diff options
Diffstat (limited to 'Coprod.thy')
| -rw-r--r-- | Coprod.thy | 40 |
1 files changed, 20 insertions, 20 deletions
@@ -14,43 +14,43 @@ section \<open>Constants and type rules\<close> axiomatization Coprod :: "[Term, Term] \<Rightarrow> Term" (infixr "+" 50) and - inl :: "Term \<Rightarrow> Term" ("(1inl'(_'))") and - inr :: "Term \<Rightarrow> Term" ("(1inr'(_'))") and - indCoprod :: "[Term, Term, Typefam, Term \<Rightarrow> Term, Term \<Rightarrow> Term, Term] \<Rightarrow> Term" ("(1ind\<^sub>+[_,/ _])") + inl :: "Term \<Rightarrow> Term" and + inr :: "Term \<Rightarrow> Term" and + indCoprod :: "[Term \<Rightarrow> Term, Term \<Rightarrow> Term, Term] \<Rightarrow> Term" ("(1ind\<^sub>+)") where - Coprod_form: "\<And>i A B. \<lbrakk>A : U(i); B : U(i)\<rbrakk> \<Longrightarrow> A + B : U(i)" + Coprod_form: "\<And>i A B. \<lbrakk>A : U(i); B : U(i)\<rbrakk> \<Longrightarrow> A + B: U(i)" and - Coprod_intro1: "\<And>A B a b. \<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> inl(a) : A + B" + Coprod_intro1: "\<And>A B a b. \<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> inl(a): A + B" and - Coprod_intro2: "\<And>A B a b. \<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> inr(b) : A + B" + Coprod_intro2: "\<And>A B a b. \<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> inr(b): A + B" and Coprod_elim: "\<And>i A B C c d e. \<lbrakk> C: A + B \<longrightarrow> U(i); - \<And>x. x : A \<Longrightarrow> c x : C inl(x); - \<And>y. y : B \<Longrightarrow> d y : C inr(y); - e : A + B - \<rbrakk> \<Longrightarrow> ind\<^sub>+[A,B] C c d e : C e" + \<And>x. x: A \<Longrightarrow> c(x): C(inl(x)); + \<And>y. y: B \<Longrightarrow> d(y): C(inr(y)); + e: A + B + \<rbrakk> \<Longrightarrow> ind\<^sub>+(c)(d)(e) : C(e)" and Coprod_comp1: "\<And>i A B C c d a. \<lbrakk> C: A + B \<longrightarrow> U(i); - \<And>x. x : A \<Longrightarrow> c x : C inl(x); - \<And>y. y : B \<Longrightarrow> d y : C inr(y); - a : A - \<rbrakk> \<Longrightarrow> ind\<^sub>+[A,B] C c d inl(a) \<equiv> c a" + \<And>x. x: A \<Longrightarrow> c(x): C(inl(x)); + \<And>y. y: B \<Longrightarrow> d(y): C(inr(y)); + a: A + \<rbrakk> \<Longrightarrow> ind\<^sub>+(c)(d)(inl(a)) \<equiv> c(a)" and Coprod_comp2: "\<And>i A B C c d b. \<lbrakk> C: A + B \<longrightarrow> U(i); - \<And>x. x : A \<Longrightarrow> c x : C inl(x); - \<And>y. y : B \<Longrightarrow> d y : C inr(y); - b : B - \<rbrakk> \<Longrightarrow> ind\<^sub>+[A,B] C c d inr(b) \<equiv> d b" + \<And>x. x: A \<Longrightarrow> c(x): C(inl(x)); + \<And>y. y: B \<Longrightarrow> d(y): C(inr(y)); + b: B + \<rbrakk> \<Longrightarrow> ind\<^sub>+(c)(d)(inr(b)) \<equiv> d(b)" text "Admissible formation inference rules:" axiomatization where - Coprod_form_cond1: "\<And>i A B. A + B : U(i) \<Longrightarrow> A : U(i)" + Coprod_form_cond1: "\<And>i A B. A + B: U(i) \<Longrightarrow> A: U(i)" and - Coprod_form_cond2: "\<And>i A B. A + B : U(i) \<Longrightarrow> B : U(i)" + Coprod_form_cond2: "\<And>i A B. A + B: U(i) \<Longrightarrow> B: U(i)" lemmas Coprod_rules [intro] = Coprod_form Coprod_intro1 Coprod_intro2 Coprod_elim Coprod_comp1 Coprod_comp2 |