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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.
Things that *should* be automated:
\<bullet> Checking that \<open>A\<close> is a well-formed type, when writing things like \<open>x : A\<close> and \<open>A : U\<close>.
\<bullet> Checking that the argument to a (dependent/non-dependent) function matches the type? Also the arguments to a pair?"
\<comment> \<open>Turn on trace for unification and the simplifier, for debugging.\<close>
declare[[unify_trace_simp, unify_trace_types, simp_trace, simp_trace_depth_limit=5]]
section \<open>\<Prod> 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."
lemma "\<^bold>\<lambda>x:A. x : A\<rightarrow>A" ..
proposition "A \<equiv> B \<Longrightarrow> \<^bold>\<lambda>x:A. x : B\<rightarrow>A"
proof -
assume "A \<equiv> B"
then have *: "A\<rightarrow>A \<equiv> B\<rightarrow>A" by simp
have "\<^bold>\<lambda>x:A. x : A\<rightarrow>A" ..
with * show "\<^bold>\<lambda>x:A. x : B\<rightarrow>A" by simp
qed
proposition "\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x : A\<rightarrow>B\<rightarrow>A"
proof
fix a
assume "a : A"
then show "\<^bold>\<lambda>y:B. a : B \<rightarrow> A" ..
qed
subsection \<open>Function application\<close>
proposition "a : A \<Longrightarrow> (\<^bold>\<lambda>x:A. x)`a \<equiv> a" by simp
text "Currying:"
term "lambda A (\<lambda>x. \<^bold>\<lambda>y:B(x). y)"
thm Prod_comp[where ?B = "\<lambda>x. \<Prod>y:B(x). B(x)"]
lemma "a : A \<Longrightarrow> (\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B(x). y)`a \<equiv> \<^bold>\<lambda>y:B(a). y"
proof (rule Prod_comp[where ?B = "\<lambda>x. \<Prod>y:B(x). B(x)"])
show "\<And>x. a : A \<Longrightarrow> x : A \<Longrightarrow> \<^bold>\<lambda>y:B x. y : B x \<rightarrow> B x"
lemma "\<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B(x). y)`a`b \<equiv> b" by simp
lemma "a : A \<Longrightarrow> (\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B(x). f x y)`a \<equiv> \<^bold>\<lambda>y:B(a). f a y" by simp
lemma "\<lbrakk>a : A; b : B(a); c : C(a)(b)\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B(x). \<^bold>\<lambda>z:C(x)(y). f x y z)`a`b`c \<equiv> f a b c" by simp
subsection \<open>Currying functions\<close>
proposition curried_function_formation:
fixes
A::Term and
B::"Term \<Rightarrow> Term" and
C::"Term \<Rightarrow> Term \<Rightarrow> Term"
assumes
"A : U" and
"B: A \<rightarrow> U" and
"\<And>x::Term. C(x): B(x) \<rightarrow> U"
shows "\<Prod>x:A. \<Prod>y:B(x). C x y : U"
proof
fix x::Term
assume *: "x : A"
show "\<Prod>y:B(x). C x y : U"
proof
show "B(x) : U" using * by (rule assms)
qed (rule assms)
qed (rule assms)
(**** GOOD CANDIDATE FOR AUTOMATION - EISBACH! ****)
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."
consts Ui::Term
definition Id where "Id \<equiv> \<^bold>\<lambda>A:Ui. \<^bold>\<lambda>x:A. 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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