Theory Polygonal_Number_Theorem_Cauchy

subsection ‹ Cauchy's Polygonal Number Theorem ›

text ‹We will use the definition of the polygonal numbers from the Gauss Theorem theory file
which also imports the Three Squares Theorem AFP entry \cite{Three_Squares-AFP}.›

theory Polygonal_Number_Theorem_Cauchy
  imports Polygonal_Number_Theorem_Gauss
begin

text‹The following lemma shows there are two consecutive odd integers in any four consecutive
integers.›

lemma two_consec_odd:
  fixes a1 a2 a3 a4 :: int
  assumes "a1-a2 = 1"
  assumes "a2-a3 = 1"
  assumes "a3-a4 = 1"
  shows "∃k1 k2 :: int. {k1, k2} ⊆ {a1, a2, a3, a4} ∧ (k2 = k1+2) ∧ odd k1"

proof -
  have c1:"∃k1 k2 :: int. {k1, k2} ⊆ {a1, a2, a3, a4} ∧ (k2 = k1+2) ∧ odd k1"
    if odd_case:"odd a4"
  proof-
    define k1 where k1_def:"k1 = a4"
    define k2 where k2_def:"k2 = k1 + 2"
    have 0:"k2 = a2" using k2_def k1_def assms by simp
    have 1:"odd k1" using k1_def odd_case by simp
    show "∃k1 k2 :: int. {k1, k2} ⊆ {a1, a2, a3, a4} ∧ (k2 = k1+2) ∧ odd k1"
      using 0 1 k1_def k2_def by auto
  qed

  have c2:"∃k1 k2 :: int. {k1, k2} ⊆ {a1, a2, a3, a4} ∧ (k2 = k1+2) ∧ odd k1"
    if even_case:"even a4"
  proof -
    define k1 where k1_def:"k1 = a3"
    define k2 where k2_def:"k2 = k1 + 2"
    have 2:"odd k1" using even_case assms k1_def by presburger
    have 3:"k2 = a1" using k1_def k2_def assms by simp
    show "∃k1 k2 :: int. {k1, k2} ⊆ {a1, a2, a3, a4} ∧ (k2 = k1+2) ∧ odd k1"
      using 2 3 k1_def k2_def by auto
  qed
  show ?thesis using c1 c2 by auto
qed

text ‹This lemma proves that for two consecutive integers $b_1$ and $b_2$, and $r \in \{0,1,\dots,m-3\}$,
numbers of the form $b_1+r$ and $b_2+r$ can cover all the congruence classes modulo $m$.›

lemma cong_classes:
  fixes b1 b2 :: int
  fixes m :: nat
  assumes "m ≥ 4"
  assumes "odd b1"
  assumes "b2 = b1 + 2"
  shows "∀N::nat. ∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"

proof -
  have first:"∀N::nat. ∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
    if first_assum:"b1 mod m ≥ 3"
  proof -
    define k1 where k1_def:"k1 = b1 mod m"
    define l where l_def:"l = m - k1"
    have k1_size:"k1≥3" using first_assum k1_def by simp
    have l_size:"l ≤ m-3" using first_assum k1_def l_def by auto
    have "(l+k1) mod m = 0" using l_def by auto
    hence "(l+b1) mod m = 0" using k1_def l_def by (metis mod_add_right_eq)
    define w where w_def:"w = m-3-l"
    have w_size:"w≥0 ∧ w≤m-3" using w_def l_size l_def k1_def first_assum
      by (smt (verit, best) Euclidean_Rings.pos_mod_bound assms(1) le_antisym numeral_neq_zero
 of_nat_0_less_iff order_trans_rules(22) verit_comp_simplify(3) zero_le_numeral)
    have "k1 = w+3" using w_def k1_def l_def w_size first_assum by linarith
    hence "w+2 = k1-1" by auto
    hence "w+2 = (b1-1) mod m" using first_assum k1_def
      by (smt (verit, del_insts) Euclidean_Rings.pos_mod_bound assms(1)
 mod_diff_eq mod_pos_pos_trivial of_nat_le_0_iff verit_comp_simplify(8))
    hence w_cover:"w+2 = k1-1" using k1_def using ‹w + 2 = k1 - 1› by fastforce

    have "∃r::nat. (r≤m-3) ∧ [N=b1+r] (mod m)" if asm1:"N mod m ≥ k1 ∧ N mod m ≤ m-1" for N
    proof -
      have "m - (N mod m) ≤ l" using asm1 l_def k1_def by linarith
      hence "∃d::nat. d≤l ∧ [N = k1+d] (mod m)" using asm1 k1_def l_def
        by (metis add.commute add_le_cancel_left cong_mod_left cong_refl diff_add_cancel
 diff_le_self le_trans of_nat_mod of_nat_mono zle_iff_zadd)
      hence "∃d::nat. d≤l ∧ [N=b1+d] (mod m)" using  k1_def
        by (metis mod_add_left_eq unique_euclidean_semiring_class.cong_def)
      thus "∃r::nat. (r≤m-3) ∧ [N=b1+r] (mod m)" using l_size
        by (smt (verit, best) nat_leq_as_int)
    qed
    hence c1:"∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)" if asm1:"N mod m ≥ k1 ∧ N mod m ≤ m-1" for N using asm1 by blast

    have c2:"∃r::nat. (r≤m-3) ∧ [N=b1+r] (mod m)" if asm2:"N mod m =0" for N using l_def k1_def
      by (smt (verit, ccfv_threshold) ‹(l + b1) mod int m = 0› add_diff_cancel_left' cong_0_iff
 cong_sym cong_trans diff_add_cancel diff_ge_0_iff_ge dvd_eq_mod_eq_0 int_dvd_int_iff nat_0_le
of_nat_le_iff that w_def w_size)
    hence c2:"∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
      if asm2:"N mod m = 0" for N using asm2 by metis

    have "∃r::nat. (r≤m-3) ∧ [N=b1+r] (mod m)" if asm3:"N mod m > 0 ∧ N mod m ≤w" for N
    proof -
      have "l+ (N mod m) ≤ m-3" using asm3 w_def by auto
      hence "∃d::nat. (d≤w) ∧ [N = k1+l+d] (mod m)" using asm3 w_def k1_def l_def
        by (smt (verit, ccfv_threshold) minus_mod_self2 mod_mod_trivial of_nat_mod
            unique_euclidean_semiring_class.cong_def)
      hence "∃d::nat. (d≤w) ∧ [N = b1+l+d] (mod m)" using k1_def by (metis (mono_tags,
              opaque_lifting) mod_add_left_eq unique_euclidean_semiring_class.cong_def)
      hence "∃r::nat. (r≤w+l) ∧ [N = b1+r] (mod m)" by (smt (verit) add.commute
            add.left_commute le_add_same_cancel2 of_nat_0_le_iff w_def w_size zero_le_imp_eq_int)
      thus "∃r::nat. (r≤m-3) ∧ [N = b1+r] (mod m)" using w_def by auto
    qed
    hence "∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
      if asm3:"N mod m > 0 ∧ N mod m ≤w" for N using asm3 by blast
    hence c3:"∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
      if asm8:"N mod m > 0 ∧ N mod m ≤k1-3" using asm8 w_cover by auto

