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-rw-r--r--hott/Nat.thy22
1 files changed, 13 insertions, 9 deletions
diff --git a/hott/Nat.thy b/hott/Nat.thy
index b88398b..59ec517 100644
--- a/hott/Nat.thy
+++ b/hott/Nat.thy
@@ -17,39 +17,43 @@ where
NatE: "\<lbrakk>
n: Nat;
- n\<^sub>0: C 0;
+ c\<^sub>0: C 0;
\<And>n. n: Nat \<Longrightarrow> C n: U i;
\<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
+ \<rbrakk> \<Longrightarrow> NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k c. f k c) n: C n" and
Nat_comp_zero: "\<lbrakk>
- n\<^sub>0: C 0;
+ c\<^sub>0: C 0;
\<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>k c. f k c) 0 \<equiv> n\<^sub>0" and
+ \<rbrakk> \<Longrightarrow> NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k c. f k c) 0 \<equiv> c\<^sub>0" and
Nat_comp_suc: "\<lbrakk>
n: Nat;
- n\<^sub>0: C 0;
+ c\<^sub>0: C 0;
\<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>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)"
+ NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k c. f k c) (suc n) \<equiv>
+ f n (NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k c. f k c) n)"
lemmas
[intros] = NatF Nat_zero Nat_suc and
[elims] = NatE and
[comps] = Nat_comp_zero Nat_comp_suc
+text \<open>Non-dependent recursion\<close>
+
+abbreviation "NatRec C \<equiv> NatInd (K C)"
+
section \<open>Basic arithmetic\<close>
definition add (infixl "+" 50) where
- [comps]: "m + n \<equiv> NatInd (K Nat) n (K suc) m"
+ [comps]: "m + n \<equiv> NatRec Nat n (K suc) m"
definition mul (infixl "*" 55) where
- [comps]: "m * n \<equiv> NatInd (K Nat) 0 (K $ add n) m"
+ [comps]: "m * n \<equiv> NatRec Nat 0 (K $ add n) m"
end