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(* Title: HoTT/HoTT_Test.thy
Author: Josh Chen
Date: Aug 2018
This is an old "test suite" from early implementations of the theory.
It is not always guaranteed to be up to date, or reflect most recent versions of the theory.
*)
theory HoTT_Test
imports HoTT
begin
text "
A bunch of theorems and other statements for sanity-checking, as well as things that should be automatically simplified.
Things that *should* be automated:
- Checking that \<open>A\<close> is a well-formed type, when writing things like \<open>x : A\<close> and \<open>A : U\<close>.
- Checking that the argument to a (dependent/non-dependent) function matches the type? Also the arguments to a pair?
"
declare[[unify_trace_simp, unify_trace_types, simp_trace, simp_trace_depth_limit=5]]
\<comment> \<open>Turn on trace for unification and the simplifier, for debugging.\<close>
section \<open>\<Pi>-type\<close>
subsection \<open>Typing functions\<close>
text "
Declaring \<open>Prod_intro\<close> with the \<open>intro\<close> attribute (in HoTT.thy) enables \<open>standard\<close> to prove the following.
"
proposition assumes "A : U(i)" shows "\<^bold>\<lambda>x. x: A \<rightarrow> A" using assms ..
proposition
assumes "A : U(i)" and "A \<equiv> B"
shows "\<^bold>\<lambda>x. x : B \<rightarrow> A"
proof -
have "A\<rightarrow>A \<equiv> B\<rightarrow>A" using assms by simp
moreover have "\<^bold>\<lambda>x. x : A \<rightarrow> A" using assms(1) ..
ultimately show "\<^bold>\<lambda>x. x : B \<rightarrow> A" by simp
qed
proposition
assumes "A : U(i)" and "B : U(i)"
shows "\<^bold>\<lambda>x y. x : A \<rightarrow> B \<rightarrow> A"
by (simple lems: assms)
subsection \<open>Function application\<close>
proposition assumes "a : A" shows "(\<^bold>\<lambda>x. x)`a \<equiv> a" by (simple lems: assms)
text "Currying:"
lemma
assumes "a : A" and "\<And>x. x: A \<Longrightarrow> B(x): U(i)"
shows "(\<^bold>\<lambda>x y. y)`a \<equiv> \<^bold>\<lambda>y. y"
proof
show "\<And>x. x : A \<Longrightarrow> \<^bold>\<lambda>y. y : B(x) \<rightarrow> B(x)" by (simple lems: assms)
qed fact
lemma "\<lbrakk>A: U(i); B: U(i); a : A; b : B\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x y. y)`a`b \<equiv> b" by (compute, simple)
lemma "\<lbrakk>A: U(i); a : A \<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x y. f x y)`a \<equiv> \<^bold>\<lambda>y. f a y"
proof compute
show "\<And>x. \<lbrakk>A: U(i); x: A\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>y. f x y: \<Prod>y:B(x). C x y"
proof
oops
lemma "\<lbrakk>a : A; b : B(a); c : C(a)(b)\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. \<^bold>\<lambda>y. \<^bold>\<lambda>z. f x y z)`a`b`c \<equiv> f a b c"
oops
subsection \<open>Currying functions\<close>
proposition curried_function_formation:
fixes A B C
assumes
"A : U(i)" and
"B: A \<longrightarrow> U(i)" and
"\<And>x. C(x): B(x) \<longrightarrow> U(i)"
shows "\<Prod>x:A. \<Prod>y:B(x). C x y : U(i)"
by (simple lems: assms)
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? ****)
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
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.
"
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
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 "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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