    have "∃r::nat. (r≤m-3) ∧ [N=b2+r] (mod m)" if asm4:"N mod m = w+1 ∨ N mod m = w+2" for N
    proof -
      have c4_1:"[N = b2+(m-3)] (mod m)" if asm5:"N mod m=w+2" for N using asm5 w_def assms(3) l_def
        by (smt (verit) ‹w + 2 = (b1 - 1) mod int m› ‹w + 2 = k1 - 1› mod_add_self1 of_nat_mod
            unique_euclidean_semiring_class.cong_def)
      hence "[N-1 = b2+(m-4)] (mod m)" if asm5:"N mod m = w+2" for N
        by (smt (verit, ccfv_threshold) Num.of_nat_simps(2) ‹w + 2 = k1 - 1› asm5 assms(1)
 cong_iff_lin first_assum k1_def l_def mod_less_eq_dividend numeral_Bit0 of_nat_diff of_nat_le_iff
of_nat_numeral semiring_norm(172) w_def)
      hence "[N = b2+(m-4)] (mod m)" if asm6:"N mod m = w+1" for N using asm6
        by (metis ‹w + 2 = (b1 - 1) mod int m› add_diff_cancel_right' arith_special(3) int_ops(4)
            is_num_normalize(1) mod_add_left_eq mod_diff_left_eq of_nat_mod)
      thus ?thesis using c4_1 by (metis asm4 diff_le_mono2 nat_le_linear numeral_le_iff
            verit_comp_simplify(10) verit_comp_simplify(13))
    qed
    hence "∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
      if asm4:"N mod m = w+1 ∨ N mod m = w+2" for N using asm4 by blast
    hence c4:"∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
      if asm7:"N mod m = k1-2 ∨ N mod m = k1-1" for N using w_cover asm7 by auto

    have "∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
      if asm10:"N mod m ≥ 0 ∧ N mod m ≤w" for N using c2 c3 asm10
      using ‹⋀N. 0 < N mod m ∧ int (N mod m) ≤ w ⟹ ∃b r. r ≤ m - 3 ∧ [int N = b + int r]
      (mod int m) ∧ (b = b1 ∨ b = b2)› by blast
    hence c5:"∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
      if asm11:"N mod m ≥ 0 ∧ N mod m ≤k1-3" for N using w_cover using asm11 by force

    have c6:"∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
      if asm9:"(N mod m ≥0 ∧ N mod m ≤ k1-3) ∨ N mod m = k1-2 ∨ N mod m = k1-1" for N
      using c5 c4 asm9 by blast

    hence c7:"∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
      if asm12:"(N mod m ≥0 ∧ N mod m ≤ k1-3) ∨ N mod m = k1-2 ∨ N mod m = k1-1 ∨
 (N mod m ≥ k1 ∧ N mod m ≤ m-1)" for N using asm12 c1 by blast

    have "∀N::nat. (N mod m ≥0 ∧ N mod m≤ k1-3) ∨ N mod m = k1-2 ∨ N mod m = k1-1 ∨
 (N mod m ≥ k1 ∧ N mod m ≤ m-1)" using k1_def
      by (smt (verit, best) Suc_pred' assms(1) bot_nat_0.extremum le_simps(2) mod_less_divisor
not_numeral_le_zero of_nat_0_less_iff of_nat_le_0_iff)

    thus ?thesis using c7 by auto
  qed

  have second:"∀N::nat. ∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
    if second_assum:"b1 mod m ≥0 ∧ b1 mod m ≤2"
  proof -
    have case1:"∀N::nat. ∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
      if case1_assum:"b1 mod m = 0"
    proof -
      have "∃r::nat. (r ≤ m-3) ∧ [N = b1+r] (mod m)" if case1_1_assum:"N mod m ≤m-3" for N
        using case1_assum case1_1_assum
        by (metis cong_add_rcancel_0 cong_mod_left cong_refl cong_sym_eq zmod_int)
      hence case1_1:"∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
        if case1_1_assum:"N mod m ≤m-3" for N using case1_1_assum by blast

      have "[N = b1+(m-2)] (mod m)" if case1_2_assum:"N mod m = m-2" for N using case1_2_assum
        case1_assum by (metis (no_types, opaque_lifting) add.commute cong_add_lcancel_0
            cong_mod_right of_nat_mod unique_euclidean_semiring_class.cong_def)
      hence "[N = b2+(m-4)] (mod m)" if case1_2_assum:"N mod m = m-2" for N using case1_2_assum assms(3)
        by (smt (verit, best) add_leD2 assms(1) int_ops(2) numeral_Bit0 of_nat_diff of_nat_numeral
            semiring_norm(172))
      hence "∃r::nat. (r ≤ m-3) ∧ [N = b2+r] (mod m)" if case1_2_assum:"N mod m = m-2" for N
        using case1_2_assum
        by (meson diff_le_mono2 less_num_simps(2) numeral_le_iff verit_comp_simplify(15))
      hence case1_2:"∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
        if case1_2_assum:"N mod m = m-2" for N using case1_2_assum by blast

      have "[N = b1+(m-1)] (mod m)" if case1_3_assum:"N mod m = m-1" for N using case1_3_assum
        case1_assum by (metis (no_types, opaque_lifting) add.commute cong_add_lcancel_0
            cong_mod_right of_nat_mod unique_euclidean_semiring_class.cong_def)
      hence "[N = b2+(m-3)] (mod m)" if case1_3_assum:"N mod m = m-1" for N using case1_3_assum assms(3)
        by (smt (verit, best) assms(1) int_ops(2) int_ops(6) numeral_Bit0 numeral_Bit1 of_nat_mono
            of_nat_numeral semiring_norm(172))
      hence "∃r::nat. (r ≤ m-3) ∧ [N = b2+r] (mod m)" if case1_3_assum:"N mod m = m-1" for N
        using case1_3_assum by blast
      hence case1_3:"∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
        if case1_3_assum:"N mod m = m-1" for N using case1_3_assum by blast

      have "∀N::nat. (N mod m = m-1) ∨ (N mod m = m-2) ∨ (N mod m ≤ m-3)"
        by (smt (verit, ccfv_threshold) Suc_pred' assms(1) bot_nat_0.not_eq_extremum diff_diff_add
diff_is_0_eq' le_simps(2) mod_less_divisor nat_1_add_1 nat_less_le not_numeral_le_zero
numeral.simps(3) semiring_norm(172))
      thus ?thesis using case1_1 case1_2 case1_3 by blast
    qed

    have case2:"∀N::nat. ∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
      if case2_assum:"b1 mod m = 1"
    proof -
      have case2b2:"b2 mod m = 3" using case2_assum assms(3) by (smt (verit) assms(1) int_ops(2)
 mod_add_eq mod_pos_pos_trivial numeral_Bit0 of_nat_mono of_nat_numeral semiring_norm(172))

