diff options
| -rw-r--r-- | hott/Nat.thy | 19 |
1 files changed, 11 insertions, 8 deletions
diff --git a/hott/Nat.thy b/hott/Nat.thy index 269d6bb..b88398b 100644 --- a/hott/Nat.thy +++ b/hott/Nat.thy @@ -19,23 +19,23 @@ where n: Nat; n\<^sub>0: C 0; \<And>n. n: Nat \<Longrightarrow> C n: U i; - \<And>n c. \<lbrakk>n: Nat; c: C n\<rbrakk> \<Longrightarrow> f n c: C (suc n) - \<rbrakk> \<Longrightarrow> NatInd (\<lambda>n. C n) n\<^sub>0 (\<lambda>n c. f n c) n: C n" and + \<And>k c. \<lbrakk>k: Nat; c: C k\<rbrakk> \<Longrightarrow> f k c: C (suc k) + \<rbrakk> \<Longrightarrow> NatInd (\<lambda>n. C n) n\<^sub>0 (\<lambda>k c. f k c) n: C n" and Nat_comp_zero: "\<lbrakk> n\<^sub>0: C 0; - \<And>n c. \<lbrakk>n: Nat; c: C n\<rbrakk> \<Longrightarrow> f n c: C (suc n); + \<And>k c. \<lbrakk>k: Nat; c: C k\<rbrakk> \<Longrightarrow> f k c: C (suc k); \<And>n. n: Nat \<Longrightarrow> C n: U i - \<rbrakk> \<Longrightarrow> NatInd (\<lambda>n. C n) n\<^sub>0 (\<lambda>n c. f n c) 0 \<equiv> n\<^sub>0" and + \<rbrakk> \<Longrightarrow> NatInd (\<lambda>n. C n) n\<^sub>0 (\<lambda>k c. f k c) 0 \<equiv> n\<^sub>0" and Nat_comp_suc: "\<lbrakk> n: Nat; n\<^sub>0: C 0; - \<And>n c. \<lbrakk>n: Nat; c: C n\<rbrakk> \<Longrightarrow> f n c: C (suc n); + \<And>k c. \<lbrakk>k: Nat; c: C k\<rbrakk> \<Longrightarrow> f k c: C (suc k); \<And>n. n: Nat \<Longrightarrow> C n: U i \<rbrakk> \<Longrightarrow> - NatInd (\<lambda>n. C n) n\<^sub>0 (\<lambda>n c. f n c) (suc n) \<equiv> - f n (NatInd (\<lambda>n. C n) n\<^sub>0 (\<lambda>n c. f n c) n)" + NatInd (\<lambda>n. C n) n\<^sub>0 (\<lambda>k c. f k c) (suc n) \<equiv> + f n (NatInd (\<lambda>n. C n) n\<^sub>0 (\<lambda>k c. f k c) n)" lemmas [intros] = NatF Nat_zero Nat_suc and @@ -46,7 +46,10 @@ lemmas section \<open>Basic arithmetic\<close> definition add (infixl "+" 50) where - [comps]: "m + n \<equiv> NatInd (K Nat) n (K $ \<lambda>n. suc n) m" + [comps]: "m + n \<equiv> NatInd (K Nat) n (K suc) m" + +definition mul (infixl "*" 55) where + [comps]: "m * n \<equiv> NatInd (K Nat) 0 (K $ add n) m" end |