author  blanchet 
Mon, 20 Dec 2010 14:17:49 +0100  
changeset 41317  fc48faccd77b 
parent 35762  af3ff2ba4c54 
child 41959  b460124855b8 
permissions  rwrr 
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(* Title: CTT/bool 
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

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Copyright 1991 University of Cambridge 
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*) 

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header {* The twoelement type (booleans and conditionals) *} 
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theory Bool 

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imports CTT 

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begin 

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definition 
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Bool :: "t" where 
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"Bool == T+T" 
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21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
19762
diff
changeset

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definition 
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
19762
diff
changeset

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true :: "i" where 
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"true == inl(tt)" 
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21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
19762
diff
changeset

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definition 
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
19762
diff
changeset

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false :: "i" where 
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"false == inr(tt)" 
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21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
19762
diff
changeset

24 
definition 
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
19762
diff
changeset

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cond :: "[i,i,i]=>i" where 
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"cond(a,b,c) == when(a, %u. b, %u. c)" 
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lemmas bool_defs = Bool_def true_def false_def cond_def 

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subsection {* Derivation of rules for the type Bool *} 

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(*formation rule*) 

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lemma boolF: "Bool type" 

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apply (unfold bool_defs) 

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apply (tactic "typechk_tac []") 

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done 

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(*introduction rules for true, false*) 

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lemma boolI_true: "true : Bool" 

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apply (unfold bool_defs) 

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apply (tactic "typechk_tac []") 

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done 

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lemma boolI_false: "false : Bool" 

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apply (unfold bool_defs) 

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apply (tactic "typechk_tac []") 

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done 

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(*elimination rule: typing of cond*) 
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lemma boolE: 

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"[ p:Bool; a : C(true); b : C(false) ] ==> cond(p,a,b) : C(p)" 

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apply (unfold bool_defs) 

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apply (tactic "typechk_tac []") 

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apply (erule_tac [!] TE) 

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apply (tactic "typechk_tac []") 

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done 

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lemma boolEL: 

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"[ p = q : Bool; a = c : C(true); b = d : C(false) ] 

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==> cond(p,a,b) = cond(q,c,d) : C(p)" 

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apply (unfold bool_defs) 

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apply (rule PlusEL) 

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apply (erule asm_rl refl_elem [THEN TEL])+ 

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done 

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(*computation rules for true, false*) 

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lemma boolC_true: 

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"[ a : C(true); b : C(false) ] ==> cond(true,a,b) = a : C(true)" 

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apply (unfold bool_defs) 

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apply (rule comp_rls) 

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apply (tactic "typechk_tac []") 

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apply (erule_tac [!] TE) 

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apply (tactic "typechk_tac []") 

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done 

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lemma boolC_false: 

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"[ a : C(true); b : C(false) ] ==> cond(false,a,b) = b : C(false)" 

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apply (unfold bool_defs) 

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apply (rule comp_rls) 

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apply (tactic "typechk_tac []") 

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apply (erule_tac [!] TE) 

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apply (tactic "typechk_tac []") 

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done 

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end 