      have "∃r::nat. (r≤m-3) ∧ [N = b2+r] (mod m)" if case2_1_assum:"N mod m = m-1" for N
      proof -
        have "[N = 3+(m-4)] (mod m)" using case2_1_assum
          by (metis (mono_tags, lifting) Suc_eq_plus1 Suc_numeral add_diff_cancel_left
             arithmetic_simps(1) arithmetic_simps(7) assms(1)mod_mod_trivial
             ordered_cancel_comm_monoid_diff_class.diff_add_assoc unique_euclidean_semiring_class.cong_def)
        hence "[N = b2+(m-4)] (mod m)" using case2b2
          by (metis (mono_tags, lifting) Num.of_nat_simps(4) mod_add_left_eq
              of_nat_mod of_nat_numeral unique_euclidean_semiring_class.cong_def)
        thus ?thesis using le_diff_conv by fastforce
      qed
      hence case2_1:"∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
        if case2_1_assum:"N mod m = m-1" for N using case2_1_assum by blast

      have "∃r::nat. (r≤m-3) ∧ [N = b2+r] (mod m)" if case2_2_assum:"N mod m =0" for N
      proof -
        have "(3+(m-3)) mod m = 0" using assms(1) by fastforce
        hence "(b2+(m-3)) mod m = 0" using case2b2 by (metis Num.of_nat_simps(1)
              Num.of_nat_simps(4) mod_add_left_eq of_nat_mod of_nat_numeral)
        thus ?thesis using case2_2_assum
          by (metis int_ops(1) nat_le_linear of_nat_mod unique_euclidean_semiring_class.cong_def)
      qed
      hence case2_2:"∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
        if case2_2_assum:"N mod m =0" for N using case2_2_assum by metis

      have "∃r::nat. (r ≤ m-3) ∧ [N = b1+r] (mod m)" if
        case2_3_assum:"N mod m ≤m-2 ∧ N mod m ≥1" for N
      proof -
        have "∃r::nat. (r≤m-3)∧((b1+r) mod m = l)" if asml:"l≥1 ∧ l≤m-2" for l
        proof -
          define r where r_def:"r = l-1"
          from asml have r_range:"r≥0 ∧ r≤m-3" using r_def by linarith
          have "(1+r) mod m = l" using asml r_def r_range by fastforce
          hence "(b1+r) mod m = l" using case2_assum
            by (metis Num.of_nat_simps(3) int_ops(9) mod_add_left_eq plus_1_eq_Suc)
          thus ?thesis using asml r_range by blast
        qed
        thus ?thesis using case2_3_assum
          by (metis case2_3_assum of_nat_mod unique_euclidean_semiring_class.cong_def)
      qed
      hence case2_3:"∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
        if  case2_3_assum:"N mod m ≤m-2 ∧ N mod m ≥1" for N using case2_3_assum by blast

      have "∀N::nat. N mod m = 0 ∨ (N mod m ≥1 ∧ N mod m ≤m-1)" by (metis One_nat_def Suc_pred
assms(1) bot_nat_0.extremum_uniqueI leI less_Suc_eq_le mod_less_divisor not_numeral_le_zero)
      hence "∀N::nat. N mod m = 0 ∨ (N mod m ≥1 ∧ N mod m ≤m-2) ∨ N mod m = m-1"
        by (smt (verit) arithmetic_simps(68) diff_diff_eq le_add_diff_inverse le_neq_implies_less le_simps(2)
           le_trans plus_1_eq_Suc)
      thus ?thesis using case2_1 case2_2 case2_3 by (metis ‹⋀N. N mod m ≤ m - 2 ∧ 1 ≤ N mod m
   ⟹ ∃r≤m - 3. [int N = b1 + int r] (mod int m)›)
    qed

    have case3:"∀N::nat. ∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
      if case3_assum:"b1 mod m = 2"
    proof -
      have case3b2:"b2 mod m = 4"
        using assms case3_assum
        by (smt (verit, ccfv_SIG) Euclidean_Rings.pos_mod_sign dvd_mod_imp_dvd even_numeral int_ops(2)
                int_ops(4) mod_diff_eq mod_pos_pos_trivial nat_1_add_1 numeral_Bit0 of_nat_le_iff
                of_nat_numeral plus_1_eq_Suc)

      have "∃r::nat. (r≤m-3)∧ [N = b2+r] (mod m)" if case3_1_assum:"N mod m = 0 ∨ N mod m =1" for N
      proof -
        have "(4+(m-3)) mod m = (4+m-3) mod m" using assms(1) by auto
        have "(4+m-3) mod m = (1+m) mod m" by simp
        hence "(4+(m-3)) mod m = 1" using ‹(4+(m-3)) mod m = (4+m-3) mod m›
          by (smt (verit, best) Euclidean_Rings.pos_mod_bound add_lessD1 arith_special(2) assms(1) case3b2
              landau_product_preprocess(4) mod_add_self2 mod_less numeral_Bit0 of_nat_numeral order_le_less)
        hence caseone:"(b2+(m-3)) mod m = 1" using case3b2
          by (metis Num.of_nat_simps(2) Num.of_nat_simps(4) mod_add_left_eq of_nat_mod of_nat_numeral)

        have "(4+(m-4)) mod m = 0" using assms(1) by auto
        hence casezero:"(b2+(m-4)) mod m = 0" using case3b2
          by (metis (full_types) Num.of_nat_simps(1) Num.of_nat_simps(4) mod_add_left_eq of_nat_mod of_nat_numeral)

        show ?thesis using caseone casezero case3_1_assum
          by (metis cong_int cong_mod_right cong_refl diff_le_mono2 nat_le_linear numeral_le_iff
              of_nat_0 of_nat_1 semiring_norm(69) semiring_norm(72))
      qed
      hence case3_1:"∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
        if case3_1_assum:"N mod m = 0 ∨ N mod m =1" for N using case3_1_assum by metis

      have "∃r::nat.(r≤m-3)∧ [N = b1+r] (mod m)" if case3_2_assum:"N mod m ≥2∧N mod m ≤ m-1" for N
      proof -
        have "∃r::nat. (r≤m-3)∧((b1+r) mod m = l)" if asml1:"l≥2 ∧ l≤m-1" for l
        proof -
          define r1 where r1_def:"r1 = l-2"
          from asml1 have r1_range:"r1≥0 ∧ r1≤m-2" using r1_def by linarith
          have "(2+r1) mod m = l" using asml1 r1_def r1_range by fastforce
          hence "(b1+r1) mod m = l" using case3_assum
            by (metis Num.of_nat_simps(4) mod_add_left_eq of_nat_mod of_nat_numeral)
          thus ?thesis using asml1 r1_range by (metis One_nat_def diff_diff_add diff_le_mono
                nat_1_add_1 numeral_3_eq_3 plus_1_eq_Suc r1_def)
        qed
        thus ?thesis using case3_2_assum
          by (metis case3_2_assum of_nat_mod unique_euclidean_semiring_class.cong_def)
      qed
      hence case3_2:"∃b::int. ∃r::nat. (r ≤ m-3) ∧ [N=b+r] (mod m) ∧ (b = b1 ∨ b = b2)"
        if case3_2_assum:"N mod m ≥2∧N mod m ≤ m-1" for N using case3_2_assum by blast

      have "∀N::nat. N mod m = 0 ∨ (N mod m ≥1 ∧ N mod m ≤m-1)" by (metis Suc_pred' assms(1)
bot_nat_0.not_eq_extremum less_one mod_Suc_le_divisor rel_simps(76) verit_comp_simplify1(3))
      hence "∀N::nat. N mod m = 0 ∨ N mod m = 1 ∨ (N mod m ≥2 ∧ N mod m ≤m-1)"
        by (metis Suc_eq_plus1 le_neq_implies_less le_simps(3) nat_1_add_1)

