--- a/src/Cube/Cube.thy Sat Oct 22 16:44:34 2011 +0200
+++ b/src/Cube/Cube.thy Sat Oct 22 16:57:24 2011 +0200
@@ -21,42 +21,36 @@
MT_context :: "context" and
Context :: "[typing, context] => context" and
star :: "term" ("*") and
- box :: "term" ("[]") and
+ box :: "term" ("\<box>") and
app :: "[term, term] => term" (infixl "^" 20) and
Has_type :: "[term, term] => typing"
-notation (xsymbols)
- box ("\<box>")
-
nonterminal context' and typing'
syntax
- "_Trueprop" :: "[context', typing'] => prop" ("(_/ |- _)")
+ "_Trueprop" :: "[context', typing'] => prop" ("(_/ \<turnstile> _)")
"_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)
+ "_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)
translations
"_Trueprop(G, t)" == "CONST Trueprop(G, t)"
- ("prop") "x:X" == ("prop") "|- x:X"
+ ("prop") "x:X" == ("prop") "\<turnstile> x:X"
"_MT_context" == "CONST MT_context"
"_Context" == "CONST Context"
"_Has_type" == "CONST Has_type"
- "Lam x:A. B" == "CONST Abs(A, %x. B)"
- "Pi x:A. B" => "CONST Prod(A, %x. B)"
- "A -> B" => "CONST Prod(A, %_. B)"
+ "\<Lambda> x:A. B" == "CONST Abs(A, %x. B)"
+ "\<Pi> x:A. B" => "CONST Prod(A, %x. B)"
+ "A \<rightarrow> B" => "CONST Prod(A, %_. B)"
syntax (xsymbols)
- "_Trueprop" :: "[context', typing'] => prop" ("(_/ \<turnstile> _)")
- "_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 {*
[(@{const_syntax Prod},
@@ -64,10 +58,10 @@
*}
axiomatization where
- s_b: "*: []" and
+ s_b: "*: \<box>" 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
+ strip_s: "[| A:*; a:A ==> G \<turnstile> x:X |] ==> a:A G \<turnstile> x:X" and
+ strip_b: "[| A:\<box>; a:A ==> G \<turnstile> x:X |] ==> a:A G \<turnstile> x:X" and
app: "[| F:Prod(A, B); C:A |] ==> F^C: B(C)" and
@@ -82,7 +76,7 @@
lemmas rules = simple
lemma imp_elim:
- assumes "f:A->B" and "a:A" and "f^a:B ==> PROP P"
+ assumes "f:A\<rightarrow>B" and "a:A" and "f^a:B ==> PROP P"
shows "PROP P" by (rule app assms)+
lemma pi_elim:
@@ -91,25 +85,31 @@
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):* |]
+ assumes pi_bs: "[| A:\<box>; !!x. x:A ==> B(x):* |] ==> Prod(A,B):*"
+ and lam_bs: "[| A:\<box>; !!x. x:A ==> f(x):B(x); !!x. x:A ==> B(x):* |]
==> Abs(A,f) : Prod(A,B)"
+begin
-lemmas (in L2) rules = simple lam_bs pi_bs
+lemmas rules = simple 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):[] |]
+ pi_bb: "[| A:\<box>; !!x. x:A ==> B(x):\<box> |] ==> Prod(A,B):\<box>"
+ and lam_bb: "[| A:\<box>; !!x. x:A ==> f(x):B(x); !!x. x:A ==> B(x):\<box> |]
==> Abs(A,f) : Prod(A,B)"
+begin
-lemmas (in Lomega) rules = simple lam_bb pi_bb
+lemmas rules = simple 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):[] |]
+ assumes pi_sb: "[| A:*; !!x. x:A ==> B(x):\<box> |] ==> Prod(A,B):\<box>"
+ and lam_sb: "[| A:*; !!x. x:A ==> f(x):B(x); !!x. x:A ==> B(x):\<box> |]
==> Abs(A,f) : Prod(A,B)"
begin
--- a/src/Cube/Example.thy Sat Oct 22 16:44:34 2011 +0200
