst z)) ⊗⇘BIJ⇙ τ"
        unfolding τ_def BIJ_def by auto
      show ?thesis 
        apply(rule generate.simps[THEN iffD2])
        using inv_ρ_τ inv_ρ_gen τ_gen[folded τ_def] by auto
    qed
  qed 
qed

lemma "comm_group (BIJ⦇carrier := (generate BIJ {ρ, τ})⦈)" 
proof-
  have ρτ_ss_BIJ: "{ρ, τ} ⊆ carrier BIJ" 
    using bij_ρ bij_τ unfolding BIJ_def by simp
  interpret ρτ_sg: subgroup "(generate BIJ {ρ, τ})" BIJ 
    using ρτ_ss_BIJ by (rule bij.generate_is_subgroup)
  interpret ρτ_g: group "BIJ⦇carrier := (generate BIJ {ρ, τ})⦈"
    by (rule ρτ_sg.subgroup_is_group[OF bij.group_axioms])
  have car_S: "carrier (BIJ⦇carrier := S⦈) = S" by simp
  have BIJ_comp: "x ⊗⇘BIJ⦇carrier := S⦈⇙ y = x ∘ y" for x y
    unfolding BIJ_def by auto
  from ρτ_g.group_comm_groupI[
      unfolded generate_ρτ car_S BIJ_comp, OF comm_S, simplified
      ]
  show ?thesis unfolding generate_ρτ by assumption 
qed

lemma id_pair_def: "(λx. x) = (λz. (fst z, snd z))" by simp

lemma distinct_single: "distinct [x] = True" by simp

lemma ne_ff'_gg'_imp_ne_fgf'g':
  assumes "f ≠ f' ∨ g ≠ g'"
  shows 
    "(λz. (f (fst z) (snd z), g (fst z) (snd z))) ≠ 
    (λz. (f' (fst z) (snd z), g' (fst z) (snd z)))"
  using assms
proof(rule disjE)
  assume "f ≠ f'"
  then obtain x y where "f x y ≠ f' x y" by blast  
  then show ?thesis by (metis (hide_lams) fst_eqD snd_eqD)
next
  assume "g ≠ g'"
  then obtain x y where "g x y ≠ g' x y" by blast  
  then show ?thesis by (metis (hide_lams) fst_eqD snd_eqD)
qed

lemma id_ne_hyp: "(λa. a) ≠ (λa. 1/(t*a))" 
proof(rule ccontr, simp)
  assume id_eq_hyp: "(λa. a) = (λa. 1/(t*a))"
  {
    fix a :: real assume "a > 0"
    define b where "b = sqrt(a)"
    from ‹a > 0› have "a = b*b" and "b > 0" unfolding b_def by auto
    from id_eq_hyp have "b = 1/(t*b)" by metis
    with ‹b > 0› have "b div b =(1/(t*b)) div b" by simp
    with ‹b > 0› have "1 = (1/(t*a))" unfolding ‹a = b*b› by simp
    with ‹a > 0› nz_t have "t*a = 1" by simp
  }
  note ta_eq_one = this
  define t2 where "t2 = (if t > 0 then 2/t else -2/t)" 
  with nz_t have "t2 > 0" unfolding t2_def by auto
  from nz_t have "t*t2 = 2 ∨ t*t2 = -2" unfolding t2_def by auto
  from ta_eq_one ‹t2 > 0› this show False by auto
qed

lemma id_ne_mhyp: "(λa. a) ≠ (λa. -1/(t*a))"
proof(rule ccontr, simp)
  assume id_eq_hyp: "(λa. a) = (λa. -(1/(t*a)))"
  {
    fix a :: real assume "a > 0"
    define b where "b = sqrt(a)"
    from ‹a > 0› have "a = b*b" and "b > 0" unfolding b_def by auto
    from id_eq_hyp have "b = -(1/(t*b))" by metis
    with ‹b > 0› have "b div b =-1/(t*b) div b" by simp
    with ‹b > 0› have "1 = -1/(t*a)" unfolding ‹a = b*b› by simp
    with ‹a > 0› nz_t have "t*a = -1" by (metis divide_eq_1_iff)
  }
  note ta_eq_one = this
  define t2 where "t2 = (if t > 0 then 2/t else -2/t)" 
  with nz_t have "t2 > 0" unfolding t2_def by auto
  from nz_t have "t*t2 = 2 ∨ t*t2 = -2" unfolding t2_def by auto
  from ta_eq_one ‹t2 > 0› this show False by auto
qed

lemma mid_ne_hyp: "(λa. -a) ≠ (λa. 1 / (t*a))"
  using id_ne_mhyp by (metis minus_divide_left minus_equation_iff)

lemma mid_ne_mhyp: "(λa. -a) ≠ (λa. -1 / (t*a))"
  using id_ne_hyp by (metis divide_minus_left minus_equation_iff)

lemma hyp_neq_hyp_1: "(λa. - 1/(t*a)) ≠ (λa. 1/(t*a))"
  using nz_t 
  by (metis divide_cancel_right id_ne_mhyp mult_cancel_right1 mult_left_cancel 
      one_neq_neg_one)

lemma distinct:
  "distinct 
    [
      id,
      (λz. (-snd z, fst z)),
      (λz. (-fst z, -snd z)),
      (λz. (snd z, -fst z)),
      (λz. (1/(t*fst z), 1/(t*snd z))),
      (λz. (-1/(t*snd z), 1/(t*fst z))),
      (λz. (-1/(t*fst z), -1/(t*snd z))),
      (λz. (1/(t*snd z), -1/(t*fst z)))
    ]"
  apply(unfold distinct_length_2_or_more)+
  unfolding 
    distinct_length_2_or_more
    distinct_single
    id_def id_pair_def
    HOL.simp_thms(21)
  by 
    (intro conjI) 
    (
      rule ne_ff'_gg'_imp_ne_fgf'g', 
      metis one_neq_neg_one id_ne_hyp id_ne_mhyp
      mid_ne_hyp mid_ne_mhyp hyp_neq_hyp_1
    )+ 

lemma "card S = 8"
  using distinct unfolding S_def using card_empty card_insert_disjoint by auto

end

end

备注

  • 我依靠sledgehammer来证明许多内容,并且有一些不必要的代码重复。因此,就像我大多数关于SO的答案一样,从编码风格的角度来看,这个答案也不是完美的。
  • 我想知道该解决方案是否有更好的整体方法。某种程度上,我开始相信大多数更周到的方法(例如,使用关于循环组的定理来确定ρτ的顺序,然后使用|HK|=|H||K|/|H∩K|来确定{ {1}}需要证明G的许多其他定理,但是在做此评论之前我没有与AFP核实过,并且我不定期使用HOL-Algebra。因此,我可能错过了一些东西。