      thus ?thesis using case3_1 case3_2 by blast
    qed

    show ?thesis using case1 case2 case3 using that by fastforce
  qed

  show ?thesis using first second using assms(1) by force
qed

text‹The strong form of Cauchy's polygonal number theorem shows for a natural number $N$ satisfying
certain conditions, it may be written as the sum of $m+1$ polygonal numbers of order $m+2$, at most four
of which are different from 0 or 1. This corresponds to Theorem 1.9 in \cite{nathanson1996}.›

theorem Strong_Form_of_Cauchy_Polygonal_Number_Theorem_1:
  fixes m N :: nat
  assumes "m≥4"
  assumes "N≥108*m"
  shows "∃ xs :: nat list. (length xs = m+1) ∧ (sum_list xs = N) ∧ (∀k≤3. ∃a. xs! k = polygonal_number m a)
  ∧ (∀ k ∈ {4..m} . xs! k = 0 ∨ xs! k = 1)"

proof -
  define L where L_def:"L = (2/3 + sqrt (8*N/m - 8)) - (1/2 + sqrt (6*N/m - 3))"
  from assms L_def have "L>4" using interval_length_greater_than_four
    apply(rule_tac N = "of_nat N" and m = "of_nat m" in interval_length_greater_than_four)
    by auto
  define c1 where c1_def:"c1 = ⌈1/2 + sqrt (6*N/m - 3)⌉"
  define c2 where c2_def:"c2 = c1+1"
  define c3 where c3_def:"c3 = c1+2"
  define c4 where c4_def:"c4 = c1+3"
  from ‹L>4› c1_def c2_def c3_def c4_def L_def have "c4<(2/3 + sqrt (8*N/m - 8))" by linarith

  have "N/m ≥ 108" using assms using le_divide_eq by fastforce
  hence "sqrt(6*N/m - 3)≥1" by simp
  hence "1/2 + sqrt(6*N/m - 3) ≥1" by linarith
  hence "c1 ≥1" using c1_def by simp

  obtain b1 b2 where bproperties:"{b1, b2} ⊆ {c1, c2, c3, c4} ∧ (b2 = b1+2) ∧ odd b1"
    using two_consec_odd c1_def c2_def c3_def c4_def by (metis (no_types, opaque_lifting) Groups.add_ac(2)
empty_subsetI even_plus_one_iff insert_commute insert_mono nat_arith.add1 numeral.simps(2) numeral.simps(3))
  have b1andb2:"odd b1 ∧ b2 = b1+2" using bproperties by auto
  have b1pos:"b1 ≥1" using ‹c1≥1› c2_def c3_def c4_def bproperties by auto
  hence b2pos:"b2 ≥3" using bproperties by simp
  have b2odd:"odd b2" using bproperties by simp

  obtain b r where b_r:"r≤m-3 ∧ (b = b1 ∨ b = b2) ∧ [int N = b+r] (mod m)" using b1andb2 assms(1)
      cong_classes by meson
  have bpos:"b≥1" using b1pos b2pos b_r by auto
  have bodd:"odd b" using b_r bproperties by auto

  define a where a_def:"a = b+2*(N-b-r) div m"
  have m_div_num:"m dvd (N-b-r)" using b_r
    by (simp add: diff_diff_add mod_eq_dvd_iff unique_euclidean_semiring_class.cong_def)
  hence "(N-b-r)/m = (N-b-r) div m" by (simp add: real_of_int_div)
  hence a_def1:"a = b+2*(N-b-r)/m" using a_def by (metis ‹int m dvd int N - b - int r›
        dvd_add_right_iff mult_2 of_int_add of_int_of_nat_eq real_of_int_div)
  have "N-m>0" using assms by linarith
  hence "N-r>0" using b_r by force
  hence "(N-b-r) = (N-r)-b" by linarith
  hence "(N-b-r)/m = (N-r)/m - b/m" by (metis diff_divide_distrib int_of_reals(3) of_int_of_nat_eq)
  hence "a = b+2*((N-r)/m - b/m)" using a_def1 by (metis int_of_reals(6) of_int_mult times_divide_eq_right)
  hence a_def2:"a = b- b*2/m+2*(N-r)/m " by simp
  have "b*(1-2/m) = b*1-b*(2/m)" by (simp add: Rings.ring_distribs(4))
  hence a_def3:"a = b*(1-2/m) + 2*(N-r)/m" using a_def2 by simp
  have "1-2/m>0" using assms(1) by simp
  hence size1:"b*(1-2/m)>0" using bpos by simp
  have "N-r>0" using b_r assms by auto
  hence size2:"2*(N-r)/m>0" using assms(1) by simp
  have apos:"a≥1" using size1 size2 a_def3
    by linarith

  have "odd (b+2*(N-b-r) div m)" using m_div_num b_r b2odd bproperties
    by (metis div_mult_swap zdvd_reduce)
  hence aodd:"odd a" using a_def by simp

  from a_def1 have "a-b = 2*(N-b-r)/m" by simp
  hence "m*(a-b)/2 = N-b-r" using assms(1) by simp
  hence N_expr:"N = r+b+m*(a-b)/2" by simp

  have "b1 ≥ c1" using bproperties c2_def c3_def c4_def by force
  hence "b1 ≥ 1/2 + sqrt (6*N/m - 3)" using c1_def using ceiling_le_iff by blast
  have b_ineq1:"b ≥ 1/2 + sqrt (6*N/m - 3)" using b_r bproperties
    using ‹1 / 2 + sqrt (real (6 * N) / real m - 3) ≤ real_of_int b1› by fastforce

  have "b2 ≤ c4" using bproperties c1_def c2_def c3_def c4_def by fastforce
  hence "b2 ≤ (2/3 + sqrt (8*N/m - 8))"
    using ‹real_of_int c4 < 2 / 3 + sqrt (real (8 * N) / real m - 8)› by linarith
  hence b_ineq2:"b≤(2/3 + sqrt (8*N/m - 8))" using b_r bproperties by linarith

  define Nr where "Nr = real_of_nat N"
  define mr where "mr = real m"
  define ar where "ar = real_of_int a"
  define br where "br = real_of_int b"
  define rr where "rr = real_of_nat r"
  from assms(1) have "mr ≥3" using mr_def by auto
  from assms(2) have "N≥2*m" by simp
  hence "Nr ≥ 2*mr" using Nr_def mr_def ‹N ≥ 2 * m› by auto
  moreover have "br≥0" using br_def bpos by auto
  moreover have "mr≥3" using mr_def assms by auto
  moreover have "ar≥0" using ar_def apos by auto
  moreover have "rr≥0" using rr_def b_r by auto
  moreover have "mr > rr" using mr_def rr_def b_r assms(1) by linarith
  moreover have "Nr = mr*(ar-br)/2+br+rr" using Nr_def mr_def ar_def br_def N_expr rr_def by auto
  moreover have "1/2+sqrt(6*Nr/mr-3)≤br ∧ br≤2/3+sqrt(8*Nr/mr-8)" using Nr_def mr_def br_def b_ineq1 b_ineq2 by auto
  ultimately have "br^2<4*ar ∧ 3*ar<br^2+2*br+4" using Cauchy_lemma by auto
  hence real_ineq:"(real_of_int b)^2 < 4*(real_of_int a) ∧ 3*(real_of_int a) < (real_of_int b)^2 + 2*(real_of_int b) + 4"
    using br_def ar_def by auto
  hence int_ineq1: "b^2<4*a" using of_int_less_iff by fastforce
  from real_ineq have int_ineq2: "3*a<b^2+2*b+4" using of_int_less_iff by fastforce

