More rigorous rules for Product type. Propositions and proofs all working, but have to think about maybe relaxing the computation rule, or else automating the currying of dependent functions.

2 files changed, 72 insertions, 24 deletions

diff --git a/HoTT.thy b/HoTT.thy
index 4713a0d..96bd3c1 100644
--- a/HoTT.thy
+++ b/HoTT.thy
@@ -54,24 +54,27 @@ subsubsection \<open>Dependent function/product\<close>
 
 consts
   Prod :: "[Term, (Term \<Rightarrow> Term)] \<Rightarrow> Term"
+  lambda :: "[Term, (Term \<Rightarrow> Term)] \<Rightarrow> Term"
 syntax
-  "_Prod" :: "[idt, Term, Term] \<Rightarrow> Term"  ("(3\<Prod>_:_./ _)" 10)
+  "_Prod" :: "[idt, Term, Term] \<Rightarrow> Term"     ("(3\<Prod>_:_./ _)" 10)
+  "__lambda" :: "[idt, Term, Term] \<Rightarrow> Term"  ("(3\<^bold>\<lambda>_:_./ _)" 10)
 translations
   "\<Prod>x:A. B" \<rightleftharpoons> "CONST Prod A (\<lambda>x. B)"
-(* The above syntax translation binds the x in the expression B *)
+  "\<^bold>\<lambda>x:A. b" \<rightleftharpoons> "CONST lambda A (\<lambda>x. b)"
+(* The above syntax translations bind the x in the expressions B, b. *)
 
 abbreviation Function :: "[Term, Term] \<Rightarrow> Term"  (infixr "\<rightarrow>" 30)
   where "A\<rightarrow>B \<equiv> \<Prod>_:A. B"
 
 axiomatization
-  lambda :: "(Term \<Rightarrow> Term) \<Rightarrow> Term"  (binder "\<^bold>\<lambda>" 10) and
-  appl :: "[Term, Term] \<Rightarrow> Term"      ("(3_`/_)" [10, 10] 60)
+  appl :: "[Term, Term] \<Rightarrow> Term"  (infixl "`" 60)
 where
   Prod_form: "\<And>(A::Term) (B::Term \<Rightarrow> Term). \<lbrakk>A : U; B : A \<rightarrow> U\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x) : U" and
-  Prod_intro [intro]: "\<And>(A::Term) (B::Term \<Rightarrow> Term) (b::Term \<Rightarrow> Term). \<lbrakk>A : U; B : A \<rightarrow> U; \<And>x::Term. x : A \<Longrightarrow> b(x) : B(x)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b(x) : \<Prod>x:A. B(x)" and
+  Prod_intro [intro]: "\<And>(A::Term) (B::Term \<Rightarrow> Term) (b::Term \<Rightarrow> Term). \<lbrakk>A : U; B : A \<rightarrow> U; \<And>x::Term. x : A \<Longrightarrow> b(x) : B(x)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x:A. b(x) : \<Prod>x:A. B(x)" and
   Prod_elim [elim]: "\<And>(A::Term) (B::Term \<Rightarrow> Term) (f::Term) (a::Term). \<lbrakk>f : \<Prod>x:A. B(x); a : A\<rbrakk> \<Longrightarrow> f`a : B(a)" and
-  Prod_comp [simp]: "\<And>(b::Term \<Rightarrow> Term) (a::Term). (\<^bold>\<lambda>x. b(x))`a \<equiv> b(a)" and
-  Prod_uniq [simp]: "\<And>(A::Term) (B::Term \<Rightarrow> Term) (f::Term). f : \<Prod>x:A. B(x) \<Longrightarrow> \<^bold>\<lambda>x. (f`x) \<equiv> f"
+  Prod_comp [simp]: "\<And>(A::Term) (B::Term \<Rightarrow> Term) (b::Term \<Rightarrow> Term) (a::Term). \<lbrakk>A : U; B : A \<rightarrow> U; \<And>x::Term. x : A \<Longrightarrow> b(x) : B(x); a : A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x:A. b(x))`a \<equiv> b(a)" and
+  Prod_uniq [simp]: "\<And>(A::Term) (B::Term \<Rightarrow> Term) (f::Term). f : \<Prod>x:A. B(x) \<Longrightarrow> \<^bold>\<lambda>x:A. (f`x) \<equiv> f"
+(* Thinking about the premises for the computation rule... they make simplification rather cumbersome, should I remove them? Would this potentially result in logical problems with being able to state untrue statements? (But probably not prove them?) *)
 
