Quelle  Abe.thy

theory Abe
  importsHOL."HOL.eal_Vector_Spaces VectorSpa
begin



locale GGV ==
  fixes fi ::"a:gyrocommutative_gyrogroup> 'b::real_inner"
es "\forallx. (y∈ "
 x\<n dom ∧ x≠> (∃
  
locale " 1 
   fi<'b::real_inner"
  fixes sal ::"real 'a ==>
  fixes (*r2a= r1 r2
  fixes smult'::"realas "a<>gyrozerofi(scaler∣R (norm (fi (scale r a))) = (fi a) /<^sub>R (norm ( a))"
  (*fixes zero'::"real"*)
  assumes " fi
  assumes vector_space_with_domain>. x = norm (fi a) ∨'"
  es ""scale 1a = a"
  assumes ( a\>b) plus) norm
  assumes "scale (r1*r2) a = scale r1 (scale r2 a)"
  gyrocommutative_gyrogroup
  assumes  ale
  assumes scale_1 a :: 'a. 1 \otimes
  assumes "vector_space_with_domain {x.\exists nrm(fi a \or - orm(f a)} pls' 0smult'"
  assumes " assumesscal_assc: "\And r1real) (r2real) (a : ) ( * r2<otimes a = r1 ⊗ a)"
 es"norm \oplus b) plus (norm (fi a)) ((norm (fib))"
begin
  
end

class gyrolinear_space =
  gyrocommutative_gyrogroup +
  fixes scale :: "real ==> 'a" (infixl "⊗
  assumes scale_1: "∧ a :: 'a. 1 ⊗
  assumesscale_distrib: "∧ ( ::real) (r2: reala::a)(1r2otimes a = r1 ⊗\oplus> r2 a"
  umesseasso: "<>  (r2 ⊗
  assumes gyroauto_property: "∧ real"
  assumes: "<> (r1 :: rea) r2 :: real) (v
  
begin

end

locale normed_gyrolinear_space =
  fixesas "bij_betwUNIV)x:real \ge>0"
  fixes f::"real real"
  assumes "∀ norm y>z)⟶
  assumes\forally::real. (y∈ (y\>
  assumesnormaler∣
  mes<y::real. ∀ norm' ` UNIV ∧
z∈ yzlongrightarrow (f y) > (f z))"

  assumes
  assumes "f(' (scale>r∣
  assumes "norm' (gyr u v x) = norm' x"
  assumes "∀"
begin
  
definition norms::"real set" where 
  "norms = norm' ` UNIV"

definition norms_neg::"real set" where 
  "norms_neg = (λx. -1 * norm' x) ` UNIV"

definition norms_all::"real set" where 
  "norms_all = norms ∪ norms_neg"

lemma norms_neq_not_empty:
  shows "norms_neg ≠ {}"
  using add\mbda. -*rmV"


lemma zero_only_norms_norm> norms_neg"
  assumes
  shows0
  by ( "norms_neg ≠


lemma a1_a2:
  shows "∃ realforallx::real. ∀norms_all ∧norms_all ∧ (f' x) > (f' yjava.lang.StringIndexOutOfBoundsException: Index 173 out of bounds for length 173
 ∧ (f' 0) = 0 ∧ (f' 0) = 0 ∧
proof-
  let ?f' = "λ3?f' = 0"
  ave?  =0"
    by auto
  moreover have fact1: "(∀x::real. ∀y::real. ( x∈norms_all \   have: "(∀y::real. ( x∈> ∈ ∧ (?f' x) > (?f' y))"
  proof-
    {fix x y 
    assume "∈ y ∈ x>y"
    have "(?f' x) > (?f' y)"
    proof-
      havex < x≠
       {
        assume "x=0"
        then have ?thesis 
          by (smtmoreover
          
      }
      moreover {
        assume "x≠x ∈ norms_all ∧ y ∈ norms_all ∧ f_inv_into_f mem_Collect_eq normed_gyrolinear_space.norms_neg_def normed_gyrolinear_space_axioms normed_gyrolinear_space_def norms_all_def normsdef rangeI)
        assume x🚫
          using <>x \<in \in> norms_all ∧ norms_all_def by force
        moreover {
          assume "x∈ have y≠
          thenjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
            by blast
          moreover {
            assume "y=0"
            then have ?hesis 
              by (smt (z3) ‹{
 } moreover {
 assumesume "y≠
 have "y∈norms ∨norms_neg"
 using ‹ norms_all ∧ norms_all_def by auto
 moreover {
 assume "y∈norms"
 then have ?thesis
 using 🚫
 
 } moreover {
 assume "y∈norms_neg"  {
 then have "?f' y = - (f (-y))"
 using ‹
 moreover have "-y ∈ norms"
  uin\open<> 
 moreover have "?f' y ≤ 0"
 
 by (sbysmt (z3) \openx ∈ norms› ‹ < norms_all ‹ normed_gyrolinear_space_axioms normed_gyrolinear_space_def norms_def)
 
 moreover have "?f' y ≠0"
 proof(ule ccontr)
 assume "¬
 then show False
 (smt (verit, d) \open \<noteq\
 qed
 ultimately have ?thesis
 by (smt (z3) ‹x ∈ norms› normed_gyrolinear_space_axioms normed_gyrolinear_space_def norms_def)
 }
 ultimately have ?thesis by blast
 }
 ultimately have ?thesis by blast
 } moreover {
 assume "x∈norms_neg"
 then have ?thesis
 by (smt (verit, del_insts) Un_def ‹x ∈ norms_all ∧= - (f(-y))"

 }
 ultimately have ?thesis by blast
 } ultimately show ?thesis by blast
 qed
 }
 then show ?thesis by blast
 
 moreover have fact2: " bij_betw ?f' norms_all UNIV"
 proof-
 have moreove have "-y \<>  nnorms_neg by force
 by (smt (verit, ccfv_threshold) calculation(2))
 moreover have **:"∀x::real. ∃y. (y∈ norms_all ∧ ?f' y = x)"
 proof-
 have "∀x::real. (x≥' y \le 0"
 by (metis (no_types, opaque_lifting) bij_betw_iff_bijections mem_Collect_eq normed_gyrolinear_space_axioms normed_gyrolinear_space_def norms_def)
 moreover have "∀) calculatio2) norme normed_gyroli norms_def)
 by (simp add: calculation)
 moreover have "∀x::real. (x≥
 by (s (smt((z3 calculation1) f_inv_into_f normed_gyr normed_gy norms_def)
 moreover have "∀x::real. (x<0 ⟶ (∃y. (y ∈ norms_neg ∧
 using calculation(2) norms_def norms_neg_def by auto
 moreover have "∀
 by (smt (z3) calculation(1by (smt (ver, del) \open ≠)
 moreover have "∀x::real. (x≥ 0 ∨ x<0)"
 by (simp add: linorder_le_less_linear)
 ultimately show ?thesis
 proof -
 { fix rr :: real
 have ff1: "∀r. (r::real) < 0 ∨qed
 by (smt (z3))
 have ff2: "\<forallr
 by (smt (z3) inf_sup_aci(5))
 have ff3: "∀R Ra. (Ra::real set) ∪ R = R ∪ Ra"
 by (smt (z3) Un_commute)
 have ff4: "∀r ra. (r::real) ≤> normed_gyrolinear_space_axioms normed_gyrolinear_sp nor)
 by simp
 have ff5: "∀R Ra. (R::real set) ⊆ Ra ∪ R"
 by (smt (z3) inf_sup_ord(4))
 have ff6: "∀r. (r::real) ≤ r"
 by (smt (z3))
 have ff7: "∀r R Ra. (r::real) ∉ }
 by blast
 have ff8: "∀r. - (- (r::real)) = r"
 using verit_minus_simplify(4) by blast
 have ff9: "- (0::real) = 0"
 by (smt (z3))
 have "∀r ra. r ∉ norms_all ∨
 using ‹
 then have "∀t
 using ff9 by (smt (z3))
 then have "(∃smt ((verit, ) Un_ \<x\ <\<
 using ff9 ff8 ff7 ff6 ff5 ff4 ff3 ff2 ff1 ‹∀x<0. ∃y. y ∈ norms_neg ∧ f (- y) = - x› ‹∀x≥0. ∃y. y ∈ norms ∧ (if y = 0 then 0 else if y ∈ norms then f y else if y ∈ norms_neg then - f (- y) else undefined) = x› ‹by blas
 then show ?thesis
 by blast
 qed
 
