diff options
Diffstat (limited to 'hott/Equivalence.thy')
| -rw-r--r-- | hott/Equivalence.thy | 48 |
1 files changed, 6 insertions, 42 deletions
diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy index a57ed44..99300a0 100644 --- a/hott/Equivalence.thy +++ b/hott/Equivalence.thy @@ -144,8 +144,6 @@ Lemma (def) commute_homotopy: apply (transport eq: pathcomp_refl, transport eq: refl_pathcomp) by refl -\<comment> \<open>TODO: *Really* need normalization during type-checking and better equality - type rewriting to do the proof below properly\<close> Corollary (def) commute_homotopy': assumes "A: U i" @@ -154,21 +152,18 @@ Corollary (def) commute_homotopy': "H: f ~ (id A)" shows "H (f x) = f [H x]" proof - + (*FIXME: Bug; if the following type declaration isn't made then bad things + happen on the next lines.*) from \<open>H: f ~ id A\<close> have [type]: "H: \<Prod>x: A. f x = x" by (reduce add: homotopy_def) - have *: "(id A)[H x]: f x = x" - by (rewrite at "\<hole> = _" id_comp[symmetric], - rewrite at "_ = \<hole>" id_comp[symmetric]) - have "H (f x) \<bullet> H x = H (f x) \<bullet> (id A)[H x]" - by (rule left_whisker, fact *, transport eq: ap_id) (reduce+, refl) + by (rule left_whisker, transport eq: ap_id, refl) also have [simplified id_comp]: "H (f x) \<bullet> (id A)[H x] = f[H x] \<bullet> H x" by (rule commute_homotopy) - finally have *: "{}" using * by this + finally have "{}" by this - show "H (f x) = f [H x]" - by pathcomp_cancelr (fact, typechk+) + thus "H (f x) = f [H x]" by pathcomp_cancelr (fact, typechk+) qed @@ -309,23 +304,11 @@ Lemma (def) is_qinv_if_is_biinv: assumes "A: U i" "B: U i" "f: A \<rightarrow> B" shows "is_biinv f \<rightarrow> is_qinv f" apply intro - - 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 for calculational reasoning. - \<close> \<^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]) @@ -437,7 +420,7 @@ Lemma (def) equiv_if_equal: by (rule lift_universe_codomain, rule Ui_in_USi) \<^enum> vars A using [[solve_side_conds=1]] - apply (subst transport_comp) + apply (rewrite transport_comp) \<circ> by (rule Ui_in_USi) \<circ> by reduce (rule in_USi_if_in_Ui) \<circ> by reduce (rule id_is_biinv) @@ -452,24 +435,5 @@ Lemma (def) equiv_if_equal: by (rule lift_universe_codomain, rule Ui_in_USi) 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" - "is_qinv f" \<leftharpoondown> "CONST Equivalence.is_qinv A B f" - "is_biinv f" \<leftharpoondown> "CONST Equivalence.is_biinv A B f" -*) - end |