author  desharna 
Tue, 28 Sep 2021 10:47:18 +0200  
changeset 74370  d8dc8fdc46fc 
parent 60770  240563fbf41d 
permissions  rwrr 
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(* Title: CCL/ex/Nat.thy 
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Author: Martin Coen, Cambridge University Computer Laboratory 
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Copyright 1993 University of Cambridge 
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*) 

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section \<open>Programs defined over the natural numbers\<close> 
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theory Nat 

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imports "../Wfd" 
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begin 
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definition not :: "i\<Rightarrow>i" 
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where "not(b) == if b then false else true" 
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definition add :: "[i,i]\<Rightarrow>i" (infixr "#+" 60) 
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where "a #+ b == nrec(a, b, \<lambda>x g. succ(g))" 

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definition mult :: "[i,i]\<Rightarrow>i" (infixr "#*" 60) 
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where "a #* b == nrec(a, zero, \<lambda>x g. b #+ g)" 

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definition sub :: "[i,i]\<Rightarrow>i" (infixr "#" 60) 
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"a # b == 
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letrec sub x y be ncase(y, x, \<lambda>yy. ncase(x, zero, \<lambda>xx. sub(xx,yy))) 
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in sub(a,b)" 
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definition le :: "[i,i]\<Rightarrow>i" (infixr "#<=" 60) 
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where 
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"a #<= b == 
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letrec le x y be ncase(x, true, \<lambda>xx. ncase(y, false, \<lambda>yy. le(xx,yy))) 
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in le(a,b)" 
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definition lt :: "[i,i]\<Rightarrow>i" (infixr "#<" 60) 
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where "a #< b == not(b #<= a)" 
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definition div :: "[i,i]\<Rightarrow>i" (infixr "##" 60) 
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where 
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"a ## b == 
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letrec div x y be if x #< y then zero else succ(div(x#y,y)) 
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in div(a,b)" 
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definition ackermann :: "[i,i]\<Rightarrow>i" 
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"ackermann(a,b) == 
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letrec ack n m be ncase(n, succ(m), \<lambda>x. 
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ncase(m,ack(x,succ(zero)), \<lambda>y. ack(x,ack(succ(x),y)))) 

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in ack(a,b)" 
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lemmas nat_defs = not_def add_def mult_def sub_def le_def lt_def ackermann_def napply_def 
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lemma natBs [simp]: 

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"not(true) = false" 

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"not(false) = true" 

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"zero #+ n = n" 

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"succ(n) #+ m = succ(n #+ m)" 

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"zero #* n = zero" 

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"succ(n) #* m = m #+ (n #* m)" 

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"f^zero`a = a" 

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"f^succ(n)`a = f(f^n`a)" 

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by (simp_all add: nat_defs) 

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lemma napply_f: "n:Nat \<Longrightarrow> f^n`f(a) = f^succ(n)`a" 
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apply (erule Nat_ind) 
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apply simp_all 

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done 

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lemma addT: "\<lbrakk>a:Nat; b:Nat\<rbrakk> \<Longrightarrow> a #+ b : Nat" 
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apply (unfold add_def) 
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apply typechk 
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done 
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lemma multT: "\<lbrakk>a:Nat; b:Nat\<rbrakk> \<Longrightarrow> a #* b : Nat" 
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apply (unfold add_def mult_def) 
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apply typechk 
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done 
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(* Defined to return zero if a<b *) 

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lemma subT: "\<lbrakk>a:Nat; b:Nat\<rbrakk> \<Longrightarrow> a # b : Nat" 
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apply (unfold sub_def) 
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apply typechk 
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apply clean_ccs 

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apply (erule NatPRI [THEN wfstI, THEN NatPR_wf [THEN wmap_wf, THEN wfI]]) 
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done 

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lemma leT: "\<lbrakk>a:Nat; b:Nat\<rbrakk> \<Longrightarrow> a #<= b : Bool" 
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apply (unfold le_def) 
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apply typechk 
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apply clean_ccs 

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apply (erule NatPRI [THEN wfstI, THEN NatPR_wf [THEN wmap_wf, THEN wfI]]) 
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done 

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lemma ltT: "\<lbrakk>a:Nat; b:Nat\<rbrakk> \<Longrightarrow> a #< b : Bool" 
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apply (unfold not_def lt_def) 
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apply (typechk leT) 
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done 
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subsection \<open>Termination Conditions for Ackermann's Function\<close> 
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lemmas relI = NatPR_wf [THEN NatPR_wf [THEN lex_wf, THEN wfI]] 

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lemma "\<lbrakk>a:Nat; b:Nat\<rbrakk> \<Longrightarrow> ackermann(a,b) : Nat" 
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apply (unfold ackermann_def) 
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apply gen_ccs 
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apply (erule NatPRI [THEN lexI1 [THEN relI]] NatPRI [THEN lexI2 [THEN relI]])+ 
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done 

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end 