src/ZF/Bin.thy
author wenzelm
Thu Dec 14 11:24:26 2017 +0100 (20 months ago)
changeset 67198 694f29a5433b
parent 61798 27f3c10b0b50
child 68233 5e0e9376b2b0
permissions -rw-r--r--
merged
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(*  Title:      ZF/Bin.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1994  University of Cambridge
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   The sign Pls stands for an infinite string of leading 0's.
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   The sign Min stands for an infinite string of leading 1's.
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A number can have multiple representations, namely leading 0's with sign
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Pls and leading 1's with sign Min.  See twos-compl.ML/int_of_binary for
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the numerical interpretation.
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The representation expects that (m mod 2) is 0 or 1, even if m is negative;
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For instance, ~5 div 2 = ~3 and ~5 mod 2 = 1; thus ~5 = (~3)*2 + 1
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*)
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section\<open>Arithmetic on Binary Integers\<close>
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theory Bin
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imports Int_ZF Datatype_ZF
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begin
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consts  bin :: i
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datatype
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  "bin" = Pls
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        | Min
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        | Bit ("w \<in> bin", "b \<in> bool")     (infixl "BIT" 90)
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consts
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  integ_of  :: "i=>i"
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  NCons     :: "[i,i]=>i"
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  bin_succ  :: "i=>i"
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  bin_pred  :: "i=>i"
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  bin_minus :: "i=>i"
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  bin_adder :: "i=>i"
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  bin_mult  :: "[i,i]=>i"
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primrec
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  integ_of_Pls:  "integ_of (Pls)     = $# 0"
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  integ_of_Min:  "integ_of (Min)     = $-($#1)"
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  integ_of_BIT:  "integ_of (w BIT b) = $#b $+ integ_of(w) $+ integ_of(w)"
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    (** recall that cond(1,b,c)=b and cond(0,b,c)=0 **)
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primrec (*NCons adds a bit, suppressing leading 0s and 1s*)
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  NCons_Pls: "NCons (Pls,b)     = cond(b,Pls BIT b,Pls)"
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  NCons_Min: "NCons (Min,b)     = cond(b,Min,Min BIT b)"
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  NCons_BIT: "NCons (w BIT c,b) = w BIT c BIT b"
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primrec (*successor.  If a BIT, can change a 0 to a 1 without recursion.*)
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  bin_succ_Pls:  "bin_succ (Pls)     = Pls BIT 1"
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  bin_succ_Min:  "bin_succ (Min)     = Pls"
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  bin_succ_BIT:  "bin_succ (w BIT b) = cond(b, bin_succ(w) BIT 0, NCons(w,1))"
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primrec (*predecessor*)
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  bin_pred_Pls:  "bin_pred (Pls)     = Min"
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  bin_pred_Min:  "bin_pred (Min)     = Min BIT 0"
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  bin_pred_BIT:  "bin_pred (w BIT b) = cond(b, NCons(w,0), bin_pred(w) BIT 1)"
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primrec (*unary negation*)
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  bin_minus_Pls:
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    "bin_minus (Pls)       = Pls"
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  bin_minus_Min:
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    "bin_minus (Min)       = Pls BIT 1"
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  bin_minus_BIT:
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    "bin_minus (w BIT b) = cond(b, bin_pred(NCons(bin_minus(w),0)),
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                                bin_minus(w) BIT 0)"
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primrec (*sum*)
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  bin_adder_Pls:
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    "bin_adder (Pls)     = (\<lambda>w\<in>bin. w)"
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  bin_adder_Min:
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    "bin_adder (Min)     = (\<lambda>w\<in>bin. bin_pred(w))"
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  bin_adder_BIT:
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    "bin_adder (v BIT x) =
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       (\<lambda>w\<in>bin.