 qed
 moreover have "inj_on ?f' norms_all"
 using "*" inj_on_def by blast
 moreover have ***:"∀ \<>y
 using "**" by blast
 moreover have "?f' ` norms_all = UNIV"
 proof-
 have "?f' ` norms_all ⊆ UNIV"
 by blast
 moreover have "UNIV ⊆ ?f' ` norms_all"
 proof-
 fix x::real
 have "∃y∈norms_all. (?f' y = x)"
 using "**" by blast
 then have "x ∈ (?f' ` norms_all)"
 by blast
 then have "∀x::real. (x ∈ (?f' ` norms_all))"
 by (smt (verit, del_insts) "**" image_iff)
 then show ?thesis
 by blast
 qed
 ultimately show ?thesis
 by force
 qed
 ultimately show " bij_betw ?f' norms_all UNIV"
 using bij_betw_def by blast
 qed
 
 moreover have fact_fin: " ((∀x::real. ∀y::real. ( x∈norms_all ∧ y ∈norms_all ∧ x>y)⟶ (?f' x) > (?f' y))
 ∧ccfv_threshold) calculation(2))
 using fact1 fact2 by argo
 
 ultimately show ?thesis
 using fact_fin
 by (smt (verit, del_insts))
 

 

  normed_gyrolinear_space' =
 fixes nor moreover have ****:"∀' = x)"
 fixes f'::"real ==> real"
 assumes "∀a::'a. (norm' a ≥ 0)"
 assumes "bij_betw f' ((norm' ` UNIV) ∪ ((λx. -1 * norm' x) ` UNIV)) UNIV"
 assumes "∀y::real. ∀z::real. (( y∈ ((norm' ` UNIV) ∪ ((λx. -1 * norm' x) ` UNIV)) ∧
 ∈ ((norm' ` UNIV) ∪ ((λx. -1 * norm' x) ` UNIV)) ∧y= x)))"
 assumes "f' 0 = 0"
 assumes "∀x::'a. ∀y::'a. f'(norm' (gyroplus x y)) ≤ (f' (norm' x)) + (f' (norm' y))"
 assumes "f' (norm' (scale r x)) = ∣r∣ * (f' (norm' x))"
 assumes "norm' (gyr u v x) = norm' x"
 assumes "∀x::'a. ((norm' x) = 0 ⟷ x = gyrozero)"
 

  norms::"real set" where
 "norms = norm' ` UNIV"

  norms_neg::"real set" where
 "norms_neg = (λx. -1 * norm' x) ` UNIV"

  norms_all::"real set" where
 "norms_all = norms ∪ norms_neg"

  norms_neq_not_empty:
 shows "norms_neg ≠ {}"
 using add.inverse_inverse norms_neg_def by fastforce


  zero_only_norms_norms_neg:
 assumes "x\ x∈
 shows "x=0"
 by (smt (verit, ccfv_threshold) assms(1) assms(2) f_inv_into_f normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_def norms_neg_def)
 

  norm_oplus_f::"real ==> real ==> real" (infixl " ⊕
 where "a ⊕_f b = (if (a∈norms_all ∧ b∈norms_all) then (inv_into norms_all f') ((f' a) + (f' b))
  undefined)"


  norm_otimes_f::"real ==> real ==> real" (infixl "⊗ nor)
 where "r ⊗_f a = (if (a∈norms_all) then (inv_into norms_all f') (r * (f' a))
  undefined)"

  vector_space_of_norms:
 shows "vector_space_with_domain norms_all norm_oplus_f 0 norm_otimes_f"
 
 fix x y
 show "x ∈ "\<forallx norms_ne \and (f (-y))= -x)))"
 proof-
 assume "x∈norms_all"
 show "y ∈ norms_all ==> x ⊕_f y ∈ norms_all"
 proof-
 assume "y∈norms_all"
java.lang.NullPointerException
 by (smt (verit, del_insts) UNIV_I ‹ (-y)) = -x)-)))"
 qed
 qed
 
 show "0 ∈ norms_all"
 by (metis Un_iff normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_all_def norms_def rangeI)
 
 fix x y z
 show " x ∈ norms_all ==>
 y ∈ norms_all ==> z ∈ norms_all ==> x ⊕_f y ⊕(z3) calculation(1) clculation(4) f_inv_into_f normed_ normed_gyrolinear_space_def norms_d norms_neg_de rangeI)
 proof-
 assume "x∈norms_all"
 show " y ∈ norms_all ==>\<>x
 proof-
 assume "y ∈ norms_all"
 show "z ∈ norms_all ==> x ⊕₎
 proof-
 assume "z ∈ norms_all"
 show " x ⊕_f y ⊕
 proof-
 have " x ⊕_f y = (inv_into norms_all f') ((f' x) + (f' y))"
 by (simp add: ‹x ∈ norms_all› \<by 
 moreover have "x ⊕_realset) ∪
 f' ( (inv_into norms_all f') ((f' x) + (f' y)))) + (f' z))"
 by (metis (no_types, lifting) UNIV_I ‹z ∈ norms_all› bij_betw_imp_surj_on calculation inv_into_into norm_oplus_f_def normed_gyrolinear_space'.norms_neg_def normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_all_def norms_def)
 moreover have "x ⊕_rra
 by (metis (mono_tags, lifting) UNIV_I bij_betw_imp_surj_on calculation(2) f_inv_into_f normed_gyrolinear_space'.norms_neg_def normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_all_def norms_def)
 moreover have " (y ⊕_f z) = (inv_into norms_all f') ((f' y) + (f' z))"
 by (simp add: ‹
 moreover have " x ⊕_f (y ⊕_f z) = (inv_into norms_all f') ((f' x) +
 (f' ((inv_into norms_all f') ((f' y) + (f' z)))))"
 by (metis (ono_ta ifting) UNIV_I \<openx
 moreover have " x ⊕_f (y ⊕_f z) = (inv_into norms_all f') ((f' x) +
 ((f' y) + (f' z)))"
 by (metis (mono_tags, lifting) UNIV_I bij_betw_imp_surj_on calculation(5) f_inv_into_f normed_gyrolinear_space'.norms_neg_def normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_all_def norms_def)
 ultimately show ?thesis
 by argo
 qed
 qed
 qed
 qed
 
 fix x y
 show "x ∈ norms_all ==> y ∈ norms_all ==> x ⊕_f y = y ⊕r.(r::real) \le r"
 proof-
 assume "x∈ norms_all"
 in> no \Longrightarrow x \<oplus\
 proof-
 assume "y ∈ norms_all"
 show " x ⊕_f y = y ⊕_f x"
 by (s a: a.commutenorm_)
 qed
 qed
 
 fix x
 show " x ∈ norms_all ==> x ⊕
 proof-
 assume "x∈norms_all"
 show "x ⊕\<^usingverit_minus_simplify
 proof-
java.lang.NullPointerException
 by (metis (mono_tags, lifting) Un_iff ‹x ∈ norms_all› norm_oplus_f_def normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_all_def norms_def rangeI)
 then show ?thesis
 ‹
 qed
 qed
 
 fix x
 show "x ∈ norms_all ==> ∃y∈norms_all. x ⊕_f y = 0"
 proof-
 assume "x∈norms_all"
 show " ∃y∈norms_all. x ⊕_f y = 0"
 
