Theory Round

(*  Title:      HOL/SPARK/Examples/RIPEMD-160/Round.thy
    Author:     Fabian Immler, TU Muenchen

Verification of the RIPEMD-160 hash function
*)

theory Round
imports RMD_Specification
begin

spark_open ‹rmd/round›

abbreviation from_chain :: "chain ⇒ RMD.chain" where
  "from_chain c ≡ (
    word_of_int (h0 c),
    word_of_int (h1 c),
    word_of_int (h2 c),
    word_of_int (h3 c),
    word_of_int (h4 c))"

abbreviation from_chain_pair :: "chain_pair ⇒ RMD.chain × RMD.chain" where
  "from_chain_pair cc ≡ (
    from_chain (left cc),
    from_chain (right cc))"

abbreviation to_chain :: "RMD.chain ⇒ chain" where
  "to_chain c ≡
    (let (h0, h1, h2, h3, h4) = c in
      (|h0 = uint h0,
        h1 = uint h1,
        h2 = uint h2,
        h3 = uint h3,
        h4 = uint h4|))"

abbreviation to_chain_pair :: "RMD.chain × RMD.chain ⇒ chain_pair" where
  "to_chain_pair c == (let (c1, c2) = c in
    (| left = to_chain c1,
      right = to_chain c2 |))"

abbreviation steps' :: "chain_pair ⇒ int ⇒ block ⇒ chain_pair" where
  "steps' cc i b == to_chain_pair (steps
    (λn. word_of_int (b (int n)))
    (from_chain_pair cc)
    (nat i))"

abbreviation round_spec :: "chain ⇒ block ⇒ chain" where
  "round_spec c b == to_chain (round (λn. word_of_int (b (int n))) (from_chain c))"

spark_proof_functions
  steps = steps'
  round_spec = round_spec

lemma uint_word_of_int_id:
  assumes "0 <= (x::int)"
  assumes "x <= 4294967295"
  shows"uint(word_of_int x::word32) = x"
  using assms by (simp add: take_bit_int_eq_self unsigned_of_int)

lemma steps_step: "steps X cc (Suc i) = step_both X (steps X cc i) i"
  unfolding steps_def
  by (induct i) simp_all

lemma from_to_id: "from_chain_pair (to_chain_pair CC) = CC"
proof (cases CC)
  fix a::RMD.chain
  fix b c d e f::word32
  assume "CC = (a, b, c, d, e, f)"
  thus ?thesis by (cases a) simp
qed

lemma steps_to_steps':
  "F A (steps X cc i) B =
   F A (from_chain_pair (to_chain_pair (steps X cc i))) B"
  unfolding from_to_id ..

lemma steps'_step:
  assumes "0 <= i"
  shows
  "steps' cc (i + 1) X = to_chain_pair (
     step_both
       (λn. word_of_int (X (int n)))
       (from_chain_pair (steps' cc i X))
       (nat i))"
proof -
  have "nat (i + 1) = Suc (nat i)" using assms by simp
  show ?thesis
    unfolding ‹nat (i + 1) = Suc (nat i)› steps_step steps_to_steps'
    ..
qed