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         bin_case (v BIT x, bin_pred(v BIT x),
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                   %w y. NCons(bin_adder (v) ` cond(x and y, bin_succ(w), w),
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                               x xor y),
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                   w))"
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(*The bin_case above replaces the following mutually recursive function:
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primrec
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  "adding (v,x,Pls)     = v BIT x"
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  "adding (v,x,Min)     = bin_pred(v BIT x)"
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  "adding (v,x,w BIT y) = NCons(bin_adder (v, cond(x and y, bin_succ(w), w)),
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                                x xor y)"
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*)
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definition
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  bin_add   :: "[i,i]=>i"  where
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    "bin_add(v,w) == bin_adder(v)`w"
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primrec
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  bin_mult_Pls:
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    "bin_mult (Pls,w)     = Pls"
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  bin_mult_Min:
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    "bin_mult (Min,w)     = bin_minus(w)"
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  bin_mult_BIT:
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    "bin_mult (v BIT b,w) = cond(b, bin_add(NCons(bin_mult(v,w),0),w),
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                                 NCons(bin_mult(v,w),0))"
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syntax
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  "_Int0" :: i  ("#()0")
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  "_Int1" :: i  ("#()1")
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  "_Int2" :: i  ("#()2")
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  "_Neg_Int1" :: i  ("#-()1")
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  "_Neg_Int2" :: i  ("#-()2")
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translations
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  "#0" \<rightleftharpoons> "CONST integ_of(CONST Pls)"
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  "#1" \<rightleftharpoons> "CONST integ_of(CONST Pls BIT 1)"
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  "#2" \<rightleftharpoons> "CONST integ_of(CONST Pls BIT 1 BIT 0)"
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  "#-1" \<rightleftharpoons> "CONST integ_of(CONST Min)"
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  "#-2" \<rightleftharpoons> "CONST integ_of(CONST Min BIT 0)"
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syntax
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  "_Int" :: "num_token => i"  ("#_" 1000)
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  "_Neg_Int" :: "num_token => i"  ("#-_" 1000)
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ML_file "Tools/numeral_syntax.ML"
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declare bin.intros [simp,TC]
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lemma NCons_Pls_0: "NCons(Pls,0) = Pls"
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by simp
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lemma NCons_Pls_1: "NCons(Pls,1) = Pls BIT 1"
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by simp
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lemma NCons_Min_0: "NCons(Min,0) = Min BIT 0"
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by simp
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lemma NCons_Min_1: "NCons(Min,1) = Min"
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by simp
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lemma NCons_BIT: "NCons(w BIT x,b) = w BIT x BIT b"
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by (simp add: bin.case_eqns)
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lemmas NCons_simps [simp] =
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    NCons_Pls_0 NCons_Pls_1 NCons_Min_0 NCons_Min_1 NCons_BIT
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(** Type checking **)
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lemma integ_of_type [TC]: "w \<in> bin ==> integ_of(w) \<in> int"
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apply (induct_tac "w")
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apply (simp_all add: bool_into_nat)
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done
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lemma NCons_type [TC]: "[| w \<in> bin; b \<in> bool |] ==> NCons(w,b) \<in> bin"
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by (induct_tac "w", auto)
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lemma bin_succ_type [TC]: "w \<in> bin ==> bin_succ(w) \<in> bin"
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by (induct_tac "w", auto)
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lemma bin_pred_type [TC]: "w \<in> bin ==> bin_pred(w) \<in> bin"
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by (induct_tac "w", auto)
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lemma bin_minus_type [TC]: "w \<in> bin ==> bin_minus(w) \<in> bin"
