diff options
Diffstat (limited to 'spartan/lib')
| -rw-r--r-- | spartan/lib/List.thy | 12 | ||||
| -rw-r--r-- | spartan/lib/Maybe.thy | 13 | ||||
| -rw-r--r-- | spartan/lib/More_Types.thy | 12 |
3 files changed, 17 insertions, 20 deletions
diff --git a/spartan/lib/List.thy b/spartan/lib/List.thy index 11b8406..a755859 100644 --- a/spartan/lib/List.thy +++ b/spartan/lib/List.thy @@ -26,13 +26,13 @@ where c\<^sub>0: C (nil A); \<And>x xs rec. \<lbrakk>x: A; xs: List A; rec: C xs\<rbrakk> \<Longrightarrow> f x xs rec: C (cons A x xs); \<And>xs. xs: List A \<Longrightarrow> C xs: U i - \<rbrakk> \<Longrightarrow> ListInd A (\<lambda>xs. C xs) c\<^sub>0 (\<lambda>x xs rec. f x xs rec) xs: C xs" and + \<rbrakk> \<Longrightarrow> ListInd A (fn xs. C xs) c\<^sub>0 (fn x xs rec. f x xs rec) xs: C xs" and List_comp_nil: "\<lbrakk> c\<^sub>0: C (nil A); \<And>x xs rec. \<lbrakk>x: A; xs: List A; rec: C xs\<rbrakk> \<Longrightarrow> f x xs rec: C (cons A x xs); \<And>xs. xs: List A \<Longrightarrow> C xs: U i - \<rbrakk> \<Longrightarrow> ListInd A (\<lambda>xs. C xs) c\<^sub>0 (\<lambda>x xs rec. f x xs rec) (nil A) \<equiv> c\<^sub>0" and + \<rbrakk> \<Longrightarrow> ListInd A (fn xs. C xs) c\<^sub>0 (fn x xs rec. f x xs rec) (nil A) \<equiv> c\<^sub>0" and List_comp_cons: "\<lbrakk> xs: List A; @@ -40,15 +40,15 @@ where \<And>x xs rec. \<lbrakk>x: A; xs: List A; rec: C xs\<rbrakk> \<Longrightarrow> f x xs rec: C (cons A x xs); \<And>xs. xs: List A \<Longrightarrow> C xs: U i \<rbrakk> \<Longrightarrow> - ListInd A (\<lambda>xs. C xs) c\<^sub>0 (\<lambda>x xs rec. f x xs rec) (cons A x xs) \<equiv> - f x xs (ListInd A (\<lambda>xs. C xs) c\<^sub>0 (\<lambda>x xs rec. f x xs rec) xs)" + ListInd A (fn xs. C xs) c\<^sub>0 (fn x xs rec. f x xs rec) (cons A x xs) \<equiv> + f x xs (ListInd A (fn xs. C xs) c\<^sub>0 (fn x xs rec. f x xs rec) xs)" lemmas [intros] = ListF List_nil List_cons and [elims "?xs"] = ListE and [comps] = List_comp_nil List_comp_cons -abbreviation "ListRec A C \<equiv> ListInd A (\<lambda>_. C)" +abbreviation "ListRec A C \<equiv> ListInd A (fn _. C)" Lemma list_cases [cases]: assumes @@ -56,7 +56,7 @@ Lemma list_cases [cases]: nil_case: "c\<^sub>0: C (nil A)" and cons_case: "\<And>x xs. \<lbrakk>x: A; xs: List A\<rbrakk> \<Longrightarrow> f x xs: C (cons A x xs)" and "\<And>xs. xs: List A \<Longrightarrow> C xs: U i" - shows "?List_cases A (\<lambda>xs. C xs) c\<^sub>0 (\<lambda>x xs. f x xs) xs: C xs" + shows "C xs" by (elim xs) (fact nil_case, rule cons_case) diff --git a/spartan/lib/Maybe.thy b/spartan/lib/Maybe.thy index 6729812..d821920 100644 --- a/spartan/lib/Maybe.thy +++ b/spartan/lib/Maybe.thy @@ -24,7 +24,7 @@ Definition MaybeInd: "c\<^sub>0: C (none A)" "\<And>a. a: A \<Longrightarrow> f a: C (some A a)" "m: Maybe A" - shows "?MaybeInd A (\<lambda>m. C m) c\<^sub>0 (\<lambda>a. f a) m: C m" + shows "C m" supply assms[unfolded Maybe_def none_def some_def] apply (elim m) \<guillemotright> unfolding Maybe_def . @@ -38,7 +38,7 @@ Lemma Maybe_comp_none: "c\<^sub>0: C (none A)" "\<And>a. a: A \<Longrightarrow> f a: C (some A a)" "\<And>m. m: Maybe A \<Longrightarrow> C m: U i" - shows "MaybeInd A (\<lambda>m. C m) c\<^sub>0 (\<lambda>a. f a) (none A) \<equiv> c\<^sub>0" + shows "MaybeInd A C c\<^sub>0 f (none A) \<equiv> c\<^sub>0" supply assms[unfolded Maybe_def some_def