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Diffstat (limited to 'HoTT_Theorems.thy')
-rw-r--r-- | HoTT_Theorems.thy | 68 |
1 files changed, 68 insertions, 0 deletions
diff --git a/HoTT_Theorems.thy b/HoTT_Theorems.thy new file mode 100644 index 0000000..bea3dfe --- /dev/null +++ b/HoTT_Theorems.thy @@ -0,0 +1,68 @@ +theory HoTT_Theorems + imports HoTT +begin + +text "A bunch of theorems and other statements for sanity-checking, as well as things that should be automatically simplified." + +section \<open>Foundational stuff\<close> + +theorem "\<lbrakk>A : U; A \<equiv> B\<rbrakk> \<Longrightarrow> B : U" by simp + +section \<open>Functions\<close> + +lemma "A : U \<Longrightarrow> \<^bold>\<lambda>x. x : A\<rightarrow>A" + by (rule Prod_intro) + +text "Note that there is no provision for declaring the type of bound variables outside of the scope of a lambda expression. +Hence a statement like \<open>x : A\<close> is not needed (nor possible!) in the premises of the following lemma." + +lemma "\<lbrakk>A : U; A \<equiv> B\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. x : B\<rightarrow>A" +proof - + assume + 0: "A : U" and + 1: "A \<equiv> B" + from 0 have 2: "\<^bold>\<lambda>x. x : A\<rightarrow>A" by (rule Prod_intro) + from 1 have 3: "A\<rightarrow>A \<equiv> B\<rightarrow>A" by simp + from 3 and 2 show "\<^bold>\<lambda>x. x : B\<rightarrow>A" by (rule equal_types) + qed + +lemma "\<lbrakk>A : U; B : U; x : A\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>y. x : B\<rightarrow>A" +proof - +assume + 1: "A : U" and + 2: "B : U" and + 3: "x : A" +then show "\<^bold>\<lambda>y. x : B\<rightarrow>A" +proof - +from 3 have "\<^bold>\<lambda>y. x : B\<rightarrow>A" by (rule Prod_intro) +qed +qed + +lemma "\<lbrakk>A : U; B : U\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. \<^bold>\<lambda>y. x : A\<rightarrow>B\<rightarrow>A" +proof - + fix x + assume + "A : U" and + "B : U" and + "x : A" + then have "\<^bold>\<lambda>y. x : B\<rightarrow>A" by (rule Prod_intro) + +qed + +section \<open>Nats\<close> + +text "Here's a dumb proof that 2 is a natural number." + +lemma "succ(succ 0) : Nat" +proof - + have "0 : Nat" by (rule Nat_intro1) + from this have "(succ 0) : Nat" by (rule Nat_intro2) + thus "succ(succ 0) : Nat" by (rule Nat_intro2) +qed + +text "We can of course iterate the above for as many applications of \<open>succ\<close> as we like. +The next thing to do is to implement induction to automate such proofs. + +When we get more stuff working, I'd like to aim for formalizing the encode-decode method to be able to prove the only naturals are 0 and those obtained from it by \<open>succ\<close>." + +end
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