diff options
Diffstat (limited to 'hott')
| -rw-r--r-- | hott/Base.thy | 2 | ||||
| -rw-r--r-- | hott/Equivalence.thy | 416 | ||||
| -rw-r--r-- | hott/Identity.thy | 464 |
3 files changed, 881 insertions, 1 deletions
diff --git a/hott/Base.thy b/hott/Base.thy index 2a4ff9c..610a373 100644 --- a/hott/Base.thy +++ b/hott/Base.thy @@ -1,5 +1,5 @@ theory Base -imports Spartan.Equivalence +imports Equivalence begin diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy new file mode 100644 index 0000000..2975738 --- /dev/null +++ b/hott/Equivalence.thy @@ -0,0 +1,416 @@ +theory Equivalence +imports Identity + +begin + +section \<open>Homotopy\<close> + +definition "homotopy A B f g \<equiv> \<Prod>x: A. f `x =\<^bsub>B x\<^esub> g `x" + +definition homotopy_i (infix "~" 100) + where [implicit]: "f ~ g \<equiv> homotopy ? ? f g" + +translations "f ~ g" \<leftharpoondown> "CONST homotopy A B f g" + +Lemma homotopy_type [typechk]: + assumes + "A: U i" + "\<And>x. x: A \<Longrightarrow> B x: U i" + "f: \<Prod>x: A. B x" "g: \<Prod>x: A. B x" + shows "f ~ g: U i" + unfolding homotopy_def by typechk + +Lemma (derive) homotopy_refl [refl]: + assumes + "A: U i" + "f: \<Prod>x: A. B x" + shows "f ~ f" + unfolding homotopy_def by intros + +Lemma (derive) hsym: + assumes + "f: \<Prod>x: A. B x" + "g: \<Prod>x: A. B x" + "A: U i" + "\<And>x. x: A \<Longrightarrow> B x: U i" + shows "H: f ~ g \<Longrightarrow> g ~ f" + unfolding homotopy_def + apply intros + apply (rule pathinv) + \<guillemotright> by (elim H) + \<guillemotright> by typechk + done + +Lemma (derive) htrans: + assumes + "f: \<Prod>x: A. B x" + "g: \<Prod>x: A. B x" + "h: \<Prod>x: A. B x" + "A: U i" + "\<And>x. x: A \<Longrightarrow> B x: U i" + shows "\<lbrakk>H1: f ~ g; H2: g ~ h\<rbrakk> \<Longrightarrow> f ~ h" + unfolding homotopy_def + apply intro + \<guillemotright> vars x + apply (rule pathcomp[where ?y="g x"]) + \<^item> by (elim H1) + \<^item> by (elim H2) + done + \<guillemotright> by typechk + done + +text \<open>For calculations:\<close> + +lemmas + homotopy_sym [sym] = hsym[rotated 4] and + homotopy_trans [trans] = htrans[rotated 5] + +Lemma (derive) commute_homotopy: + assumes + "A: U i" "B: U i" + "x: A" "y: A" + "p: x =\<^bsub>A\<^esub> y" + "f: A \<rightarrow> B" "g: A \<rightarrow> B" + "H: homotopy A (\<lambda>_. B) f g" + shows "(H x) \<bullet> g[p] = f[p] \<bullet> (H y)" + \<comment> \<open>Need this assumption unfolded for typechecking:\<close> + supply assms(8)[unfolded homotopy_def] + apply (eq p) + focus vars x + apply reduce + \<comment> \<open>Here it would really be nice to have automation for transport and + propositional equality.\<close> + apply (rule use_transport[where ?y="H x \<bullet> refl (g x)"]) + \<guillemotright> by (rule pathcomp_right_refl) + \<guillemotright> by (rule pathinv) (rule pathcomp_left_refl) + \<guillemotright> by typechk + done + done + +Corollary (derive) commute_homotopy': + assumes + "A: U i" + "x: A" + "f: A \<rightarrow> A" + "H: homotopy A (\<lambda>_. A) f (id A)" + shows "H (f x) = f [H x]" +oops + +Lemma homotopy_id_left [typechk]: + assumes "A: U i" "B: U i" "f: A \<rightarrow> B" + shows "homotopy_refl A f: (id B) \<circ> f ~ f" + unfolding homotopy_refl_def homotopy_def by reduce + +Lemma homotopy_id_right [typechk]: + assumes "A: U i" "B: U i" "f: A \<rightarrow> B" + shows "homotopy_refl A f: f \<circ> (id A) ~ f" + unfolding homotopy_refl_def homotopy_def by reduce + +Lemma homotopy_funcomp_left: + assumes + "H: homotopy B C g g'" + "f: A \<rightarrow> B" + "g: \<Prod>x: B. C x" + "g': \<Prod>x: B. C x" + "A: U i" "B: U i" + "\<And>x. x: B \<Longrightarrow> C x: U i" + shows "homotopy A {} (g \<circ>\<^bsub>A\<^esub> f) (g' \<circ>\<^bsub>A\<^esub> f)" + unfolding homotopy_def + apply (intro; reduce) + apply (insert \<open>H: _\<close>[unfolded homotopy_def]) + apply (elim H) + done + +Lemma homotopy_funcomp_right: + assumes + "H: homotopy A (\<lambda>_. B) f f'" + "f: A \<rightarrow> B" + "f': A \<rightarrow> B" + "g: B \<rightarrow> C" + "A: U i" "B: U i" "C: U i" + shows "homotopy A {} (g \<circ>\<^bsub>A\<^esub> f) (g \<circ>\<^bsub>A\<^esub> f')" + unfolding homotopy_def + apply (intro; reduce) + apply (insert \<open>H: _\<close>[unfolded homotopy_def]) + apply (dest PiE, assumption) + apply (rule ap, assumption) + done + + +section \<open>Quasi-inverse and bi-invertibility\<close> + +subsection \<open>Quasi-inverses\<close> + +definition "qinv A B f \<equiv> \<Sum>g: B \<rightarrow> A. + homotopy A (\<lambda>_. A) (g \<circ>\<^bsub>A\<^esub> f) (id A) \<times> homotopy B (\<lambda>_. B) (f \<circ>\<^bsub>B\<^esub> g) (id B)" + +lemma qinv_type [typechk]: + assumes "A: U i" "B: U i" "f: A \<rightarrow> B" + shows "qinv A B f: U i" + unfolding qinv_def by typechk + +definition qinv_i ("qinv") + where [implicit]: "qinv f \<equiv> Equivalence.qinv ? ? f" + +translations "qinv f" \<leftharpoondown> "CONST Equivalence.qinv A B f" + +Lemma (derive) id_qinv: + assumes "A: U i" + shows "qinv (id A)" + unfolding qinv_def + apply intro defer + apply intro defer + apply (rule homotopy_id_right) + apply (rule homotopy_id_left) + done + +Lemma (derive) quasiinv_qinv: + assumes "A: U i" "B: U i" "f: A \<rightarrow> B" + shows "prf: qinv f \<Longrightarrow> qinv (fst prf)" + unfolding qinv_def + apply intro + \<guillemotright> by (rule \<open>f:_\<close>) + \<guillemotright> apply (elim "prf") + focus vars g HH + apply intro + \<^item> by reduce (snd HH) + \<^item> by reduce (fst HH) + done + done + done + +Lemma (derive) funcomp_qinv: + assumes + "A: U i" "B: U i" "C: U i" + "f: A \<rightarrow> B" "g: B \<rightarrow> C" + shows "qinv f \<rightarrow> qinv g \<rightarrow> qinv (g \<circ> f)" + apply (intros, unfold qinv_def, elims) + focus + prems prms + vars _ _ finv _ ginv _ HfA HfB HgB HgC + + apply intro + apply (rule funcompI[where ?f=ginv and ?g=finv]) + + text \<open>Now a whole bunch of rewrites and we're done.\<close> +oops + +subsection \<open>Bi-invertible maps\<close> + +definition "biinv A B f \<equiv> + (\<Sum>g: B \<rightarrow> A. homotopy A (\<lambda>_. A) (g \<circ>\<^bsub>A\<^esub> f) (id A)) + \<times> (\<Sum>g: B \<rightarrow> A. homotopy B (\<lambda>_. B) (f \<circ>\<^bsub>B\<^esub> g) (id B))" + +lemma biinv_type [typechk]: + assumes "A: U i" "B: U i" "f: A \<rightarrow> B" + shows "biinv A B f: U i" + unfolding biinv_def by typechk + +definition biinv_i ("biinv") + where [implicit]: "biinv f \<equiv> Equivalence.biinv ? ? f" + +translations "biinv f" \<leftharpoondown> "CONST Equivalence.biinv A B f" + +Lemma (derive) qinv_imp_biinv: + assumes + "A: U i" "B: U i" + "f: A \<rightarrow> B" + shows "?prf: qinv f \<rightarrow> biinv f" + apply intros + unfolding qinv_def biinv_def + by (rule Sig_dist_exp) + +text \<open> + Show that bi-invertible maps are quasi-inverses, as a demonstration of how to + work in this system. +\<close> + +Lemma (derive) biinv_imp_qinv: + assumes "A: U i" "B: U i" "f: A \<rightarrow> B" + shows "biinv f \<rightarrow> qinv f" + + text \<open>Split the hypothesis \<^term>\<open>biinv f\<close> into its components:\<close> + apply intro + unfolding biinv_def + apply elims + + text \<open>Name the components:\<close> + focus prems vars _ _ _ g H1 h H2 + thm \<open>g:_\<close> \<open>h:_\<close> \<open>H1:_\<close> \<open>H2:_\<close> + + text \<open> + \<^term>\<open>g\<close> is a quasi-inverse to \<^term>\<open>f\<close>, so the proof will be a triple + \<^term>\<open><g, <?H1, ?H2>>\<close>. + \<close> + unfolding qinv_def + apply intro + \<guillemotright> by (rule \<open>g: _\<close>) + \<guillemotright> apply intro + text \<open>The first part \<^prop>\<open>?H1: g \<circ> f ~ id A\<close> is given by \<^term>\<open>H1\<close>.\<close> + apply (rule \<open>H1: _\<close>) + + text \<open> + It remains to prove \<^prop>\<open>?H2: f \<circ> g ~ id B\<close>. First show that \<open>g ~ h\<close>, + then the result follows from @{thm \<open>H2: f \<circ> h ~ id B\<close>}. Here a proof + block is used to calculate "forward". + \<close> + proof - + have "g ~ g \<circ> (id B)" by reduce refl + also have "g \<circ> (id B) ~ g \<circ> f \<circ> h" + by (rule homotopy_funcomp_right) (rule \<open>H2:_\<close>[symmetric]) + also have "g \<circ> f \<circ> h ~ (id A) \<circ> h" + by (subst funcomp_assoc[symmetric]) + (rule homotopy_funcomp_left, rule \<open>H1:_\<close>) + also have "(id A) \<circ> h ~ h" by reduce refl + finally have "g ~ h" by this + + then have "f \<circ> g ~ f \<circ> h" by (rule homotopy_funcomp_right) + + with \<open>H2:_\<close> + show "f \<circ> g ~ id B" + by (rule homotopy_trans) (assumption+, typechk) + qed + done + done + +Lemma (derive) id_biinv: + "A: U i \<Longrightarrow> biinv (id A)" + by (rule qinv_imp_biinv) (rule id_qinv) + +Lemma (derive) funcomp_biinv: + assumes + "A: U i" "B: U i" "C: U i" + "f: A \<rightarrow> B" "g: B \<rightarrow> C" + shows "biinv f \<rightarrow> biinv g \<rightarrow> biinv (g \<circ> f)" + apply intros + focus vars biinv\<^sub>f biinv\<^sub>g + + text \<open>Follows from \<open>funcomp_qinv\<close>.\<close> +oops + + +section \<open>Equivalence\<close> + +text \<open> + Following the HoTT book, we first define equivalence in terms of + bi-invertibility. +\<close> + +definition equivalence (infix "\<simeq>" 110) + where "A \<simeq> B \<equiv> \<Sum>f: A \<rightarrow> B. Equivalence.biinv A B f" + +lemma equivalence_type [typechk]: + assumes "A: U i" "B: U i" + shows "A \<simeq> B: U i" + unfolding equivalence_def by typechk + +Lemma (derive) equivalence_refl: + assumes "A: U i" + shows "A \<simeq> A" + unfolding equivalence_def + apply intro defer + apply (rule qinv_imp_biinv) defer + apply (rule id_qinv) + done + +text \<open> + The following could perhaps be easier with transport (but then I think we need + univalence?)