diff options
Diffstat (limited to 'hott/Equivalence.thy')
| -rw-r--r-- | hott/Equivalence.thy | 58 |
1 files changed, 25 insertions, 33 deletions
diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy index 379dc81..9fe11a7 100644 --- a/hott/Equivalence.thy +++ b/hott/Equivalence.thy @@ -88,12 +88,12 @@ lemmas 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" - by reduce + by compute 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" - by reduce + by compute Lemma funcomp_left_htpy: assumes @@ -105,7 +105,7 @@ Lemma funcomp_left_htpy: "H: g ~ g'" shows "(g \<circ> f) ~ (g' \<circ> f)" unfolding homotopy_def - apply (intro, reduce) + apply (intro, compute) apply (htpy H) done @@ -118,12 +118,12 @@ Lemma funcomp_right_htpy: "H: f ~ f'" shows "(g \<circ> f) ~ (g \<circ> f')" unfolding homotopy_def - proof (intro, reduce) + proof (intro, compute) fix x assuming "x: A" have *: "f x = f' x" by (htpy H) show "g (f x) = g (f' x)" - by (transport eq: *) refl + by (rewr eq: *) refl qed method lhtpy = rule funcomp_left_htpy[rotated 6] @@ -139,8 +139,8 @@ Lemma (def) commute_homotopy: 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) + apply (eq p, compute) + apply (rewr eq: pathcomp_refl, rewr eq: refl_pathcomp) by refl Corollary (def) commute_homotopy': @@ -151,21 +151,13 @@ Corollary (def) commute_homotopy': "H: f ~ (id A)" shows "H (f x) = f [H x]" proof - - (* Rewrite the below proof - have *: "H (f x) \<bullet> (id A)[H x] = f[H x] \<bullet> H x" - using \<open>H:_\<close> unfolding homotopy_def by (rule commute_homotopy) - - thus "H (f x) = f[H x]" - apply (pathcomp_cancelr) - ... - *) - (*FUTURE: Because we don't have very good normalization integrated into + (*FUTURE: Because we don't have very good normalization integrated into things yet, we need to manually unfold the type of H.*) from \<open>H: f ~ id A\<close> have [type]: "H: \<Prod>x: A. f x = x" - by (reduce add: homotopy_def) + by (compute add: homotopy_def) have "H (f x) \<bullet> H x = H (f x) \<bullet> (id A)[H x]" - by (rule left_whisker, transport eq: ap_id, refl) + by (rule left_whisker, rewr 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 "?" by this @@ -198,7 +190,7 @@ Lemma (def) id_is_qinv: unfolding is_qinv_def proof intro show "id A: A \<rightarrow> A" by typechk - qed (reduce, intro; refl) + qed (compute, intro; refl) Lemma is_qinvI: assumes @@ -245,17 +237,17 @@ Lemma (def) funcomp_is_qinv: \<^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 + have "(finv \<circ> ginv) \<circ> g \<circ> f ~ finv \<circ> (ginv \<circ> g) \<circ> f" by compute refl also have ".. ~ finv \<circ> id B \<circ> f" by (rhtpy, lhtpy) fact - also have ".. ~ id A" by reduce fact + also have ".. ~ id A" by compute 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 + have "(g \<circ> f) \<circ> finv \<circ> ginv ~ g \<circ> (f \<circ> finv) \<circ> ginv" by compute refl also have ".. ~ g \<circ> id B \<circ> ginv" by (rhtpy, lhtpy) fact - also have ".. ~ id C" by reduce fact + also have ".. ~ id C" by compute fact finally show "?" by this qed qed @@ -317,10 +309,10 @@ Lemma (def) is_qinv_if_is_biinv: \<^item> by (fact \<open>g: _\<close>) \<^item> by (fact \<open>H1: _\<close>) \<^item> proof - - have "g ~ g \<circ> (id B)" by reduce refl + have "g ~ g \<circ> (id B)" by compute 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 + also have ".. ~ (id A) \<circ> h" by (comp funcomp_assoc[symmetric]) (lhtpy, fact) + also have ".. ~ h" by compute refl finally have "g ~ h" by this then have "f \<circ> g ~ f \<circ> h" by (rhtpy, this) also note \<open>H2:_\<close> @@ -420,24 +412,24 @@ Lemma (def) equiv_if_equal: apply intro defer \<^item> apply (eq p) \<^enum> vars A B - apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric]) + apply (comp at A in "A \<rightarrow> B" id_comp[symmetric]) using [[solve_side_conds=1]] - apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric]) + apply (comp at B in "_ \<rightarrow> B" id_comp[symmetric]) apply (rule transport, rule Ui_in_USi) by (rule lift_universe_codomain, rule Ui_in_USi) \<^enum> vars A using [[solve_side_conds=1]] - apply (rewrite transport_comp) + apply (comp transport_comp) \<circ> by (rule Ui_in_USi) - \<circ> by reduce (rule U_lift) - \<circ> by reduce (rule id_is_biinv) + \<circ> by compute (rule U_lift) + \<circ> by compute (rule id_is_biinv) done done \<^item> \<comment> \<open>Similar proof as in the first subgoal above\<close> - apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric]) + apply (comp at A in "A \<rightarrow> B" id_comp[symmetric]) using [[solve_side_conds=1]] - apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric]) + apply (comp at B in "_ \<rightarrow> B" id_comp[symmetric]) apply (rule transport, rule Ui_in_USi) by (rule lift_universe_codomain, rule Ui_in_USi) done |