lemma step_from_hyp:
  assumes
  step_hyp:
  "⦇left =
      ⦇h0 = a, h1 = b, h2 = c, h3 = d, h4 = e⦈,
    right =
      ⦇h0 = a', h1 = b', h2 = c', h3 = d', h4 = e'⦈⦈ =
   steps'
     (⦇left =
         ⦇h0 = a_0, h1 = b_0, h2 = c_0,
          h3 = d_0, h4 = e_0⦈,
       right =
         ⦇h0 = a_0, h1 = b_0, h2 = c_0,
          h3 = d_0, h4 = e_0⦈⦈)
     j x"
  assumes "a <= 4294967295" (is "_ <= ?M")
  assumes                "b  <= ?M" and "c  <= ?M" and "d  <= ?M" and "e  <= ?M"
  assumes "a' <= ?M" and "b' <= ?M" and "c' <= ?M" and "d' <= ?M" and "e' <= ?M"
  assumes "0 <= a " and "0 <= b " and "0 <= c " and "0 <= d " and "0 <= e "
  assumes "0 <= a'" and "0 <= b'" and "0 <= c'" and "0 <= d'" and "0 <= e'"
  assumes "0 <= x (r_l_spec j)" and "x (r_l_spec j) <= ?M"
  assumes "0 <= x (r_r_spec j)" and "x (r_r_spec j) <= ?M"
  assumes "0 <= j" and "j <= 79"
  shows
  "⦇left =
      ⦇h0 = e,
         h1 =
           (rotate_left (s_l_spec j)
             ((((a + f_spec j b c d) mod 4294967296 +
                x (r_l_spec j)) mod
               4294967296 +
               k_l_spec j) mod
              4294967296) +
            e) mod
           4294967296,
         h2 = b, h3 = rotate_left 10 c,
         h4 = d⦈,
      right =
        ⦇h0 = e',
           h1 =
             (rotate_left (s_r_spec j)
               ((((a' + f_spec (79 - j) b' c' d') mod
                  4294967296 +
                  x (r_r_spec j)) mod
                 4294967296 +
                 k_r_spec j) mod
                4294967296) +
              e') mod
             4294967296,
           h2 = b', h3 = rotate_left 10 c',
           h4 = d'⦈⦈ =
   steps'
    (⦇left =
        ⦇h0 = a_0, h1 = b_0, h2 = c_0,
           h3 = d_0, h4 = e_0⦈,
        right =
          ⦇h0 = a_0, h1 = b_0, h2 = c_0,
             h3 = d_0, h4 = e_0⦈⦈)
    (j + 1) x"
  using step_hyp
proof -
  let ?MM = 4294967296
  have AL: "uint(word_of_int e::word32) = e"
    by (rule uint_word_of_int_id[OF ‹0 <= e› ‹e <= ?M›])
  have CL: "uint(word_of_int b::word32) = b"
    by (rule uint_word_of_int_id[OF ‹0 <= b› ‹b <= ?M›])
  have DL: "True" ..
  have EL: "uint(word_of_int d::word32) = d"
    by (rule uint_word_of_int_id[OF ‹0 <= d› ‹d <= ?M›])
  have AR: "uint(word_of_int e'::word32) = e'"
    by (rule uint_word_of_int_id[OF ‹0 <= e'› ‹e' <= ?M›])
  have CR: "uint(word_of_int b'::word32) = b'"
    by (rule uint_word_of_int_id[OF ‹0 <= b'› ‹b' <= ?M›])
  have DR: "True" ..
  have ER: "uint(word_of_int d'::word32) = d'"
    by (rule uint_word_of_int_id[OF ‹0 <= d'› ‹d' <= ?M›])
  have BL:
    "(uint
      (word_rotl (s (nat j))
        ((word_of_int::int⇒word32)
          ((((a + f_spec j b c d) mod ?MM +
             x (r_l_spec j)) mod ?MM +
            k_l_spec j) mod ?MM))) +
        e) mod ?MM
    =
    uint
              (word_rotl (s (nat j))
                (word_of_int a +
                 f (nat j) (word_of_int b)
                  (word_of_int c) (word_of_int d) +
                 word_of_int (x (r_l_spec j)) +
                 K (nat j)) +
               word_of_int e)"
    (is "(uint (word_rotl _ (_ ((((_ + ?F) mod _ + ?X) mod _ + _) mod _))) + _) mod _ = _")
  proof -
    have "a mod ?MM = a" using ‹0 <= a› ‹a <= ?M›
      by simp
    have "?X mod ?MM = ?X" using ‹0 <= ?X› ‹?X <= ?M›
      by simp
    have "e mod ?MM = e" using ‹0 <= e› ‹e <= ?M›
      by simp
    have "(?MM::int) = 2 ^ LENGTH(32)" by simp
    show ?thesis
      unfolding
        word_add_def
        uint_word_of_int_id[OF ‹0 <= a› ‹a <= ?M›]
        uint_word_of_int_id[OF ‹0 <= ?X› ‹?X <= ?M›]
      using ‹a mod ?MM = a›
        ‹e mod ?MM = e›
        ‹?X mod ?MM = ?X›
      unfolding ‹?MM = 2 ^ LENGTH(32)›
      apply (simp only: flip: take_bit_eq_mod add: uint_word_of_int_eq)
      apply (metis (mono_tags, opaque_lifting) of_int_of_nat_eq ucast_id uint_word_of_int_eq unsigned_of_int)
      done
  qed

