src/Cube/Example.thy

 *}
 
 
 subsection {* Simple types *}
 
-schematic_lemma "A:* |- A->A : ?T"
-  by (depth_solve rules)
-
-schematic_lemma "A:* |- Lam a:A. a : ?T"
-  by (depth_solve rules)
-
-schematic_lemma "A:* B:* b:B |- Lam x:A. b : ?T"
-  by (depth_solve rules)
-
-schematic_lemma "A:* b:A |- (Lam a:A. a)^b: ?T"
-  by (depth_solve rules)
-
-schematic_lemma "A:* B:* c:A b:B |- (Lam x:A. b)^ c: ?T"
-  by (depth_solve rules)
-
-schematic_lemma "A:* B:* |- Lam a:A. Lam b:B. a : ?T"
+schematic_lemma "A:* \<turnstile> A\<rightarrow>A : ?T"
+  by (depth_solve rules)
+
+schematic_lemma "A:* \<turnstile> \<Lambda> a:A. a : ?T"
+  by (depth_solve rules)
+
+schematic_lemma "A:* B:* b:B \<turnstile> \<Lambda> x:A. b : ?T"
+  by (depth_solve rules)
+
+schematic_lemma "A:* b:A \<turnstile> (\<Lambda> a:A. a)^b: ?T"
+  by (depth_solve rules)
+
+schematic_lemma "A:* B:* c:A b:B \<turnstile> (\<Lambda> x:A. b)^ c: ?T"
+  by (depth_solve rules)
+
+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"
-  by (depth_solve rules)
-
-schematic_lemma (in L2) "A:* |- (Lam B:*.Lam 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"
-  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> A:*. \<Lambda> a:A. a : ?T"
+  by (depth_solve rules)
+
+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 \<turnstile> (\<Lambda> B:*.\<Lambda> b:B. b) ^ A ^ b: ?T"
+  by (depth_solve rules)
+
+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"
-  by (depth_solve rules)
-
-schematic_lemma (in Lomega) "B:* |- (Lam A:*.A->A) ^ B : ?T"
-  by (depth_solve rules)
-
-schematic_lemma (in Lomega) "B:* b:B |- (Lam y:B. b): ?T"
-  by (depth_solve rules)
-
-schematic_lemma (in Lomega) "A:* F:*->* |- F^(F^A): ?T"
-  by (depth_solve rules)
-
-schematic_lemma (in Lomega) "A:* |- Lam F:*->*.F^(F^A): ?T"
+schematic_lemma (in Lomega) "\<turnstile> \<Lambda> A:*.A\<rightarrow>A : ?T"
+  by (depth_solve rules)
+
+schematic_lemma (in Lomega) "B:* \<turnstile> (\<Lambda> A:*.A\<rightarrow>A) ^ B : ?T"
+  by (depth_solve rules)
+
+schematic_lemma (in Lomega) "B:* b:B \<turnstile> (\<Lambda> y:B. b): ?T"
+  by (depth_solve rules)
+
+schematic_lemma (in Lomega) "A:* F:*\<rightarrow>* \<turnstile> F^(F^A): ?T"
+  by (depth_solve rules)
+
+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"
-  by (depth_solve rules)
-
-schematic_lemma (in LP) "A:* P:A->* a:A |- P^a: ?T"
-  by (depth_solve rules)
-
-schematic_lemma (in LP) "A:* P:A->A->* a:A |- 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"
-  by (depth_solve rules)
-
-schematic_lemma (in LP) "A:* P:A->* |- Pi a:A. P^a -> P^a: ?T"
-  by (depth_solve rules)
-
-schematic_lemma (in LP) "A:* P:A->* |- Lam a:A. Lam 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"
-  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:* \<turnstile> A \<rightarrow> * : ?T"
+  by (depth_solve rules)
+
+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\<rightarrow>A\<rightarrow>* a:A \<turnstile> \<Pi> a:A. P^a^a: ?T"
+  by (depth_solve rules)
+
+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\<rightarrow>* \<turnstile> \<Pi> a:A. P^a \<rightarrow> P^a: ?T"
+  by (depth_solve rules)
+
+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\<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\<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"
-  by (depth_solve rules)
-
-schematic_lemma (in Lomega2) "|- Lam A:*.Lam B:*.Pi C:*.(A->B->C)->C : ?T"
-  by (depth_solve rules)
-
-schematic_lemma (in Lomega2) "|- Lam A:*.Lam B:*.Lam x:A. Lam 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 L2) "A:* B:* \<turnstile> \<Pi> C:*.(A\<rightarrow>B\<rightarrow>C)\<rightarrow>C : ?T"
+  by (depth_solve rules)
+
+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) "\<turnstile> \<Lambda> A:*.\<Lambda> B:*.\<Lambda> x:A. \<Lambda> y:B. x : ?T"
+  by (depth_solve rules)
+
+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)
    prefer 2
    apply (depth_solve1 rules)

   done
 
 
 subsection {* Second-order Predicate Logic *}
 
-schematic_lemma (in LP2) "A:* P:A->* |- Lam a:A. P^a->(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"
-  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>* \<turnstile> \<Lambda> a:A. P^a\<rightarrow>(\<Pi> A:*.A) : ?T"
+  by (depth_solve rules)
+
+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\<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)
     apply (depth_solve1 rules)
    prefer 2

   done
 
 
 subsection {* LPomega *}
 
-schematic_lemma (in LPomega) "A:* |- Lam P:A->A->*.Lam 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) "A:* \<turnstile> \<Lambda> P:A\<rightarrow>A\<rightarrow>*.\<Lambda> a:A. P^a^a : ?T"
+  by (depth_solve rules)
+
+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"
-  by (depth_solve rules)
-
-schematic_lemma (in CC) "|- Lam A:*.Lam P:A->*.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) "\<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) "\<turnstile> \<Lambda> A:*.\<Lambda> P:A\<rightarrow>*.\<Pi> a:A. P^a: ?T"
+  by (depth_solve rules)
+
+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)
    prefer 2
    apply (depth_solve1 rules)

   done
 
 
 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"
-  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 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) "\<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)
    apply (rule lam_bs)
      apply (depth_solve1 rules)