Theory Cube

(*  Title:      Cube/Cube.thy
    Author:     Tobias Nipkow
*)

section Barendregt's Lambda-Cube

theory Cube
imports Pure
begin

setup Pure_Thy.old_appl_syntax_setup

named_theorems rules "Cube inference rules"

typedecl "term"
typedecl "context"
typedecl typing

axiomatization
  Abs :: "[term, term  term]  term" and
  Prod :: "[term, term  term]  term" and
  Trueprop :: "[context, typing]  prop" and
  MT_context :: "context" and
  Context :: "[typing, context]  context" and
  star :: "term"  ("*") and
  box :: "term"  ("") and
  app :: "[term, term]  term"  (infixl "" 20) and
  Has_type :: "[term, term]  typing"

nonterminal context' and 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"  ("(3λ_:_./ _)" [0, 0, 0] 10)
  "_Pi" :: "[idt, term, term]  term"  ("(3_:_./ _)" [0, 0] 10)
  "_arrow" :: "[term, term]  term"  (infixr "" 10)
translations
  "_Trueprop(G, t)"  "CONST Trueprop(G, t)"
  ("prop") "x:X"  ("prop") " x:X"
  "_MT_context"  "CONST MT_context"
  "_Context"  "CONST Context"
  "_Has_type"  "CONST Has_type"
  "λx:A. B"  "CONST Abs(A, λx. B)"
  "x:A. B"  "CONST Prod(A, λx. B)"
  "A  B"  "CONST Prod(A, λ_. B)"
print_translation 
  [(const_syntaxProd,
    fn _ => Syntax_Trans.dependent_tr' (syntax_const‹_Pi›, syntax_const‹_arrow›))]


axiomatization where
  s_b: "*: "  and

  strip_s: "A:*; a:A  G  x:X  a:A G  x:X" and
  strip_b: "A:; a:A  G  x:X  a:A G  x:X" and

  app: "F:Prod(A, B); C:A  FC: B(C)" and

  pi_ss: "A:*; x. x:A  B(x):*  Prod(A, B):*" and

  lam_ss: "A:*; x. x:A  f(x):B(x); x. x:A  B(x):* 
             Abs(A, f) : Prod(A, B)" and

  beta: "Abs(A, f)a  f(a)"

lemmas [rules] = s_b strip_s strip_b app lam_ss pi_ss

lemma imp_elim:
  assumes "f:AB" and "a:A" and "fa:B  PROP P"
  shows "PROP P" by (rule app assms)+

lemma pi_elim:
  assumes "F:Prod(A,B)" and "a:A" and "Fa:B(a)  PROP P"
  shows "PROP P" by (rule app assms)+


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)"
begin

lemmas [rules] = lam_bs pi_bs

end


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)"
begin

lemmas [rules] = lam_bb pi_bb

end


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)"
begin

lemmas [rules] = lam_sb pi_sb

end


locale LP2 = LP + L2
begin

lemmas [rules] = lam_bs pi_bs lam_sb pi_sb

end


locale Lomega2 = L2 + Lomega
begin

lemmas [rules] = lam_bs pi_bs lam_bb pi_bb

end


locale LPomega = LP + Lomega
begin

lemmas [rules] = lam_bb pi_bb lam_sb pi_sb

end


locale CC = L2 + LP + Lomega
begin

lemmas [rules] = lam_bs pi_bs lam_bb pi_bb lam_sb pi_sb

end

end