 let ?y = "(inv_into norms_all f') (-(f' x))"
 have " x ⊕_f ?y = (inv_into norms_all f') ((f' x) + (f' ?y))"
 by (smt (verit, ccfv_SIG) ‹ 0 else if r ∈ norms_n - (- r ) els undefined) ≠ 0 \in normsls undefined) <> 
 moreover have " x ⊕_f ?y = (inv_into norms_all f') ((f' x) + (-(f' x)))"
 by (smt (verit, ccfv_SIG) bij_betw_inv_into_right calculation iso_tuple_UNIV_I normed_gyrolinear_space'.norms_neg_def normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_all_def norms_def)
 moreover have "x ⊕_f ?y =(inv_into norms_all f') 0"
 using calculation(2) by force
 moreover have "x ⊕_f ?y = 0"
 by (metis (no_types, lifting) Un_iff bij_betw_def calculation(3) inv_into_f_eq normed_gyrolinear_space'.norms_neg_def normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_all_def norms_def rangeI)
 moreover have "?y ∈ norms_all"
 by (metis (no_types, lifting) UNIV_I bij_betw_imp_surj_on inv_into_into normed_gyrolinear_space'.norms_neg_def normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_all_def norms_def)
 ultimately show ?thesis
 by blast
 qed
 qed
 
 fix x a
 show "x ∈ norms_all ==> a ⊗_f x ∈ norms_all"
 proof-
 assume "x∈norms_all"
 show " a ⊗_f x ∈ norms_all"
 by (smt (verit, best) ‹x ∈ norms_all› bij_betw_imp_surj_on bij_betw_inv_into norm_otimes_f_def normed_gyrolinear_space'.norms_neg_def normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_all_def norms_def rangeI)
 qed
 
 fix x a b
 show "x ∈ norms_all ==> (a + b) ⊗_f x = a ⊗_f x ⊕_f (b ⊗_f x)"
 proof-
 assume "x∈norms_all"
 show "(a + b) ⊗_f x = a ⊗_f x ⊕_f (b ⊗_f x)"
 proof-
 have "(a + b) ⊗_f x = (inv_into norms_all f') ((a+b) * (f' x))"
 using ‹
 moreover have "(a + b) ⊗_f x = (inv_into norms_all f') ( then have "(∃ else if r \<in norms_neg th -f (- r else undefi) = rr \and r \inn) \<>\- r) else undefined)= r \<and 
 using calculation by argo
 moreover have *:" a ⊗_f x ⊕_f (b ⊗_f x) = (inv_into norms_all f')
 ((f' (a ⊗_f x)) + (f' (b ⊗_f x)))"
 proof -
 have "∧f. ¬ normed_gyrolinear_space' norm' f ∨ bij_betw f norms_all UNIV"
 by (metis (no_types) normed_gyrolinear_space'.norms_neg_def normed_gyrolinear_space'_def norms_all_def norms_def)
 then show ?thesis
 by (metis (full_types) UNIV_I ‹x ∈ norms_all› bij_betw_imp_surj_on inv_into_into norm_oplus_f_def norm_otimes_f_def normed_gyrolinear_space'_axioms)
 qed
 moreover have **:" (inv_into norms_all f')
 ((f' (a ⊗_f x)) + (f' (b ⊗_f x))) = (inv_into norms_all f')
 ((f' ((inv_into norms_all f') (a*(f' x)))) +
 (f' ((inv_into norms_all f') (b*(f' x)))))"
 using ‹x ∈ norms_all› norm_otimes_f_def by presburger
 moreover have "a ⊗_f x ⊕_f (b ⊗_f x) = (inv_into norms_all f') ((a*(f' x)) + (b*(f' x)))"
 using * **
 by (smt (verit, ccfv_threshold) UNIV_I bij_betw_imp_surj_on f_inv_into_f normed_gyrolinear_space'.norms_neg_def normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_all_def norms_def)
 ultimately show ?thesis
 by presburger
 qed
 qed
 
 fix x a b
 show " x ∈ norms_all ==> a ⊗_f (b ⊗_f x) = (a * b) ⊗_f x"
 proof-
 assume "x∈norms_all"
 show "a ⊗_f (b ⊗_f x) = (a * b) ⊗_f x"
 by (smt (verit, best) UNIV_I ‹x ∈ norms_all› ab_semigroup_mult_class.mult_ac(1) bij_betw_imp_surj_on f_inv_into_f inv_into_into norm_otimes_f_def normed_gyrolinear_space'.norms_neg_def normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def nusing ff9 ff8 ff7 ff6f6 ff5 f ff3 f2 f 🚫>0 ∃<> 
 qed
 
 fix x
 show "x ∈ norms_all ==> 1 ⊗_f x = x"
 proof-
 assume "x∈norms_all"
 show " 1 ⊗_f x = x"
 proof-
 have " 1 ⊗_f x = (inv_into norms_all f') (1*(f' x))"
 using ‹x ∈ norms_all› norm_otimes_f_def by presburger
 then show ?thesis
 by (metis (no_types, lifting) ‹x ∈ norms_all› bij_betw_inv_into_left lambda_one normed_gyrolinear_space'.norms_neg_def normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_all_def norms_def)
 qed
 qed
 
 show "∧x y a.
 x ∈ norms_all ==>
 y ∈ norms_all ==> a ⊗_f (x ⊕_f y) = a ⊗_f x ⊕_f (a ⊗_f y)"
 proof-
 {
 fix x y a
 assume "x∈ norms_all ∧ y∈ norms_all"
 have "a ⊗_f (x ⊕_f y) = (inv_into norms_all f') (a * f' ((inv_into norms_all f') ((f' x) + (f' y))))"
 by (smt (verit, best) UNIV_I ‹x ∈ norms_all ∧ y ∈ norms_all› bij_betw_imp_surj_on inv_into_into norm_oplus_f_def norm_otimes_f_def normed_gyrolinear_space'.norms_neg_def normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_all_def norms_def)
 moreover have "a ⊗_f x ⊕_f (a ⊗_f y) = (inv_into norms_all f') ((f' (inv_into norms_all f' (a * (f' x))))+(f' (inv_into norms_all f' (a * (f' y)))))"
 by (smt (verit) ‹x ∈ norms_all ∧ y ∈ norms_all› bij_betw_def inv_into_into iso_tuple_UNIV_I normed_gyrolinear_space'.norm_oplus_f_def normed_gyrolinear_space'.norm_otimes_f_def normed_gyrolinear_space'.norms_neg_def normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_all_def norms_def)
 ultimately have "a ⊗_f (x ⊕_f y) = a ⊗_f x ⊕_f (a ⊗_f y)"
 using UNIV_I bij_betw_imp_surj_on f_inv_into_f normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_all_def norms_def norms_neg_def ring_class.ring_distribs(1)
 by (smt (verit, best) normed_gyrolinear_space'.norms_neg_def)
 }
 show "∧x y a.
 x ∈ norms_all ==>
 y ∈ norms_all ==> a ⊗_f (x ⊕_f y) = a ⊗_f x ⊕_f (a ⊗_f y)"
 using ‹∧y x a. x ∈ norms_all ∧ y ∈ norms_all ==> a ⊗_f (x ⊕_f y) = a ⊗_f x ⊕_f (a ⊗_f y)› by blast
 qed
 


  r2:
 shows "norm' (x ⊕ y) ≤ (norm' x) ⊕_f (norm' y)"
 -
 have " (f' (norm' (x ⊕ y))) ≤ (f' (norm' x)) + (f' (norm' y))"
 using normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def by blast
 moreover have "(inv_into norms_all f' (f' (norm' (x ⊕ y)))) ≤
 inv_into norms_all f' ((f' (norm' x)) + (f' (norm' y))))"
 by (smt (verit, ccfv_SIG) UNIV_I bij_betw_def f_inv_into_f inv_into_into normed_gyrolinear_space'.norms_neg_def normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_all_def norms_def)
 ultimately show ?thesis
 by (metis (no_types, lifting) UnI1 bij_betw_def inv_into_f_eq normed_gyrolinear_space'.norm_oplus_f_def normed_gyrolinear_space'.norms_neg_def normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_all_def norms_def rangeI)
 

  r3:
 shows "norm' (r ⊗ x) = ∣r∣ ⊗_f (norm' x)"
 by (smt (verit, best) bij_betw_inv_into_left in_mono inf_sup_ord(3) norm_otimes_f_def normed_gyrolinear_space'.norms_neg_def normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_all_def norms_def rangeI)

