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-chapter \<open>The identity type\<close>
-
-text \<open>
- The identity type, the higher groupoid structure of types,
- and type families as fibrations.
-\<close>
-
-theory Identity
-imports Spartan
-
-begin
-
-axiomatization
- Id :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> and
- refl :: \<open>o \<Rightarrow> o\<close> and
- IdInd :: \<open>o \<Rightarrow> (o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close>
-
-syntax "_Id" :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2_ =\<^bsub>_\<^esub>/ _)" [111, 0, 111] 110)
-
-translations "a =\<^bsub>A\<^esub> b" \<rightleftharpoons> "CONST Id A a b"
-
-axiomatization where
- IdF: "\<lbrakk>A: U i; a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^bsub>A\<^esub> b: U i" and
-
- IdI: "a: A \<Longrightarrow> refl a: a =\<^bsub>A\<^esub> a" and
-
- IdE: "\<lbrakk>
- p: a =\<^bsub>A\<^esub> b;
- a: A;
- b: A;
- \<And>x y p. \<lbrakk>p: x =\<^bsub>A\<^esub> y; x: A; y: A\<rbrakk> \<Longrightarrow> C x y p: U i;
- \<And>x. x: A \<Longrightarrow> f x: C x x (refl x)
- \<rbrakk> \<Longrightarrow> IdInd A (\<lambda>x y p. C x y p) f a b p: C a b p" and
-
- Id_comp: "\<lbrakk>
- a: A;
- \<And>x y p. \<lbrakk>x: A; y: A; p: x =\<^bsub>A\<^esub> y\<rbrakk> \<Longrightarrow> C x y p: U i;
- \<And>x. x: A \<Longrightarrow> f x: C x x (refl x)
- \<rbrakk> \<Longrightarrow> IdInd A (\<lambda>x y p. C x y p) f a a (refl a) \<equiv> f a"
-
-lemmas
- [intros] = IdF IdI and
- [elims "?p" "?a" "?b"] = IdE and
- [comps] = Id_comp and
- [refl] = IdI
-
-
-section \<open>Path induction\<close>
-
-method_setup eq = \<open>
-Args.term >> (fn tm => fn ctxt => CONTEXT_METHOD (K (
- CONTEXT_SUBGOAL (fn (goal, i) =>
- let
- val facts = Proof_Context.facts_of ctxt
- val prems = Logic.strip_assums_hyp goal
- val template = \<^const>\<open>has_type\<close>
- $ tm
- $ (\<^const>\<open>Id\<close> $ Var (("*?A", 0), \<^typ>\<open>o\<close>)
- $ Var (("*?x", 0), \<^typ>\<open>o\<close>)
- $ Var (("*?y", 0), \<^typ>\<open>o\<close>))
- val types =
- map (Thm.prop_of o #1) (Facts.could_unify facts template)
- @ filter (fn prem => Term.could_unify (template, prem)) prems
- |> map Lib.type_of_typing
- in case types of
- (\<^const>\<open>Id\<close> $ _ $ x $ y)::_ =>
- elim_context_tac [tm, x, y] ctxt i
- | _ => Context_Tactic.CONTEXT_TACTIC no_tac
- end) 1)))
-\<close>
-
-
-section \<open>Symmetry and transitivity\<close>
-
-Lemma (derive) pathinv:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
- shows "y =\<^bsub>A\<^esub> x"
- by (eq p) intro
-
-lemma pathinv_comp [comps]:
- assumes "x: A" "A: U i"
- shows "pathinv A x x (refl x) \<equiv> refl x"
- unfolding pathinv_def by reduce
-
-Lemma (derive) pathcomp:
- assumes
- "A: U i" "x: A" "y: A" "z: A"
- "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z"
- shows
- "x =\<^bsub>A\<^esub> z"
- apply (eq p)
- focus prems vars x p
- apply (eq p)
- apply intro
- done
- done
-
-lemma pathcomp_comp [comps]:
- assumes "a: A" "A: U i"
- shows "pathcomp A a a a (refl a) (refl a) \<equiv> refl a"
- unfolding pathcomp_def by reduce
-
-
-section \<open>Notation\<close>
-
-definition Id_i (infix "=" 110)
- where [implicit]: "Id_i x y \<equiv> x =\<^bsub>?\<^esub> y"
-
-definition pathinv_i ("_\<inverse>" [1000])
- where [implicit]: "pathinv_i p \<equiv> pathinv ? ? ? p"
-
-definition pathcomp_i (infixl "\<bullet>" 121)
- where [implicit]: "pathcomp_i p q \<equiv> pathcomp ? ? ? ? p q"
-
-translations
- "x = y" \<leftharpoondown> "x =\<^bsub>A\<^esub> y"
- "p\<inverse>" \<leftharpoondown> "CONST pathinv A x y p"
- "p \<bullet> q" \<leftharpoondown> "CONST pathcomp A x y z p q"
-
-
-section \<open>Calculational reasoning\<close>
-
-consts rhs :: \<open>'a\<close> ("''''")
-
-ML \<open>
-local fun last_rhs ctxt =
- let
- val this_name = Name_Space.full_name (Proof_Context.naming_of ctxt)
- (Binding.name Auto_Bind.thisN)
- val this = #thms (the (Proof_Context.lookup_fact ctxt this_name))
- handle Option => []
- val rhs = case map Thm.prop_of this of
- [\<^const>\<open>has_type\<close> $ _ $ (\<^const>\<open>Id\<close> $ _ $ _ $ y)] => y
- | _ => Term.dummy