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by (induct_tac "w", auto)
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(*This proof is complicated by the mutual recursion*)
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lemma bin_add_type [rule_format,TC]:
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     "v \<in> bin ==> \<forall>w\<in>bin. bin_add(v,w) \<in> bin"
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apply (unfold bin_add_def)
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apply (induct_tac "v")
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apply (rule_tac [3] ballI)
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apply (rename_tac [3] "w'")
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apply (induct_tac [3] "w'")
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apply (simp_all add: NCons_type)
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done
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lemma bin_mult_type [TC]: "[| v \<in> bin; w \<in> bin |] ==> bin_mult(v,w) \<in> bin"
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by (induct_tac "v", auto)
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subsubsection\<open>The Carry and Borrow Functions,
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            @{term bin_succ} and @{term bin_pred}\<close>
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(*NCons preserves the integer value of its argument*)
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lemma integ_of_NCons [simp]:
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     "[| w \<in> bin; b \<in> bool |] ==> integ_of(NCons(w,b)) = integ_of(w BIT b)"
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apply (erule bin.cases)
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apply (auto elim!: boolE)
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done
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lemma integ_of_succ [simp]:
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     "w \<in> bin ==> integ_of(bin_succ(w)) = $#1 $+ integ_of(w)"
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apply (erule bin.induct)
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apply (auto simp add: zadd_ac elim!: boolE)
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done
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lemma integ_of_pred [simp]:
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     "w \<in> bin ==> integ_of(bin_pred(w)) = $- ($#1) $+ integ_of(w)"
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apply (erule bin.induct)
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apply (auto simp add: zadd_ac elim!: boolE)
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done
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subsubsection\<open>@{term bin_minus}: Unary Negation of Binary Integers\<close>
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lemma integ_of_minus: "w \<in> bin ==> integ_of(bin_minus(w)) = $- integ_of(w)"
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apply (erule bin.induct)
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apply (auto simp add: zadd_ac zminus_zadd_distrib  elim!: boolE)
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done
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subsubsection\<open>@{term bin_add}: Binary Addition\<close>
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lemma bin_add_Pls [simp]: "w \<in> bin ==> bin_add(Pls,w) = w"
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by (unfold bin_add_def, simp)
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lemma bin_add_Pls_right: "w \<in> bin ==> bin_add(w,Pls) = w"
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apply (unfold bin_add_def)
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apply (erule bin.induct, auto)
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done
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lemma bin_add_Min [simp]: "w \<in> bin ==> bin_add(Min,w) = bin_pred(w)"
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by (unfold bin_add_def, simp)
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lemma bin_add_Min_right: "w \<in> bin ==> bin_add(w,Min) = bin_pred(w)"
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apply (unfold bin_add_def)
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apply (erule bin.induct, auto)
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done
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lemma bin_add_BIT_Pls [simp]: "bin_add(v BIT x,Pls) = v BIT x"
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by (unfold bin_add_def, simp)
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lemma bin_add_BIT_Min [simp]: "bin_add(v BIT x,Min) = bin_pred(v BIT x)"
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by (unfold bin_add_def, simp)
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lemma bin_add_BIT_BIT [simp]:
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     "[| w \<in> bin;  y \<in> bool |]
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      ==> bin_add(v BIT x, w BIT y) =
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          NCons(bin_add(v, cond(x and y, bin_succ(w), w)), x xor y)"
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by (unfold bin_add_def, simp)
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lemma integ_of_add [rule_format]:
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     "v \<in> bin ==>
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          \<forall>w\<in>bin. integ_of(bin_add(v,w)) = integ_of(v) $+ integ_of(w)"