  have BR:
    "(uint
      (word_rotl (s' (nat j))
        ((word_of_int::int⇒word32)
          ((((a' + f_spec (79 - j) b' c' d') mod ?MM +
             x (r_r_spec j)) mod ?MM +
            k_r_spec j) mod ?MM))) +
        e') mod ?MM
    =
    uint
              (word_rotl (s' (nat j))
                (word_of_int a' +
                 f (79 - nat j) (word_of_int b')
                  (word_of_int c') (word_of_int d') +
                 word_of_int (x (r_r_spec j)) +
                 K' (nat j)) +
               word_of_int e')"
    (is "(uint (word_rotl _ (_ ((((_ + ?F) mod _ + ?X) mod _ + _) mod _))) + _) mod _ = _")
  proof -
    have "a' mod ?MM = a'" using ‹0 <= a'› ‹a' <= ?M›
      by simp
    have "?X mod ?MM = ?X" using ‹0 <= ?X› ‹?X <= ?M›
      by simp
    have "e' mod ?MM = e'" using ‹0 <= e'› ‹e' <= ?M›
      by simp
    have "(?MM::int) = 2 ^ LENGTH(32)" by simp
    have nat_transfer: "79 - nat j = nat (79 - j)"
      using nat_diff_distrib ‹0 <= j›  ‹j <= 79›
      by simp
    show ?thesis
      unfolding
        word_add_def
        uint_word_of_int_id[OF ‹0 <= a'› ‹a' <= ?M›]
        uint_word_of_int_id[OF ‹0 <= ?X› ‹?X <= ?M›]
        nat_transfer
      using ‹a' mod ?MM = a'›
        ‹e' mod ?MM = e'›
        ‹?X mod ?MM = ?X›
      unfolding ‹?MM = 2 ^ LENGTH(32)›
      apply (simp only: flip: take_bit_eq_mod add: uint_word_of_int_eq)
      apply (metis (mono_tags, opaque_lifting) of_nat_nat_take_bit_eq ucast_id unsigned_of_int)
      done
  qed

  show ?thesis
    unfolding steps'_step[OF ‹0 <= j›] step_hyp[symmetric]
      step_both_def step_r_def step_l_def
    using AL CL EL AR CR ER
    by (simp add: BL DL BR DR take_bit_int_eq_self_iff take_bit_int_eq_self)
qed

spark_vc procedure_round_61
proof -
  let ?M = "4294967295::int"
  have step_hyp:
  "⦇left =
      ⦇h0 = ca, h1 = cb, h2 = cc,
         h3 = cd, h4 = ce⦈,
      right =
        ⦇h0 = ca, h1 = cb, h2 = cc,
           h3 = cd, h4 = ce⦈⦈ =
   steps'
    (⦇left =
        ⦇h0 = ca, h1 = cb, h2 = cc,
           h3 = cd, h4 = ce⦈,
        right =
          ⦇h0 = ca, h1 = cb, h2 = cc,
             h3 = cd, h4 = ce⦈⦈)
    0 x"
    unfolding steps_def
    using
      uint_word_of_int_id[OF ‹0 <= ca› ‹ca <= ?M›]
      uint_word_of_int_id[OF ‹0 <= cb› ‹cb <= ?M›]
      uint_word_of_int_id[OF ‹0 <= cc› ‹cc <= ?M›]
      uint_word_of_int_id[OF ‹0 <= cd› ‹cd <= ?M›]
      uint_word_of_int_id[OF ‹0 <= ce› ‹ce <= ?M›]
    by (simp add: take_bit_int_eq_self_iff take_bit_int_eq_self)
  let ?rotate_arg_l =
    "((((ca + f 0 cb cc cd) mod 4294967296 +
        x (r_l 0)) mod 4294967296 + k_l 0) mod 4294967296)"
  let ?rotate_arg_r =
    "((((ca + f 79 cb cc cd) mod 4294967296 +
        x (r_r 0)) mod 4294967296 + k_r 0) mod 4294967296)"
  note returns =
    ‹wordops__rotate (s_l 0) ?rotate_arg_l =
     rotate_left (s_l 0) ?rotate_arg_l›
    ‹wordops__rotate (s_r 0) ?rotate_arg_r =
     rotate_left (s_r 0) ?rotate_arg_r›
    ‹wordops__rotate 10 cc = rotate_left 10 cc›
    ‹f 0 cb cc cd = f_spec 0 cb cc cd›
    ‹f 79 cb cc cd = f_spec 79 cb cc cd›
    ‹k_l 0 = k_l_spec 0›
    ‹k_r 0 = k_r_spec 0›
    ‹r_l 0 = r_l_spec 0›
    ‹r_r 0 = r_r_spec 0›
    ‹s_l 0 = s_l_spec 0›
    ‹s_r 0 = s_r_spec 0›

  note x_borders = ‹∀i. 0 ≤ i ∧ i ≤ 15 ⟶ 0 ≤ x i ∧ x i ≤ ?M›

  from ‹0 <= r_l 0› ‹r_l 0 <= 15› x_borders
  have "0 ≤ x (r_l 0)" by blast
  hence x_lower: "0 <= x (r_l_spec 0)" unfolding returns .