- in
- map_aterms (fn t => case t of Const (\<^const_name>\<open>rhs\<close>, _) => rhs | _ => t)
- end
-in val _ = Context.>>
- (Syntax_Phases.term_check 5 "" (fn ctxt => map (last_rhs ctxt)))
-end
-\<close>
-
-lemmas
- [sym] = pathinv[rotated 3] and
- [trans] = pathcomp[rotated 4]
-
-
-section \<open>Basic propositional equalities\<close>
-
-Lemma (derive) pathcomp_left_refl:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
- shows "(refl x) \<bullet> p = p"
- apply (eq p)
- apply (reduce; intros)
- done
-
-Lemma (derive) pathcomp_right_refl:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
- shows "p \<bullet> (refl y) = p"
- apply (eq p)
- apply (reduce; intros)
- done
-
-Lemma (derive) pathcomp_left_inv:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
- shows "p\<inverse> \<bullet> p = refl y"
- apply (eq p)
- apply (reduce; intros)
- done
-
-Lemma (derive) pathcomp_right_inv:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
- shows "p \<bullet> p\<inverse> = refl x"
- apply (eq p)
- apply (reduce; intros)
- done
-
-Lemma (derive) pathinv_pathinv:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
- shows "p\<inverse>\<inverse> = p"
- apply (eq p)
- apply (reduce; intros)
- done
-
-Lemma (derive) pathcomp_assoc:
- assumes
- "A: U i" "x: A" "y: A" "z: A" "w: A"
- "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z" "r: z =\<^bsub>A\<^esub> w"
- shows "p \<bullet> (q \<bullet> r) = p \<bullet> q \<bullet> r"
- apply (eq p)
- focus prems vars x p
- apply (eq p)
- focus prems vars x p
- apply (eq p)
- apply (reduce; intros)
- done
- done
- done
-
-
-section \<open>Functoriality of functions\<close>
-
-Lemma (derive) ap:
- assumes
- "A: U i" "B: U i"
- "x: A" "y: A"
- "f: A \<rightarrow> B"
- "p: x =\<^bsub>A\<^esub> y"
- shows "f x = f y"
- apply (eq p)
- apply intro
- done
-
-definition ap_i ("_[_]" [1000, 0])
- where [implicit]: "ap_i f p \<equiv> ap ? ? ? ? f p"
-
-translations "f[p]" \<leftharpoondown> "CONST ap A B x y f p"
-
-Lemma ap_refl [comps]:
- assumes "f: A \<rightarrow> B" "x: A" "A: U i" "B: U i"
- shows "f[refl x] \<equiv> refl (f x)"
- unfolding ap_def by reduce
-
-Lemma (derive) ap_pathcomp:
- assumes
- "A: U i" "B: U i"
- "x: A" "y: A" "z: A"
- "f: A \<rightarrow> B"
- "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z"
- shows
- "f[p \<bullet> q] = f[p] \<bullet> f[q]"
- apply (eq p)
- focus prems vars x p
- apply (eq p)
- apply (reduce; intro)
- done
- done
-
-Lemma (derive) ap_pathinv:
- assumes
- "A: U i" "B: U i"
- "x: A" "y: A"
- "f: A \<rightarrow> B"
- "p: x =\<^bsub>A\<^esub> y"
- shows "f[p\<inverse>] = f[p]\<inverse>"
- by (eq p) (reduce; intro)
-
-text \<open>The next two proofs currently use some low-level rewriting.\<close>
-
-Lemma (derive) ap_funcomp:
- assumes
- "A: U i" "B: U i" "C: U i"
- "x: A" "y: A"
- "f: A \<rightarrow> B" "g: B \<rightarrow> C"
- "p: x =\<^bsub>A\<^esub> y"
- shows "(g \<circ> f)[p] = g[f[p]]"
- apply (eq p)
- apply (simp only: funcomp_apply_comp[symmetric])
- apply (reduce; intro)
- done
-
-Lemma (derive) ap_id:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
- shows "(id A)[p] = p"
- apply (eq p)
- apply (rewrite at "\<hole> = _" id_comp[symmetric])
- apply (rewrite at "_ = \<hole>" id_comp[symmetric])
- apply (reduce; intros)
- done
-
-
-section \<open>Transport\<close>
-
-Lemma (derive) transport:
- assumes
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- "x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
- shows "P x \<rightarrow> P y"
- by (eq p) intro
-
-definition transport_i ("trans")
- where [implicit]: "trans P p \<equiv> transport ? P ? ? p"
-
-translations "trans P p" \<leftharpoondown> "CONST transport A P x y p"
-
-Lemma transport_comp [comps]:
- assumes
- "a: A"
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- shows "trans P (refl a) \<equiv> id (P a)"
- unfolding transport_def by reduce
-
-\<comment> \<open>TODO: Build transport automation!\<close>
-
-Lemma use_transport:
- assumes
- "p: y =\<^bsub>A\<^esub> x"
- "u: P x"
- "x: A" "y: A"
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- shows "trans P p\<inverse> u: P y"
- by typechk
-
