author  hoelzl 
Wed, 09 Apr 2014 09:37:48 +0200  
changeset 56480  093ea91498e6 
parent 52143  36ffe23b25f8 
child 57945  cacb00a569e0 
permissions  rwrr 
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(* Title: Cube/Cube.thy 
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Author: Tobias Nipkow 

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*) 

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header {* Barendregt's LambdaCube *} 
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theory Cube 

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imports Pure 

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begin 

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setup Pure_Thy.old_appl_syntax_setup 
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setup PureThy.old_appl_syntax_setup  theory Pure provides regular application syntax by default;
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ML {* 
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structure Rules = Named_Thms 
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( 
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val name = @{binding rules} 
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val description = "Cube inference rules" 
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) 
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*} 
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setup Rules.setup 
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typedecl "term" 
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typedecl "context" 

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typedecl typing 

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axiomatization 
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Abs :: "[term, term => term] => term" and 

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Prod :: "[term, term => term] => term" and 

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Trueprop :: "[context, typing] => prop" and 

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MT_context :: "context" and 

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Context :: "[typing, context] => context" and 

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star :: "term" ("*") and 

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box :: "term" ("\<box>") and 
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app :: "[term, term] => term" (infixl "^" 20) and 
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Has_type :: "[term, term] => typing" 

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nonterminal context' and typing' 
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syntax 

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"_Trueprop" :: "[context', typing'] => prop" ("(_/ \<turnstile> _)") 
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"_Trueprop1" :: "typing' => prop" ("(_)") 
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"" :: "id => context'" ("_") 

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"" :: "var => context'" ("_") 

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"_MT_context" :: "context'" ("") 

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"_Context" :: "[typing', context'] => context'" ("_ _") 

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"_Has_type" :: "[term, term] => typing'" ("(_:/ _)" [0, 0] 5) 

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"_Lam" :: "[idt, term, term] => term" ("(3\<Lambda> _:_./ _)" [0, 0, 0] 10) 
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"_Pi" :: "[idt, term, term] => term" ("(3\<Pi> _:_./ _)" [0, 0] 10) 

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"_arrow" :: "[term, term] => term" (infixr "\<rightarrow>" 10) 

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translations 

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"_Trueprop(G, t)" == "CONST Trueprop(G, t)" 
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("prop") "x:X" == ("prop") "\<turnstile> x:X" 
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"_MT_context" == "CONST MT_context" 
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"_Context" == "CONST Context" 

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"_Has_type" == "CONST Has_type" 

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"\<Lambda> x:A. B" == "CONST Abs(A, %x. B)" 
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"\<Pi> x:A. B" => "CONST Prod(A, %x. B)" 

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"A \<rightarrow> B" => "CONST Prod(A, %_. B)" 

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syntax (xsymbols) 

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"_Pi" :: "[idt, term, term] => term" ("(3\<Pi> _:_./ _)" [0, 0] 10) 
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print_translation {* 
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[(@{const_syntax Prod}, 
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fn _ => Syntax_Trans.dependent_tr' (@{syntax_const "_Pi"}, @{syntax_const "_arrow"}))] 
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*} 
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axiomatization where 
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s_b: "*: \<box>" and 
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strip_s: "[ A:*; a:A ==> G \<turnstile> x:X ] ==> a:A G \<turnstile> x:X" and 
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strip_b: "[ A:\<box>; a:A ==> G \<turnstile> x:X ] ==> a:A G \<turnstile> x:X" and 

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app: "[ F:Prod(A, B); C:A ] ==> F^C: B(C)" and 
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pi_ss: "[ A:*; !!x. x:A ==> B(x):* ] ==> Prod(A, B):*" and 
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lam_ss: "[ A:*; !!x. x:A ==> f(x):B(x); !!x. x:A ==> B(x):* ] 
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==> Abs(A, f) : Prod(A, B)" and 

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beta: "Abs(A, f)^a == f(a)" 
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lemmas [rules] = s_b strip_s strip_b app lam_ss pi_ss 
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lemma imp_elim: 

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assumes "f:A\<rightarrow>B" and "a:A" and "f^a:B ==> PROP P" 
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shows "PROP P" by (rule app assms)+ 
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lemma pi_elim: 

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assumes "F:Prod(A,B)" and "a:A" and "F^a:B(a) ==> PROP P" 

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shows "PROP P" by (rule app assms)+ 
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locale L2 = 

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assumes pi_bs: "[ A:\<box>; !!x. x:A ==> B(x):* ] ==> Prod(A,B):*" 
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and lam_bs: "[ A:\<box>; !!x. x:A ==> f(x):B(x); !!x. x:A ==> B(x):* ] 

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==> Abs(A,f) : Prod(A,B)" 
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begin 
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lemmas [rules] = lam_bs pi_bs 
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end 

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locale Lomega = 
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assumes 

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pi_bb: "[ A:\<box>; !!x. x:A ==> B(x):\<box> ] ==> Prod(A,B):\<box>" 
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and lam_bb: "[ A:\<box>; !!x. x:A ==> f(x):B(x); !!x. x:A ==> B(x):\<box> ] 

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==> Abs(A,f) : Prod(A,B)" 
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begin 
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lemmas [rules] = lam_bb pi_bb 
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end 

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locale LP = 

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assumes pi_sb: "[ A:*; !!x. x:A ==> B(x):\<box> ] ==> Prod(A,B):\<box>" 
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and lam_sb: "[ A:*; !!x. x:A ==> f(x):B(x); !!x. x:A ==> B(x):\<box> ] 

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==> Abs(A,f) : Prod(A,B)" 
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begin 
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lemmas [rules] = lam_sb pi_sb 
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end 

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locale LP2 = LP + L2 
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begin 
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lemmas [rules] = lam_bs pi_bs lam_sb pi_sb 
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end 

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locale Lomega2 = L2 + Lomega 
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begin 
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lemmas [rules] = lam_bs pi_bs lam_bb pi_bb 
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end 

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locale LPomega = LP + Lomega 
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begin 
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lemmas [rules] = lam_bb pi_bb lam_sb pi_sb 
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end 

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locale CC = L2 + LP + Lomega 
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begin 
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lemmas [rules] = lam_bs pi_bs lam_bb pi_bb lam_sb pi_sb 
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end 

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end 