diff options
Diffstat (limited to 'hott/Equivalence.thy')
| -rw-r--r-- | hott/Equivalence.thy | 20 |
1 files changed, 6 insertions, 14 deletions
diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy index b29a213..a4eea93 100644 --- a/hott/Equivalence.thy +++ b/hott/Equivalence.thy @@ -40,7 +40,7 @@ Lemma (def) homotopy_refl [refl]: "f: \<Prod>x: A. B x" shows "f ~ f" unfolding homotopy_def - by intros + by intros fact Lemma (def) hsym: assumes @@ -205,11 +205,8 @@ Lemma is_qinv_components [type]: 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]] + using \<open>pf:_\<close>[unfolded is_qinv_def] \<comment> \<open>Should be unfolded by the typechecker\<close> 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 sec_of_is_qinv) \<^item> by (rule ret_of_is_qinv) @@ -233,7 +230,7 @@ Lemma (def) funcomp_is_qinv: 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 @@ -363,11 +360,6 @@ Lemma (def) equivalence_refl: show "is_biinv (id A)" by (rule is_biinv_if_is_qinv) (rule id_is_qinv) qed typechk -text \<open> - The following could perhaps be easier with transport (but then I think we need - univalence?)... -\<close> - Lemma (def) equivalence_symmetric: assumes "A: U i" "B: U i" shows "A \<simeq> B \<rightarrow> B \<simeq> A" @@ -408,8 +400,8 @@ Lemma 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. + The following proof is wordy because (1) the typechecker doesn't normalize, + and (2) we don't yet have universe level inference. \<close> Lemma (def) equiv_if_equal: @@ -417,7 +409,7 @@ Lemma (def) equiv_if_equal: "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 + apply intro defer \<^item> apply (eq p) \<^enum> vars A B apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric]) |