none_def] unfolding MaybeInd_def none_def by reduce @@ -49,7 +49,7 @@ Lemma Maybe_comp_some: "c\<^sub>0: C (none A)" "\<And>a. a: A \<Longrightarrow> f a: C (some A a)" "\<And>m. m: Maybe A \<Longrightarrow> C m: U i" - shows "MaybeInd A (\<lambda>m. C m) c\<^sub>0 (\<lambda>a. f a) (some A a) \<equiv> f a" + shows "MaybeInd A C c\<^sub>0 f (some A a) \<equiv> f a" supply assms[unfolded Maybe_def some_def none_def] unfolding MaybeInd_def some_def by (reduce add: Maybe_def) @@ -62,11 +62,8 @@ lemmas abbreviation "MaybeRec A C \<equiv> MaybeInd A (K C)" -definition none_i ("none") - where [implicit]: "none \<equiv> Maybe.none ?" - -definition some_i ("some") - where [implicit]: "some a \<equiv> Maybe.some ? a" +definition none_i ("none") where [implicit]: "none \<equiv> Maybe.none ?" +definition some_i ("some") where [implicit]: "some a \<equiv> Maybe.some ? a" translations "none" \<leftharpoondown> "CONST Maybe.none A" diff --git a/spartan/lib/More_Types.thy b/spartan/lib/More_Types.thy index 38ba2aa..0d7096f 100644 --- a/spartan/lib/More_Types.thy +++ b/spartan/lib/More_Types.thy @@ -25,21 +25,21 @@ axiomatization where \<And>s. s: A \<or> B \<Longrightarrow> C s: U i; \<And>a. a: A \<Longrightarrow> c a: C (inl A B a); \<And>b. b: B \<Longrightarrow> d b: C (inr A B b) - \<rbrakk> \<Longrightarrow> SumInd A B (\<lambda>s. C s) (\<lambda>a. c a) (\<lambda>b. d b) s: C s" and + \<rbrakk> \<Longrightarrow> SumInd A B (fn s. C s) (fn a. c a) (fn b. d b) s: C s" and Sum_comp_inl: "\<lbrakk> a: A; \<And>s. s: A \<or> B \<Longrightarrow> C s: U i; \<And>a. a: A \<Longrightarrow> c a: C (inl A B a); \<And>b. b: B \<Longrightarrow> d b: C (inr A B b) - \<rbrakk> \<Longrightarrow> SumInd A B (\<lambda>s. C s) (\<lambda>a. c a) (\<lambda>b. d b) (inl A B a) \<equiv> c a" and + \<rbrakk> \<Longrightarrow> SumInd A B (fn s. C s) (fn a. c a) (fn b. d b) (inl A B a) \<equiv> c a" and Sum_comp_inr: "\<lbrakk> b: B; \<And>s. s: A \<or> B \<Longrightarrow> C s: U i; \<And>a. a: A \<Longrightarrow> c a: C (inl A B a); \<And>b. b: B \<Longrightarrow> d b: C (inr A B b) - \<rbrakk> \<Longrightarrow> SumInd A B (\<lambda>s. C s) (\<lambda>a. c a) (\<lambda>b. d b) (inr A B b) \<equiv> d b" + \<rbrakk> \<Longrightarrow> SumInd A B (fn s. C s) (fn a. c a) (fn b. d b) (inr A B b) \<equiv> d b" lemmas [intros] = SumF Sum_inl Sum_inr and @@ -67,13 +67,13 @@ axiomatization where TopI: "tt: \<top>" and - TopE: "\<lbrakk>a: \<top>; \<And>x. x: \<top> \<Longrightarrow> C x: U i; c: C tt\<rbrakk> \<Longrightarrow> TopInd (\<lambda>x. C x) c a: C a" and + TopE: "\<lbrakk>a: \<top>; \<And>x. x: \<top> \<Longrightarrow> C x: U i; c: C tt\<rbrakk> \<Longrightarrow> TopInd (fn x. C x) c a: C a" and - Top_comp: "\<lbrakk>\<And>x. x: \<top> \<Longrightarrow> C x: U i; c: C tt\<rbrakk> \<Longrightarrow> TopInd (\<lambda>x. C x) c tt \<equiv> c" + Top_comp: "\<lbrakk>\<And>x. x: \<top> \<Longrightarrow> C x: U i; c: C tt\<rbrakk> \<Longrightarrow> TopInd (fn x. C x) c tt \<equiv> c" and BotF: "\<bottom>: U i" and - BotE: "\<lbrakk>x: \<bottom>; \<And>x. x: \<bottom> \<Longrightarrow> C x: U i\<rbrakk> \<Longrightarrow> BotInd (\<lambda>x. C x) x: C x" + BotE: "\<lbrakk>x: \<bottom>; \<And>x. x: \<bottom> \<Longrightarrow> C x: U i\<rbrakk> \<Longrightarrow> BotInd (fn x. C x) x: C x" lemmas [intros] = TopF TopI BotF and |