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Diffstat (limited to 'spartan/theories/Equivalence.thy')
-rw-r--r-- | spartan/theories/Equivalence.thy | 416 |
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diff --git a/spartan/theories/Equivalence.thy b/spartan/theories/Equivalence.thy deleted file mode 100644 index 2975738..0000000 --- a/spartan/theories/Equivalence.thy +++ /dev/null @@ -1,416 +0,0 @@ -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 |