 text "Note that the syntax \<open>\<^bold>\<lambda>\<close> (bold lambda) used for dependent functions clashes with the proof term syntax (cf. \<section>2.5.2 of the Isabelle/Isar Implementation)."
 
@@ -90,17 +93,19 @@ abbreviation Pair :: "[Term, Term] \<Rightarrow> Term"   (infixr "\<times>" 50)
   where "A\<times>B \<equiv> \<Sum>_:A. B"
 
 axiomatization
-  pair :: "[Term, Term] \<Rightarrow> Term"     ("(1'(_,/ _'))") and
+  pair :: "[Term, Term] \<Rightarrow> Term"  ("(1'(_,/ _'))") and
   indSum :: "[Term \<Rightarrow> Term, Term \<Rightarrow> Term, Term] \<Rightarrow> Term"
 where
   Sum_form: "\<And>(A::Term) (B::Term \<Rightarrow> Term). \<lbrakk>A : U; B: A \<rightarrow> U\<rbrakk> \<Longrightarrow> \<Sum>x:A. B(x) : U" and
-  Sum_intro: "\<And>(A::Term) (B::Term \<Rightarrow> Term) (a::Term) (b::Term). \<lbrakk>A : U; B: A \<rightarrow> U; a : A; b : B(a)\<rbrakk> \<Longrightarrow> (a, b) : \<Sum>x:A. B(x)" and
-  Sum_elim: "\<And>(A::Term) (B::Term \<Rightarrow> Term) (C::Term \<Rightarrow> Term) (f::Term \<Rightarrow> Term) (p::Term). \<lbrakk>A : U; B: A \<rightarrow> U; C: \<Sum>x:A. B(x) \<rightarrow> U; \<And>x y::Term. \<lbrakk>x : A; y : B(x)\<rbrakk> \<Longrightarrow> f((x,y)) : C((x,y)); p : \<Sum>x:A. B(x)\<rbrakk> \<Longrightarrow> (indSum C f p) : C(p)"
-  Sum_comp: ""
+  Sum_intro [intro]: "\<And>(A::Term) (B::Term \<Rightarrow> Term) (a::Term) (b::Term). \<lbrakk>A : U; B: A \<rightarrow> U; a : A; b : B(a)\<rbrakk> \<Longrightarrow> (a, b) : \<Sum>x:A. B(x)" and
+  Sum_elim [elim]: "\<And>(A::Term) (B::Term \<Rightarrow> Term) (C::Term \<Rightarrow> Term) (f::Term \<Rightarrow> Term) (p::Term). \<lbrakk>A : U; B: A \<rightarrow> U; C: \<Sum>x:A. B(x) \<rightarrow> U; \<And>x y::Term. \<lbrakk>x : A; y : B(x)\<rbrakk> \<Longrightarrow> f((x,y)) : C((x,y)); p : \<Sum>x:A. B(x)\<rbrakk> \<Longrightarrow> (indSum C f p) : C(p)" and
+  Sum_comp [simp]: "\<And>(A::Term) (B::Term \<Rightarrow> Term) (C::Term \<Rightarrow> Term) (f::Term \<Rightarrow> Term) (a::Term) (b::Term). \<lbrakk>A : U; B: A \<rightarrow> U; C: \<Sum>x:A. B(x) \<rightarrow> U; \<And>x y::Term. \<lbrakk>x : A; y : B(x)\<rbrakk> \<Longrightarrow> f((x,y)) : C((x,y)); a : A; b : B(a)\<rbrakk> \<Longrightarrow> (indSum C f (a,b)) \<equiv> f((a,b))"
 