  have con1:"nat a ≥1" using apos by auto
  have con2:"nat b ≥1" using bpos by auto
  have con3:"odd (nat a)" using aodd apos even_nat_iff by auto
  have con4:"odd (nat b)" using bodd bpos even_nat_iff by auto
  have "(nat b)^2 = b^2" using ‹nat b ≥1› by auto
  hence con5:"(nat b)^2<4*(nat a)" using int_ineq1 by linarith
  have con6:"3*(nat a)<(nat b)^2+2*(nat b)+4" using ‹(nat b)^2 = b^2› int_ineq2 by linarith
  obtain s t u v where stuv:"s ≥ 0 ∧ t ≥ 0 ∧ u ≥ 0 ∧ v ≥ 0 ∧ int(nat a) = s^2 + t^2 + u^2 + v^2 ∧
  int(nat b) = s+t+u+v" using four_nonneg_int_sum con1 con2 con3 con4 con5 con6 by presburger
  have a_expr:"a = s^2 + t^2 + u^2 + v^2" using apos stuv by linarith
  have b_expr:"b = s+t+u+v" using bpos stuv by linarith

  from N_expr have "N = m/2*(s^2-s+t^2-t+u^2-u+v^2-v)+r+(s+t+u+v)" using a_expr b_expr by simp
  hence N_expr2:"N = m/2*(s^2-s)+ m/2*(t^2-t)+ m/2*(u^2-u)+ m/2*(v^2-v)+ r+(s+t+u+v)"
    by (metis (no_types, opaque_lifting) add_diff_eq nat_distrib(2) of_int_add)
  have s_div2:"m/2*(s^2-s) = m*(s^2-s) div 2" using real_of_int_div by auto
  have t_div2:"m/2*(t^2-t) = m*(t^2-t) div 2" using real_of_int_div by auto
  have u_div2:"m/2*(u^2-u) = m*(u^2-u) div 2" using real_of_int_div by auto
  have v_div2:"m/2*(v^2-v) = m*(v^2-v) div 2" using real_of_int_div by auto
  have N_expr3:"N = (m*(s^2-s) div 2+s)+(m*(t^2-t) div 2+t)+(m*(u^2-u) div 2+u)+(m*(v^2-v) div 2+v)+r"
    using s_div2 t_div2 u_div2 v_div2 N_expr2 by simp

  define sn where "sn = nat s"
  define tn where "tn = nat t"
  define un where "un = nat u"
  define vn where "vn = nat v"
  have seqsn:"s^2-s = sn^2 - sn" using stuv sn_def
    by (metis int_nat_eq le_refl of_nat_diff of_nat_power power2_nat_le_imp_le)
  have teqtn:"t^2-t = tn^2 - tn" using stuv tn_def
    by (metis int_nat_eq le_refl of_nat_diff of_nat_power power2_nat_le_imp_le)
  have uequn:"u^2-u = un^2 - un" using stuv un_def
    by (metis int_nat_eq le_refl of_nat_diff of_nat_power power2_nat_le_imp_le)
  have veqvn:"v^2-v = vn^2 - vn" using stuv vn_def
    by (metis int_nat_eq le_refl of_nat_diff of_nat_power power2_nat_le_imp_le)

  from N_expr3 have
    "N = (m*(sn^2-sn) div 2+s)+(m*(tn^2-tn) div 2+t)+(m*(un^2-un) div 2+u)+(m*(vn^2-vn) div 2+ v)+r"
    using seqsn teqtn uequn veqvn by (metis (mono_tags, lifting) int_ops(2) int_ops(4) int_ops(7)
        numeral_Bit0 numeral_code(1) plus_1_eq_Suc zdiv_int)
  hence "N = (m*(sn^2-sn) div 2+sn)+(m*(tn^2-tn) div 2+tn)+(m*(un^2-un) div 2+un)+(m*(vn^2-vn) div 2+ v)+r"
    using sn_def tn_def un_def  stuv int_nat_eq int_ops(5) by presburger
  hence "N = (m*(sn^2-sn) div 2+sn)+(m*(tn^2-tn) div 2+tn)+(m*(un^2-un) div 2+un)+(m*(vn^2-vn) div 2+ vn)+r"
    using vn_def stuv by (smt (verit, del_insts) Num.of_nat_simps(4) int_nat_eq of_nat_eq_iff)
  hence "N = (m* sn*(sn-1) div 2+sn)+(m*tn*(tn-1) div 2+tn)+(m*un*(un-1) div 2+un)+(m* vn*(vn-1) div 2+ vn)+r"
    by (smt (verit, ccfv_threshold) more_arith_simps(11) mult.right_neutral power2_eq_square right_diff_distrib')
  hence N_expr4:"N = polygonal_number m sn + polygonal_number m tn + polygonal_number m un + polygonal_number m vn +r"
    using Polygonal_Number_Theorem_Gauss.polygonal_number_def by presburger

  define T where T_def:"T = [polygonal_number m sn,polygonal_number m tn,polygonal_number m un,polygonal_number m vn]"
  define ones where ones_def:"ones = replicate r (1::nat)"
  define zeros where zeros_def:"zeros = replicate (m+1-4-r) (0::nat)"
  define final where final_def:"final = T@ones@zeros"

  have "m+1-4-r≥0" using assms(1) b_r by force
  hence "4+r+(m+1-4-r) = m+1" using assms(1) b_r by force
  have "length final = 4+r+(m+1-4-r)" using final_def T_def ones_def zeros_def by auto
  hence final_length:"length final = m+1" using ‹4+r+(m+1-4-r) = m+1› by simp
  have T_sum:"sum_list T = polygonal_number m sn + polygonal_number m tn + polygonal_number m un + polygonal_number m vn" by (simp add: T_def)
  have ones_sum:"sum_list ones = r" using ones_def by (simp add: sum_list_replicate)
  have zeros_sum:"sum_list zeros = 0" using zeros_def by simp
  have "sum_list final = sum_list T + sum_list ones + sum_list zeros" using final_def by simp
  hence final_sum:"sum_list final = N" using N_expr4 by (simp add: T_sum ones_sum zeros_sum)

  have final_0th:"final! 0 = polygonal_number m sn" using final_def T_def by simp
  have final_1st:"final! 1 = polygonal_number m tn" using final_def T_def by simp
  have final_2nd:"final! 2 = polygonal_number m un" using final_def T_def by simp
  have final_3rd:"final! 3 = polygonal_number m vn" using final_def T_def by simp