+++ b/src/Cube/Example.thy Sat Oct 22 16:57:24 2011 +0200
@@ -30,98 +30,98 @@
subsection {* Simple types *}
-schematic_lemma "A:* |- A->A : ?T"
+schematic_lemma "A:* \<turnstile> A\<rightarrow>A : ?T"
by (depth_solve rules)
-schematic_lemma "A:* |- Lam a:A. a : ?T"
+schematic_lemma "A:* \<turnstile> \<Lambda> a:A. a : ?T"
by (depth_solve rules)
-schematic_lemma "A:* B:* b:B |- Lam x:A. b : ?T"
+schematic_lemma "A:* B:* b:B \<turnstile> \<Lambda> x:A. b : ?T"
by (depth_solve rules)
-schematic_lemma "A:* b:A |- (Lam a:A. a)^b: ?T"
+schematic_lemma "A:* b:A \<turnstile> (\<Lambda> a:A. a)^b: ?T"
by (depth_solve rules)
-schematic_lemma "A:* B:* c:A b:B |- (Lam x:A. b)^ c: ?T"
+schematic_lemma "A:* B:* c:A b:B \<turnstile> (\<Lambda> x:A. b)^ c: ?T"
by (depth_solve rules)
-schematic_lemma "A:* B:* |- Lam a:A. Lam b:B. a : ?T"
+schematic_lemma "A:* B:* \<turnstile> \<Lambda> a:A. \<Lambda> b:B. a : ?T"
by (depth_solve rules)
subsection {* Second-order types *}
-schematic_lemma (in L2) "|- Lam A:*. Lam a:A. a : ?T"
+schematic_lemma (in L2) "\<turnstile> \<Lambda> A:*. \<Lambda> a:A. a : ?T"
by (depth_solve rules)
-schematic_lemma (in L2) "A:* |- (Lam B:*.Lam b:B. b)^A : ?T"
+schematic_lemma (in L2) "A:* \<turnstile> (\<Lambda> B:*.\<Lambda> b:B. b)^A : ?T"
by (depth_solve rules)
-schematic_lemma (in L2) "A:* b:A |- (Lam B:*.Lam b:B. b) ^ A ^ b: ?T"
+schematic_lemma (in L2) "A:* b:A \<turnstile> (\<Lambda> B:*.\<Lambda> b:B. b) ^ A ^ b: ?T"
by (depth_solve rules)
-schematic_lemma (in L2) "|- Lam B:*.Lam a:(Pi A:*.A).a ^ ((Pi A:*.A)->B) ^ a: ?T"
+schematic_lemma (in L2) "\<turnstile> \<Lambda> B:*.\<Lambda> a:(\<Pi> A:*.A).a ^ ((\<Pi> A:*.A)\<rightarrow>B) ^ a: ?T"
by (depth_solve rules)
subsection {* Weakly higher-order propositional logic *}
-schematic_lemma (in Lomega) "|- Lam A:*.A->A : ?T"
+schematic_lemma (in Lomega) "\<turnstile> \<Lambda> A:*.A\<rightarrow>A : ?T"
by (depth_solve rules)
-schematic_lemma (in Lomega) "B:* |- (Lam A:*.A->A) ^ B : ?T"
+schematic_lemma (in Lomega) "B:* \<turnstile> (\<Lambda> A:*.A\<rightarrow>A) ^ B : ?T"
by (depth_solve rules)
-schematic_lemma (in Lomega) "B:* b:B |- (Lam y:B. b): ?T"
+schematic_lemma (in Lomega) "B:* b:B \<turnstile> (\<Lambda> y:B. b): ?T"
by (depth_solve rules)
-schematic_lemma (in Lomega) "A:* F:*->* |- F^(F^A): ?T"
+schematic_lemma (in Lomega) "A:* F:*\<rightarrow>* \<turnstile> F^(F^A): ?T"
by (depth_solve rules)
-schematic_lemma (in Lomega) "A:* |- Lam F:*->*.F^(F^A): ?T"
+schematic_lemma (in Lomega) "A:* \<turnstile> \<Lambda> F:*\<rightarrow>*.F^(F^A): ?T"
by (depth_solve rules)
subsection {* LP *}
-schematic_lemma (in LP) "A:* |- A -> * : ?T"
+schematic_lemma (in LP) "A:* \<turnstile> A \<rightarrow> * : ?T"
by (depth_solve rules)
-schematic_lemma (in LP) "A:* P:A->* a:A |- P^a: ?T"
+schematic_lemma (in LP) "A:* P:A\<rightarrow>* a:A \<turnstile> P^a: ?T"
by (depth_solve rules)
-schematic_lemma (in LP) "A:* P:A->A->* a:A |- Pi a:A. P^a^a: ?T"
