diff options
author | Josh Chen | 2020-05-25 17:09:54 +0200 |
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committer | Josh Chen | 2020-05-25 17:09:54 +0200 |
commit | 80edbd08e13200d2c080ac281d19948bbbcd92e0 (patch) | |
tree | 95cc00c52c846406e04cd985ef9df2d5a1e9996c /spartan/theories/Identity.thy | |
parent | 22c5b895a4a2ba0ecb97a5c7ccab4b13c42c24e3 (diff) |
Reorganize theory structure. In particular, the identity type moves out from under Spartan to HoTT. Spartan now only has Pi and Sigma.
Diffstat (limited to 'spartan/theories/Identity.thy')
-rw-r--r-- | spartan/theories/Identity.thy | 464 |
1 files changed, 0 insertions, 464 deletions
diff --git a/spartan/theories/Identity.thy b/spartan/theories/Identity.thy deleted file mode 100644 index 3a982f6..0000000 --- a/spartan/theories/Identity.thy +++ /dev/null @@ -1,464 +0,0 @@ -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 |