  one_dim_vs:
 shows "one_dim_vector_space_with_domain norms_all norm_oplus_f 0 norm_otimes_f"
 -
 have step1: "vector_space_with_domain norms_all norm_oplus_f 0 norm_otimes_f"
 using vector_space_of_norms by auto
 moreover have step2: "∀y. ∀x. (y∈ norms_all ∧
 x∈ norms_all ∧ x≠0 ⟶ (∃!r::real. y = r ⊗_f x))"
 proof
 fix y
 show " ∀x. (y∈ norms_all ∧
 x∈
 proof
 fix x
 show "y∈
 x∈ norms_all ∧
 proof
 assume "y∈ norms_all ∧
 x∈
 show "(∃!r::real. y = r ⊗_f x)"
 proof-
 have "(∃r::real. y = r ⊗ ha ***:"∀?f y = )"
 proof-
 let ?r = "f'(y)/f'(x)"
 have "?r ⊗_f x = (inv_into norms_all f') (?r * (f' x))"
 by (simp add: ‹y ∈ norms_all ∧
 then show ?thesis
 by (smt (verit, ccfv_SIG) ‹y ∈ norms_all ∧ x ∈ norms_all ∧
 qed
 
java.lang.NullPointerException
 proof
 fix r1
 show "∀r2. y = r1 ⊗
 proof
 fix r2
 show "y = r1 \<otimes\ \<>\
 proof
 assume "y = r1 ⊗_f x ∧ y = r2 ⊗_f x "
 show "r1=r2"
 proof-
 have "r1 ⊗_f x = (inv_into norms_all f') (r1 * (f' x))"
 by (simp add: ‹y ∈ norms_all ∧ x ∈ norms_all ∧ x ≠ 0›
 moreover have "r2 ⊗₎
 using ‹y ∈ norms_all ∧ x ∈ norms_all ∧ x ≠by bl blas
 moreover
 have "(inv_into norms_all f') (r1 *(f' xx)) (inv_into norms_all f')(r2* (' x)"
 using ‹y = r1 ⊗_f x ∧ y = r2 ⊗_f x› calculation(1) calculation(2) by fastforce
 moreover have" f' ( (inv_into norms_all f') (r1 * (f' x))) =
 f'( (inv_into norms f') (r2 * (f' x)))"
 using calculation by presburger
 moreover have "r1* (f' x) = r2* (f' x)"
 by (metis (mono_tags, lifting) UNIV_I bij_betw_imp_surj_on calculation(3) inv_into_injective normed_gyrolinear_space'.norms_neg_def normed_gyrolinear_space'_axioms normed_gyrolinear_space'_def norms_all_def norms_def)
 ultimately show ?thesis
 by (metis (no_types, opaque_lifting) ‹
 qed
 
 qed
 qed
 qed
 ultimately show ?thesis
 by blast
 qed
 
 
 
 ultimately show ?thesis
 by (simp add: one_dim_vector_space_with_domain.intro one_dim_vectorultim show " bij_b ?f' nnorms_all UNIV"  
 

 

  normed_gyrolinear_space'' =
 fixes norm'::"'a::gyrolinear_space ==>
 fixes oplus'::"real ==> real ==>
 fixes otimes'::"real==>
 assumes "∀a::'a. (norm' a ≥∀and> y ∈(f' ) > (?f'y))
 assumes ax_space: "one_dim_vector_space_with_domain ((norm' ` UNIV) ∪ ((λx. -1 * norm' x) ` UNIV))
 oplus' 0 otimes'"
 assumes ax3: "∀x::'a. ∀y::'a. (norm' (gyroplus x y)) ≤ oplus' (norm' x) (norm' y)"
 assumes "(norm' (scale r x)) = otimes' ∣r∣ (norm' x)"
 assumes "norm' (gyr u v x) = norm' x"
 assumes "∀x::'a. ((norm' x) = 0 ⟷ x = gyrozero)"
 

  norms::"real set" where
 "nor = norm' ` UNIV""

  norms_neg::"real set" where
 "norms_neg = (λx. -1 * norm' x) ` UNIV"

  norms_all::"real set" where
 "norms_all = norms ∪ norms_neg"

  norms_neq_not_empty:
 shows "norms_neg ≠ {}"
 using add.inverse_inverse norms_neg_def by fastforce


  zero_only_norms_norms_neg:
 assumes "x∈norms" "x∈
 shows "x=0"
 by (smt (verit, ccfv_threshold) assms(1) assms(2) f_inv_into_f normed_gyrolinear_space''_axioms normed_gyrolinear_space''_def norms_def norms_neg_def)

  not_trivial_domen_has_pos:
 assumes "∃x. (x∈norms_all ∧ x≠0)"
 shows "∃x. (x∈norms ∧'::"real ==>
 using assms norms_all_def norms_def norms_neg_def by auto

 iso_with_real:
  assumes "\<exists>x. (x\<in>norms_all \<and> x\<noteq>0)" (* not trivial domain *)
  shows "∃g. (bij_betw g norms_all UNIV ∧ (g 0) = 0 ∧ assumes "'norm<union\lambda-1*'x)` )) UNIV
  (∀u.∀v. (u∈norms_all ∧ v∈norms_all ⟶ g (oplus' u v) = (g u) + (g v)))
 ∧ (∀u.∀r::real. (u∈forall. <forall:real( y∈ -1 * norm' x) ` UNIV) java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
)" (*∧ (∀u. (u∈norms ⟶ (g u)≥0))*)
proof-
  obtain "x" where "x∈norms ∧assumesnorm( u vx =norm"
    using assms not_trivial_domen_has_pos by presburger
  moreover have "x∈ norms_all"
    by (simp add: calculation norms_all_def)
  have "< \
    using ax_space  one_dim_vector_space_with_domain_axioms_def
    by (metis ‹x ∈ norms_all› calculation norms_all_def norms_def norms_neg_def one_dim_vector_space_with_domain.axioms(2))
  let ?g = "λy. (THE r. y = otimes' r x)"
  have "bij_betw ?g norms_all UNIV"
  proof-
    have "inj_on ?g norms_all"
      by (smt (verit, best) ‹∀y. y ∈
 moreover have "\<efinition 
 by (metis ‹x ∈ norms_all› * norm' x) ` UNIV"
 moreover have "∀r::real.∃y∈norms_all. ?g y = r"
 using ‹
 ultimately show ?thesis
 by (smt (verit, ccfv_threshold) UNIV_eq_I bij_betw_apply inj_on_imp_bij_betw)
 qed
 moreover have "?g 0 = 0"
 proof-
 obtain "r" where "0 = otimes' r x"
 by (metis ‹∀
 
 moreover obtain "xx" where "x=norm' xx "
 using norms_all_def
 using no shows " "norms_neg \<noteq 
 using ‹ by fastfo
 