-text "A choice had to be made for the elimination rule: we formalize the function \<open>f\<close> taking \<open>a : A\<close> and \<open>b : B(x)\<close> and returning \<open>C((a,b))\<close> as a meta-lambda \<open>f::Term \<Rightarrow> Term\<close> instead of an object dependent function \<open>f : \<Prod>x:A. B(x)\<close>.
+text "A choice had to be made for the elimination rule: we formalize the function \<open>f\<close> taking \<open>a : A\<close> and \<open>b : B(x)\<close> and returning \<open>C((a,b))\<close> as a meta level \<open>f::Term \<Rightarrow> Term\<close> instead of an object logic dependent function \<open>f : \<Prod>x:A. B(x)\<close>.
 However we should be able to later show the equivalence of the formalizations."
 
+
+
 \<comment> \<open>Projection onto first component\<close>
 (*
 definition proj1 :: "Term \<Rightarrow> Term \<Rightarrow> Term" ("(proj1\<langle>_,_\<rangle>)") where
diff --git a/HoTT_Theorems.thy b/HoTT_Theorems.thy
index 33b0957..d83a08c 100644
--- a/HoTT_Theorems.thy
+++ b/HoTT_Theorems.thy
@@ -6,23 +6,25 @@ text "A bunch of theorems and other statements for sanity-checking, as well as t
 
 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]]
+declare[[unify_trace_simp, unify_trace_types, simp_trace, simp_trace_depth_limit=2]]
 
 section \<open>Functions\<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 id_function: "A : U \<Longrightarrow> \<^bold>\<lambda>x. x : A\<rightarrow>A" ..
+lemma id_function: "A : U \<Longrightarrow> \<^bold>\<lambda>x:A. x : A\<rightarrow>A" ..
 
 text "Here is the same result, stated and proved differently.
 The standard method invoked after the keyword \<open>proof\<close> is applied to the goal \<open>\<^bold>\<lambda>x. x: A\<rightarrow>A\<close>, and so we need to show the prover how to continue, as opposed to the previous lemma."
 
 lemma
   assumes "A : U"
-  shows "\<^bold>\<lambda>x. x : A\<rightarrow>A"
+  shows "\<^bold>\<lambda>x:A. x : A\<rightarrow>A"
 proof
   show "A : U" using assms .
   show "\<lambda>x. A : A \<rightarrow> U" using assms ..
@@ -31,29 +33,29 @@ qed
 text "Note that there is no provision for declaring the type of bound variables outside of the scope of a lambda expression.
 More generally, we cannot write an assumption stating 'Let \<open>x\<close> be a variable of type \<open>A\<close>'."
 
-proposition "\<lbrakk>A : U; A \<equiv> B\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. x : B\<rightarrow>A"
+proposition "\<lbrakk>A : U; A \<equiv> B\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x:A. x : B\<rightarrow>A"
 proof -
   assume
     1: "A : U" and
     2: "A \<equiv> B"
-  from id_function[OF 1] have 3: "\<^bold>\<lambda>x. x : A\<rightarrow>A" .
+  from id_function[OF 1] have 3: "\<^bold>\<lambda>x:A. x : A\<rightarrow>A" .
   from 2 have "A\<rightarrow>A \<equiv> B\<rightarrow>A" by simp
-  with 3 show "\<^bold>\<lambda>x. x : B\<rightarrow>A" ..
+  with 3 show "\<^bold>\<lambda>x:A. x : B\<rightarrow>A" ..
 qed
 
 text "It is instructive to try to prove \<open>\<lbrakk>A : U; B : U\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. \<^bold>\<lambda>y. x : A\<rightarrow>B\<rightarrow>A\<close>.
 First we prove an intermediate step."
 