+schematic_lemma (in LP) "A:* P:A\<rightarrow>A\<rightarrow>* a:A \<turnstile> \<Pi> a:A. P^a^a: ?T"
by (depth_solve rules)
-schematic_lemma (in LP) "A:* P:A->* Q:A->* |- Pi a:A. P^a -> Q^a: ?T"
+schematic_lemma (in LP) "A:* P:A\<rightarrow>* Q:A\<rightarrow>* \<turnstile> \<Pi> a:A. P^a \<rightarrow> Q^a: ?T"
by (depth_solve rules)
-schematic_lemma (in LP) "A:* P:A->* |- Pi a:A. P^a -> P^a: ?T"
+schematic_lemma (in LP) "A:* P:A\<rightarrow>* \<turnstile> \<Pi> a:A. P^a \<rightarrow> P^a: ?T"
by (depth_solve rules)
-schematic_lemma (in LP) "A:* P:A->* |- Lam a:A. Lam x:P^a. x: ?T"
+schematic_lemma (in LP) "A:* P:A\<rightarrow>* \<turnstile> \<Lambda> a:A. \<Lambda> x:P^a. x: ?T"
by (depth_solve rules)
-schematic_lemma (in LP) "A:* P:A->* Q:* |- (Pi a:A. P^a->Q) -> (Pi a:A. P^a) -> Q : ?T"
+schematic_lemma (in LP) "A:* P:A\<rightarrow>* Q:* \<turnstile> (\<Pi> a:A. P^a\<rightarrow>Q) \<rightarrow> (\<Pi> a:A. P^a) \<rightarrow> Q : ?T"
by (depth_solve rules)
-schematic_lemma (in LP) "A:* P:A->* Q:* a0:A |-
- Lam x:Pi a:A. P^a->Q. Lam y:Pi a:A. P^a. x^a0^(y^a0): ?T"
+schematic_lemma (in LP) "A:* P:A\<rightarrow>* Q:* a0:A \<turnstile>
+ \<Lambda> x:\<Pi> a:A. P^a\<rightarrow>Q. \<Lambda> y:\<Pi> a:A. P^a. x^a0^(y^a0): ?T"
by (depth_solve rules)
subsection {* Omega-order types *}
-schematic_lemma (in L2) "A:* B:* |- Pi C:*.(A->B->C)->C : ?T"
+schematic_lemma (in L2) "A:* B:* \<turnstile> \<Pi> C:*.(A\<rightarrow>B\<rightarrow>C)\<rightarrow>C : ?T"
by (depth_solve rules)
-schematic_lemma (in Lomega2) "|- Lam A:*.Lam B:*.Pi C:*.(A->B->C)->C : ?T"
+schematic_lemma (in Lomega2) "\<turnstile> \<Lambda> A:*.\<Lambda> B:*.\<Pi> C:*.(A\<rightarrow>B\<rightarrow>C)\<rightarrow>C : ?T"
by (depth_solve rules)
-schematic_lemma (in Lomega2) "|- Lam A:*.Lam B:*.Lam x:A. Lam y:B. x : ?T"
+schematic_lemma (in Lomega2) "\<turnstile> \<Lambda> A:*.\<Lambda> B:*.\<Lambda> x:A. \<Lambda> y:B. x : ?T"
by (depth_solve rules)
-schematic_lemma (in Lomega2) "A:* B:* |- ?p : (A->B) -> ((B->Pi P:*.P)->(A->Pi P:*.P))"
+schematic_lemma (in Lomega2) "A:* B:* \<turnstile> ?p : (A\<rightarrow>B) \<rightarrow> ((B\<rightarrow>\<Pi> P:*.P)\<rightarrow>(A\<rightarrow>\<Pi> P:*.P))"
apply (strip_asms rules)
apply (rule lam_ss)
apply (depth_solve1 rules)
@@ -145,15 +145,15 @@
subsection {* Second-order Predicate Logic *}
-schematic_lemma (in LP2) "A:* P:A->* |- Lam a:A. P^a->(Pi A:*.A) : ?T"
+schematic_lemma (in LP2) "A:* P:A\<rightarrow>* \<turnstile> \<Lambda> a:A. P^a\<rightarrow>(\<Pi> A:*.A) : ?T"
by (depth_solve rules)
-schematic_lemma (in LP2) "A:* P:A->A->* |-
- (Pi a:A. Pi b:A. P^a^b->P^b^a->Pi P:*.P) -> Pi a:A. P^a^a->Pi P:*.P : ?T"
+schematic_lemma (in LP2) "A:* P:A\<rightarrow>A\<rightarrow>* \<turnstile>
+ (\<Pi> a:A. \<Pi> b:A. P^a^b\<rightarrow>P^b^a\<rightarrow>\<Pi> P:*.P) \<rightarrow> \<Pi> a:A. P^a^a\<rightarrow>\<Pi> P:*.P : ?T"
by (depth_solve rules)
