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-rw-r--r--HoTT_Test.thy132
1 files changed, 19 insertions, 113 deletions
diff --git a/HoTT_Test.thy b/HoTT_Test.thy
index 67284cf..da044b4 100644
--- a/HoTT_Test.thy
+++ b/HoTT_Test.thy
@@ -86,130 +86,36 @@ proposition curried_function_formation:
proposition higher_order_currying_formation:
- fixes
- A::Term and
- B::"Term \<Rightarrow> Term" and
- C::"[Term, Term] \<Rightarrow> Term" and
- D::"[Term, Term, Term] \<Rightarrow> Term"
assumes
- "A : U" and
- "B: A \<rightarrow> U" and
- "\<And>x y::Term. y : B(x) \<Longrightarrow> C(x)(y) : U" and
- "\<And>x y z::Term. z : C(x)(y) \<Longrightarrow> D(x)(y)(z) : U"
- shows "\<Prod>x:A. \<Prod>y:B(x). \<Prod>z:C(x)(y). D(x)(y)(z) : U"
-proof
- fix x::Term assume 1: "x : A"
- show "\<Prod>y:B(x). \<Prod>z:C(x)(y). D(x)(y)(z) : U"
- proof
- show "B(x) : U" using 1 by (rule assms)
-
- fix y::Term assume 2: "y : B(x)"
- show "\<Prod>z:C(x)(y). D(x)(y)(z) : U"
- proof
- show "C x y : U" using 2 by (rule assms)
- show "\<And>z::Term. z : C(x)(y) \<Longrightarrow> D(x)(y)(z) : U" by (rule assms)
- qed
- qed
-qed (rule assms)
-
-(**** AND PROBABLY THIS TOO? ****)
+ "A: U(i)" and
+ "B: A \<longrightarrow> U(i)" and
+ "\<And>x y. y: B(x) \<Longrightarrow> C(x)(y): U(i)" and
+ "\<And>x y z. z : C(x)(y) \<Longrightarrow> D(x)(y)(z): U(i)"
+ shows "\<Prod>x:A. \<Prod>y:B(x). \<Prod>z:C(x)(y). D(x)(y)(z): U(i)"
+ by (simple lems: assms)
+
+
lemma curried_type_judgment:
- fixes
- a b A::Term and
- B::"Term \<Rightarrow> Term" and
- f C::"[Term, Term] \<Rightarrow> Term"
- assumes "\<And>x y::Term. \<lbrakk>x : A; y : B(x)\<rbrakk> \<Longrightarrow> f x y : C x y"
- shows "\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B(x). f x y : \<Prod>x:A. \<Prod>y:B(x). C x y"
-proof
- fix x::Term
- assume *: "x : A"
- show "\<^bold>\<lambda>y:B(x). f x y : \<Prod>y:B(x). C x y"
- proof
- fix y::Term
- assume **: "y : B(x)"
- show "f x y : C x y" using * ** by (rule assms)
- qed
-qed
+ assumes "A: U(i)" "B: A \<longrightarrow> U(i)" "\<And>x y. \<lbrakk>x : A; y : B(x)\<rbrakk> \<Longrightarrow> f x y : C x y"
+ shows "\<^bold>\<lambda>x y. f x y : \<Prod>x:A. \<Prod>y:B(x). C x y"
+ by (simple lems: assms)
+
text "
- Note that the propositions and proofs above often say nothing about the well-formedness of the types, or the well-typedness of the lambdas involved; one has to be very explicit and prove such things separately!
- This is the result of the choices made regarding the premises of the type rules.
+ Polymorphic identity function. Trivial due to lambda expression polymorphism.
+ (Was more involved in previous monomorphic incarnations.)
"
+definition Id :: "Term" where "Id \<equiv> \<^bold>\<lambda>x. x"
-section \<open>\<Sum> type\<close>
-
-text "The following shows that the dependent sum inductor has the type we expect it to have:"
-
-lemma
- assumes "C: \<Sum>x:A. B(x) \<rightarrow> U"
- shows "indSum(C) : \<Prod>f:(\<Prod>x:A. \<Prod>y:B(x). C((x,y))). \<Prod>p:(\<Sum>x:A. B(x)). C(p)"
-proof -
- define F and P where
- "F \<equiv> \<Prod>x:A. \<Prod>y:B(x). C((x,y))" and
- "P \<equiv> \<Sum>x:A. B(x)"
-
- have "\<^bold>\<lambda>f:F. \<^bold>\<lambda>p:P. indSum(C)`f`p : \<Prod>f:F. \<Prod>p:P. C(p)"
- proof (rule curried_type_judgment)
- fix f p::Term
- assume "f : F" and "p : P"
- with assms show "indSum(C)`f`p : C(p)" unfolding F_def P_def ..
- qed
+lemma "\<lbrakk>x: A\<rbrakk> \<Longrightarrow> Id`x \<equiv> x"
+unfolding Id_def by (compute, simple)
- then show "indSum(C) : \<Prod>f:F. \<Prod>p:P. C(p)" by simp
-qed
-
-(**** AUTOMATION CANDIDATE ****)
-text "Propositional uniqueness principle for dependent sums:"
-
-text "We would like to eventually automate proving that 'a given type \<open>A\<close> is inhabited', i.e. search for an element \<open>a:A\<close>.
-
-A good starting point would be to automate the application of elimination rules."
-notepad begin
-
-fix A B assume "A : U" and "B: A \<rightarrow> U"
-
-define C where "C \<equiv> \<lambda>p. p =[\<Sum>x:A. B(x)] (fst[A,B]`p, snd[A,B]`p)"
-have *: "C: \<Sum>x:A. B(x) \<rightarrow> U"
-proof -
- fix p assume "p : \<Sum>x:A. B(x)"
- have "(fst[A,B]`p, snd[A,B]`p) : \<Sum>x:A. B(x)"
-
-define f where "f \<equiv> \<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B(x). refl((x,y))"
-have "f`x`y : C((x,y))"
-sorry
-
-have "p : \<Sum>x:A. B(x) \<Longrightarrow> indSum(C)`f`p : C(p)" using * ** by (rule Sum_elim)
-
-end
-
-
-section \<open>Universes and polymorphism\<close>
-
-text "Polymorphic identity function. Uses universe types."
-
-
-
-definition Id :: "Ord \<Rightarrow> Term" where "Id(i) \<equiv> \<^bold>\<lambda>A x. x"
-
-
-(*
section \<open>Natural numbers\<close>
-text "Here's a dumb proof that 2 is a natural number."
-
-proposition "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 "Automatic proof methods recognize natural numbers."
-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>."
-*)
+proposition "succ(succ(succ 0)): Nat" by simple
end