-lemma constant_function: "\<lbrakk>A : U; B : U; x : A\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>y. x : B\<rightarrow>A" ..
+lemma constant_function: "\<lbrakk>A : U; B : U; x : A\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>y:B. x : B\<rightarrow>A" ..
 
 text "And now the actual result:"
 
 proposition
   assumes 1: "A : U" and 2: "B : U"
-  shows "\<^bold>\<lambda>x. \<^bold>\<lambda>y. x : A\<rightarrow>B\<rightarrow>A"
+  shows "\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x : A\<rightarrow>B\<rightarrow>A"
 proof
   show "A : U" using assms(1) .
-  show "\<And>x. x : A \<Longrightarrow> \<^bold>\<lambda>y. x : B \<rightarrow> A" using assms by (rule constant_function)
+  show "\<And>x. x : A \<Longrightarrow> \<^bold>\<lambda>y:B. x : B \<rightarrow> A" using assms by (rule constant_function)
 
   from assms have "B \<rightarrow> A : U" by (rule Prod_formation)
   then show "\<lambda>x. B \<rightarrow> A: A \<rightarrow> U" using assms(1) by (rule constant_type_family)
@@ -61,13 +63,54 @@ qed
 
 text "Maybe a nicer way to write it:"
 
-proposition "\<lbrakk>A : U; B: U\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. \<^bold>\<lambda>y. x : A\<rightarrow>B\<rightarrow>A"
+proposition alternating_function: "\<lbrakk>A : U; B: U\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x : A\<rightarrow>B\<rightarrow>A"
 proof
   fix x
-  show "\<lbrakk>A : U;  B : U; x : A\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>y. x : B \<rightarrow> A" by (rule constant_function)
+  show "\<lbrakk>A : U;  B : U; x : A\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>y:B. x : B \<rightarrow> A" by (rule constant_function)
   show "\<lbrakk>A : U; B : U\<rbrakk> \<Longrightarrow> B\<rightarrow>A : U" by (rule Prod_formation)
 qed
 
+subsection \<open>Function application\<close>
+
+lemma "\<lbrakk>A : U; a : A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x:A. x)`a \<equiv> a" by simp
+
+lemma
+  assumes
+    "A:U" and
+    "B:U" and
+    "a:A" and
+    "b:B"
+  shows "(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x)`a`b \<equiv> a"
+proof -
+  have "(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x)`a \<equiv> \<^bold>\<lambda>y:B. a" 
+    proof (rule Prod_comp[of A "\<lambda>_. B\<rightarrow>A"])
+      have "B \<rightarrow> A : U" using constant_type_family[OF assms(1) assms(2)] assms(2) by (rule Prod_formation)
+      then show "\<lambda>x. B \<rightarrow> A: A \<rightarrow> U" using assms(1) by (rule constant_type_family[of "B\<rightarrow>A"])
+      
+      show "\<And>x. x : A \<Longrightarrow> \<^bold>\<lambda>y:B. x : B \<rightarrow> A" using assms(2) assms(1) ..
+      show "A:U" using assms(1) .
+      show "a:A" using assms(3) .
+    qed  (* Why do I need to do the above for the last two goals? Can't Isabelle do it automatically? *)
+
+  then have "(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x)`a`b \<equiv> (\<^bold>\<lambda>y:B. a)`b" by simp
+
+  also have "(\<^bold>\<lambda>y:B. a)`b \<equiv> a"
+    proof (rule Prod_comp[of B "\<lambda>_. A"])
+      show "\<lambda>y. A: B \<rightarrow> U" using assms(1) assms(2) by (rule constant_type_family)
+      show "\<And>y. y : B \<Longrightarrow> a : A" using assms(3) .
+      show "B:U" using assms(2) .
+      show "b:B" using assms(4) .
+    qed
+
+  finally show "(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x)`a`b \<equiv> a" .
+qed
+
+text "Polymorphic identity function."
+
+consts Ui::Term
+definition Id where "Id \<equiv> \<^bold>\<lambda>A:Ui. \<^bold>\<lambda>x:A. x"
+(* Have to think about universes... *)
+
 section \<open>Nats\<close>
 
 text "Here's a dumb proof that 2 is a natural number."