  from ‹0 <= r_l 0› ‹r_l 0 <= 15› x_borders
  have "x (r_l 0) <= ?M" by blast
  hence x_upper: "x (r_l_spec 0) <= ?M" unfolding returns .

  from ‹0 <= r_r 0› ‹r_r 0 <= 15› x_borders
  have "0 ≤ x (r_r 0)" by blast
  hence x_lower': "0 <= x (r_r_spec 0)" unfolding returns .

  from ‹0 <= r_r 0› ‹r_r 0 <= 15› x_borders
  have "x (r_r 0) <= ?M" by blast
  hence x_upper': "x (r_r_spec 0) <= ?M" unfolding returns .

  have "0 <= (0::int)" by simp
  have "0 <= (79::int)" by simp
  note step_from_hyp [OF
    step_hyp
    H2 H4 H6 H8 H10 H2 H4 H6 H8 H10 ― ‹upper bounds›
    H1 H3 H5 H7 H9  H1 H3 H5 H7 H9  ― ‹lower bounds›
  ]
  from this[OF x_lower x_upper x_lower' x_upper' ‹0 <= 0› ‹0 <= 79›]
    ‹0 ≤ ca› ‹0 ≤ ce› x_lower x_lower'
  show ?thesis unfolding returns(1) returns(2) unfolding returns
    by (simp del: mod_pos_pos_trivial mod_neg_neg_trivial)
qed

spark_vc procedure_round_62
proof -
  let ?M = "4294967295::int"
  let ?rotate_arg_l =
    "((((cla + f (loop__1__j + 1) clb clc cld) mod 4294967296 +
         x (r_l (loop__1__j + 1))) mod 4294967296 +
         k_l (loop__1__j + 1)) mod 4294967296)"
  let ?rotate_arg_r =
    "((((cra + f (79 - (loop__1__j + 1)) crb crc crd) mod
         4294967296 + x (r_r (loop__1__j + 1))) mod 4294967296 +
         k_r (loop__1__j + 1)) mod 4294967296)"

  have s: "78 - loop__1__j = (79 - (loop__1__j + 1))" by simp
  note returns =
    ‹wordops__rotate (s_l (loop__1__j + 1)) ?rotate_arg_l =
     rotate_left (s_l (loop__1__j + 1)) ?rotate_arg_l›
    ‹wordops__rotate (s_r (loop__1__j + 1)) ?rotate_arg_r =
     rotate_left (s_r (loop__1__j + 1)) ?rotate_arg_r›
    ‹f (loop__1__j + 1) clb clc cld =
     f_spec (loop__1__j + 1) clb clc cld›
    ‹f (78 - loop__1__j) crb crc crd =
     f_spec (78 - loop__1__j) crb crc crd›[simplified s]
    ‹wordops__rotate 10 clc = rotate_left 10 clc›
    ‹wordops__rotate 10 crc = rotate_left 10 crc›
    ‹k_l (loop__1__j + 1) = k_l_spec (loop__1__j + 1)›
    ‹k_r (loop__1__j + 1) = k_r_spec (loop__1__j + 1)›
    ‹r_l (loop__1__j + 1) = r_l_spec (loop__1__j + 1)›
    ‹r_r (loop__1__j + 1) = r_r_spec (loop__1__j + 1)›
    ‹s_l (loop__1__j + 1) = s_l_spec (loop__1__j + 1)›
    ‹s_r (loop__1__j + 1) = s_r_spec (loop__1__j + 1)›

  note x_borders = ‹∀i. 0 ≤ i ∧ i ≤ 15 ⟶ 0 ≤ x i ∧ x i ≤ ?M›

  from ‹0 <= r_l (loop__1__j + 1)› ‹r_l (loop__1__j + 1) <= 15› x_borders
  have "0 ≤ x (r_l (loop__1__j + 1))" by blast
  hence x_lower: "0 <= x (r_l_spec (loop__1__j + 1))" unfolding returns .