 moreover have "otimes' 0 x = norm' (0 ⊗
 by (metis (no_types, lifting) calculation(2) norm_zero normed_gyrolinear_space''_axioms normed_gyrolinear_space''_def real_norm_def)
 moreover have "otimes' 0 x = 0"
 by (smt (ver ccfv_threshld) \open🚫
 ultimately show ?thesis
 by (smt (verit) ‹∀y. y ∈ norms_all ⟶ (∃!r. y = otimes' r x)› ax_space norms_all_def norms_def norms_neg_def one_dim_vector_space_with_domain.axioms(1) the1_equality vector_space_with_domain.zero_in_dom)
 qed
 moreover have "∀u.∀v. (u∈norms_all ∧ v∈norms_all ⟶ ?g (oplus' u v) = (?g u) + (?g v))"
 proof
 fix u
 show "∀v. (u∈norms_all ∧ v∈norms_all ⟶ ?g (oplus' u v) = (?g u) + (?g v))"
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 9
 fix v
 show "u∈norms_all ∧ v∈norms_all ⟶ ?g (oplus' u v) = (?g u) + (?g v)"
 proof
 assume "u∈norms_all ∧ v∈norms_all"
 show " ?g (oplus' u v) = (?g u) + (?g v)"
 proof-
 obtain "a" where "u = otimes' a x"
 using ‹
 moreover obtain "b" where "v = otimes' b' b xb x"
 using ‹∀y. y ∈ norms_all ⟶ (∃!r. y = otimes' r x)› ‹u ∈ norms_all ∧ v ∈ norms_all› by blast
 moreover have *:"oplus' u v = otimes' (a+b) x"
 by (metis ‹
 moreover have "oplus' u v ∈ norms_all"
 by (metis "*" ‹x ∈ norms_all› ax_space norms_all_def norms_def norms_neg_def one_dim_vector_space_with_domain.axioms(1) vector_space_with_domain.smult_closed)
 moreover have "?g (oplus' u v) = (a+b)"
 using *
 using ‹∀y. y ∈ norms_all ⟶ (∃!r. y = otimes' r x)› calculation(4) by auto
 ultimately show ?thesis
 by (smt (verit, del_insts) ‹
 qed
 qed
 qed
 qed
 moreover h have "(\forall.∀'r u) = r*(?g u)))"
 proof
 fix u
 show "∀r::real. (u∈norms_all ⟶ ?g (otimes' r u) = r*(?g u))"
 proof
 fix r
 show "u∈norms_all ⟶ ?g (otimes' r u) = r*(?g u)"
 proof
 assume "u∈norms_all"
 show "?g (otimes' r u) = r*(?g u)"
 proof-
 obtain "a" where "u = otimes' a x"
 using ‹norms_all f' f') (r * (f' a))
 moreover have "otimes' r u = otimes' (r*a) x"
 by (metis ‹x ∈ norms_all› ax_space calculation norms_all_def norms_def norms_neg_def one_dim_vector_space_with_domain.axioms(1) vector_space_with_domain.smult_assoc)eund)"
 moreover have "otimes' r u ∈ norms_all"
 by (metis ‹vector_space_of_norms:
 moreover have "?g (otimes' r u) = (r*a)"
 using ‹
 ultimately show ?thesis
 by (smt (verit, ccfv_threshold) ‹y
 qed
 qed
 qed
 qed
 
 ultimately show ?thesis
 by bla
 

  g_iso::"(real\\Rightarrow>re)\<RightarrowRightarrow
 "g_iso g ⟷ (bij_betw g norms_all UNIV ∧ (g 0) = 0 ∧
 (∀u.∀v. (u∈norms_all ∧ v∈
 ∧

  iso_neg_with_real:
  assumes "\<exists>x. (x\<in>norms_all \<and> x\<noteq>0)" (* not trivial domain *)
  shows "g_iso g ⟶ g_iso (λx. -1 * (g x))" 
proof
  assume "g_iso(verit, del_insts)UNIV_I🚫
  show " g_iso (λx. -1 * (g x))"
  proof-
    have "bij_betw (λx. -1 * (g x)) norms_all UNIV"
    proof-
      have "inj_on (λx. -1 * (g x)) norms_all"
        by (smt (verit, ccfv_threshold) ‹g_iso g› bij_betw_imp_inj_on g_iso_def inj_on_def)
      moreover have "∀r::real.∃y∈norms_all. ((λ    qed
        by   by ( UNIV_I›
      ultimately show ?thesis 
        by (metis (mono_tags, lifting) UNIV_eq_I bij_betwE bij_betw_imageI)
    qed
    moreover have " (λx. -1 * (g x)) 0 = 0"
      using ‹
    moreover have "(\<forall>u.\<forall>v. (u\<in>norms_all \<and> v\<in>norms_all \<longrightarrow    by (metis Un_iff normed_gyrolinear_space'axiomsnormed_gyrolinear_space_defnorms_all_def norms_def rangeI)
 = ( (\<lambda>x. -1 * (g x)) u) + ( (\<lambda>x. -1 * (g x)) v)))"
      using \<open>g_iso g\<close> g_iso_def by auto
    moreover have "(\<forall>u.\<forall>r::real. (u\<in>norms_all \<longrightarrow>  (\<lambda>x. -1 * (g x)) (otimes' r u) = r*( (\<lambda>x. -1 * (g x)) u)))"
      using \<open>g_iso g\<close> g_iso_def by auto
    ultimately show ?thesis 
      using g_iso_def by presburger
  qed
qed

lemma iso_with_real_positive_on_norms:
  assumes "\<exists>x. (x\<in>norms_all \<and> x\<noteq>0)" (* not trivial domain *)
  shows<>.x<>orms
∧ bij_betw (λx. if x ∈ norms then (g x) else undefined) norms {r::real. r≥0})"
proof-
  obtain "xx" where "xx∈
    using assms not_trivial_domen_has_pos by blast
  moreover" where "norm xx
    using calculation norms_def by auto
  moreover obtain "g" where "g_iso g"
    using iso_with_real
    using assms g_iso_def by blast
  let ?g = "if (g xx) < 0 then (λx. -1 * (g x)) else g"
  have *:"?g xx ≥ 0"
    by force
  moreover have "?g xx ≠0"
  proof (rule ccontr)
    assume "¬(?g xx ≠0)"
    have "?g show "x    ⊕sub"
      using ‹¬ (if g xx < 0 then λx. - 1 * g x else g) xx ≠ 0› by blast
    then have "?g xx = g xx"
      by (smt (verit, ccfv_threshold))
    then have "g xx = 0"
      by (simp add: ‹(if g xx < 0 then λx. - 1 * g x else g) xx = 0›
    then ha have"="
      by (metis ‹g_iso g› ax_space bij_betw_iff_bijections calculation(1) g_iso_def in_mono inf_sup_ord(3) norms_all_def norms_def norms_neg_def one_dim_vector_space_with_domain.axioms(1) vector_space_with_domain.zero_in_dom)
    alse
      using calculation(1) by blast
  
  moreover have "g_iso ?g"
    using ‹g_iso g›+(f' z)))"
  moreover have "∀x.(x∈norms ⟶ (?g x)≥0)"
  proof(rule ccontr)
    assume "¬(∀x.(x∈ li) UNIV_I \<openz)
    have "∃x. (x∈norms ∧ (?g x) < 0)"
      using ‹¬ (∀x. x ∈ norms ⟶ 0 ≤ (if g xx < 0 then λx. - 1 * g x else g) x)› by fastforce
    moreover obtain "yy" where "yy ∈ norms ∧ (?g yy) <0"
      using calculation by blast
    moreover obtain "y" where "norm' y = yy"
      using calculation(2) norms_def by auto
    let ?A = "{norm' (r ⊗ x) | r::real. True}"
    let let ? = "norm ×
    have "?A ∪ ?B ⊆ norms"
      using norms_def by auto
    let ?gA = "{(?g a)|a. a∈?A}"
    have "?gA = {r::real. r≥0}"
    proof-
      have "∀a. (a∈?A ⟶ ?g a ≥0)"
      proof
        fix a
        show "(a∈?A ⟶ ?g a ≥0)"
        proof
            assume "a∈?A"
            show "?g a ≥0"
            proof-
              obtain "r" where "a = norm' (r ⊗ x) "
                using ‹\^sub>f z) = (inv_into n f') ((f' y)+ (f' z))))"
 moreover have "?g a = ?g (norm' (r ⊗ x) )"
 using calculation by presburger
 moreover have "?g a = ?g ( otimes' ∣r∣ (norm' x))"
 by (metis calculation(1) normed_gyrolinear_space''_axioms normed_gyrolinear_space''_def)
 moreover have "?g a = ∣r∣ * ?g (norm' x)"
 using ‹g_iso (if g xx < 0
 ultimately show ?thesis
 by (simp add: ‹norm' x = xx›)
 qed
 qed
 qed
 
 using calculation by fastforce
 moreover have "{r::real. r≥0} ⊆ ?gA"
 proof-
 have "bij_betw ?g norms_all UNIV"
 using ‹g_iso (if g xx < 0 then λx. - 1 * g x else g)› g_iso_def by blast
 moreover have "∀r::real. (r≥0 ⟶ r∈?gA)"
 proof
 fix r
 show "r≥0 ⟶ r∈?gA"
 proof
 assume "r≥0"
 show "r∈ mono_tags lift) NIV_I\openx \<in .nrms_neg_ normed_gyroli'_axioms normed_gyr'_def norms_all_def normsdef)
 proof-
 obtain "r'" where "∣r'∣ = r / (?g xx)"
 using *
 by (meson ‹0 ≤ r› abs_of_nonneg divide_nonneg_nonneg)
 moreover have "r = ∣r'∣ * (?g xx)"
 by (simp add: ‹(if g xx < 0
 moreover have "r = ∣r'∣ * (?g (norm' x))"
 using ‹norm' x = xx› calculation(2) by blast
 moreover have "r = ?g (otimes' ∣r'∣ (norm' x))"
 using ‹g_iso (if g xx < 0 then λx. - 1 * g x else g)› ‹norm' x = xx›(f' z))"
 moreover have "r = ?g (norm' (∣r'∣ ⊗ x))"
 by (smt (verit, del_insts) calculation(4) normed_gyrolinear_space''_axioms normed_gyrolinear_space''_def)
 ultimately show ?thesis
 by blast
 qed
java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
 qed
 ultimately show ?thesis
 by blast
 qed
 
