diff options
Diffstat (limited to '')
| -rw-r--r-- | hott/Equivalence.thy | 23 |
1 files changed, 13 insertions, 10 deletions
diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy index d976677..d844b59 100644 --- a/hott/Equivalence.thy +++ b/hott/Equivalence.thy @@ -33,13 +33,16 @@ Lemma (derive) hsym: "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 intro - apply (rule pathinv) - \<guillemotright> by (elim H) - \<guillemotright> by typechk - done + "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: assumes @@ -71,9 +74,9 @@ Lemma (derive) commute_homotopy: assumes "A: U i" "B: U i" "x: A" "y: A" - "p: x =\<^bsub>A\<^esub> y" + "p: x = y" "f: A \<rightarrow> B" "g: A \<rightarrow> B" - "H: homotopy A (fn _. B) f g" + "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] @@ -94,7 +97,7 @@ Corollary (derive) commute_homotopy': "A: U i" "x: A" "f: A \<rightarrow> A" - "H: homotopy A (fn _. A) f (id A)" + "H: f ~ (id A)" shows "H (f x) = f [H x]" oops |