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-rw-r--r-- | spartan/lib/More_Types.thy | 104 |
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diff --git a/spartan/lib/More_Types.thy b/spartan/lib/More_Types.thy new file mode 100644 index 0000000..1d7abb9 --- /dev/null +++ b/spartan/lib/More_Types.thy @@ -0,0 +1,104 @@ +chapter \<open>Some standard types\<close> + +theory More_Types +imports Spartan + +begin + +section \<open>Sum type\<close> + +axiomatization + Sum :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> and + inl :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> and + inr :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> and + SumInd :: \<open>o \<Rightarrow> o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o\<close> + +notation Sum (infixl "\<or>" 50) + +axiomatization where + SumF: "\<lbrakk>A: U i; B: U i\<rbrakk> \<Longrightarrow> A \<or> B: U i" and + + Sum_inl: "\<lbrakk>a: A; B: U i\<rbrakk> \<Longrightarrow> inl A B a: A \<or> B" and + + Sum_inr: "\<lbrakk>b: B; A: U i\<rbrakk> \<Longrightarrow> inr A B b: A \<or> B" and + + SumE: "\<lbrakk> + s: A \<or> 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) 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 + + 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" + +lemmas + [intros] = SumF Sum_inl Sum_inr and + [elims ?s] = SumE and + [comps] = Sum_comp_inl Sum_comp_inr + +method left = rule Sum_inl +method right = rule Sum_inr + + +section \<open>Empty and unit types\<close> + +axiomatization + Top :: \<open>o\<close> and + tt :: \<open>o\<close> and + TopInd :: \<open>(o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> +and + Bot :: \<open>o\<close> and + BotInd :: \<open>(o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o\<close> + +notation Top ("\<top>") and Bot ("\<bottom>") + +axiomatization where + TopF: "\<top>: U i" and + + 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 + + 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" +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" + +lemmas + [intros] = TopF TopI BotF and + [elims ?a] = TopE and + [elims ?x] = BotE and + [comps] = Top_comp + + +section \<open>Booleans\<close> + +definition "Bool \<equiv> \<top> \<or> \<top>" +definition "true \<equiv> inl \<top> \<top> tt" +definition "false \<equiv> inr \<top> \<top> tt" + +Lemma + BoolF: "Bool: U i" and + Bool_true: "true: Bool" and + Bool_false: "false: Bool" + unfolding Bool_def true_def false_def by typechk+ + +lemmas [intros] = BoolF Bool_true Bool_false + +\<comment> \<open>Can define if-then-else etc.\<close> + + +end |