diff options
Diffstat (limited to '')
-rw-r--r-- | HoTT_Base.thy | 21 |
1 files changed, 12 insertions, 9 deletions
diff --git a/HoTT_Base.thy b/HoTT_Base.thy index cf71813..c2bb092 100644 --- a/HoTT_Base.thy +++ b/HoTT_Base.thy @@ -9,6 +9,10 @@ theory HoTT_Base imports Pure begin +text "Use the syntax \<open>f(x)\<close> instead of \<open>f x\<close> for function application." + +setup Pure_Thy.old_appl_syntax_setup + section \<open>Foundational definitions\<close> @@ -20,12 +24,12 @@ typedecl Term section \<open>Judgments\<close> text " - Formalize the typing judgment \<open>a : A\<close>. + Formalize the typing judgment \<open>a: A\<close>. For judgmental/definitional equality we use the existing Pure equality \<open>\<equiv>\<close> and hence do not need to define a separate judgment for it. " consts - has_type :: "[Term, Term] \<Rightarrow> prop" ("(1_ : _)" [0, 0] 1000) + has_type :: "[Term, Term] \<Rightarrow> prop" ("(1_: _)" [0, 0] 1000) section \<open>Universe hierarchy\<close> @@ -39,11 +43,11 @@ axiomatization S :: "Ord \<Rightarrow> Ord" ("S_" [0] 1000) and lt :: "[Ord, Ord] \<Rightarrow> prop" (infix "<-" 999) where - Ord_lt_min: "\<And>n. O <- S n" + Ord_min: "\<And>n. O <- S(n)" and - Ord_lt_trans: "\<And>m n. m <- n \<Longrightarrow> S m <- S n" + Ord_monotone: "\<And>m n. m <- n \<Longrightarrow> S(m) <- S(n)" -lemmas Ord_rules [intro] = Ord_lt_min Ord_lt_trans +lemmas Ord_rules [intro] = Ord_min Ord_monotone \<comment> \<open>Enables \<open>standard\<close> to automatically solve inequalities.\<close> text "Define the universe types." @@ -51,12 +55,11 @@ text "Define the universe types." axiomatization U :: "Ord \<Rightarrow> Term" where - U_hierarchy: "\<And>i j. i <- j \<Longrightarrow> U(i) : U(j)" + U_hierarchy: "\<And>i j. i <- j \<Longrightarrow> U(i): U(j)" and - U_cumulative: "\<And>A i j. \<lbrakk>A : U(i); i <- j\<rbrakk> \<Longrightarrow> A : U(j)" + U_cumulative: "\<And>A i j. \<lbrakk>A: U(i); i <- j\<rbrakk> \<Longrightarrow> A: U(j)" \<comment> \<open>WARNING: \<open>rule Universe_cumulative\<close> can result in an infinite rewrite loop!\<close> - section \<open>Type families\<close> text " @@ -64,7 +67,7 @@ text " " abbreviation (input) constrained :: "[Term \<Rightarrow> Term, Term, Term] \<Rightarrow> prop" ("(1_: _ \<longrightarrow> _)") - where "f: A \<longrightarrow> B \<equiv> (\<And>x. x : A \<Longrightarrow> f x : B)" + where "f: A \<longrightarrow> B \<equiv> (\<And>x. x : A \<Longrightarrow> f(x): B)" text " The above is used to define type families, which are constrained meta-lambdas \<open>P: A \<longrightarrow> B\<close> where \<open>A\<close> and \<open>B\<close> are small types. |