  from ‹0 <= r_l (loop__1__j + 1)› ‹r_l (loop__1__j + 1) <= 15› x_borders
  have "x (r_l (loop__1__j + 1)) <= ?M" by blast
  hence x_upper: "x (r_l_spec (loop__1__j + 1)) <= ?M" unfolding returns .

  from ‹0 <= r_r (loop__1__j + 1)› ‹r_r (loop__1__j + 1) <= 15› x_borders
  have "0 ≤ x (r_r (loop__1__j + 1))" by blast
  hence x_lower': "0 <= x (r_r_spec (loop__1__j + 1))" unfolding returns .

  from ‹0 <= r_r (loop__1__j + 1)› ‹r_r (loop__1__j + 1) <= 15› x_borders
  have "x (r_r (loop__1__j + 1)) <= ?M" by blast
  hence x_upper': "x (r_r_spec (loop__1__j + 1)) <= ?M" unfolding returns .

  from ‹0 <= loop__1__j› have "0 <= loop__1__j + 1" by simp
  from ‹loop__1__j <= 78› have "loop__1__j + 1 <= 79" by simp

  have "loop__1__j + 1 + 1 = loop__1__j + 2" by simp

  note step_from_hyp[OF H1
    ‹cla <= ?M›
    ‹clb <= ?M›
    ‹clc <= ?M›
    ‹cld <= ?M›
    ‹cle <= ?M›
    ‹cra <= ?M›
    ‹crb <= ?M›
    ‹crc <= ?M›
    ‹crd <= ?M›
    ‹cre <= ?M›

    ‹0 <= cla›
    ‹0 <= clb›
    ‹0 <= clc›
    ‹0 <= cld›
    ‹0 <= cle›
    ‹0 <= cra›
    ‹0 <= crb›
    ‹0 <= crc›
    ‹0 <= crd›
    ‹0 <= cre›]
  from this[OF
    x_lower x_upper x_lower' x_upper'
    ‹0 <= loop__1__j + 1› ‹loop__1__j + 1 <= 79›]
    ‹0 ≤ cla› ‹0 ≤ cle› ‹0 ≤ cra› ‹0 ≤ cre› x_lower x_lower'
  show ?thesis unfolding ‹loop__1__j + 1 + 1 = loop__1__j + 2›
    unfolding returns(1) returns(2) unfolding returns
    by (simp del: mod_pos_pos_trivial mod_neg_neg_trivial)

qed

spark_vc procedure_round_76
proof -
  let ?M = "4294967295 :: int"
  let ?INIT_CHAIN =
     "⦇h0 = ca_init, h1 = cb_init,
         h2 = cc_init, h3 = cd_init,
         h4 = ce_init⦈"
  have steps_to_steps':
    "steps
       (λn::nat. word_of_int (x (int n)))
       (from_chain ?INIT_CHAIN, from_chain ?INIT_CHAIN)
       80 =
    from_chain_pair (
      steps'
      (⦇left = ?INIT_CHAIN, right = ?INIT_CHAIN⦈)
      80
      x)"
    unfolding from_to_id by simp
  from
    ‹0 ≤ ca_init› ‹ca_init ≤ ?M›
    ‹0 ≤ cb_init› ‹cb_init ≤ ?M›
    ‹0 ≤ cc_init› ‹cc_init ≤ ?M›
    ‹0 ≤ cd_init› ‹cd_init ≤ ?M›
    ‹0 ≤ ce_init› ‹ce_init ≤ ?M›
    ‹0 ≤ cla› ‹cla ≤ ?M›
    ‹0 ≤ clb› ‹clb ≤ ?M›
    ‹0 ≤ clc› ‹clc ≤ ?M›
    ‹0 ≤ cld› ‹cld ≤ ?M›
    ‹0 ≤ cle› ‹cle ≤ ?M›
    ‹0 ≤ cra› ‹cra ≤ ?M›
    ‹0 ≤ crb› ‹crb ≤ ?M›
    ‹0 ≤ crc› ‹crc ≤ ?M›
    ‹0 ≤ crd› ‹crd ≤ ?M›
    ‹0 ≤ cre› ‹cre ≤ ?M›
  show ?thesis
    unfolding round_def
    unfolding steps_to_steps'
    unfolding H1[symmetric]
    by (simp add: uint_word_ariths(1) mod_simps
      uint_word_of_int_id take_bit_int_eq_self)
qed

spark_end

end