diff options
| -rw-r--r-- | HoTT.thy | 5 | ||||
| -rw-r--r-- | HoTT_Theorems.thy | 99 |
2 files changed, 29 insertions, 75 deletions
@@ -70,11 +70,10 @@ axiomatization 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:A. b(x) : \<Prod>x:A. B(x)" and + Prod_intro [intro]: "\<And>(A::Term) (B::Term \<Rightarrow> Term) (b::Term \<Rightarrow> Term). (\<And>x::Term. x : A \<Longrightarrow> b(x) : B(x)) \<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>(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_comp [simp]: "\<And>(A::Term) (B::Term \<Rightarrow> Term) (b::Term \<Rightarrow> Term) (a::Term). \<lbrakk>\<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)." diff --git a/HoTT_Theorems.thy b/HoTT_Theorems.thy index d83a08c..10a0d2c 100644 --- a/HoTT_Theorems.thy +++ b/HoTT_Theorems.thy @@ -6,10 +6,11 @@ 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?" + \<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=2]] +declare[[unify_trace_simp, unify_trace_types, simp_trace, simp_trace_depth_limit=1]] section \<open>Functions\<close> @@ -17,97 +18,51 @@ 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:A. x : A\<rightarrow>A" .. +lemma "\<^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:A. x : A\<rightarrow>A" -proof - show "A : U" using assms . - show "\<lambda>x. A : A \<rightarrow> U" using assms .. -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:A. x : B\<rightarrow>A" +proposition "A \<equiv> B \<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:A. x : A\<rightarrow>A" . - from 2 have "A\<rightarrow>A \<equiv> B\<rightarrow>A" by simp - 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:B. x : B\<rightarrow>A" .. - -text "And now the actual result:" - -proposition - assumes 1: "A : U" and 2: "B : U" - 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: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) + assume assm: "A \<equiv> B" + have id: "\<^bold>\<lambda>x:A. x : A\<rightarrow>A" .. + from assm have "A\<rightarrow>A \<equiv> B\<rightarrow>A" by simp + with id show "\<^bold>\<lambda>x:A. x : B\<rightarrow>A" .. qed -text "Maybe a nicer way to write it:" - -proposition alternating_function: "\<lbrakk>A : U; B: U\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x : A\<rightarrow>B\<rightarrow>A" +proposition "\<^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: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) + fix a + assume "a : A" + then show "\<^bold>\<lambda>y:B. a : B \<rightarrow> A" .. qed subsection \<open>Function application\<close> -lemma "\<lbrakk>A : U; a : A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x:A. x)`a \<equiv> a" by simp +proposition "\<lbrakk>A : U; a : A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x:A. x)`a \<equiv> a" by simp + +text "Two arguments:" lemma - assumes - "A:U" and - "B:U" and - "a:A" and - "b:B" + assumes "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? *) - + proof (rule Prod_comp[of A _ "\<lambda>_. B\<rightarrow>A"]) + fix x + assume "x:A" + then show "\<^bold>\<lambda>y:B. x : B \<rightarrow> A" .. + qed (rule assms) 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 - + also have "(\<^bold>\<lambda>y:B. a)`b \<equiv> a" using assms by (simp add: Prod_comp[of B _ "\<lambda>_. A"]) finally show "(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x)`a`b \<equiv> a" . 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." + 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... *) |