 ultimately show ?thesis
 by fastforce
 qed
 let ?gB = "{(?g b)|b. b∈?B}"
 have "?gB = {r::real. r≤0}"


 proof-
 have "∀
 proof
 fix a
 show " show "x ∈y= \<\^
 proof
 assume "a∈?B"
 show "?g a≤0"
 proof-
 obtain "r" where "a = norm' (r ⊗ y) "
 using ‹a ∈ {norm' (r ⊗ y) |r. True}›
  "g a == ?g (n' r <otimes 
 using calculation by presburger
 moreover have "?g a = ?g ( otimes' ∣r∣ (norm' y))"
 by (metis calculation(1) normed_gyrolinear_space''_axioms normed_gyrolinear_space''_def)
 moreover have "?g a = ∣r∣ * ?g (norm' y)"
 using ‹g_iso (if g xx < 0 then λx. - 1 * g x else g)› ‹
 
 ultimately show ?thesis
 by (simp add: ‹norm' y = yy› ‹yy ∈ norms ∧ (if g xx < 0 then λx. - 1 * g x else g) yy < 0›
 qed
 qed
 qed
 moreover have "?gB ⊆ {r::real. r≤
 using calculation by fastforce
 moreover have "{r::real. r≤0} ⊆> x \<>\
 proof-
 have "bij_betw ?g norms_all UNIV"
 using ‹
 moreover have "∀r::real. (r≤0 ⟶ r∈?gB)"
 proof
 fix rhave "x \<>\
 show "r≤0 ⟶ r∈?gB"
 proof
 assume "r≤0"
 show "r∈metis (mono_tags lifting Un_i ‹
 proof-
 obtain "r'" where "∣r'∣ = r / (?g yy)"
 using *
 by (metis ‹r ≤ 0›
 moreover have "r = ∣r'∣ (veverit, del_inst) \<> 
 using ‹yy ∈ norms ∧ (if g xx < 0 then λx. - 1 * g x else g) yy < 0› calculation by auto
 moreover "r = ∣
 using ‹norm' y = yy› calculation(2) by blast
 moreover have "r = ?g (otimes' ∣r'∣
 using ‹
java.lang.NullPointerException
 by (smt (verit, del_insts) calculation(4) normed_gyrolinear_space''_axioms normed_gyrolinear_space''_def)
 ultimately show ?thesis
 by blast
 qed
 qed
 qed
 ultimately show ?thesis
 by blast
 qed
 
 ultimately show ?thesis
 by fastforce
 qed

 let ?gX_norms = "{(?gll"
 let ?gX_norms_all = "{(?g x)|x. x∈norms_all}"
 let ?gA_union_B = "{(?g x)|x. x∈ ?A∪?B}"
 have "?gA_union_B ⊆ ?gX_norms"
 using ‹{norm' (r ⊗ x) |r. True} ∪ {norm' (r ⊗
 moreover hmoreoh "?gA_ = gA \<union 
 proof-
 have "?gA_union_B ⊆ ?gA ∪ ?gB"
 by blast
java.lang.NullPointerException
 by blast
 ultimately show ?thesis
 by force
 qed
 moreover have "?gA_union_B = UNIV"
 using ‹{(if g xx < 0 then λx. - 1 * g x else g) a |a. a ∈ {norm' (r ⊗ x) |r. True}} = {r. 0 ≤ r}›norms_all›axioms normed_gyrolinear_space'_ normsall_def n rangeI
 moreover have "UNIV ⊆ ?gX_norms"
 using calculation(3) calculation(5) by argo
 (* moreover have "?gX_norms \ ?gX_norms_all"
 proof-
 have "∀a. (a∈ ?gX_norms ⟶ a∈ ?gX_norms_all)"
        using norms_all_def by fastforce*)
      (*moreover have "\<exists>a. (a\<in>?gX_norms_all \<and> \<not>a\<in>?gX_norms)"
      proof-*)
        obtain "a" where "a∈moreo have "x  \<^ubf
          by (metis (mono_tags, lifting) Un_iff add.inverse_inverse assms mult_minus1 norms_all_def norms_def norms_neg_def rangeE rangeI zero_only_norms_norms_neg)
java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 23
        have "?a ∈ ?gX_norms_all "

          using ‹using calculat(2) by force

 moreover have "¬ ⊕
 proof(rule ccontr)
 assume "¬(¬?a∈ ?gX_norms)"
 have "?a∈?gX_norms"
  then <>xclose> by blast
 then obtain "b" where "b∈norms ∧ ?g b = ?a"
 by force
 
 then show False using ‹a ∈ norms_all ∧ a ∉ norms› ‹g_iso (if g xx < 0 then λx. - 1 * g x else g)› bij_betw_inv_into_left g_iso_def inf_sup_ord(3) norms_all_def subsetD
 by (smt (verit, ccfv_threshold) ‹g_iso g›)
 qed
 moreover have "False"

 using ‹UNIV ⊆ {(if g xx < 0 then λx. - 1 * g x else g) x |x. x ∈ norms}› calculation(7) by blast
 
 ultimately show False
 by auto
 qed
 

 moreover have " bij_betw (λx. if x ∈ norms then (?g x) else undefined) norms {r::real. r≥0}"
 proof-
 let ?f = "(λx. if x ∈ norms then (?g x) else undefined)"
 let ?A = "{norm' (r ⊗ x) | r::real. True}"
 let ?gA = "{(?g a)|a. a∈?A}"
 have s1:"?gA = {r::real. r≥0}"
 proof-
 have "∀a. (a∈?A ⟶blast
 proof
 fix a
 show "(a∈
 proof
 assume "a∈
 show "?g a ≥0"
 proof-
 obtain "r" where "a = norm' (r ⊗fix x
 using ‹Longrightarrow> a \<otimes\
 moreover have "?g a = ?g (norm' (r ⊗ x) )"
 using calculation by presburger
 moreover have "?g a = ?g ( otimes' ∣r∣ (norm' x))"
 by (metis calculation(1) normed_gyrolinear_space''_axioms normed_gyrolinear_space''_def)
 moreover have "?g a = ∣
 using ‹
 ultimately show ?thesis
 by (simp add: ‹show " a ×
 qed
 qed
 qed
 moreover have "?gA ⊆ {r::real. r≥0}"
 using calculati by fastforce
 moreover have "{r::real. r≥0} ⊆ ?gA"
 proof-
 have "bij_betw ?g norms_all UNIV"
 using ‹g_iso (if g xx < 0 then λx. - 1 * g x else g)›
 moreover have "∀r::real. (r≥
 proof
 fix r
 show "r≥0 ⟶ r∈?gA"
 proof
 assume "r≥0"
 show "r∈?gA"
 proof-
 obtain "r'" where "∣r'∣ = r / (?g xx)"
 using *
 by (meson ‹
 have" = \bar<>*
 by (simp add: ‹(if g xx < 0 then λx. - 1 * g x proof-
 moreover have "r = ∣\<^ubf
 using ‹norm' x = xx› calculation(2) by blast
 moreover have "r = ?g (otimes' ∣r'∣ (norm' x))"
 n>g (if g <0 close> calculation(3 g_iso_def norms_all y auto
 moreover have "r = ?g (norm' (∣r'∣ ⊗ x))"
 by (smt (verit, del_insts) calculation(4) normed_gyrolinear_space''_axioms normed_gyrolinear_space''_def)
 ultimately show ?thesis
 by blast
 qed
 qed
 qed
 ultimately show ?thesis
 by blast
 qed
 
