diff options
Diffstat (limited to 'hott/Equivalence.thy')
| -rw-r--r-- | hott/Equivalence.thy | 456 |
1 files changed, 249 insertions, 207 deletions
diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy index d844b59..66c64f6 100644 --- a/hott/Equivalence.thy +++ b/hott/Equivalence.thy @@ -16,53 +16,68 @@ 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" + "f: \<Prod>x: A. B x" + "g: \<Prod>x: A. B x" shows "f ~ g: U i" - unfolding homotopy_def by typechk + unfolding homotopy_def + by typechk -Lemma (derive) homotopy_refl [refl]: +Lemma apply_homotopy: + 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" + "H: f ~ g" + "x: A" + shows "H x: f x = g x" + using \<open>H:_\<close> unfolding homotopy_def + by typechk + +method htpy for H::o = rule apply_homotopy[where ?H=H] + +Lemma (def) homotopy_refl [refl]: assumes "A: U i" "f: \<Prod>x: A. B x" shows "f ~ f" - unfolding homotopy_def by intros + unfolding homotopy_def + by intros -Lemma (derive) hsym: +Lemma (def) 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" + "f: \<Prod>x: A. B x" + "g: \<Prod>x: A. B x" "H: f ~ g" shows "g ~ f" -unfolding homotopy_def -proof intro - fix x assume "x: A" then have "H x: f x = g x" - using \<open>H:_\<close>[unfolded homotopy_def] - \<comment> \<open>this should become unnecessary when definitional unfolding is implemented\<close> - by typechk - thus "g x = f x" by (rule pathinv) fact -qed typechk - -Lemma (derive) htrans: + unfolding homotopy_def + proof intro + fix x assume "x: A" then have "f x = g x" + by (htpy H) + thus "g x = f x" + by (rule pathinv) fact + qed + +Lemma (def) htrans: 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" "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" + "H1: f ~ g" + "H2: g ~ h" + shows "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 + proof intro + fix x assume "x: A" + have *: "f x = g x" "g x = h x" + by (htpy H1, htpy H2) + show "f x = h x" + by (rule pathcomp; (rule *)?) typechk + qed -text \<open>For calculations:\<close> +section \<open>Rewriting homotopies\<close> congruence "f ~ g" rhs g @@ -70,229 +85,258 @@ 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 = y" - "f: A \<rightarrow> B" "g: A \<rightarrow> B" - "H: 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_refl) - \<guillemotright> by (rule pathinv) (rule refl_pathcomp) - \<guillemotright> by typechk - done - done - -Corollary (derive) commute_homotopy': - assumes - "A: U i" - "x: A" - "f: A \<rightarrow> A" - "H: f ~ (id A)" - shows "H (f x) = f [H x]" -oops - -Lemma homotopy_id_left [typechk]: +Lemma id_funcomp_htpy: 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 + by reduce -Lemma homotopy_id_right [typechk]: +Lemma funcomp_id_htpy: 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 + by reduce -Lemma homotopy_funcomp_left: +Lemma funcomp_left_htpy: assumes - "H: homotopy B C g g'" + "A: U i" "B: U i" + "\<And>x. x: B \<Longrightarrow> C x: U i" "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)" + "H: g ~ g'" + shows "(g \<circ> f) ~ (g' \<circ> f)" unfolding homotopy_def - apply (intro; reduce) - apply (insert \<open>H: _\<close>[unfolded homotopy_def]) - apply (elim H) + apply (intro, reduce) + apply (htpy H) done -Lemma homotopy_funcomp_right: +Lemma funcomp_right_htpy: assumes - "H: homotopy A (fn _. B) f f'" + "A: U i" "B: U i" "C: U i" "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')" + "H: f ~ f'" + shows "(g \<circ> f) ~ (g \<circ> f')" unfolding homotopy_def - apply (intro; reduce) - apply (insert \<open>H: _\<close>[unfolded homotopy_def]) - apply (dest PiE, assumption) - apply (rule ap, assumption) - done + proof (intro, reduce) + fix x assume "x: A" + have *: "f x = f' x" + by (htpy H) + show "g (f x) = g (f' x)" + by (transport eq: *) refl + qed -method id_htpy = (rule homotopy_id_left) -method htpy_id = (rule homotopy_id_right) -method htpy_o = (rule homotopy_funcomp_left) -method o_htpy = (rule homotopy_funcomp_right) +method lhtpy = rule funcomp_left_htpy[rotated 6] +method rhtpy = rule funcomp_right_htpy[rotated 6] + + +Lemma (def) commute_homotopy: + assumes + "A: U i" "B: U i" + "x: A" "y: A" + "p: x = y" + "f: A \<rightarrow> B" "g: A \<rightarrow> B" + "H: f ~ g" + shows "(H x) \<bullet> g[p] = f[p] \<bullet> (H y)" + using \<open>H:_\<close> + unfolding homotopy_def + apply (eq p, reduce) + apply (transport eq: pathcomp_refl, transport eq: refl_pathcomp) + by refl + +Corollary (def) commute_homotopy': + assumes + "A: U i" + "x: A" + "f: A \<rightarrow> A" + "H: f ~ (id A)" + shows "H (f x) = f [H x]" +oops section \<open>Quasi-inverse and bi-invertibility\<close> subsection \<open>Quasi-inverses\<close> -definition "qinv A B f \<equiv> \<Sum>g: B \<rightarrow> A. +definition "is_qinv A B f \<equiv> \<Sum>g: B \<rightarrow> A. homotopy A (fn _. A) (g \<circ>\<^bsub>A\<^esub> f) (id A) \<times> homotopy B (fn _. B) (f \<circ>\<^bsub>B\<^esub> g) (id B)" -lemma qinv_type [typechk]: +lemma is_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 + shows "is_qinv A B f: U i" + unfolding is_qinv_def + by typechk -definition qinv_i ("qinv") - where [implicit]: "qinv f \<equiv> Equivalence.qinv ? ? f" +definition is_qinv_i ("is'_qinv") + where [implicit]: "is_qinv f \<equiv> Equivalence.is_qinv ? ? f" -translations "qinv f" \<leftharpoondown> "CONST Equivalence.qinv A B f" +no_translations "is_qinv f" \<leftharpoondown> "CONST Equivalence.is_qinv A B f" -Lemma (derive) id_qinv: +Lemma (def) id_is_qinv: assumes "A: U i" - shows "qinv (id A)" - unfolding qinv_def + shows "is_qinv (id A)" + unfolding is_qinv_def proof intro show "id A: A \<rightarrow> A" by typechk qed (reduce, intro; refl) -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 +Lemma is_qinvI: + assumes + "A: U i" "B: U i" "f: A \<rightarrow> B" + "g: B \<rightarrow> A" + "H1: g \<circ> f ~ id A" + "H2: f \<circ> g ~ id B" + shows "is_qinv f" + unfolding is_qinv_def + proof intro + show "g: B \<rightarrow> A" by fact + show "g \<circ> f ~ id A \<and> f \<circ> g ~ id B" by (intro; fact) + qed + +Lemma is_qinv_components [typechk]: + assumes "A: U i" "B: U i" "f: A \<rightarrow> B" "pf: is_qinv f" + shows + fst_of_is_qinv: "fst pf: B \<rightarrow> A" and + p\<^sub>2\<^sub>1_of_is_qinv: "p\<^sub>2\<^sub>1 pf: fst pf \<circ> f ~ id A" and + p\<^sub>2\<^sub>2_of_is_qinv: "p\<^sub>2\<^sub>2 pf: f \<circ> fst pf ~ id B" + using assms unfolding is_qinv_def + by typechk+ + +Lemma (def) qinv_is_qinv: + assumes "A: U i" "B: U i" "f: A \<rightarrow> B" "pf: is_qinv f" + shows "is_qinv (fst pf)" +using [[debug_typechk]] + using [[solve_side_conds=2]] + apply (rule is_qinvI) + back \<comment> \<open>Typechecking/inference goes too far here. Problem would likely be + solved with definitional unfolding\<close> + \<^item> by (fact \<open>f:_\<close>) + \<^item> by (rule p\<^sub>2\<^sub>2_of_is_qinv) + \<^item> by (rule p\<^sub>2\<^sub>1_of_is_qinv) done -Lemma (derive) funcomp_qinv: +Lemma (def) funcomp_is_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 vars _ _ finv _ ginv - apply (intro, rule funcompI[where ?f=ginv and ?g=finv]) - proof (reduce, intro) - have "finv \<circ> ginv \<circ> g \<circ> f ~ finv \<circ> (ginv \<circ> g) \<circ> f" by reduce refl - also have ".. ~ finv \<circ> id B \<circ> f" by (o_htpy, htpy_o) fact - also have ".. ~ id A" by reduce fact - finally show "finv \<circ> ginv \<circ> g \<circ> f ~ id A" by this - - have "g \<circ> f \<circ> finv \<circ> ginv ~ g \<circ> (f \<circ> finv) \<circ> ginv" by reduce refl - also have ".. ~ g \<circ> id B \<circ> ginv" by (o_htpy, htpy_o) fact - also have ".. ~ id C" by reduce fact - finally show "g \<circ> f \<circ> finv \<circ> ginv ~ id C" by this - qed + shows "is_qinv f \<rightarrow> is_qinv g \<rightarrow> is_qinv (g \<circ> f)" + apply intros + unfolding is_qinv_def apply elims + focus vars _ _ finv _ ginv + apply intro + \<^item> by (rule funcompI[where ?f=ginv and ?g=finv]) + \<^item> proof intro + show "(finv \<circ> ginv) \<circ> g \<circ> f ~ id A" + proof - + have "(finv \<circ> ginv) \<circ> g \<circ> f ~ finv \<circ> (ginv \<circ> g) \<circ> f" by reduce refl + also have ".. ~ finv \<circ> id B \<circ> f" by (rhtpy, lhtpy) fact + also have ".. ~ id A" by reduce fact + finally show "{}" by this + qed + + show "(g \<circ> f) \<circ> finv \<circ> ginv ~ id C" + proof - + have "(g \<circ> f) \<circ> finv \<circ> ginv ~ g \<circ> (f \<circ> finv) \<circ> ginv" by reduce refl + also have ".. ~ g \<circ> id B \<circ> ginv" by (rhtpy, lhtpy) fact + also have ".. ~ id C" by reduce fact + finally show "{}" by this + qed + qed + done done subsection \<open>Bi-invertible maps\<close> -definition "biinv A B f \<equiv> +definition "is_biinv A B f \<equiv> (\<Sum>g: B \<rightarrow> A. homotopy A (fn _. A) (g \<circ>\<^bsub>A\<^esub> f) (id A)) \<times> (\<Sum>g: B \<rightarrow> A. homotopy B (fn _. B) (f \<circ>\<^bsub>B\<^esub> g) (id B))" -lemma biinv_type [typechk]: +lemma is_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 + shows "is_biinv A B f: U i" + unfolding is_biinv_def by typechk -definition biinv_i ("biinv") - where [implicit]: "biinv f \<equiv> Equivalence.biinv ? ? f" +definition is_biinv_i ("is'_biinv") + where [implicit]: "is_biinv f \<equiv> Equivalence.is_biinv ? ? f" -translations "biinv f" \<leftharpoondown> "CONST Equivalence.biinv A B f" +translations "is_biinv f" \<leftharpoondown> "CONST Equivalence.is_biinv A B f" -Lemma (derive) qinv_imp_biinv: +Lemma is_biinvI: 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> + "A: U i" "B: U i" "f: A \<rightarrow> B" + "g: B \<rightarrow> A" "h: B \<rightarrow> A" + "H1: g \<circ> f ~ id A" "H2: f \<circ> h ~ id B" + shows "is_biinv f" + unfolding is_biinv_def + proof intro + show "<g, H1>: {}" by typechk + show "<h, H2>: {}" by typechk + qed -Lemma (derive) biinv_imp_qinv: +Lemma is_biinv_components [typechk]: + assumes "A: U i" "B: U i" "f: A \<rightarrow> B" "pf: is_biinv f" + shows + p\<^sub>1\<^sub>1_of_is_biinv: "p\<^sub>1\<^sub>1 pf: B \<rightarrow> A" and + p\<^sub>2\<^sub>1_of_is_biinv: "p\<^sub>2\<^sub>1 pf: B \<rightarrow> A" and + p\<^sub>1\<^sub>2_of_is_biinv: "p\<^sub>1\<^sub>2 pf: p\<^sub>1\<^sub>1 pf \<circ> f ~ id A" and + p\<^sub>2\<^sub>2_of_is_biinv: "p\<^sub>2\<^sub>2 pf: f \<circ> p\<^sub>2\<^sub>1 pf ~ id B" + using assms unfolding is_biinv_def + by typechk+ + +Lemma (def) is_qinv_imp_is_biinv: assumes "A: U i" "B: U i" "f: A \<rightarrow> B" - shows "biinv f \<rightarrow> qinv f" + shows "is_qinv f \<rightarrow> is_biinv f" + apply intros + unfolding is_qinv_def is_biinv_def + by (rule distribute_Sig) - 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 +Lemma (def) is_biinv_imp_is_qinv: + assumes "A: U i" "B: U i" "f: A \<rightarrow> B" + shows "is_biinv f \<rightarrow> is_qinv f" apply intro - \<guillemotright> by (fact \<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 (fact \<open>H1: _\<close>) + text \<open>Split the hypothesis \<^term>\<open>is_biinv f\<close> into its components and name them.\<close> + unfolding is_biinv_def apply elims + focus vars _ _ _ g H1 h H2 + text \<open>Need to give the required function and homotopies.\<close> + apply (rule is_qinvI) + text \<open>We claim that \<^term>\<open>g\<close> is a quasi-inverse to \<^term>\<open>f\<close>.\<close> + \<^item> by (fact \<open>g: _\<close>) + + text \<open>The first part \<^prop>\<open>?H1: g \<circ> f ~ id A\<close> is given by \<^term>\<open>H1\<close>.\<close> + \<^item> by (fact \<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". + block is used for calculational reasoning. \<close> - proof - - have "g ~ g \<circ> (id B)" by reduce refl - also have ".. ~ g \<circ> f \<circ> h" by o_htpy (rule \<open>H2:_\<close>[symmetric]) - also have ".. ~ (id A) \<circ> h" by (subst funcomp_assoc[symmetric]) (htpy_o, fact) - also have ".. ~ h" by reduce refl - finally have "g ~ h" by this - then have "f \<circ> g ~ f \<circ> h" by (o_htpy, this) - also note \<open>H2:_\<close> - finally show "f \<circ> g ~ id B" by this - qed + \<^item> proof - + have "g ~ g \<circ> (id B)" by reduce refl + also have ".. ~ g \<circ> f \<circ> h" by rhtpy (rule \<open>H2:_\<close>[symmetric]) + also have ".. ~ (id A) \<circ> h" by (rewrite funcomp_assoc[symmetric]) (lhtpy, fact) + also have ".. ~ h" by reduce refl + finally have "g ~ h" by this + then have "f \<circ> g ~ f \<circ> h" by (rhtpy, this) + also note \<open>H2:_\<close> + finally show "f \<circ> g ~ id B" by this + qed done done -Lemma (derive) id_biinv: - "A: U i \<Longrightarrow> biinv (id A)" - by (rule qinv_imp_biinv) (rule id_qinv) +Lemma (def) id_is_biinv: + "A: U i \<Longrightarrow> is_biinv (id A)" + by (rule is_qinv_imp_is_biinv) (rule id_is_qinv) -Lemma (derive) funcomp_biinv: +Lemma (def) funcomp_is_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)" + shows "is_biinv f \<rightarrow> is_biinv g \<rightarrow> is_biinv (g \<circ> f)" apply intros - focus vars biinv\<^sub>f biinv\<^sub>g - by (rule qinv_imp_biinv) (rule funcomp_qinv; rule biinv_imp_qinv) + focus vars pf pg + by (rule is_qinv_imp_is_biinv) + (rule funcomp_is_qinv; rule is_biinv_imp_is_qinv, fact) done @@ -304,19 +348,19 @@ text \<open> \<close> definition equivalence (infix "\<simeq>" 110) - where "A \<simeq> B \<equiv> \<Sum>f: A \<rightarrow> B. Equivalence.biinv A B f" + where "A \<simeq> B \<equiv> \<Sum>f: A \<rightarrow> B. Equivalence.is_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: +Lemma (def) equivalence_refl: assumes "A: U i" shows "A \<simeq> A" unfolding equivalence_def proof intro - show "biinv (id A)" by (rule qinv_imp_biinv) (rule id_qinv) + show "is_biinv (id A)" by (rule is_qinv_imp_is_biinv) (rule id_is_qinv) qed typechk text \<open> @@ -324,32 +368,30 @@ text \<open> univalence?)... \<close> -Lemma (derive) equivalence_symmetric: +Lemma (def) 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 + apply (dest (4) is_biinv_imp_is_qinv) + apply intro + \<^item> by (rule fst_of_is_qinv) facts + \<^item> by (rule is_qinv_imp_is_biinv) (rule qinv_is_qinv) done -Lemma (derive) equivalence_transitive: +Lemma (def) equivalence_transitive: assumes "A: U i" "B: U i" "C: U i" shows "A \<simeq> B \<rightarrow> B \<simeq> C \<rightarrow> A \<simeq> C" proof intros fix AB BC assume *: "AB: A \<simeq> B" "BC: B \<simeq> C" then have - "fst AB: A \<rightarrow> B" and 1: "snd AB: biinv (fst AB)" - "fst BC: B \<rightarrow> C" and 2: "snd BC: biinv (fst BC)" + "fst AB: A \<rightarrow> B" and 1: "snd AB: is_biinv (fst AB)" + "fst BC: B \<rightarrow> C" and 2: "snd BC: is_biinv (fst BC)" unfolding equivalence_def by typechk+ then have "fst BC \<circ> fst AB: A \<rightarrow> C" by typechk - moreover have "biinv (fst BC \<circ> fst AB)" - using * unfolding equivalence_def by (rules funcomp_biinv 1 2) + moreover have "is_biinv (fst BC \<circ> fst AB)" + using * unfolding equivalence_def by (rule funcomp_is_biinv 1 2) facts ultimately show "A \<simeq> C" unfolding equivalence_def by intro facts qed @@ -370,14 +412,14 @@ text \<open> (2) we don't yet have universe automation. \<close> -Lemma (derive) id_imp_equiv: +Lemma (def) 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 - \<guillemotright> apply (eq p) - \<^item> prems vars A B + \<^item> apply (eq p) + \<^enum> vars A B apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric]) using [[solve_side_conds=1]] apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric]) @@ -385,16 +427,16 @@ Lemma (derive) id_imp_equiv: apply (rule lift_universe_codomain, rule Ui_in_USi) apply (typechk, rule Ui_in_USi) done - \<^item> prems vars A + \<^enum> vars A using [[solve_side_conds=1]] apply (subst transport_comp) - \<^enum> by (rule Ui_in_USi) - \<^enum> by reduce (rule in_USi_if_in_Ui) - \<^enum> by reduce (rule id_biinv) + \<circ> by (rule Ui_in_USi) + \<circ> by reduce (rule in_USi_if_in_Ui) + \<circ> by reduce (rule id_is_biinv) done done - \<guillemotright> \<comment> \<open>Similar proof as in the first subgoal above\<close> + \<^item> \<comment> \<open>Similar proof as in the first subgoal above\<close> apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric]) using [[solve_side_conds=1]] apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric]) @@ -419,8 +461,8 @@ no_translations "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" + "is_qinv f" \<leftharpoondown> "CONST Equivalence.is_qinv A B f" + "is_biinv f" \<leftharpoondown> "CONST Equivalence.is_biinv A B f" *) |