diff options
| -rw-r--r-- | HoTT_Test.thy | 132 |
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 |