author  wenzelm 
Sun, 26 Nov 2006 23:43:53 +0100  
changeset 21539  c5cf9243ad62 
parent 21404  eb85850d3eb7 
child 29751  e2756594c414 
permissions  rwrr 
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(* Title: FOL/ex/NatClass.thy 
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ID: $Id$ 

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Author: Markus Wenzel, TU Muenchen 

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*) 

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theory NatClass 
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imports FOL 

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begin 

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text {* 

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This is an abstract version of theory @{text "Nat"}. Instead of 

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axiomatizing a single type @{text nat} we define the class of all 

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these types (up to isomorphism). 

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Note: The @{text rec} operator had to be made \emph{monomorphic}, 

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because class axioms may not contain more than one type variable. 

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*} 

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consts 

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0 :: 'a ("0") 
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Suc :: "'a => 'a" 

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rec :: "['a, 'a, ['a, 'a] => 'a] => 'a" 

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axclass 

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nat < "term" 
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induct: "[ P(0); !!x. P(x) ==> P(Suc(x)) ] ==> P(n)" 

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Suc_inject: "Suc(m) = Suc(n) ==> m = n" 

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Suc_neq_0: "Suc(m) = 0 ==> R" 

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rec_0: "rec(0, a, f) = a" 

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rec_Suc: "rec(Suc(m), a, f) = f(m, rec(m, a, f))" 

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definition 
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
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diff
changeset

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add :: "['a::nat, 'a] => 'a" (infixl "+" 60) where 
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"m + n = rec(m, n, %x y. Suc(y))" 
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lemma Suc_n_not_n: "Suc(k) ~= (k::'a::nat)" 

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apply (rule_tac n = k in induct) 

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apply (rule notI) 

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apply (erule Suc_neq_0) 

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apply (rule notI) 

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apply (erule notE) 

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apply (erule Suc_inject) 

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done 

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lemma "(k+m)+n = k+(m+n)" 

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apply (rule induct) 

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back 

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back 

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back 

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back 

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back 

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back 

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oops 

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lemma add_0 [simp]: "0+n = n" 

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apply (unfold add_def) 

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apply (rule rec_0) 

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done 

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lemma add_Suc [simp]: "Suc(m)+n = Suc(m+n)" 
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apply (unfold add_def) 

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apply (rule rec_Suc) 

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done 

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lemma add_assoc: "(k+m)+n = k+(m+n)" 

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apply (rule_tac n = k in induct) 

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apply simp 

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apply simp 

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done 

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lemma add_0_right: "m+0 = m" 

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apply (rule_tac n = m in induct) 

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apply simp 

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apply simp 

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done 

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lemma add_Suc_right: "m+Suc(n) = Suc(m+n)" 

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apply (rule_tac n = m in induct) 

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apply simp_all 

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done 

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lemma 

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assumes prem: "!!n. f(Suc(n)) = Suc(f(n))" 

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shows "f(i+j) = i+f(j)" 

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apply (rule_tac n = i in induct) 

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apply simp 

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apply (simp add: prem) 

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done 

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end 