(*  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