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apply (erule bin.induct, simp, simp)
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apply (rule ballI)
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apply (induct_tac "wa")
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apply (auto simp add: zadd_ac elim!: boolE)
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done
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(*Subtraction*)
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lemma diff_integ_of_eq:
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     "[| v \<in> bin;  w \<in> bin |]
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      ==> integ_of(v) $- integ_of(w) = integ_of(bin_add (v, bin_minus(w)))"
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apply (unfold zdiff_def)
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apply (simp add: integ_of_add integ_of_minus)
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done
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subsubsection\<open>@{term bin_mult}: Binary Multiplication\<close>
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lemma integ_of_mult:
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     "[| v \<in> bin;  w \<in> bin |]
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      ==> integ_of(bin_mult(v,w)) = integ_of(v) $* integ_of(w)"
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apply (induct_tac "v", simp)
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apply (simp add: integ_of_minus)
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apply (auto simp add: zadd_ac integ_of_add zadd_zmult_distrib  elim!: boolE)
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done
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subsection\<open>Computations\<close>
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(** extra rules for bin_succ, bin_pred **)
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lemma bin_succ_1: "bin_succ(w BIT 1) = bin_succ(w) BIT 0"
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by simp
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lemma bin_succ_0: "bin_succ(w BIT 0) = NCons(w,1)"
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by simp
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lemma bin_pred_1: "bin_pred(w BIT 1) = NCons(w,0)"
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by simp
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lemma bin_pred_0: "bin_pred(w BIT 0) = bin_pred(w) BIT 1"
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by simp
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(** extra rules for bin_minus **)
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lemma bin_minus_1: "bin_minus(w BIT 1) = bin_pred(NCons(bin_minus(w), 0))"
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by simp
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lemma bin_minus_0: "bin_minus(w BIT 0) = bin_minus(w) BIT 0"
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by simp
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(** extra rules for bin_add **)
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lemma bin_add_BIT_11: "w \<in> bin ==> bin_add(v BIT 1, w BIT 1) =
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                     NCons(bin_add(v, bin_succ(w)), 0)"
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by simp
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lemma bin_add_BIT_10: "w \<in> bin ==> bin_add(v BIT 1, w BIT 0) =
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                     NCons(bin_add(v,w), 1)"
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by simp
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lemma bin_add_BIT_0: "[| w \<in> bin;  y \<in> bool |]
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      ==> bin_add(v BIT 0, w BIT y) = NCons(bin_add(v,w), y)"
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by simp
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(** extra rules for bin_mult **)
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lemma bin_mult_1: "bin_mult(v BIT 1, w) = bin_add(NCons(bin_mult(v,w),0), w)"
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by simp
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lemma bin_mult_0: "bin_mult(v BIT 0, w) = NCons(bin_mult(v,w),0)"
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by simp
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(** Simplification rules with integer constants **)
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lemma int_of_0: "$#0 = #0"
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by simp
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lemma int_of_succ: "$# succ(n) = #1 $+ $#n"
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by (simp add: int_of_add [symmetric] natify_succ)
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lemma zminus_0 [simp]: "$- #0 = #0"
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by simp
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   327
lemma zadd_0_intify [simp]: "#0 $+ z = intify(z)"
wenzelm@23146
   328
by simp
wenzelm@23146
   329
wenzelm@23146
   330
lemma zadd_0_right_intify [simp]: "z $+ #0 = intify(z)"
wenzelm@23146
   331
by simp
wenzelm@23146
   332
wenzelm@23146
   333
lemma zmult_1_intify [simp]: "#1 $* z = intify(z)"
wenzelm@23146
   334
by simp
wenzelm@23146
   335
wenzelm@23146
   336
lemma zmult_1_right_intify [simp]: "z $* #1 = intify(z)"
wenzelm@23146
   337
by (subst zmult_commute, simp)
wenzelm@23146
   338
wenzelm@23146
   339
lemma zmult_0 [simp]: "#0 $* z = #0"