  have first_four:"∀k≤3. ∃a. final! k = polygonal_number m a" using final_0th final_1st final_2nd final_3rd
    by (metis Suc_eq_plus1 add_leD2 arith_simps(50) le_simps(2) numeral_Bit0 numeral_Bit1
        numeral_One verit_comp_simplify1(3) verit_la_disequality)

  have "length T = 4" using T_def by simp
  have "∀k<length (ones@zeros). (ones@zeros)! k =1 ∨  (ones@zeros)! k =0" using ones_def zeros_def
    by (simp add: nth_append)
  hence "final! k = 1 ∨ final! k = 0" if "k≥4 ∧ k<(length final)" for k
    using ‹length T = 4› final_def that by (metis add_less_cancel_left le_add_diff_inverse
        length_append nth_append verit_comp_simplify1(3))
  hence other_terms:"∀ k ∈ {4..m} . final! k = 0 ∨ final! k = 1" using final_length
    by (metis Suc_eq_plus1 atLeastAtMost_iff le_simps(2))

  show ?thesis using final_length final_sum first_four other_terms by auto
qed


theorem Strong_Form_of_Cauchy_Polygonal_Number_Theorem_2:
  fixes N :: nat
  assumes "N≥324"
  shows "∃ p1 p2 p3 p4 r ::nat. N = p1+p2+p3+p4+r ∧ (∃k1. p1 = polygonal_number 3 k1) ∧ (∃k2. p2 = polygonal_number 3 k2)
∧ (∃k3. p3 = polygonal_number 3 k3) ∧ (∃k4. p4 = polygonal_number 3 k4) ∧ (r = 0 ∨ r = 1)"

proof -
  define L where L_def:"L = (2/3 + sqrt (8*N/3 - 8)) - (1/2 + sqrt (6*N/3 - 3))" (*Now m = 3*)
  from assms L_def have "L>4" using interval_length_greater_than_four
    apply -
    apply(rule interval_length_greater_than_four[where N = "of_nat N" and m = "of_nat 3"])
    by auto
  define c1 where c1_def:"c1 = ⌈1/2 + sqrt (6*N/3 - 3)⌉"
  define c2 where c2_def:"c2 = c1+1"
  define c3 where c3_def:"c3 = c1+2"
  define c4 where c4_def:"c4 = c1+3"
  from ‹L>4› c1_def c2_def c3_def c4_def L_def have "c4<(2/3 + sqrt (8*N/3 - 8))" by linarith

  define Nn where "Nn = int N"
  have "c4<(2/3 + sqrt (8*Nn/3 - 8))" using Nn_def ‹c4<(2/3 + sqrt (8*N/3 - 8))› by simp
  have Nn3:"(Nn-3)^2 - (sqrt (8*Nn/3 - 8))^2 = Nn^2-3*Nn-3*Nn+9 - (sqrt (8*Nn/3 - 8))^2"
    using assms Nn_def power2_diff by (simp add: power2_eq_square algebra_simps)
  have "(Nn-3)^2 - (sqrt (8*Nn/3 - 8))^2 = Nn^2-3*Nn-3*Nn+9 - (8*Nn/3 - 8)" using assms Nn_def Nn3 by fastforce
  hence "(Nn-3)^2 - (sqrt (8*Nn/3 - 8))^2 = Nn^2-6*Nn+9-8*Nn/3 +8" by linarith
  hence Nn4:"(Nn-3)^2 - (sqrt (8*Nn/3 - 8))^2 = Nn*(Nn-26/3)+17" by (simp add: Rings.ring_distribs(4) power2_eq_square)
  have "Nn*(Nn-26/3)+17>17" using assms Nn_def by auto
  hence "(Nn-3)^2 - (sqrt (8*Nn/3 - 8))^2 > 0" using Nn4 by auto
  hence "Nn-3 > sqrt (8*Nn/3 - 8)" using assms Nn_def by (simp add: real_less_lsqrt)
  hence "Nn-2 > sqrt (8*Nn/3 - 8)+2/3" by linarith
  hence "N > c4" using Nn_def ‹c4<(2/3 + sqrt (8*Nn/3 - 8))› by simp

  have "N/3 ≥ 108" using assms using le_divide_eq by fastforce
  hence "sqrt(6*N/3 - 3)≥1" by simp
  hence "1/2 + sqrt(6*N/3 - 3) ≥1" by linarith
  hence "c1 ≥1" using c1_def by simp

  obtain b1 b2 where bproperties:"{b1, b2} ⊆ {c1, c2, c3, c4} ∧ (b2 = b1+2) ∧ odd b1"
    using two_consec_odd c1_def c2_def c3_def c4_def by (metis (no_types, opaque_lifting) Groups.add_ac(2)
empty_subsetI even_plus_one_iff insert_commute insert_mono nat_arith.add1 numeral.simps(2) numeral.simps(3))
  have b1andb2:"odd b1 ∧ b2 = b1+2" using bproperties by auto
  have b1pos:"b1 ≥1" using ‹c1≥1› c2_def c3_def c4_def bproperties by auto
  hence b2pos:"b2 ≥3" using bproperties by simp
  have b2odd:"odd b2" using bproperties by simp
  define b1n where "b1n = nat b1"
  define b2n where "b2n = nat b2"

  from b1n_def b1pos have "b1n mod 3 = b1 mod 3" using int_ops(9) by force
  from b2n_def b2pos have "b2n mod 3 = b2 mod 3" using int_ops(9) by force

  have b_and_r:"∃b r::nat. [N = b+r] (mod 3) ∧ (b = b1n ∨ b = b2n) ∧ (r = 0 ∨ r = 1)"
  proof -
    have case1:"∃b r::nat. [N = b+r] (mod 3) ∧ (b = b1n ∨ b = b2n) ∧ (r = 0 ∨ r = 1)"
      if asm1:"b1 mod 3 = 0"
    proof -
      have "b1n mod 3 = 0" using asm1 ‹b1n mod 3 = b1 mod 3› by simp
      hence "b2n mod 3 = 2" using ‹b2n mod 3 = b2 mod 3› bproperties asm1 by fastforce
      have case1_1:"[0 = b1n+0] (mod 3)" using ‹b1n mod 3 = 0›
        by (metis mod_0 nat_arith.rule0 unique_euclidean_semiring_class.cong_def)
      have case1_2:"[1 = b1n+1] (mod 3)" using ‹b1n mod 3 = 0›
        by (metis ‹[0 = b1n + 0] (mod 3)› add.commute cong_add_lcancel_0_nat cong_sym)
      have case1_3:"[2 = b2n+0] (mod 3)" using ‹b2n mod 3 = 2›
        by (simp add: unique_euclidean_semiring_class.cong_def)
      have "∀N::nat. N mod 3 = 0 ∨ N mod 3 ≥ 1" by linarith
      hence "∀N::nat. N mod 3 = 0 ∨ N mod 3 = 1 ∨ N mod 3 = 2" by linarith
      hence "∀N. ∃b r::nat. [N = b+r] (mod 3) ∧ (b = b1n ∨ b = b2n) ∧ (r = 0 ∨ r = 1)"
        if asm1:"b1 mod 3 = 0" using case1_1 case1_2 case1_3 by (metis cong_mod_left)
      thus ?thesis using asm1 by auto
    qed

