src/Cube/Cube.thy
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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 Lambda-Cube *}
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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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typedecl "term"
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typedecl "context"
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typedecl typing
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nonterminal context' and typing'
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consts
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  Abs :: "[term, term => term] => term"
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  Prod :: "[term, term => term] => term"
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  Trueprop :: "[context, typing] => prop"
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  MT_context :: "context"
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  Context :: "[typing, context] => context"
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  star :: "term"    ("*")
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  box :: "term"    ("[]")
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  app :: "[term, term] => term"    (infixl "^" 20)
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  Has_type :: "[term, term] => typing"
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syntax
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  "\<^const>Cube.Trueprop" :: "[context', typing'] => prop"    ("(_/ |- _)")
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  "_Trueprop1" :: "typing' => prop"    ("(_)")
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  "" :: "id => context'"    ("_")
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  "" :: "var => context'"    ("_")
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  "\<^const>Cube.MT_context" :: "context'"    ("")
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  "\<^const>Cube.Context" :: "[typing', context'] => context'"    ("_ _")
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  "\<^const>Cube.Has_type" :: "[term, term] => typing'"    ("(_:/ _)" [0, 0] 5)
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  "_Lam" :: "[idt, term, term] => term"    ("(3Lam _:_./ _)" [0, 0, 0] 10)
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  "_Pi" :: "[idt, term, term] => term"    ("(3Pi _:_./ _)" [0, 0] 10)
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  "_arrow" :: "[term, term] => term"    (infixr "->" 10)
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translations
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  ("prop") "x:X" == ("prop") "|- x:X"
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  "Lam 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 -> B" => "CONST Prod(A, %_. B)"
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syntax (xsymbols)
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  "\<^const>Cube.Trueprop" :: "[context', typing'] => prop"    ("(_/ \<turnstile> _)")
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  "\<^const>Cube.box" :: "term"    ("\<box>")
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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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print_translation {*
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  [(@{const_syntax Prod}, dependent_tr' (@{syntax_const "_Pi"}, @{syntax_const "_arrow"}))]
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*}
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axioms
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  s_b:          "*: []"
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  strip_s:      "[| A:*;  a:A ==> G |- x:X |] ==> a:A G |- x:X"
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  strip_b:      "[| A:[]; a:A ==> G |- x:X |] ==> a:A G |- x:X"
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  app:          "[| F:Prod(A, B); C:A |] ==> F^C: B(C)"
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  pi_ss:        "[| A:*; !!x. x:A ==> B(x):* |] ==> Prod(A, B):*"
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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)"
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  beta:          "Abs(A, f)^a == f(a)"
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lemmas simple = s_b strip_s strip_b app lam_ss pi_ss
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lemmas rules = simple
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lemma imp_elim:
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  assumes "f:A->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:[]; !!x. x:A ==> B(x):* |] ==> Prod(A,B):*"
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    and lam_bs: "[| A:[]; !!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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lemmas (in L2) rules = simple lam_bs pi_bs
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locale Lomega =
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  assumes
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    pi_bb: "[| A:[]; !!x. x:A ==> B(x):[] |] ==> Prod(A,B):[]"
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    and lam_bb: "[| A:[]; !!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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lemmas (in Lomega) rules = simple lam_bb pi_bb
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locale LP =
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  assumes pi_sb: "[| A:*; !!x. x:A ==> B(x):[] |] ==> Prod(A,B):[]"
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    and lam_sb: "[| A:*; !!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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lemmas (in LP) rules = simple lam_sb pi_sb
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locale LP2 = LP + L2
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lemmas (in LP2) rules = simple lam_bs pi_bs lam_sb pi_sb
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locale Lomega2 = L2 + Lomega
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lemmas (in Lomega2) rules = simple lam_bs pi_bs lam_bb pi_bb
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locale LPomega = LP + Lomega
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lemmas (in LPomega) rules = simple lam_bb pi_bb lam_sb pi_sb
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locale CC = L2 + LP + Lomega
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lemmas (in CC) rules = simple lam_bs pi_bs lam_bb pi_bb lam_sb pi_sb
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end