 ultimately show ?thesis
 by fastforce
 qed
 moreover have s2:"∀y. (?g (norm' y) ≥0)"
 using ‹∀x. x ∈ norms ⟶ 0 ≤ (if g xx < 0 then λx. - 1 * g x else g) x› norms_def by blast
 moreover have "norms = ?A"
 proof-
 have "∀y. (?g (norm' y) ∈ ?gA)"
 using s1 s2 by blast
 moreover have "norms ⊆ ?A"
 proof-
 have "∀y. (y∈norms ⟶ y∈?A)"
 proof
 fix y
 show "y∈norms ⟶ have (a + b) ⊗')(a*(f' x) + b*(f' x))"
 proof
 assume "y∈norms"
 show "y∈?A"
 proof-
 obtain "yy" where "y=norm' yy"
 using ‹
 moreover have "?g (norm' yy) ∈?gA"
 usingusing ‹norm' r \> |r. True}}›
 moreover have "norm' yy ∈ ?A"
 proof-
 obtain "h" where "h ∈ ?A ∧ ?g h = ?g (norm' yy)"
 using calculation(2) by fastforce
 moreover have "?g h ≥0"
 using calculation s2 by blast
 
 moreover {
 assume "?g = g"
 have " g h = g (norm' yy)"
 by (smt (verit, ccfv_SIG) calculation(1))
 
 moreover have "h=norm' yy"
 proof-
 have "h∈norms"
 using ‹h ∈ {norm' (r ⊗ x) |r. True} ∧ (if g xx < 0 then λx. - 1 * g x else g) h = (if g xx < 0 then λx. - 1 * g x else g) (norm' yy)› norms_def by force
 moreover have "norm' yy ∈ norms"
 using ‹y = norm' yy› ‹y ∈ norms› by blast
 ultimately show ?thesis
 by (metis ‹g h = g (norm' yy)› ‹g_iso g› bij_betw_inv_into_left g_iso_def inf_sup_ord(3) norms_all_def subset_iff)
 qed
 ultimately have ?thesis
 using ‹h ∈ {norm' (r ⊗ x) |r. True} ∧ (if g xx < 0 then λx. - 1 * g x else g) h = (if g xx < 0 then λx. - 1 * g x else g) (norm' yy)› by blast
 }
 moreover {
 assume "?g = (λg x))"
 have " g h = g (norm' yy)"
 by (smt (verit, ccfv_SIG) calculation(1))
 
 moreover have "h=norm' yy"
 proof-
 have "h∈norms"
 using ‹h ∈ {norm' (r ⊗ x) |r. True} ∧ (if g xx < 0 then λx. - 1 * g x else g) h = (if g xx < 0 then λx. - 1 * g x else g) (norm' yy)› norms_def by force
 moreover have "norm' yy ∈\>b f no UNIV"
 using ‹y = norm' yy›
 ultimately show ?thesis
 by (metis ‹g h = g (norm' yy)› ‹g_iso g› bij_betw_inv_into_left g_iso_def inf_sup_ord(3) norms_all_de then s ?the
 qed
 ultimately have ?thesis
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 }
 ultimately show ?thesis
 by argo
 qed
 ultimately show ?thesis
 by fastforce
 qed
 qed
 qed
 show ?thesis
 using ‹∀y. y ∈ norms ⟶ y ∈ {norm' (r \<    (
 moreover have "a \<otimes\)= (inv_int norms_all f) ((a(f' x)) + (b*(f' x)))"
 ultimately show ?thesis
 using norms_def by fastforce
 qed
 moreover have step1:"inj_on ?f norms"
 proof-
 have "∀x.∀y. (x∈ norms ∧ y∈ norms ∧ (?f x) = (?f y) ⟶ x=y)"
 proof
 fix x
 show "∀y. (x∈ norms ∧ y∈ norms ∧ (?f x) = (?f y) ⟶ x=y)"
 proof
 fix y
 show " (x∈ norms ∧ y∈ norms ∧(smt (verit, ccf) UNIV_I bij_betw_imp_surj_on f normed_g.norms_ normed_gyrolinear_spce'_'_axioms normed_gyroline_defnorms norms_)
 by (metis ‹g_iso (if g xx < 0 then λx. - 1 * g x else g)› bij_betw_imp_inj_on g_iso_def inf_sup_ord(3) inj_on_def norms_all_def subsetD)
 qed
 qed
 then show ?thesis
 usinginj_on_d by blast
 qed
 moreover have "∀r::real. (r≥0 ⟶ (∃
 by (smt (verit) calculation(3) mem_Collect_eq s1)
 
 moreover h s:"\forall::real. (r\<>0
 
 using calculation(5) by blast
 moreover have "∀r∈{x::real. x≥0}. (∃x∈norms. (?f x = r))"
 using step2
 by blast
 moreover have **:"?f=(λx. if x ∈ norms then (?g x) else undefined)"
 by meson
 moreover have "?f ` norms = {r: ssume "x\innorms_"
 by (smt (verit) Collect_cong Setcompr_eq_image calculation(3) s1)
 ultimately show ?thesis
 by (simp : bij_betw_def)
 
 qed
 
 ultimately show ?thesis
 by blast
 




 smt (verit, best) \<>x'_def norm norms_def)
 assumes "x∈norms" "y∈norms_all"
 "x≤y"
  "y∈ norms"
 -
 have "x < y ∨ x=y"
 using assms(3) by argo
 moreover {
 assume "x<y"
 have ?thesis
 by (smt (verit) Un_iff ‹
 }
 moreover {
 assume "x=y"
 