    have case2:"∃b r::nat. [N = b+r] (mod 3) ∧ (b = b1n ∨ b = b2n) ∧ (r = 0 ∨ r = 1)"
      if asm2:"b1 mod 3 = 1"
    proof -
      have "b1n mod 3 = 1" using asm2 ‹b1n mod 3 = b1 mod 3› by simp
      hence "b2n mod 3 = 0" using ‹b2n mod 3 = b2 mod 3› bproperties asm2
        by (smt (verit, best) Euclidean_Rings.pos_mod_bound Euclidean_Rings.pos_mod_sign
            int_ops(1) mod_diff_eq mod_pos_pos_trivial of_nat_eq_iff)
      have case2_1:"[0 = b2n+0] (mod 3)" using ‹b2n mod 3 = 0›
        by (metis mod_0 nat_arith.rule0 unique_euclidean_semiring_class.cong_def)
      have case2_2:"[1 = b1n+0] (mod 3)" using ‹b1n mod 3 = 1›
        by (simp add: unique_euclidean_semiring_class.cong_def)
      have case2_3:"[2 = b1n+1] (mod 3)" using ‹b1n mod 3 = 1›
        by (metis case2_2 cong_add_rcancel_nat nat_1_add_1 nat_arith.rule0)
      have "∀N::nat. N mod 3 = 0 ∨ N mod 3 ≥ 1" by linarith
      hence "∀N::nat. N mod 3 = 0 ∨ N mod 3 = 1 ∨ N mod 3 = 2" by linarith
      hence "∀N. ∃b r::nat. [N = b+r] (mod 3) ∧ (b = b1n ∨ b = b2n) ∧ (r = 0 ∨ r = 1)"
        if asm2:"b1 mod 3 = 1" using case2_1 case2_2 case2_3 by (metis cong_mod_left)
      thus ?thesis using asm2 by auto
    qed

    have case3:"∃b r::nat. [N = b+r] (mod 3) ∧ (b = b1n ∨ b = b2n) ∧ (r = 0 ∨ r = 1)"
      if asm3:"b1 mod 3 = 2"
    proof -
      have "b1n mod 3 = 2" using asm3 ‹b1n mod 3 = b1 mod 3› by simp
      have "(b1+2) mod 3 = (2+2) mod 3" using asm3 by (metis Groups.add_ac(2) mod_add_right_eq)
      hence "b2n mod 3 = 1" using ‹b2n mod 3 = b2 mod 3› bproperties by simp
      have case3_1:"[0 = b1n+1] (mod 3)" using ‹b1n mod 3 = 2›
        by (metis One_nat_def add.commute mod_0 mod_add_right_eq mod_self nat_1_add_1 numeral_3_eq_3
            plus_1_eq_Suc unique_euclidean_semiring_class.cong_def)
      have case3_2:"[1 = b2n+0] (mod 3)" using ‹b2n mod 3 = 1›
        by (simp add: unique_euclidean_semiring_class.cong_def)
      have case3_3:"[2 = b1n+0] (mod 3)" using ‹b1n mod 3 = 2›
        by (simp add: unique_euclidean_semiring_class.cong_def)
      have "∀N::nat. N mod 3 = 0 ∨ N mod 3 ≥ 1" by linarith
      hence "∀N::nat. N mod 3 = 0 ∨ N mod 3 = 1 ∨ N mod 3 = 2" by linarith
      hence "∀N. ∃b r::nat. [N = b+r] (mod 3) ∧ (b = b1n ∨ b = b2n) ∧ (r = 0 ∨ r = 1)"
        if asm3:"b1 mod 3 = 2" using case3_1 case3_2 case3_3 by (metis cong_mod_left)
      thus ?thesis using asm3 by auto
    qed

    have "b1 mod 3 = 0 ∨ b1 mod 3 = 1 ∨ b1 mod 3 = 2" by auto
    thus ?thesis using case1 case2 case3 by auto
  qed

  obtain b r where b_r:"[N = b+r] (mod 3) ∧ (b = b1n ∨ b = b2n) ∧ (r = 0 ∨ r = 1)"
    using b_and_r by auto

  have bpos:"b≥1" using b1pos b2pos b_r b1n_def b2n_def by auto
  have bodd:"odd b"
    using b_r bproperties by (metis b1n_def b2n_def b2odd bpos even_nat_iff nat_eq_iff2 rel_simps(45))

  define a where a_def:"a = b+2*(N-b-r) div 3"
  have "int b1n = b1" using b1n_def b1pos by linarith
  have "int b2n = b2" using b2n_def b2pos by linarith
  have m_div_num:"3 dvd (N-b-r)" using b_r
    by (metis cong_altdef_nat diff_diff_left diff_is_0_eq' dvd_0_right nat_le_linear)
  hence a_def1:"a = b+2*(N-b-r)/3" using a_def m_div_num real_of_nat_div by auto
  from ‹N>c4› have "N>b" using b_r bproperties b1n_def b2n_def
    by (smt (verit, del_insts) ‹int b1n = b1› ‹int b2n = b2› c2_def c3_def c4_def empty_iff insert_iff insert_subset of_nat_less_imp_less)
  hence "(N-b-r)/3 = (N-r)/3 - b/3" using ‹b < N› b_r by force
  hence "a = b- b*2/3+2*(N-r)/3" using a_def1 by linarith
  hence a_def3:"a = b*(1-2/3) + 2*(N-r)/3" by simp

  have size1:"b*(1-2/3)>0" using bpos by simp
  have "N-r>0" using b_r assms by auto
  hence size2:"2*(N-r)/3>0" using assms(1) by simp
  have apos:"a≥1" using size1 size2 a_def3 by linarith

  have "odd (b+2*(N-b-r) div 3)" using m_div_num b_r b2odd bproperties by (simp add: bodd mult_2)
  hence aodd:"odd a" using a_def by simp
  from a_def1 have "a-b = 2*(N-b-r)/3" by simp
  hence "(a-b)/2 = (N-b-r)/3" by simp
  hence "3*(a-b)/2 = N-b-r"  by simp
  have "N-b-r≥0" using b_r by simp
  hence N_expr:"N = r+b+3*(a-b)/2" using ‹N-b-r≥0›  ‹b < N› b_r ‹real (3 * (a - b)) / 2 = real (N - b - r)› by linarith
  from a_def ‹N-b-r≥0› have "a≥b" using a_def le_add1 by blast

  have "b1 ≥ c1" using bproperties c2_def c3_def c4_def by force
  hence "b1 ≥ 1/2 + sqrt (6*N/3 - 3)" using c1_def using ceiling_le_iff by blast
  hence b1ngreater:"b1n ≥ 1/2 + sqrt (6*N/3 - 3)" using b1n_def by simp
  hence b2ngreater:"b2n ≥ 1/2 + sqrt (6*N/3 - 3)" using bproperties b1n_def b2n_def by linarith
  hence b_ineq1:"b ≥ 1/2 + sqrt (6*N/3 - 3)" using b_r b1ngreater by auto

