Theory IsPrime

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(*
 * IsPrime-Test 
 *
 * Authors : Burkhart Wolff, Frédéric Tuong
 *)

chapter ‹ Clean Semantics : Another Clean Example›


theory IsPrime
  imports (* "../src/Clean"
             "../src/Hoare_Clean"
              "../src/Clean_Symbex"
              "HOL-Computational_Algebra.Primes"
           *)
          Clean.Clean
          Clean.Hoare_MonadSE
          Clean.Clean_Symbex 
          "HOL-Computational_Algebra.Primes"
begin

section‹The Primality-Test Example at a Glance›

definition "SQRT_UINT_MAX = (65536::nat)"
definition "UINT_MAX = (2^32::nat) - 1"


function_spec isPrime(n :: nat) returns bool
pre          "‹n ≤ SQRT_UINT_MAX›" 
post         "‹λres. res ⟷ prime n ›"
local_vars   i :: nat
defines " if⇩C ‹n < 2›  
            then return⇘local_isPrime_state.result_value_update⇙ ‹False›
            else skip⇩S⇩E 
          fi ;-
          ‹i := 2› ;- 
          while⇩C ‹i < SQRT_UINT_MAX ∧ i*i ≤ n  › 
            do if⇩C ‹n mod i = 0›  
                  then return⇘local_isPrime_state.result_value_update⇙ ‹False›
                  else skip⇩S⇩E 
                fi ;-
                ‹i := i + 1 › 
            od ;-
         return⇘local_isPrime_state.result_value_update⇙ ‹True›"


find_theorems name:isPrime name:core
term‹isPrime_core›

lemma XXX : 
"isPrime_core n ≡
     if⇩C (λσ. n < 2) then (return⇘result_value_update⇙ (λσ. False)) 
                     else skip⇩S⇩E fi;-
     i_update :==⇩L (λσ. 2) ;-
     while⇩C (λσ. (hd∘i)σ < SQRT_UINT_MAX ∧ (hd∘i)σ * (hd∘i)σ ≤ n) 
     do
        (if⇩C (λσ. n mod (hd ∘ i) σ = 0) 
         then (return⇘result_value_update⇙ (λσ. False)) 
         else skip⇩S⇩E fi ;-
        i_update :==⇩L (λσ. (hd ∘ i) σ + 1)) 
     od ;-
     return⇘result_value_update⇙ (λσ. True)"

  by(simp add: isPrime_core_def)

lemma YYY:
"isPrime n ≡ block⇩C push_local_isPrime_state 
                    (isPrime_core n) 
                    pop_local_isPrime_state"
  by(simp add: isPrime_def)

lemma isPrime_correct : 
  "⦃λσ.   ▹ σ ∧ isPrime_pre (n)(σ) ∧ σ = σ⇩p⇩r⇩e ⦄ 
     isPrime n 
   ⦃λr σ. ▹ σ ∧ isPrime_post(n) (σ⇩p⇩r⇩e)(σ)(r) ⦄"
   oops



end