... +\<close> + +Lemma (derive) equivalence_symmetric: + assumes "A: U i" "B: U i" + shows "A \<simeq> B \<rightarrow> B \<simeq> A" + apply intros + unfolding equivalence_def + apply elim + \<guillemotright> vars _ f "prf" + apply (dest (4) biinv_imp_qinv) + apply intro + \<^item> unfolding qinv_def by (rule fst[of "(biinv_imp_qinv A B f) prf"]) + \<^item> by (rule qinv_imp_biinv) (rule quasiinv_qinv) + done + done + +Lemma (derive) equivalence_transitive: + assumes "A: U i" "B: U i" "C: U i" + shows "A \<simeq> B \<rightarrow> B \<simeq> C \<rightarrow> A \<simeq> C" + apply intros + unfolding equivalence_def + + text \<open>Use \<open>funcomp_biinv\<close>.\<close> +oops + +text \<open> + Equal types are equivalent. We give two proofs: the first by induction, and + the second by following the HoTT book and showing that transport is an + equivalence. +\<close> + +Lemma + assumes + "A: U i" "B: U i" "p: A =\<^bsub>U i\<^esub> B" + shows "A \<simeq> B" + by (eq p) (rule equivalence_refl) + +text \<open> + The following proof is wordy because (1) the typechecker doesn't rewrite, and + (2) we don't yet have universe automation. +\<close> + +Lemma (derive) id_imp_equiv: + assumes + "A: U i" "B: U i" "p: A =\<^bsub>U i\<^esub> B" + shows "<trans (id (U i)) p, ?isequiv>: A \<simeq> B" + unfolding equivalence_def + apply intros defer + + \<comment> \<open>Switch off auto-typechecking, which messes with universe levels\<close> + supply [[auto_typechk=false]] + + \<guillemotright> apply (eq p, typechk) + \<^item> prems vars A B + apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric]) + apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric]) + apply (rule transport, rule U_in_U) + apply (rule lift_universe_codomain, rule U_in_U) + apply (typechk, rule U_in_U) + done + \<^item> prems vars A + apply (subst transport_comp) + \<^enum> by (rule U_in_U) + \<^enum> by reduce (rule lift_universe) + \<^enum> by reduce (rule id_biinv) + done + done + + \<guillemotright> \<comment> \<open>Similar proof as in the first subgoal above\<close> + apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric]) + apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric]) + apply (rule transport, rule U_in_U) + apply (rule lift_universe_codomain, rule U_in_U) + apply (typechk, rule U_in_U) + done + done + +(*Uncomment this to see all implicits from here on. +no_translations + "f x" \<leftharpoondown> "f `x" + "x = y" \<leftharpoondown> "x =\<^bsub>A\<^esub> y" + "g \<circ> f" \<leftharpoondown> "g \<circ>\<^bsub>A\<^esub> f" + "p\<inverse>" \<leftharpoondown> "CONST pathinv A x y p" + "p \<bullet> q" \<leftharpoondown> "CONST pathcomp A x y z p q" + "fst" \<leftharpoondown> "CONST Spartan.fst A B" + "snd" \<leftharpoondown> "CONST Spartan.snd A B" + "f[p]" \<leftharpoondown> "CONST ap A B x y f p" + "trans P p" \<leftharpoondown> "CONST transport A P x y p" + "trans_const B p" \<leftharpoondown> "CONST transport_const A B x y p" + "lift P p u" \<leftharpoondown> "CONST pathlift A P x y p u" + "apd f p" \<leftharpoondown> "CONST Identity.apd A P f x y p" + "f ~ g" \<leftharpoondown> "CONST homotopy A B f g" + "qinv f" \<leftharpoondown> "CONST Equivalence.qinv A B f" + "biinv f" \<leftharpoondown> "CONST Equivalence.biinv A B f" +*) + + +end diff --git a/hott/Identity.thy b/hott/Identity.thy new file mode 100644 index 0000000..3a982f6 --- /dev/null +++ b/hott/Identity.thy @@ -0,0 +1,464 @@ +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 |