wenzelm@23146
   340
by simp
wenzelm@23146
   341
wenzelm@23146
   342
lemma zmult_0_right [simp]: "z $* #0 = #0"
wenzelm@23146
   343
by (subst zmult_commute, simp)
wenzelm@23146
   344
wenzelm@23146
   345
lemma zmult_minus1 [simp]: "#-1 $* z = $-z"
wenzelm@23146
   346
by (simp add: zcompare_rls)
wenzelm@23146
   347
wenzelm@23146
   348
lemma zmult_minus1_right [simp]: "z $* #-1 = $-z"
wenzelm@23146
   349
apply (subst zmult_commute)
wenzelm@23146
   350
apply (rule zmult_minus1)
wenzelm@23146
   351
done
wenzelm@23146
   352
wenzelm@23146
   353
wenzelm@60770
   354
subsection\<open>Simplification Rules for Comparison of Binary Numbers\<close>
wenzelm@60770
   355
text\<open>Thanks to Norbert Voelker\<close>
wenzelm@23146
   356
wenzelm@23146
   357
(** Equals (=) **)
wenzelm@23146
   358
paulson@46820
   359
lemma eq_integ_of_eq:
paulson@46953
   360
     "[| v \<in> bin;  w \<in> bin |]
paulson@46821
   361
      ==> ((integ_of(v)) = integ_of(w)) \<longleftrightarrow>
wenzelm@23146
   362
          iszero (integ_of (bin_add (v, bin_minus(w))))"
wenzelm@23146
   363
apply (unfold iszero_def)
wenzelm@23146
   364
apply (simp add: zcompare_rls integ_of_add integ_of_minus)
wenzelm@23146
   365
done
wenzelm@23146
   366
wenzelm@23146
   367
lemma iszero_integ_of_Pls: "iszero (integ_of(Pls))"
wenzelm@23146
   368
by (unfold iszero_def, simp)
wenzelm@23146
   369
wenzelm@23146
   370
wenzelm@23146
   371
lemma nonzero_integ_of_Min: "~ iszero (integ_of(Min))"
wenzelm@23146
   372
apply (unfold iszero_def)
wenzelm@23146
   373
apply (simp add: zminus_equation)
wenzelm@23146
   374
done
wenzelm@23146
   375
paulson@46820
   376
lemma iszero_integ_of_BIT:
paulson@46953
   377
     "[| w \<in> bin; x \<in> bool |]
paulson@46821
   378
      ==> iszero (integ_of (w BIT x)) \<longleftrightarrow> (x=0 & iszero (integ_of(w)))"
wenzelm@23146
   379
apply (unfold iszero_def, simp)
paulson@46820
   380
apply (subgoal_tac "integ_of (w) \<in> int")
wenzelm@23146
   381
apply typecheck
wenzelm@23146
   382
apply (drule int_cases)
wenzelm@23146
   383
apply (safe elim!: boolE)
wenzelm@23146
   384
apply (simp_all (asm_lr) add: zcompare_rls zminus_zadd_distrib [symmetric]
wenzelm@23146
   385
                     int_of_add [symmetric])
wenzelm@23146
   386
done
wenzelm@23146
   387
wenzelm@23146
   388
lemma iszero_integ_of_0:
paulson@46953
   389
     "w \<in> bin ==> iszero (integ_of (w BIT 0)) \<longleftrightarrow> iszero (integ_of(w))"
paulson@46820
   390
by (simp only: iszero_integ_of_BIT, blast)
wenzelm@23146
   391
paulson@46953
   392
lemma iszero_integ_of_1: "w \<in> bin ==> ~ iszero (integ_of (w BIT 1))"
wenzelm@23146
   393
by (simp only: iszero_integ_of_BIT, blast)
wenzelm@23146
   394
wenzelm@23146
   395
wenzelm@23146
   396
wenzelm@23146
   397
(** Less-than (<) **)
wenzelm@23146
   398
paulson@46820
   399
lemma less_integ_of_eq_neg:
paulson@46953
   400
     "[| v \<in> bin;  w \<in> bin |]
paulson@46820
   401
      ==> integ_of(v) $< integ_of(w)
paulson@46821
   402
          \<longleftrightarrow> znegative (integ_of (bin_add (v, bin_minus(w))))"
wenzelm@23146
   403
apply (unfold zless_def zdiff_def)
wenzelm@23146
   404
apply (simp add: integ_of_minus integ_of_add)
wenzelm@23146
   405
done
wenzelm@23146
   406
wenzelm@23146
   407
lemma not_neg_integ_of_Pls: "~ znegative (integ_of(Pls))"
wenzelm@23146
   408
by simp
wenzelm@23146
   409
wenzelm@23146
   410
lemma neg_integ_of_Min: "znegative (integ_of(Min))"
wenzelm@23146
   411
by simp
wenzelm@23146
   412
wenzelm@23146
   413
lemma neg_integ_of_BIT:
paulson@46953
   414
     "[| w \<in> bin; x \<in> bool |]
paulson@46821
   415
      ==> znegative (integ_of (w BIT x)) \<longleftrightarrow> znegative (integ_of(w))"
wenzelm@23146
   416
apply simp
paulson@46820
   417
apply (subgoal_tac "integ_of (w) \<in> int")
wenzelm@23146
   418
apply typecheck
wenzelm@23146
   419
apply (drule int_cases)
wenzelm@23146
   420
apply (auto elim!: boolE simp add: int_of_add [symmetric]  zcompare_rls)
paulson@46820
   421
apply (simp_all add: zminus_zadd_distrib [symmetric] zdiff_def
wenzelm@23146
   422
                     int_of_add [symmetric])
wenzelm@23146
   423
apply (subgoal_tac "$#1 $- $# succ (succ (n #+ n)) = $- $# succ (n #+ n) ")
wenzelm@23146
   424
 apply (simp add: zdiff_def)
wenzelm@23146
   425
apply (simp add: equation_zminus int_of_diff [symmetric])
wenzelm@23146
   426
done
wenzelm@23146
   427
wenzelm@23146
   428
(** Less-than-or-equals (<=) **)
wenzelm@23146
   429
wenzelm@23146
   430
lemma le_integ_of_eq_not_less:
wenzelm@61395
   431
     "(integ_of(x) $\<le> (integ_of(w))) \<longleftrightarrow> ~ (integ_of(w) $< (integ_of(x)))"
wenzelm@23146
   432
by (simp add: not_zless_iff_zle [THEN iff_sym])
wenzelm@23146
   433
wenzelm@23146
   434
wenzelm@23146
   435
(*Delete the original rewrites, with their clumsy conditional expressions*)
paulson@46820
   436
declare bin_succ_BIT [simp del]
paulson@46820
   437
        bin_pred_BIT [simp del]
wenzelm@23146
   438
        bin_minus_BIT [simp del]
wenzelm@23146
   439
        NCons_Pls [simp del]
wenzelm@23146
   440
        NCons_Min [simp del]
wenzelm@23146
   441
        bin_adder_BIT [simp del]
wenzelm@23146
   442
        bin_mult_BIT [simp del]
wenzelm@23146
   443
wenzelm@23146
   444
(*Hide the binary representation of integer constants*)
wenzelm@23146
   445
declare integ_of_Pls [simp del] integ_of_Min [simp del] integ_of_BIT [simp del]
wenzelm@23146
   446
wenzelm@23146
   447
wenzelm@23146
   448
lemmas bin_arith_extra_simps =
paulson@46820
   449
     integ_of_add [symmetric]
paulson@46820
   450
     integ_of_minus [symmetric]
