src/Cube/Cube.thy
author wenzelm
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(*  Title:      Cube/Cube.thy
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    ID:         $Id$
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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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typedecl "term"
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typedecl "context"
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typedecl typing
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nonterminals
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  context_ 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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  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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  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"          ("(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"   == "Abs(A, %x. B)"
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  "Pi x:A. B"    => "Prod(A, %x. B)"
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  "A -> B"       => "Prod(A, %_. B)"
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syntax (xsymbols)
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  Trueprop      :: "[context_, typing_] => prop"        ("(_/ \<turnstile> _)")
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  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 {* [("Prod", dependent_tr' ("Pi", "arrow"))] *}
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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 prems)+
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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 prems)+
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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