diff options
| -rw-r--r-- | Coprod.thy | 20 | ||||
| -rw-r--r-- | Nat.thy | 8 |
2 files changed, 14 insertions, 14 deletions
@@ -18,27 +18,27 @@ axiomatization 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: "\<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: "\<lbrakk>a: A; B: U(i)\<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: "\<lbrakk>b: B; A: U(i)\<rbrakk> \<Longrightarrow> inr(b): A + B" and - Coprod_elim: "\<And>i A B C c d e. \<lbrakk> + Coprod_elim: "\<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>+(c)(d)(e) : C(e)" + u: A + B + \<rbrakk> \<Longrightarrow> ind\<^sub>+(c)(d)(u) : C(u)" and - Coprod_comp1: "\<And>i A B C c d a. \<lbrakk> + Coprod_comp1: "\<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>+(c)(d)(inl(a)) \<equiv> c(a)" and - Coprod_comp2: "\<And>i A B C c d b. \<lbrakk> + Coprod_comp2: "\<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)); @@ -49,9 +49,9 @@ and 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: "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: "A + B: U(i) \<Longrightarrow> B: U(i)" text "Rule declarations:" @@ -22,22 +22,22 @@ where and Nat_intro1: "0: \<nat>" and - Nat_intro2: "\<And>n. n: \<nat> \<Longrightarrow> succ(n): \<nat>" + Nat_intro2: "n: \<nat> \<Longrightarrow> succ(n): \<nat>" and - Nat_elim: "\<And>i C f a n. \<lbrakk> + Nat_elim: "\<lbrakk> C: \<nat> \<longrightarrow> U(i); \<And>n c. \<lbrakk>n: \<nat>; c: C(n)\<rbrakk> \<Longrightarrow> f(n)(c): C(succ n); a: C(0); n: \<nat> \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat>(f)(a)(n): C(n)" and - Nat_comp1: "\<And>i C f a. \<lbrakk> + Nat_comp1: "\<lbrakk> C: \<nat> \<longrightarrow> U(i); \<And>n c. \<lbrakk>n: \<nat>; c: C(n)\<rbrakk> \<Longrightarrow> f(n)(c): C(succ n); a: C(0) \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat>(f)(a)(0) \<equiv> a" and - Nat_comp2: "\<And> i C f a n. \<lbrakk> + Nat_comp2: "\<lbrakk> C: \<nat> \<longrightarrow> U(i); \<And>n c. \<lbrakk>n: \<nat>; c: C(n)\<rbrakk> \<Longrightarrow> f(n)(c): C(succ n); a: C(0); |