paulson@46820
   451
     integ_of_mult [symmetric]
paulson@46820
   452
     bin_succ_1 bin_succ_0
paulson@46820
   453
     bin_pred_1 bin_pred_0
paulson@46820
   454
     bin_minus_1 bin_minus_0
wenzelm@23146
   455
     bin_add_Pls_right bin_add_Min_right
wenzelm@23146
   456
     bin_add_BIT_0 bin_add_BIT_10 bin_add_BIT_11
wenzelm@23146
   457
     diff_integ_of_eq
wenzelm@23146
   458
     bin_mult_1 bin_mult_0 NCons_simps
wenzelm@23146
   459
wenzelm@23146
   460
wenzelm@23146
   461
(*For making a minimal simpset, one must include these default simprules
wenzelm@23146
   462
  of thy.  Also include simp_thms, or at least (~False)=True*)
wenzelm@23146
   463
lemmas bin_arith_simps =
wenzelm@23146
   464
     bin_pred_Pls bin_pred_Min
wenzelm@23146
   465
     bin_succ_Pls bin_succ_Min
wenzelm@23146
   466
     bin_add_Pls bin_add_Min
wenzelm@23146
   467
     bin_minus_Pls bin_minus_Min
paulson@46820
   468
     bin_mult_Pls bin_mult_Min
wenzelm@23146
   469
     bin_arith_extra_simps
wenzelm@23146
   470
wenzelm@23146
   471
(*Simplification of relational operations*)
wenzelm@23146
   472
lemmas bin_rel_simps =
wenzelm@23146
   473
     eq_integ_of_eq iszero_integ_of_Pls nonzero_integ_of_Min
wenzelm@23146
   474
     iszero_integ_of_0 iszero_integ_of_1
wenzelm@23146
   475
     less_integ_of_eq_neg
wenzelm@23146
   476
     not_neg_integ_of_Pls neg_integ_of_Min neg_integ_of_BIT
wenzelm@23146
   477
     le_integ_of_eq_not_less
wenzelm@23146
   478
wenzelm@23146
   479
declare bin_arith_simps [simp]
wenzelm@23146
   480
declare bin_rel_simps [simp]
wenzelm@23146
   481
wenzelm@23146
   482
wenzelm@23146
   483
(** Simplification of arithmetic when nested to the right **)
wenzelm@23146
   484
wenzelm@23146
   485
lemma add_integ_of_left [simp]:
paulson@46953
   486
     "[| v \<in> bin;  w \<in> bin |]
wenzelm@23146
   487
      ==> integ_of(v) $+ (integ_of(w) $+ z) = (integ_of(bin_add(v,w)) $+ z)"
wenzelm@23146
   488
by (simp add: zadd_assoc [symmetric])
wenzelm@23146
   489
wenzelm@23146
   490
lemma mult_integ_of_left [simp]:
paulson@46953
   491
     "[| v \<in> bin;  w \<in> bin |]
wenzelm@23146
   492
      ==> integ_of(v) $* (integ_of(w) $* z) = (integ_of(bin_mult(v,w)) $* z)"
wenzelm@23146
   493
by (simp add: zmult_assoc [symmetric])
wenzelm@23146
   494
paulson@46820
   495
lemma add_integ_of_diff1 [simp]:
paulson@46953
   496
    "[| v \<in> bin;  w \<in> bin |]
wenzelm@23146
   497
      ==> integ_of(v) $+ (integ_of(w) $- c) = integ_of(bin_add(v,w)) $- (c)"
wenzelm@23146
   498
apply (unfold zdiff_def)
wenzelm@23146
   499
apply (rule add_integ_of_left, auto)
wenzelm@23146
   500
done
wenzelm@23146
   501
wenzelm@23146
   502
lemma add_integ_of_diff2 [simp]:
paulson@46953
   503
     "[| v \<in> bin;  w \<in> bin |]
paulson@46820
   504
      ==> integ_of(v) $+ (c $- integ_of(w)) =
wenzelm@23146
   505
          integ_of (bin_add (v, bin_minus(w))) $+ (c)"
wenzelm@23146
   506
apply (subst diff_integ_of_eq [symmetric])
wenzelm@23146
   507
apply (simp_all add: zdiff_def zadd_ac)
wenzelm@23146
   508
done
wenzelm@23146
   509
wenzelm@23146
   510
wenzelm@23146
   511
(** More for integer constants **)
wenzelm@23146
   512
wenzelm@23146
   513
declare int_of_0 [simp] int_of_succ [simp]
wenzelm@23146
   514
wenzelm@23146
   515
lemma zdiff0 [simp]: "#0 $- x = $-x"
wenzelm@23146
   516
by (simp add: zdiff_def)
wenzelm@23146
   517
wenzelm@23146
   518
lemma zdiff0_right [simp]: "x $- #0 = intify(x)"
wenzelm@23146
   519
by (simp add: zdiff_def)
wenzelm@23146
   520
wenzelm@23146
   521
lemma zdiff_self [simp]: "x $- x = #0"
wenzelm@23146
   522
by (simp add: zdiff_def)
wenzelm@23146
   523
paulson@46953
   524
lemma znegative_iff_zless_0: "k \<in> int ==> znegative(k) \<longleftrightarrow> k $< #0"
wenzelm@23146
   525
by (simp add: zless_def)
wenzelm@23146
   526
paulson@46953
   527
lemma zero_zless_imp_znegative_zminus: "[|#0 $< k; k \<in> int|] ==> znegative($-k)"
wenzelm@23146
   528
by (simp add: zless_def)
wenzelm@23146
   529
wenzelm@61395
   530
lemma zero_zle_int_of [simp]: "#0 $\<le> $# n"
wenzelm@23146
   531
by (simp add: not_zless_iff_zle [THEN iff_sym] znegative_iff_zless_0 [THEN iff_sym])
wenzelm@23146
   532
wenzelm@23146
   533
lemma nat_of_0 [simp]: "nat_of(#0) = 0"
wenzelm@23146
   534
by (simp only: natify_0 int_of_0 [symmetric] nat_of_int_of)
wenzelm@23146
   535
wenzelm@61395
   536
lemma nat_le_int0_lemma: "[| z $\<le> $#0; z \<in> int |] ==> nat_of(z) = 0"
wenzelm@23146
   537
by (auto simp add: znegative_iff_zless_0 [THEN iff_sym] zle_def zneg_nat_of)
wenzelm@23146
   538
wenzelm@61395
   539
lemma nat_le_int0: "z $\<le> $#0 ==> nat_of(z) = 0"
wenzelm@23146
   540
apply (subgoal_tac "nat_of (intify (z)) = 0")
wenzelm@23146
   541
apply (rule_tac [2] nat_le_int0_lemma, auto)
wenzelm@23146
   542
done
wenzelm@23146
   543
wenzelm@23146
   544
lemma int_of_eq_0_imp_natify_eq_0: "$# n = #0 ==> natify(n) = 0"
wenzelm@23146
   545
by (rule not_znegative_imp_zero, auto)
wenzelm@23146
   546
wenzelm@23146
   547
lemma nat_of_zminus_int_of: "nat_of($- $# n) = 0"
wenzelm@23146
   548
by (simp add: nat_of_def int_of_def raw_nat_of zminus image_intrel_int)
wenzelm@23146
   549
wenzelm@61395
   550
lemma int_of_nat_of: "#0 $\<le> z ==> $# nat_of(z) = intify(z)"
wenzelm@23146
   551
apply (rule not_zneg_nat_of_intify)
wenzelm@23146
   552
apply (simp add: znegative_iff_zless_0 not_zless_iff_zle)
wenzelm@23146
   553
done
wenzelm@23146
   554
wenzelm@23146
   555
declare int_of_nat_of [simp] nat_of_zminus_int_of [simp]
wenzelm@23146
   556
wenzelm@61395
   557
lemma int_of_nat_of_if: "$# nat_of(z) = (if #0 $\<le> z then intify(z) else #0)"
