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Diffstat (limited to 'spartan/lib/Prelude.thy')
-rw-r--r-- | spartan/lib/Prelude.thy | 151 |
1 files changed, 0 insertions, 151 deletions
diff --git a/spartan/lib/Prelude.thy b/spartan/lib/Prelude.thy deleted file mode 100644 index 56adbfc..0000000 --- a/spartan/lib/Prelude.thy +++ /dev/null @@ -1,151 +0,0 @@ -theory Prelude -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>B: U i; a: A\<rbrakk> \<Longrightarrow> inl A B a: A \<or> B" and - - Sum_inr: "\<lbrakk>A: U i; b: B\<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 (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 (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 (fn s. C s) (fn a. c a) (fn b. d b) (inr A B b) \<equiv> d b" - -lemmas - [form] = SumF and - [intr] = Sum_inl Sum_inr and - [intro] = Sum_inl[rotated] Sum_inr[rotated] and - [elim ?s] = SumE and - [comp] = 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 (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 (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 (fn x. C x) x: C x" - -lemmas - [form] = TopF BotF and - [intr, intro] = TopI and - [elim ?a] = TopE and - [elim ?x] = BotE and - [comp] = 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+ - -\<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" - using assms[unfolded Bool_def true_def false_def, type] - by (elim x) (elim, fact)+ - -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 - using assms unfolding Bool_def true_def false_def - by compute - -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 - using assms unfolding Bool_def true_def false_def - by compute - -lemmas - [form] = BoolF and - [intr, intro] = Bool_true Bool_false and - [comp] = if_true if_false and - [elim ?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" - -translations "if x then a else b" \<leftharpoondown> "CONST ifelse C x a b" - -subsection \<open>Logical connectives\<close> - -definition not ("!_") where "!x \<equiv> ifelse (K Bool) x false true" - - -end |