src/HOL/ex/BinEx.thy

updated;

(*  Title:      HOL/ex/BinEx.thy
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1998  University of Cambridge
*)

header {* Binary arithmetic examples *}

theory BinEx = Main:

subsection {* The Integers *}

text {* Addition *}

lemma "(#13::int) + #19 = #32"
  by simp

lemma "(#1234::int) + #5678 = #6912"
  by simp

lemma "(#1359::int) + #-2468 = #-1109"
  by simp

lemma "(#93746::int) + #-46375 = #47371"
  by simp


text {* \medskip Negation *}

lemma "- (#65745::int) = #-65745"
  by simp

lemma "- (#-54321::int) = #54321"
  by simp


text {* \medskip Multiplication *}

lemma "(#13::int) * #19 = #247"
  by simp

lemma "(#-84::int) * #51 = #-4284"
  by simp

lemma "(#255::int) * #255 = #65025"
  by simp

lemma "(#1359::int) * #-2468 = #-3354012"
  by simp

lemma "(#89::int) * #10 \<noteq> #889"
  by simp

lemma "(#13::int) < #18 - #4"
  by simp

lemma "(#-345::int) < #-242 + #-100"
  by simp

lemma "(#13557456::int) < #18678654"
  by simp

lemma "(#999999::int) \<le> (#1000001 + #1) - #2"
  by simp

lemma "(#1234567::int) \<le> #1234567"
  by simp


text {* \medskip Quotient and Remainder *}

lemma "(#10::int) div #3 = #3"
  by simp

lemma "(#10::int) mod #3 = #1"
  by simp

text {* A negative divisor *}

lemma "(#10::int) div #-3 = #-4"
  by simp

lemma "(#10::int) mod #-3 = #-2"
  by simp

text {*
  A negative dividend\footnote{The definition agrees with mathematical
  convention but not with the hardware of most computers}
*}

lemma "(#-10::int) div #3 = #-4"
  by simp

lemma "(#-10::int) mod #3 = #2"
  by simp

text {* A negative dividend \emph{and} divisor *}

lemma "(#-10::int) div #-3 = #3"
  by simp

lemma "(#-10::int) mod #-3 = #-1"
  by simp

text {* A few bigger examples *}

lemma "(#8452::int) mod #3 = #1"
  by simp

lemma "(#59485::int) div #434 = #137"
  by simp

lemma "(#1000006::int) mod #10 = #6"
  by simp


text {* \medskip Division by shifting *}

lemma "#10000000 div #2 = (#5000000::int)"
  by simp

lemma "#10000001 mod #2 = (#1::int)"
  by simp

lemma "#10000055 div #32 = (#312501::int)"
  by simp

lemma "#10000055 mod #32 = (#23::int)"
  by simp

lemma "#100094 div #144 = (#695::int)"
  by simp

lemma "#100094 mod #144 = (#14::int)"
  by simp


subsection {* The Natural Numbers *}

text {* Successor *}

lemma "Suc #99999 = #100000"
  by (simp add: Suc_nat_number_of)
    -- {* not a default rewrite since sometimes we want to have @{text "Suc #nnn"} *}