wenzelm@23146
   558
by (simp add: int_of_nat_of znegative_iff_zless_0 not_zle_iff_zless)
wenzelm@23146
   559
paulson@46953
   560
lemma zless_nat_iff_int_zless: "[| m \<in> nat; z \<in> int |] ==> (m < nat_of(z)) \<longleftrightarrow> ($#m $< z)"
wenzelm@23146
   561
apply (case_tac "znegative (z) ")
wenzelm@23146
   562
apply (erule_tac [2] not_zneg_nat_of [THEN subst])
wenzelm@23146
   563
apply (auto dest: zless_trans dest!: zero_zle_int_of [THEN zle_zless_trans]
wenzelm@23146
   564
            simp add: znegative_iff_zless_0)
wenzelm@23146
   565
done
wenzelm@23146
   566
wenzelm@23146
   567
wenzelm@23146
   568
(** nat_of and zless **)
wenzelm@23146
   569
paulson@46820
   570
(*An alternative condition is  @{term"$#0 \<subseteq> w"}  *)
paulson@46821
   571
lemma zless_nat_conj_lemma: "$#0 $< z ==> (nat_of(w) < nat_of(z)) \<longleftrightarrow> (w $< z)"
wenzelm@23146
   572
apply (rule iff_trans)
wenzelm@23146
   573
apply (rule zless_int_of [THEN iff_sym])
wenzelm@23146
   574
apply (auto simp add: int_of_nat_of_if simp del: zless_int_of)
wenzelm@23146
   575
apply (auto elim: zless_asym simp add: not_zle_iff_zless)
wenzelm@23146
   576
apply (blast intro: zless_zle_trans)
wenzelm@23146
   577
done
wenzelm@23146
   578
paulson@46821
   579
lemma zless_nat_conj: "(nat_of(w) < nat_of(z)) \<longleftrightarrow> ($#0 $< z & w $< z)"
wenzelm@23146
   580
apply (case_tac "$#0 $< z")
wenzelm@23146
   581
apply (auto simp add: zless_nat_conj_lemma nat_le_int0 not_zless_iff_zle)
wenzelm@23146
   582
done
wenzelm@23146
   583
wenzelm@23146
   584
(*This simprule cannot be added unless we can find a way to make eq_integ_of_eq
wenzelm@23146
   585
  unconditional!
wenzelm@23146
   586
  [The condition "True" is a hack to prevent looping.
wenzelm@23146
   587
    Conditional rewrite rules are tried after unconditional ones, so a rule
wenzelm@23146
   588
    like eq_nat_number_of will be tried first to eliminate #mm=#nn.]
wenzelm@23146
   589
  lemma integ_of_reorient [simp]:
paulson@46821
   590
       "True ==> (integ_of(w) = x) \<longleftrightarrow> (x = integ_of(w))"
wenzelm@23146
   591
  by auto
wenzelm@23146
   592
*)
wenzelm@23146
   593
wenzelm@23146
   594
lemma integ_of_minus_reorient [simp]:
paulson@46821
   595
     "(integ_of(w) = $- x) \<longleftrightarrow> ($- x = integ_of(w))"
wenzelm@23146
   596
by auto
wenzelm@23146
   597
wenzelm@23146
   598
lemma integ_of_add_reorient [simp]:
paulson@46821
   599
     "(integ_of(w) = x $+ y) \<longleftrightarrow> (x $+ y = integ_of(w))"
wenzelm@23146
   600
by auto
wenzelm@23146
   601
wenzelm@23146
   602
lemma integ_of_diff_reorient [simp]:
paulson@46821
   603
     "(integ_of(w) = x $- y) \<longleftrightarrow> (x $- y = integ_of(w))"
wenzelm@23146
   604
by auto
wenzelm@23146
   605
wenzelm@23146
   606
lemma integ_of_mult_reorient [simp]:
paulson@46821
   607
     "(integ_of(w) = x $* y) \<longleftrightarrow> (x $* y = integ_of(w))"
wenzelm@23146
   608
by auto
wenzelm@23146
   609
haftmann@58022
   610
(** To simplify inequalities involving integer negation and literals,
haftmann@58022
   611
    such as -x = #3
haftmann@58022
   612
**)
haftmann@58022
   613
haftmann@58022
   614
lemmas [simp] =
haftmann@58022
   615
  zminus_equation [where y = "integ_of(w)"]
haftmann@58022
   616
  equation_zminus [where x = "integ_of(w)"]
haftmann@58022
   617
  for w
haftmann@58022
   618
haftmann@58022
   619
lemmas [iff] =
haftmann@58022
   620
  zminus_zless [where y = "integ_of(w)"]
haftmann@58022
   621
  zless_zminus [where x = "integ_of(w)"]
haftmann@58022
   622
  for w
haftmann@58022
   623
haftmann@58022
   624
lemmas [iff] =
haftmann@58022
   625
  zminus_zle [where y = "integ_of(w)"]
haftmann@58022
   626
  zle_zminus [where x = "integ_of(w)"]
haftmann@58022
   627
  for w
haftmann@58022
   628
haftmann@58022
   629
lemmas [simp] =
haftmann@58022
   630
  Let_def [where s = "integ_of(w)"] for w
haftmann@58022
   631
haftmann@58022
   632
haftmann@58022
   633
(*** Simprocs for numeric literals ***)
haftmann@58022
   634
haftmann@58022
   635
(** Combining of literal coefficients in sums of products **)
haftmann@58022
   636
haftmann@58022
   637
lemma zless_iff_zdiff_zless_0: "(x $< y) \<longleftrightarrow> (x$-y $< #0)"
haftmann@58022
   638
  by (simp add: zcompare_rls)
haftmann@58022
   639
haftmann@58022
   640
lemma eq_iff_zdiff_eq_0: "[| x \<in> int; y \<in> int |] ==> (x = y) \<longleftrightarrow> (x$-y = #0)"
haftmann@58022
   641
  by (simp add: zcompare_rls)
haftmann@58022
   642
wenzelm@61395
   643
lemma zle_iff_zdiff_zle_0: "(x $\<le> y) \<longleftrightarrow> (x$-y $\<le> #0)"
haftmann@58022
   644
  by (simp add: zcompare_rls)
haftmann@58022
   645
haftmann@58022
   646
haftmann@58022
   647
(** For combine_numerals **)
haftmann@58022
   648
haftmann@58022
   649
lemma left_zadd_zmult_distrib: "i$*u $+ (j$*u $+ k) = (i$+j)$*u $+ k"
haftmann@58022
   650
  by (simp add: zadd_zmult_distrib zadd_ac)
haftmann@58022
   651
haftmann@58022
   652
haftmann@58022
   653
(** For cancel_numerals **)
haftmann@58022
   654
haftmann@58022
   655
lemmas rel_iff_rel_0_rls =
haftmann@58022
   656
  zless_iff_zdiff_zless_0 [where y = "u $+ v"]
haftmann@58022
   657
  eq_iff_zdiff_eq_0 [where y = "u $+ v"]
haftmann@58022
   658
  zle_iff_zdiff_zle_0 [where y = "u $+ v"]
haftmann@58022
   659
  zless_iff_zdiff_zless_0 [where y = n]
haftmann@58022
   660
  eq_iff_zdiff_eq_0 [where y = n]
haftmann@58022
   661
  zle_iff_zdiff_zle_0 [where y = n]
haftmann@58022
   662
  for u v (* FIXME n (!?) *)
haftmann@58022
   663
haftmann@58022
   664
lemma eq_add_iff1: "(i$*u $+ m = j$*u $+ n) \<longleftrightarrow> ((i$-j)$*u $+ m = intify(n))"
