(* Title: Cube/Cube.thy
Author: Tobias Nipkow
*)
section \<open>Barendregt's Lambda-Cube\<close>
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 \<Rightarrow> term] \<Rightarrow> term" and
Prod :: "[term, term \<Rightarrow> term] \<Rightarrow> term" and
Trueprop :: "[context, typing] \<Rightarrow> prop" and
MT_context :: "context" and
Context :: "[typing, context] \<Rightarrow> context" and
star :: "term" ("*") and
box :: "term" ("\<box>") and
app :: "[term, term] \<Rightarrow> term" (infixl "\<cdot>" 20) and
Has_type :: "[term, term] \<Rightarrow> typing"
nonterminal context' and typing'
syntax
"_Trueprop" :: "[context', typing'] \<Rightarrow> prop" ("(_/ \<turnstile> _)")
"_Trueprop1" :: "typing' \<Rightarrow> prop" ("(_)")
"" :: "id \<Rightarrow> context'" ("_")
"" :: "var \<Rightarrow> context'" ("_")
"_MT_context" :: "context'" ("")
"_Context" :: "[typing', context'] \<Rightarrow> context'" ("_ _")
"_Has_type" :: "[term, term] \<Rightarrow> typing'" ("(_:/ _)" [0, 0] 5)
"_Lam" :: "[idt, term, term] \<Rightarrow> term" ("(3\<^bold>\<lambda>_:_./ _)" [0, 0, 0] 10)
"_Pi" :: "[idt, term, term] \<Rightarrow> term" ("(3\<Prod>_:_./ _)" [0, 0] 10)
"_arrow" :: "[term, term] \<Rightarrow> term" (infixr "\<rightarrow>" 10)
translations
"_Trueprop(G, t)" \<rightleftharpoons> "CONST Trueprop(G, t)"
("prop") "x:X" \<rightleftharpoons> ("prop") "\<turnstile> x:X"
"_MT_context" \<rightleftharpoons> "CONST MT_context"
"_Context" \<rightleftharpoons> "CONST Context"
"_Has_type" \<rightleftharpoons> "CONST Has_type"
"\<^bold>\<lambda>x:A. B" \<rightleftharpoons> "CONST Abs(A, \<lambda>x. B)"
"\<Prod>x:A. B" \<rightharpoonup> "CONST Prod(A, \<lambda>x. B)"
"A \<rightarrow> B" \<rightharpoonup> "CONST Prod(A, \<lambda>_. B)"
print_translation \<open>
[(@{const_syntax Prod},
fn _ => Syntax_Trans.dependent_tr' (@{syntax_const "_Pi"}, @{syntax_const "_arrow"}))]
\<close>
axiomatization where
s_b: "*: \<box>" and
strip_s: "\<lbrakk>A:*; a:A \<Longrightarrow> G \<turnstile> x:X\<rbrakk> \<Longrightarrow> a:A G \<turnstile> x:X" and
strip_b: "\<lbrakk>A:\<box>; a:A \<Longrightarrow> G \<turnstile> x:X\<rbrakk> \<Longrightarrow> a:A G \<turnstile> x:X" and
app: "\<lbrakk>F:Prod(A, B); C:A\<rbrakk> \<Longrightarrow> F\<cdot>C: B(C)" and
pi_ss: "\<lbrakk>A:*; \<And>x. x:A \<Longrightarrow> B(x):*\<rbrakk> \<Longrightarrow> Prod(A, B):*" and
lam_ss: "\<lbrakk>A:*; \<And>x. x:A \<Longrightarrow> f(x):B(x); \<And>x. x:A \<Longrightarrow> B(x):* \<rbrakk>
\<Longrightarrow> Abs(A, f) : Prod(A, B)" and
beta: "Abs(A, f)\<cdot>a \<equiv> f(a)"
lemmas [rules] = s_b strip_s strip_b app lam_ss pi_ss
lemma imp_elim:
assumes "f:A\<rightarrow>B" and "a:A" and "f\<cdot>a:B \<Longrightarrow> PROP P"
shows "PROP P" by (rule app assms)+
lemma pi_elim:
assumes "F:Prod(A,B)" and "a:A" and "F\<cdot>a:B(a) \<Longrightarrow> PROP P"
shows "PROP P" by (rule app assms)+
locale L2 =
assumes pi_bs: "\<lbrakk>A:\<box>; \<And>x. x:A \<Longrightarrow> B(x):*\<rbrakk> \<Longrightarrow> Prod(A,B):*"
and lam_bs: "\<lbrakk>A:\<box>; \<And>x. x:A \<Longrightarrow> f(x):B(x); \<And>x. x:A \<Longrightarrow> B(x):*\<rbrakk>
\<Longrightarrow> Abs(A,f) : Prod(A,B)"
begin
lemmas [rules] = lam_bs pi_bs
end
locale Lomega =
assumes
pi_bb: "\<lbrakk>A:\<box>; \<And>x. x:A \<Longrightarrow> B(x):\<box>\<rbrakk> \<Longrightarrow> Prod(A,B):\<box>"
and lam_bb: "\<lbrakk>A:\<box>; \<And>x. x:A \<Longrightarrow> f(x):B(x); \<And>x. x:A \<Longrightarrow> B(x):\<box>\<rbrakk>
\<Longrightarrow> Abs(A,f) : Prod(A,B)"
begin
lemmas [rules] = lam_bb pi_bb
end
locale LP =
assumes pi_sb: "\<lbrakk>A:*; \<And>x. x:A \<Longrightarrow> B(x):\<box>\<rbrakk> \<Longrightarrow> Prod(A,B):\<box>"
and lam_sb: "\<lbrakk>A:*; \<And>x. x:A \<Longrightarrow> f(x):B(x); \<And>x. x:A \<Longrightarrow> B(x):\<box>\<rbrakk>
\<Longrightarrow> 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