 using ‹x = y› assms(1) by blast
 }
 ultimately show ?thesis by blast
 

  existence_of_f:
 assumes "\<exists>x. (x\<in>norms_all \<and> x\<noteq>0)" (* not trivial domain *)
  shows "∃f. (bij_betw f norms {x::real. x≥0}
\<and> (∀y::real. ∀z::real. (( y∈ norms ∧
z∈ norms ∧ y>z)⟶ (f y) > (f z)))
  ∧ (∀x. ∀y. f(norm' (x ⊕ y)) ≤f') (1*f' x)"
∧ (∀r::real. (∀x. (f (norm' (r ⊗ x)) = ∣r∣ * (f (norm' x))))))"
proof-
  obtain "g" where "(g_iso g ∧ (∀x.(x∈norms ⟶ (g x)≥0))
∧ bij_betw (λx. if norms_all lambda_one'norms_neg_def'axioms'_defnorms_all_def)
    using  iso_with_real_positive_on_norms
    assms by blast
  let ?f = "λx. if x ∈ norms then (g x) else undefined"
  have "∀α::real. ∀β::real. ∀x. ((0 ≤ α ∧ α ≤ β) ⟶ ((otimes' α (norm' x)) ≤java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
  proof
    fix α
    show " ∀β::real. ∀x.((0 ≤ α ∧ α ≤ β) ⟶ ((otimes' α  (norm' x)) ≤ (otimes' β  (norm' x))))"
    proof
      fix β{
      show " ∀x.((0 ≤ fix
      proof
        fix x 
        show "((0 ≤ α ∧ α ≤ β) ⟶ ((otimes' α (norm' x)) ≤f ) = (i norm f') (a * ' x)+ f'y)))"
        proof
          assume "0 ≤ α ∧ α ≤ β"
          show "((otimes' α (norm' x)) ≤ (otimes' β (norm' x)))"
          proof-
            haveotimesx  norm>\otimesx"
              by (metis ‹0 ≤ α ∧ α ≤ β><sub>f x\<oplus\(inv_into no f' (a * f' x))+(f' (nv_i normf'(a * (f'' y)))))
            moreover have " norm' (α ⊗ x) = norm' (((β+α)/2 - (β-α)/2)⊗ x)"
 by (simp add: add_divide_distrib diff_divide_distrib)
 moreover have "norm' (((β+α)/2 - (β-α)/2)⊗ x) =
 norm' (((β+α)/2) ⊗ x ⊕ (- (β-α)/2) ⊗ x )"
 by (metis add.commute divide_minus_left scale_distrib uminus_add_conv_diff)
 moreover have " norm' (((β+α)/2) ⊗ x ⊕ (- (β-α)/2) ⊗ x )
 ≤ oplus' (norm' (((β+α)/2)⊗ x)) (norm' ((-(β-α)/2) ⊗ x))"
 using ax3
 by blast
 moreover have "-(β-α)/2 ≤0"
 by (simp add: ‹0 ≤ α ∧ α ≤ β›)
 moreover have "(β+α)/2 ≥0"
 using ‹0 ≤ α ∧ α ≤ β› by auto
 moreover have *:"(norm' (((β+α)/2)⊗ bij_betw_imp_surj_on f normed_gyrolinear_space'_axi normed_gyrolinear_sp'_ nor nonorms_neg_de rin.ring_di(1
 by (smt (verit, ccfv_threshold) calculation(6) normed_gyrolinear_space''_axioms normed_gyrolinear_space''_def)
 moreover have " ∣-(β-α)/2∣ = (β-\< (
 using calculation(5) by force
 moreover have **:"(norm' ((-(β-α)/2)⊗ x)) =(otimes' ((β-α>x yy .
 by (metis calculation(8) normed_gyrolinear_space''_axioms normed_gyrolinear_space''_def)
 moreover have " oplus' (norm' (((β+α)/2)⊗ x)) (norm' ((-(β-α)/2) ⊗^subf y) a \<otimes\
 oplus' (otimes' ((β+α)/2) (norm' x)) (otimes' ( (β-α)/2) (norm' x)) "
 using * **
 by presburger
 moreover have "oplus' (otimes' ((β+α)/2) (norm' x)) (otimes' ( (β-α)/2) (norm' x))
 = otimes' ((β+α)/2 + ((β-α)/2)) (norm' x)"
 by (metis Un_iff ax_space one_dim_vector_space_with_domain_def rangeI vector_space_with_domain.smult_distr_sadd)
 moreover have " otimes' ((β+α)/2 + ((β-α)/2)) (norm' x) = otimes' β (norm' x)"
 by argo
 ultimately show ?thesis
 by linarith
 qed
 qed
 qed
 qed
 qed
 moreover have "∀α::real. ∀β::real. ∀x. ((0 < \α ∧ α < \β ∧ x≠gyrozero) ⟶
 proof -
 have f1: "∀f f
 by (simp add: abs_if_raw normed_gyrolinear_space''_def)
 obtain rr :: "real ==> real" where
 : bijrrnorms_all UNIVr \in 🚫
 using assms iso_with_real by auto
 have "∀a. (0 = norm' a) = (0_g = a)"
 using f1 by (smt (z3) normed_gyrolinear_space''_axioms)
 then show ?thesis
 using f2 by (smt (z3) UnI2 bij_betw_inv_into_left calculation mult_right_cancel norms_all_def norms_def rangeI sup_commute)
 qed
 moreover obtain "xx0" where "xx0∈norms ∧ xx0≠0"
 using assms not_trivial_domen_has_pos by blast
 moreover obtain "x0" where "xx0 = norm' x0"
 using calculation(3) norms_def by auto
 moreover have mon:"(∀y z. y ∈ norms ∧ z ∈ norms ∧ z < y ⟶ ?f z < ?f y)"
 proof
 fix y
 show "∀z. (y ∈ norms ∧ z ∈ norms ∧ z < y ⟶ ?f z < ?f y)"
 proof
 fix z
 show "y ∈ norm \<> 
 proof
 assume "y ∈norms ∧ z ∈ norms ∧ z < y"
 show "?f z < ?f y"
 proof-
 let ?alpha = "(?f y)/(?f (norm' x0))"
 let ?beta = "(?f z)/(?f (norm' x0))"
 have "otimes' ?alpha (norm' x0) = y"
 by (smt (verit, del_insts) ‹
 moreover have "oti ?b (norm'x0) ) = z"
 by (smt (verit, del_insts) ‹g_iso g ∧ (∀x. x ∈ norms ⟶ 0 ≤ g x) ∧ bij_betw (λx. if x ∈ norms then g x else undefined) norms {r. 0 ≤ r}› ‹
 moreover have "?alpha ≥ 0 ∧ ?beta ≥0"
 using ‹g_iso g ∧ (∀x. x ∈ norms ⟶ 0 ≤))+ (f' no' y))
 moreover have "0 < ?
 by (smt (verit, ccfv_threshold) ‹∀α β x. 0 ≤ α ∧ α ≤ β ⟶ otimes' α (norm' x) ≤ otimes' β (norm' x)› ‹
 moreover have "0<?lpha ⟷?f y <(f
 by (smt (verit, best) ‹∀α β x. 0 ≤ α ∧ α
 ultimately show ?thesis
 using ‹g_iso g ∧'(r ⊗
 qed
 qed
 qed
 qed
 moreover have " (∀x y. ?f (norm' (x ⊕'.norms_neg_def normed'_axio n'_def norms_al_def norms_ rangeI)
 proof
 fix x
 show "∀y. (?f (norm' (x ⊕ y)) ≤ ?f (norm' x) + ?f (norm' y))"
 proof
 fix y
 show " (?f (norm' (x ⊕ y)) ≤ ?f (norm' x) + ?f (norm' y))"
 proof-
 have "norm' x∈norms"
 using norms_def by blast
 moreover have "norm' y ∈"vector_space_w normsall norm_opl 0 no
 using norms_def by blast
 moreover have "norm' (x ⊕ y)∈ norms"
 using norms_def by blast
 moreover have "norm' (x ⊕ y) ≤∀
 using ax3 by blast
 moreover have "(?f (norm' (x ⊕ y))) ≤ (?f (oplus' (norm' x) (norm' y)))"
 proof-
 have "norm' (x ⊕ y) ≤ oplus' (norm' x) (norm' y) ∨ norm' (x ⊕ y) = oplus' (norm' x) (norm' y)"
 using calculation(4) by blast
 moreover {
 assume st1:"norm' (x ⊕ y) < oplus' (norm' x) (norm' y)"
 have "norm' x ∈proof
 using norms_def by blast
 moreover have "norm' y ∈ norms"
 using norms_def by blast
 moreover have "vector_space_with_domain norms_all oplus' 0 otimes'"
 using ax_space norms_def
 one_dim_vector_space_with_domain_def
 by (metis norms_all_def norms_neg_def)
 moreover have "oplus' (norm' x) (norm' y) ∈ norms_all"
 by (metis Un_iff calculation(1) calculation(2) calculation(3) norms_all_def vector_space_with_domain.add_closed)
 moreover have st2:"norm' (x \\> y)∈
 by (simp add: ‹norm' (x ⊕ y) ∈ norms›)
 moreover have st3:"oplus' (norm' x) (norm' y) ∈ norms"
 using ax3 calculation(4) comparing_norms_help st2 by blast
 
  have "(?f (norm' (x ⊕ y))) < (?f (oplus' (norm' x) (norm' y)))"
 using mon st1 st2 st3
 by blast
 ultimately have ?thesis
  by lina
 }
 moreover {
 assume "norm' (x ⊕ y) = oplus' (norm' x) (norm' y)"
 then hav show "yn
 by auto
 }
 ultimately show ?t
 by fastforce
 qed
 moreover have " (?f (oplus' (norm' x) (norm' y))) = (?f (norm' x)) + (?f (norm' y))"
 proof-
 have f1:"norm' (x ⊕∈
 using ax3 by blast
 moreover have f2:"norm' (x ⊕ y) ∈x∈
 by (simp add: ‹norm' (x ⊕ r ×
 moreover have f3:"vector_space_wi
 using ax_space norms_def
 one_dim_vector_space_with_domain_def
 by (metis norms_all_def norms_neg_def)
 moreover have "oplus' (norm' x) (norm' y)∈ norms"
 by (metis UnI1 ‹norm' x ∈ norms›
 ultimately show ?thesis
 using ‹
 qed
 ultimately show ?thesis
 by force
 qed
 qed
 qed

 moreover have "(∀r::real. (∀x. (?f (norm' (r ⊗ x)) = ∣r∣ * (?f (norm' x)))))"
 by (smt (verit, ccfv_SIG) Un_iff ‹g_iso g ∧ (∀x. x ∈ norms ⟶ 0 ≤ g x) ∧

java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 25
 using ‹