haftmann@58022
   665
  apply (simp add: zdiff_def zadd_zmult_distrib)
haftmann@58022
   666
  apply (simp add: zcompare_rls)
haftmann@58022
   667
  apply (simp add: zadd_ac)
haftmann@58022
   668
  done
haftmann@58022
   669
haftmann@58022
   670
lemma eq_add_iff2: "(i$*u $+ m = j$*u $+ n) \<longleftrightarrow> (intify(m) = (j$-i)$*u $+ n)"
haftmann@58022
   671
  apply (simp add: zdiff_def zadd_zmult_distrib)
haftmann@58022
   672
  apply (simp add: zcompare_rls)
haftmann@58022
   673
  apply (simp add: zadd_ac)
haftmann@58022
   674
  done
haftmann@58022
   675
haftmann@58022
   676
lemma less_add_iff1: "(i$*u $+ m $< j$*u $+ n) \<longleftrightarrow> ((i$-j)$*u $+ m $< n)"
haftmann@58022
   677
  apply (simp add: zdiff_def zadd_zmult_distrib zadd_ac rel_iff_rel_0_rls)
haftmann@58022
   678
  done
haftmann@58022
   679
haftmann@58022
   680
lemma less_add_iff2: "(i$*u $+ m $< j$*u $+ n) \<longleftrightarrow> (m $< (j$-i)$*u $+ n)"
haftmann@58022
   681
  apply (simp add: zdiff_def zadd_zmult_distrib zadd_ac rel_iff_rel_0_rls)
haftmann@58022
   682
  done
haftmann@58022
   683
wenzelm@61395
   684
lemma le_add_iff1: "(i$*u $+ m $\<le> j$*u $+ n) \<longleftrightarrow> ((i$-j)$*u $+ m $\<le> n)"
haftmann@58022
   685
  apply (simp add: zdiff_def zadd_zmult_distrib)
haftmann@58022
   686
  apply (simp add: zcompare_rls)
haftmann@58022
   687
  apply (simp add: zadd_ac)
haftmann@58022
   688
  done
haftmann@58022
   689
wenzelm@61395
   690
lemma le_add_iff2: "(i$*u $+ m $\<le> j$*u $+ n) \<longleftrightarrow> (m $\<le> (j$-i)$*u $+ n)"
haftmann@58022
   691
  apply (simp add: zdiff_def zadd_zmult_distrib)
haftmann@58022
   692
  apply (simp add: zcompare_rls)
haftmann@58022
   693
  apply (simp add: zadd_ac)
haftmann@58022
   694
  done
haftmann@58022
   695
haftmann@58022
   696
ML_file "int_arith.ML"
haftmann@58022
   697
wenzelm@60770
   698
subsection \<open>examples:\<close>
wenzelm@59748
   699
wenzelm@61798
   700
text \<open>\<open>combine_numerals_prod\<close> (products of separate literals)\<close>
wenzelm@59748
   701
lemma "#5 $* x $* #3 = y" apply simp oops
wenzelm@59748
   702
wenzelm@61337
   703
schematic_goal "y2 $+ ?x42 = y $+ y2" apply simp oops
wenzelm@59748
   704
wenzelm@59748
   705
lemma "oo : int ==> l $+ (l $+ #2) $+ oo = oo" apply simp oops
wenzelm@59748
   706
wenzelm@59748
   707
lemma "#9$*x $+ y = x$*#23 $+ z" apply simp oops
wenzelm@59748
   708
lemma "y $+ x = x $+ z" apply simp oops
wenzelm@59748
   709
wenzelm@59748
   710
lemma "x : int ==> x $+ y $+ z = x $+ z" apply simp oops
wenzelm@59748
   711
lemma "x : int ==> y $+ (z $+ x) = z $+ x" apply simp oops
wenzelm@59748
   712
lemma "z : int ==> x $+ y $+ z = (z $+ y) $+ (x $+ w)" apply simp oops
wenzelm@59748
   713
lemma "z : int ==> x$*y $+ z = (z $+ y) $+ (y$*x $+ w)" apply simp oops
wenzelm@59748
   714
wenzelm@61395
   715
lemma "#-3 $* x $+ y $\<le> x $* #2 $+ z" apply simp oops
wenzelm@61395
   716
lemma "y $+ x $\<le> x $+ z" apply simp oops
wenzelm@61395
   717
lemma "x $+ y $+ z $\<le> x $+ z" apply simp oops
wenzelm@59748
   718
wenzelm@59748
   719
lemma "y $+ (z $+ x) $< z $+ x" apply simp oops
wenzelm@59748
   720
lemma "x $+ y $+ z $< (z $+ y) $+ (x $+ w)" apply simp oops
wenzelm@59748
   721
lemma "x$*y $+ z $< (z $+ y) $+ (y$*x $+ w)" apply simp oops
wenzelm@59748
   722
wenzelm@59748
   723
lemma "l $+ #2 $+ #2 $+ #2 $+ (l $+ #2) $+ (oo $+ #2) = uu" apply simp oops
wenzelm@59748
   724
lemma "u : int ==> #2 $* u = u" apply simp oops
wenzelm@59748
   725
lemma "(i $+ j $+ #12 $+ k) $- #15 = y" apply simp oops
wenzelm@59748
   726
lemma "(i $+ j $+ #12 $+ k) $- #5 = y" apply simp oops
wenzelm@59748
   727
wenzelm@59748
   728
lemma "y $- b $< b" apply simp oops
wenzelm@59748
   729
lemma "y $- (#3 $* b $+ c) $< b $- #2 $* c" apply simp oops
wenzelm@59748
   730
wenzelm@59748
   731
lemma "(#2 $* x $- (u $* v) $+ y) $- v $* #3 $* u = w" apply simp oops
wenzelm@59748
   732
lemma "(#2 $* x $* u $* v $+ (u $* v) $* #4 $+ y) $- v $* u $* #4 = w" apply simp oops
wenzelm@59748
   733
lemma "(#2 $* x $* u $* v $+ (u $* v) $* #4 $+ y) $- v $* u = w" apply simp oops
wenzelm@59748
   734
lemma "u $* v $- (x $* u $* v $+ (u $* v) $* #4 $+ y) = w" apply simp oops
wenzelm@59748
   735
wenzelm@59748
   736
lemma "(i $+ j $+ #12 $+ k) = u $+ #15 $+ y" apply simp oops
wenzelm@59748
   737
lemma "(i $+ j $* #2 $+ #12 $+ k) = j $+ #5 $+ y" apply simp oops
wenzelm@59748
   738
wenzelm@59748
   739
lemma "#2 $* y $+ #3 $* z $+ #6 $* w $+ #2 $* y $+ #3 $* z $+ #2 $* u = #2 $* y' $+ #3 $* z' $+ #6 $* w' $+ #2 $* y' $+ #3 $* z' $+ u $+ vv" apply simp oops
wenzelm@59748
   740
wenzelm@59748
   741
lemma "a $+ $-(b$+c) $+ b = d" apply simp oops
wenzelm@59748
   742
lemma "a $+ $-(b$+c) $- b = d" apply simp oops
wenzelm@59748
   743
wenzelm@60770
   744
text \<open>negative numerals\<close>
wenzelm@59748
   745
lemma "(i $+ j $+ #-2 $+ k) $- (u $+ #5 $+ y) = zz" apply simp oops
wenzelm@59748
   746
lemma "(i $+ j $+ #-3 $+ k) $< u $+ #5 $+ y" apply simp oops
wenzelm@59748
   747
lemma "(i $+ j $+ #3 $+ k) $< u $+ #-6 $+ y" apply simp oops
wenzelm@59748
   748
lemma "(i $+ j $+ #-12 $+ k) $- #15 = y" apply simp oops
wenzelm@59748
   749
lemma "(i $+ j $+ #12 $+ k) $- #-15 = y" apply simp oops
wenzelm@59748
   750
lemma "(i $+ j $+ #-12 $+ k) $- #-15 = y" apply simp oops
wenzelm@59748
   751
wenzelm@60770
   752
text \<open>Multiplying separated numerals\<close>
wenzelm@59748
   753
lemma "#6 $* ($# x $* #2) =  uu" apply simp oops
wenzelm@59748
   754
lemma "#4 $* ($# x $* $# x) $* (#2 $* $# x) =  uu" apply simp oops
wenzelm@59748
   755
wenzelm@23146
   756
end