-Lemma (derive) transport_left_inv:
- assumes
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- "x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
- shows "(trans P p\<inverse>) \<circ> (trans P p) = id (P x)"
- by (eq p) (reduce; intro)
-
-Lemma (derive) transport_right_inv:
- assumes
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- "x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
- shows "(trans P p) \<circ> (trans P p\<inverse>) = id (P y)"
- by (eq p) (reduce; intros)
-
-Lemma (derive) transport_pathcomp:
- assumes
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- "x: A" "y: A" "z: A"
- "u: P x"
- "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z"
- shows "trans P q (trans P p u) = trans P (p \<bullet> q) u"
- apply (eq p)
- focus prems vars x p
- apply (eq p)
- apply (reduce; intros)
- done
- done
-
-Lemma (derive) transport_compose_typefam:
- assumes
- "A: U i" "B: U i"
- "\<And>x. x: B \<Longrightarrow> P x: U i"
- "f: A \<rightarrow> B"
- "x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
- "u: P (f x)"
- shows "trans (\<lambda>x. P (f x)) p u = trans P f[p] u"
- by (eq p) (reduce; intros)
-
-Lemma (derive) transport_function_family:
- assumes
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- "\<And>x. x: A \<Longrightarrow> Q x: U i"
- "f: \<Prod>x: A. P x \<rightarrow> Q x"
- "x: A" "y: A"
- "u: P x"
- "p: x =\<^bsub>A\<^esub> y"
- shows "trans Q p ((f x) u) = (f y) (trans P p u)"
- by (eq p) (reduce; intros)
-
-Lemma (derive) transport_const:
- assumes
- "A: U i" "B: U i"
- "x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
- shows "\<Prod>b: B. trans (\<lambda>_. B) p b = b"
- by (intro, eq p) (reduce; intro)
-
-definition transport_const_i ("trans'_const")
- where [implicit]: "trans_const B p \<equiv> transport_const ? B ? ? p"
-
-translations "trans_const B p" \<leftharpoondown> "CONST transport_const A B x y p"
-
-Lemma transport_const_comp [comps]:
- assumes
- "x: A" "b: B"
- "A: U i" "B: U i"
- shows "trans_const B (refl x) b\<equiv> refl b"
- unfolding transport_const_def by reduce
-
-Lemma (derive) pathlift:
- assumes
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- "x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
- "u: P x"
- shows "<x, u> = <y, trans P p u>"
- by (eq p) (reduce; intros)
-
-definition pathlift_i ("lift")
- where [implicit]: "lift P p u \<equiv> pathlift ? P ? ? p u"
-
-translations "lift P p u" \<leftharpoondown> "CONST pathlift A P x y p u"
-
-Lemma pathlift_comp [comps]:
- assumes
- "u: P x"
- "x: A"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- "A: U i"
- shows "lift P (refl x) u \<equiv> refl <x, u>"
- unfolding pathlift_def by reduce
-
-Lemma (derive) pathlift_fst:
- assumes
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- "x: A" "y: A"
- "u: P x"
- "p: x =\<^bsub>A\<^esub> y"
- shows "fst[lift P p u] = p"
- apply (eq p)
- text \<open>Some rewriting needed here:\<close>
- \<guillemotright> vars x y
- (*Here an automatic reordering tactic would be helpful*)
- apply (rewrite at x in "x = y" fst_comp[symmetric])
- prefer 4
- apply (rewrite at y in "_ = y" fst_comp[symmetric])
- done
- \<guillemotright> by reduce intro
- done
-
-
-section \<open>Dependent paths\<close>
-
-Lemma (derive) apd:
- assumes
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- "f: \<Prod>x: A. P x"
- "x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
- shows "trans P p (f x) = f y"
- by (eq p) (reduce; intros; typechk)
-
-definition apd_i ("apd")
- where [implicit]: "apd f p \<equiv> Identity.apd ? ? f ? ? p"
-
-translations "apd f p" \<leftharpoondown> "CONST Identity.apd A P f x y p"
-
-Lemma dependent_map_comp [comps]:
- assumes
- "f: \<Prod>x: A. P x"
- "x: A"
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- shows "apd f (refl x) \<equiv> refl (f x)"
- unfolding apd_def by reduce
-
-Lemma (derive) apd_ap:
- assumes
- "A: U i" "B: U i"
- "f: A \<rightarrow> B"
- "x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
- shows "apd f p = trans_const B p (f x) \<bullet> f[p]"
- by (eq p) (reduce; intro)
-
-
-end
generated by cgit v1.2.3 (git 2.25.1) at 2024-07-25 12:23:03 +0000