-schematic_lemma (in LP2) "A:* P:A->A->* |-
- ?p: (Pi a:A. Pi b:A. P^a^b->P^b^a->Pi P:*.P) -> Pi a:A. P^a^a->Pi P:*.P"
+schematic_lemma (in LP2) "A:* P:A\<rightarrow>A\<rightarrow>* \<turnstile>
+ ?p: (\<Pi> a:A. \<Pi> b:A. P^a^b\<rightarrow>P^b^a\<rightarrow>\<Pi> P:*.P) \<rightarrow> \<Pi> a:A. P^a^a\<rightarrow>\<Pi> P:*.P"
-- {* Antisymmetry implies irreflexivity: *}
apply (strip_asms rules)
apply (rule lam_ss)
@@ -174,22 +174,22 @@
subsection {* LPomega *}
-schematic_lemma (in LPomega) "A:* |- Lam P:A->A->*.Lam a:A. P^a^a : ?T"
+schematic_lemma (in LPomega) "A:* \<turnstile> \<Lambda> P:A\<rightarrow>A\<rightarrow>*.\<Lambda> a:A. P^a^a : ?T"
by (depth_solve rules)
-schematic_lemma (in LPomega) "|- Lam A:*.Lam P:A->A->*.Lam a:A. P^a^a : ?T"
+schematic_lemma (in LPomega) "\<turnstile> \<Lambda> A:*.\<Lambda> P:A\<rightarrow>A\<rightarrow>*.\<Lambda> a:A. P^a^a : ?T"
by (depth_solve rules)
subsection {* Constructions *}
-schematic_lemma (in CC) "|- Lam A:*.Lam P:A->*.Lam a:A. P^a->Pi P:*.P: ?T"
+schematic_lemma (in CC) "\<turnstile> \<Lambda> A:*.\<Lambda> P:A\<rightarrow>*.\<Lambda> a:A. P^a\<rightarrow>\<Pi> P:*.P: ?T"
by (depth_solve rules)
-schematic_lemma (in CC) "|- Lam A:*.Lam P:A->*.Pi a:A. P^a: ?T"
+schematic_lemma (in CC) "\<turnstile> \<Lambda> A:*.\<Lambda> P:A\<rightarrow>*.\<Pi> a:A. P^a: ?T"
by (depth_solve rules)
-schematic_lemma (in CC) "A:* P:A->* a:A |- ?p : (Pi a:A. P^a)->P^a"
+schematic_lemma (in CC) "A:* P:A\<rightarrow>* a:A \<turnstile> ?p : (\<Pi> a:A. P^a)\<rightarrow>P^a"
apply (strip_asms rules)
apply (rule lam_ss)
apply (depth_solve1 rules)
@@ -201,23 +201,23 @@
subsection {* Some random examples *}
-schematic_lemma (in LP2) "A:* c:A f:A->A |-
- Lam a:A. Pi P:A->*.P^c -> (Pi x:A. P^x->P^(f^x)) -> P^a : ?T"
+schematic_lemma (in LP2) "A:* c:A f:A\<rightarrow>A \<turnstile>
+ \<Lambda> a:A. \<Pi> P:A\<rightarrow>*.P^c \<rightarrow> (\<Pi> x:A. P^x\<rightarrow>P^(f^x)) \<rightarrow> P^a : ?T"
by (depth_solve rules)
-schematic_lemma (in CC) "Lam A:*.Lam c:A. Lam f:A->A.
- Lam a:A. Pi P:A->*.P^c -> (Pi x:A. P^x->P^(f^x)) -> P^a : ?T"
+schematic_lemma (in CC) "\<Lambda> A:*.\<Lambda> c:A. \<Lambda> f:A\<rightarrow>A.
+ \<Lambda> a:A. \<Pi> P:A\<rightarrow>*.P^c \<rightarrow> (\<Pi> x:A. P^x\<rightarrow>P^(f^x)) \<rightarrow> P^a : ?T"
by (depth_solve rules)
schematic_lemma (in LP2)
- "A:* a:A b:A |- ?p: (Pi P:A->*.P^a->P^b) -> (Pi P:A->*.P^b->P^a)"
+ "A:* a:A b:A \<turnstile> ?p: (\<Pi> P:A\<rightarrow>*.P^a\<rightarrow>P^b) \<rightarrow> (\<Pi> P:A\<rightarrow>*.P^b\<rightarrow>P^a)"
-- {* Symmetry of Leibnitz equality *}
apply (strip_asms rules)
apply (rule lam_ss)
apply (depth_solve1 rules)
prefer 2
apply (depth_solve1 rules)
- apply (erule_tac a = "Lam x:A. Pi Q:A->*.Q^x->Q^a" in pi_elim)
+ apply (erule_tac a = "\<Lambda> x:A. \<Pi> Q:A\<rightarrow>*.Q^x\<rightarrow>Q^a" in pi_elim)
apply (depth_solve1 rules)
apply (unfold beta)
apply (erule imp_elim)