header {* Barendregt's Lambda-Cube *}
theory Cube
imports Pure
begin
typedecl "term"
typedecl "context"
typedecl typing
nonterminals
context_ typing_
consts
Abs :: "[term, term => term] => term"
Prod :: "[term, term => term] => term"
Trueprop :: "[context, typing] => prop"
MT_context :: "context"
Context :: "[typing, context] => context"
star :: "term" ("*")
box :: "term" ("[]")
app :: "[term, term] => term" (infixl "^" 20)
Has_type :: "[term, term] => typing"
syntax
Trueprop :: "[context_, typing_] => prop" ("(_/ |- _)")
Trueprop1 :: "typing_ => prop" ("(_)")
"" :: "id => context_" ("_")
"" :: "var => context_" ("_")
MT_context :: "context_" ("")
Context :: "[typing_, context_] => context_" ("_ _")
Has_type :: "[term, term] => typing_" ("(_:/ _)" [0, 0] 5)
Lam :: "[idt, term, term] => term" ("(3Lam _:_./ _)" [0, 0, 0] 10)
Pi :: "[idt, term, term] => term" ("(3Pi _:_./ _)" [0, 0] 10)
arrow :: "[term, term] => term" (infixr "->" 10)
translations
("prop") "x:X" == ("prop") "|- x:X"
"Lam x:A. B" == "Abs(A, %x. B)"
"Pi x:A. B" => "Prod(A, %x. B)"
"A -> B" => "Prod(A, %_. B)"
syntax (xsymbols)
Trueprop :: "[context_, typing_] => prop" ("(_/ \<turnstile> _)")
box :: "term" ("\<box>")
Lam :: "[idt, term, term] => term" ("(3\<Lambda> _:_./ _)" [0, 0, 0] 10)
Pi :: "[idt, term, term] => term" ("(3\<Pi> _:_./ _)" [0, 0] 10)
arrow :: "[term, term] => term" (infixr "\<rightarrow>" 10)
print_translation {* [("Prod", dependent_tr' ("Pi", "arrow"))] *}
axioms
s_b: "*: []"
strip_s: "[| A:*; a:A ==> G |- x:X |] ==> a:A G |- x:X"
strip_b: "[| A:[]; a:A ==> G |- x:X |] ==> a:A G |- x:X"
app: "[| F:Prod(A, B); C:A |] ==> F^C: B(C)"
pi_ss: "[| A:*; !!x. x:A ==> B(x):* |] ==> Prod(A, B):*"
lam_ss: "[| A:*; !!x. x:A ==> f(x):B(x); !!x. x:A ==> B(x):* |]
==> Abs(A, f) : Prod(A, B)"
beta: "Abs(A, f)^a == f(a)"
lemmas simple = s_b strip_s strip_b app lam_ss pi_ss
lemmas rules = simple
lemma imp_elim:
assumes "f:A->B" and "a:A" and "f^a:B ==> PROP P"
shows "PROP P" by (rule app prems)+
lemma pi_elim:
assumes "F:Prod(A,B)" and "a:A" and "F^a:B(a) ==> PROP P"
shows "PROP P" by (rule app prems)+
locale L2 =
assumes pi_bs: "[| A:[]; !!x. x:A ==> B(x):* |] ==> Prod(A,B):*"
and lam_bs: "[| A:[]; !!x. x:A ==> f(x):B(x); !!x. x:A ==> B(x):* |]
==> Abs(A,f) : Prod(A,B)"
lemmas (in L2) rules = simple lam_bs pi_bs
locale Lomega =
assumes
pi_bb: "[| A:[]; !!x. x:A ==> B(x):[] |] ==> Prod(A,B):[]"
and lam_bb: "[| A:[]; !!x. x:A ==> f(x):B(x); !!x. x:A ==> B(x):[] |]
==> Abs(A,f) : Prod(A,B)"
lemmas (in Lomega) rules = simple lam_bb pi_bb
locale LP =
assumes pi_sb: "[| A:*; !!x. x:A ==> B(x):[] |] ==> Prod(A,B):[]"
and lam_sb: "[| A:*; !!x. x:A ==> f(x):B(x); !!x. x:A ==> B(x):[] |]
==> Abs(A,f) : Prod(A,B)"
lemmas (in LP) rules = simple lam_sb pi_sb
locale LP2 = LP + L2
lemmas (in LP2) rules = simple lam_bs pi_bs lam_sb pi_sb
locale Lomega2 = L2 + Lomega
lemmas (in Lomega2) rules = simple lam_bs pi_bs lam_bb pi_bb
locale LPomega = LP + Lomega
lemmas (in LPomega) rules = simple lam_bb pi_bb lam_sb pi_sb
locale CC = L2 + LP + Lomega
lemmas (in CC) rules = simple lam_bs pi_bs lam_bb pi_bb lam_sb pi_sb
end