text {* \medskip Addition *}

lemma "(#13::nat) + #19 = #32"
  by simp

lemma "(#1234::nat) + #5678 = #6912"
  by simp

lemma "(#973646::nat) + #6475 = #980121"
  by simp


text {* \medskip Subtraction *}

lemma "(#32::nat) - #14 = #18"
  by simp

lemma "(#14::nat) - #15 = #0"
  by simp

lemma "(#14::nat) - #1576644 = #0"
  by simp

lemma "(#48273776::nat) - #3873737 = #44400039"
  by simp


text {* \medskip Multiplication *}

lemma "(#12::nat) * #11 = #132"
  by simp

lemma "(#647::nat) * #3643 = #2357021"
  by simp


text {* \medskip Quotient and Remainder *}

lemma "(#10::nat) div #3 = #3"
  by simp

lemma "(#10::nat) mod #3 = #1"
  by simp

lemma "(#10000::nat) div #9 = #1111"
  by simp

lemma "(#10000::nat) mod #9 = #1"
  by simp

lemma "(#10000::nat) div #16 = #625"
  by simp

lemma "(#10000::nat) mod #16 = #0"
  by simp


text {* \medskip Testing the cancellation of complementary terms *}

lemma "y + (x + -x) = (#0::int) + y"
  by simp

lemma "y + (-x + (- y + x)) = (#0::int)"
  by simp

lemma "-x + (y + (- y + x)) = (#0::int)"
  by simp

lemma "x + (x + (- x + (- x + (- y + - z)))) = (#0::int) - y - z"
  by simp

lemma "x + x - x - x - y - z = (#0::int) - y - z"
  by simp

lemma "x + y + z - (x + z) = y - (#0::int)"
  by simp

lemma "x + (y + (y + (y + (-x + -x)))) = (#0::int) + y - x + y + y"
  by simp

lemma "x + (y + (y + (y + (-y + -x)))) = y + (#0::int) + y"
  by simp

lemma "x + y - x + z - x - y - z + x < (#1::int)"
  by simp


subsection {* Normal form of bit strings *}

text {*
  Definition of normal form for proving that binary arithmetic on
  normalized operands yields normalized results.  Normal means no
  leading 0s on positive numbers and no leading 1s on negatives.
*}

consts normal :: "bin set"
inductive "normal"
  intros [simp]
    Pls: "Pls: normal"
    Min: "Min: normal"
    BIT_F: "w: normal ==> w \<noteq> Pls ==> w BIT False : normal"
    BIT_T: "w: normal ==> w \<noteq> Min ==> w BIT True : normal"

text {*
  \medskip Binary arithmetic on normalized operands yields normalized
  results.
*}

lemma normal_BIT_I [simp]: "w BIT b \<in> normal ==> w BIT b BIT c \<in> normal"
  apply (case_tac c)
   apply auto
  done

lemma normal_BIT_D: "w BIT b \<in> normal ==> w \<in> normal"
  apply (erule normal.cases)
     apply auto
  done

lemma NCons_normal [simp]: "w \<in> normal ==> NCons w b \<in> normal"
  apply (induct w)
    apply (auto simp add: NCons_Pls NCons_Min)
  done

lemma NCons_True: "NCons w True \<noteq> Pls"
  apply (induct w)
    apply auto
  done

lemma NCons_False: "NCons w False \<noteq> Min"
  apply (induct w)
    apply auto
  done

lemma bin_succ_normal [simp]: "w \<in> normal ==> bin_succ w \<in> normal"
  apply (erule normal.induct)
     apply (case_tac [4] w)
  apply (auto simp add: NCons_True bin_succ_BIT)
  done

lemma bin_pred_normal [simp]: "w \<in> normal ==> bin_pred w \<in> normal"
  apply (erule normal.induct)
     apply (case_tac [3] w)
  apply (auto simp add: NCons_False bin_pred_BIT)
  done

lemma bin_add_normal [rule_format]:
  "w \<in> normal --> (\<forall>z. z \<in> normal --> bin_add w z \<in> normal)"
  apply (induct w)
    apply simp
   apply simp
  apply (rule impI)
  apply (rule allI)
  apply (induct_tac z)
    apply (simp_all add: bin_add_BIT)
  apply (safe dest!: normal_BIT_D)
    apply simp_all
  done

lemma normal_Pls_eq_0: "w \<in> normal ==> (w = Pls) = (number_of w = (#0::int))"
  apply (erule normal.induct)
     apply auto
  done

lemma bin_minus_normal: "w \<in> normal ==> bin_minus w \<in> normal"
  apply (erule normal.induct)
     apply (simp_all add: bin_minus_BIT)
  apply (rule normal.intros)
  apply assumption
  apply (simp add: normal_Pls_eq_0)
  apply (simp only: number_of_minus iszero_def zminus_equation [of _ "int 0"])
  apply (rule not_sym)
  apply simp
  done

lemma bin_mult_normal [rule_format]:
    "w \<in> normal ==> z \<in> normal --> bin_mult w z \<in> normal"
  apply (erule normal.induct)
     apply (simp_all add: bin_minus_normal bin_mult_BIT)
  apply (safe dest!: normal_BIT_D)
  apply (simp add: bin_add_normal)
  done

end