1. Initial `Definition` keyword. 2. ifelse.

3 files changed, 55 insertions, 15 deletions

diff --git a/spartan/lib/List.thy b/spartan/lib/List.thy
index 1798a23..11b8406 100644
--- a/spartan/lib/List.thy
+++ b/spartan/lib/List.thy
@@ -50,7 +50,7 @@ lemmas
 
 abbreviation "ListRec A C \<equiv> ListInd A (\<lambda>_. C)"
 
-Lemma (derive) ListCase:
+Lemma list_cases [cases]:
   assumes
     "xs: List A" and
     nil_case: "c\<^sub>0: C (nil A)" and
@@ -59,8 +59,6 @@ Lemma (derive) ListCase:
   shows "?List_cases A (\<lambda>xs. C xs) c\<^sub>0 (\<lambda>x xs. f x xs) xs: C xs"
   by (elim xs) (fact nil_case, rule cons_case)
 
-lemmas List_cases [cases] = ListCase[unfolded ListCase_def]
-
 
 section \<open>Notation\<close>
 
@@ -84,7 +82,7 @@ section \<open>Standard functions\<close>
 
 subsection \<open>Head and tail\<close>
 
-Lemma (derive) head:
+Definition head:
   assumes "A: U i" "xs: List A"
   shows "Maybe A"
 proof (cases xs)
@@ -92,7 +90,7 @@ proof (cases xs)
   show "\<And>x. x: A \<Longrightarrow> some x: Maybe A" by intro
 qed
 
-Lemma (derive) tail:
+Definition tail:
   assumes "A: U i" "xs: List A"
   shows "List A"
 proof (cases xs)
@@ -115,7 +113,7 @@ Lemma head_type [typechk]:
 Lemma head_of_cons [comps]:
   assumes "A: U i" "x: A" "xs: List A"
   shows "head (x # xs) \<equiv> some x"
-  unfolding head_def ListCase_def by reduce
+  unfolding head_def by reduce
 
 Lemma tail_type [typechk]:
   assumes "A: U i" "xs: List A"
@@ -125,11 +123,11 @@ Lemma tail_type [typechk]:
 Lemma tail_of_cons [comps]:
   assumes "A: U i" "x: A" "xs: List A"
   shows "tail (x # xs) \<equiv> xs"
-  unfolding tail_def ListCase_def by reduce
+  unfolding tail_def by reduce
 
 subsection \<open>Append\<close>
 
-Lemma (derive) app:
+Definition app:
   assumes "A: U i" "xs: List A" "ys: List A"
   shows "List A"
   apply (elim xs)
@@ -144,7 +142,7 @@ translations "app" \<leftharpoondown> "CONST List.app A"
 
 subsection \<open>Map\<close>
 
-Lemma (derive) map:
+Definition map:
   assumes "A: U i" "B: U i" "f: A \<rightarrow> B" "xs: List A"
   shows "List B"
 proof (elim xs)
@@ -166,7 +164,7 @@ Lemma map_type [typechk]:
 
 subsection \<open>Reverse\<close>
 
-Lemma (derive) rev:
+Definition rev:
   assumes "A: U i" "xs: List A"
   shows "List A"
   apply (elim xs)
diff --git a/spartan/lib/Maybe.thy b/spartan/lib/Maybe.thy
index 1efbb95..6729812 100644
--- a/spartan/lib/Maybe.thy
+++ b/spartan/lib/Maybe.thy
@@ -17,7 +17,7 @@ lemma
   Maybe_some: "a: A \<Longrightarrow> some A a: Maybe A"
   unfolding Maybe_def none_def some_def by typechk+
 
-Lemma (derive) MaybeInd:
+Definition MaybeInd:
   assumes
     "A: U i"
     "\<And>m. m: Maybe A \<Longrightarrow> C m: U i"
diff --git a/spartan/lib/More_Types.thy b/spartan/lib/More_Types.thy
index 1d7abb9..38ba2aa 100644
--- a/spartan/lib/More_Types.thy
+++ b/spartan/lib/More_Types.thy
@@ -1,5 +1,3 @@
-chapter \<open>Some standard types\<close>
-
 theory More_Types
 imports Spartan
 
@@ -96,9 +94,53 @@ Lemma
   Bool_false: "false: Bool"
   unfolding Bool_def true_def false_def by typechk+
 
-lemmas [intros] = BoolF Bool_true Bool_false
+\<comment> \<open>Definitions like these should be handled by a future function package\<close>
+Definition ifelse [rotated 1]:
+  assumes *[unfolded Bool_def true_def false_def]:
+    "\<And>x. x: Bool \<Longrightarrow> C x: U i"
+    "x: Bool"
+    "a: C true"
+    "b: C false"
+  shows "C x"
+  by (elim x) (elim, rule *)+
+
+Lemma if_true:
+  assumes
+    "\<And>x. x: Bool \<Longrightarrow> C x: U i"
+    "a: C true"
+    "b: C false"
+  shows "ifelse C true a b \<equiv> a"
+  unfolding ifelse_def true_def
+  supply assms[unfolded Bool_def true_def false_def]
+  by reduce
+
+Lemma if_false:
+  assumes
+    "\<And>x. x: Bool \<Longrightarrow> C x: U i"
+    "a: C true"
+    "b: C false"
+  shows "ifelse C false a b \<equiv> b"
+  unfolding ifelse_def false_def
+  supply assms[unfolded Bool_def true_def false_def]
+  by reduce
+
+lemmas
+  [intros] = BoolF Bool_true Bool_false and
+  [comps] = if_true if_false and
+  [elims ?x] = ifelse
+lemmas
+  BoolE = ifelse
+
+subsection \<open>Notation\<close>
+
+definition ifelse_i ("if _ then _ else _")
+  where [implicit]: "if x then a else b \<equiv> ifelse ? x a b"
+
+no_translations "if x then a else b" \<leftharpoondown> "CONST ifelse C x a b"
+
+subsection \<open>Logical connectives\<close>
 
-\<comment> \<open>Can define if-then-else etc.\<close>
+definition not ("!_") where "not x \<equiv> ifelse (K Bool) x false true"
 
 
 end