  have "b2 ≤ c4" using bproperties c1_def c2_def c3_def c4_def by fastforce
  hence "b2 ≤ (2/3 + sqrt (8*N/3 - 8))"
    using ‹real_of_int c4 < 2 / 3 + sqrt (real (8 * N) / 3 - 8)› by linarith
  hence b2nsmaller:"b2n ≤ (2/3 + sqrt (8*N/3 - 8))" using b2n_def by (metis ‹int b2n = b2› of_int_of_nat_eq)
  hence "b1n ≤ (2/3 + sqrt (8*N/3 - 8))" using b1n_def bproperties using ‹int b2n = b2› by linarith
  hence b_ineq2:"b≤(2/3 + sqrt (8*N/3 - 8))" using b_r b2nsmaller by auto

  define Nr where "Nr = real_of_nat N"
  define ar where "ar = real_of_int a"
  define br where "br = real_of_int b"
  define rr where "rr = real_of_nat r"
  define m where "m = real_of_nat 3"
  from assms have "N≥2*m" using m_def by simp
  then have "Nr ≥ 2*m" using Nr_def  ‹N ≥ 2 * m› by auto
  moreover have "br≥0" using br_def bpos by auto
  moreover have "ar≥0" using ar_def apos by auto
  moreover have "rr≥0" using rr_def b_r by auto
  moreover have "m≥3" using m_def by auto
  moreover have "m>rr" using m_def rr_def b_r by auto
  moreover have "Nr = m*(ar-br)/2+br+rr" using Nr_def ar_def br_def N_expr rr_def m_def ‹a≥b› by auto
  moreover have "1/2+sqrt(6*Nr/m-3)≤br ∧ br≤2/3+sqrt(8*Nr/m-8)" using Nr_def  br_def b_ineq1 b_ineq2 m_def by auto
  ultimately have "br^2<4*ar ∧ 3*ar<br^2+2*br+4" using Cauchy_lemma by auto
  hence real_ineq:"(real_of_int b)^2 < 4*(real_of_int a) ∧ 3*(real_of_int a) < (real_of_int b)^2 + 2*(real_of_int b) + 4"
    using br_def ar_def by auto
  hence nat_ineq1: "b^2<4*a" using br_def  by (smt (verit, del_insts) Num.of_nat_simps(4)
mult.commute mult_2_right nat_distrib(1) numeral_Bit0 of_int_of_nat_eq of_nat_less_of_nat_power_cancel_iff)
  from real_ineq have nat_ineq2: "3*a<b^2+2*b+4" using ar_def br_def of_nat_less_iff by fastforce

  obtain s t u v where stuv:"s ≥ 0 ∧ t ≥ 0 ∧ u ≥ 0 ∧ v ≥ 0 ∧ int a = s^2 + t^2 + u^2 + v^2 ∧
 int b = s+t+u+v" using apos bpos aodd bodd nat_ineq1 nat_ineq2 four_nonneg_int_sum by presburger
  have a_expr:"a = s^2 + t^2 + u^2 + v^2" using apos stuv by linarith
  have b_expr:"b = s+t+u+v" using bpos stuv by linarith

  have "N = r + (s+t+u+v)+ 3*(a-(s+t+u+v))/2" using b_expr N_expr
    by (metis Num.of_nat_simps(4) Num.of_nat_simps(5) ‹b ≤ a› of_int_of_nat_eq of_nat_diff of_nat_numeral)
  hence "N = 3/2*(s^2-s+t^2-t+u^2-u+v^2-v)+r+(s+t+u+v)" using a_expr by simp
  hence N_expr2:"N = 3/2*(s^2-s)+ 3/2*(t^2-t)+ 3/2*(u^2-u)+ 3/2*(v^2-v)+ r+(s+t+u+v)"
    by (metis (no_types, opaque_lifting) add_diff_eq nat_distrib(2) of_int_add)

  have s_div2:"3/2*(s^2-s) = 3*(s^2-s) div 2" using real_of_int_div by auto
  have t_div2:"3/2*(t^2-t) = 3*(t^2-t) div 2" using real_of_int_div by auto
  have u_div2:"3/2*(u^2-u) = 3*(u^2-u) div 2" using real_of_int_div by auto
  have v_div2:"3/2*(v^2-v) = 3*(v^2-v) div 2" using real_of_int_div by auto
  have N_expr3:"N = (3*(s^2-s) div 2+s)+(3*(t^2-t) div 2+t)+(3*(u^2-u) div 2+u)+(3*(v^2-v) div 2+v)+r"
    using N_expr2 s_div2 t_div2 u_div2 v_div2 by simp

  define sn where "sn = nat s"
  define tn where "tn = nat t"
  define un where "un = nat u"
  define vn where "vn = nat v"
  have seqsn:"s^2-s = sn^2 - sn" using stuv sn_def
    by (metis int_nat_eq le_refl of_nat_diff of_nat_power power2_nat_le_imp_le)
  have teqtn:"t^2-t = tn^2 - tn" using stuv tn_def
    by (metis int_nat_eq le_refl of_nat_diff of_nat_power power2_nat_le_imp_le)
  have uequn:"u^2-u = un^2 - un" using stuv un_def
    by (metis int_nat_eq le_refl of_nat_diff of_nat_power power2_nat_le_imp_le)
  have veqvn:"v^2-v = vn^2 - vn" using stuv vn_def
    by (metis int_nat_eq le_refl of_nat_diff of_nat_power power2_nat_le_imp_le)

  from N_expr3 have
    "N = (3*(sn^2-sn) div 2+s)+(3*(tn^2-tn) div 2+t)+(3*(un^2-un) div 2+u)+(3*(vn^2-vn) div 2+ v)+r"
    using seqsn teqtn uequn veqvn
    by (metis (mono_tags, lifting) Num.of_nat_simps(5) of_nat_numeral zdiv_int)
  hence "N = (3*(sn^2-sn) div 2+sn)+(3*(tn^2-tn) div 2+tn)+(3*(un^2-un) div 2+un)+(3*(vn^2-vn) div 2+ v)+r"
    using sn_def tn_def un_def  stuv int_nat_eq int_ops(5) by presburger
  hence "N = (3*(sn^2-sn) div 2+sn)+(3*(tn^2-tn) div 2+tn)+(3*(un^2-un) div 2+un)+(3*(vn^2-vn) div 2+ vn)+r"
    using vn_def stuv by (smt (verit, del_insts) Num.of_nat_simps(4) int_nat_eq of_nat_eq_iff)
  hence "N = (3* sn*(sn-1) div 2+sn)+(3*tn*(tn-1) div 2+tn)+(3*un*(un-1) div 2+un)+(3* vn*(vn-1) div 2+ vn)+r"
    by (smt (verit, ccfv_threshold) more_arith_simps(11) mult.right_neutral power2_eq_square right_diff_distrib')
  hence N_expr4:"N = polygonal_number 3 sn + polygonal_number 3 tn + polygonal_number 3 un + polygonal_number 3 vn +r"
    using Polygonal_Number_Theorem_Gauss.polygonal_number_def by presburger

  define p1 where "p1 = polygonal_number 3 sn"
  define p2 where "p2 = polygonal_number 3 tn"
  define p3 where "p3 = polygonal_number 3 un"
  define p4 where "p4 = polygonal_number 3 vn"
  have N_expr5:"N = p1 + p2 + p3 + p4 + r" using N_expr4 p1_def p2_def p3_def p4_def by auto
  thus ?thesis using p1_def p2_def p3_def p4_def b_r by blast
qed
end