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-rw-r--r--hott/Equivalence.thy24
-rw-r--r--hott/Identity.thy17
2 files changed, 22 insertions, 19 deletions
diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy
index 88adc8b..d976677 100644
--- a/hott/Equivalence.thy
+++ b/hott/Equivalence.thy
@@ -338,18 +338,18 @@ Lemma (derive) equivalence_symmetric:
Lemma (derive) 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"
- let "?f: {}" = "(fst AB) :: o" *)
- apply intros
- unfolding equivalence_def
- focus vars p q apply (elim p, elim q)
- focus vars f biinv\<^sub>f g biinv\<^sub>g apply intro
- \<guillemotright> apply (rule funcompI) defer by assumption+ known
- \<guillemotright> by (rule funcomp_biinv)
- done
- done
- done
+ 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)"
+ 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)
+ ultimately show "A \<simeq> C"
+ unfolding equivalence_def by intro facts
+ qed
text \<open>
Equal types are equivalent. We give two proofs: the first by induction, and
diff --git a/hott/Identity.thy b/hott/Identity.thy
index 1cb3946..29ce26a 100644
--- a/hott/Identity.thy
+++ b/hott/Identity.thy
@@ -49,6 +49,9 @@ lemmas
section \<open>Path induction\<close>
+\<comment> \<open>With `p: x = y` in the context the invokation `eq p` is essentially
+ `elim p x y`, with some extra bits before and after.\<close>
+
method_setup eq =
\<open>Args.term >> (fn tm => K (CONTEXT_METHOD (
CHEADGOAL o SIDE_CONDS (
@@ -155,12 +158,12 @@ translations
Lemma lu_refl [comp]:
assumes "A: U i" "x: A"
shows "lu (refl x) \<equiv> refl (refl x)"
- unfolding refl_pathcomp_def by reduce+
+ unfolding refl_pathcomp_def by reduce
Lemma ru_refl [comp]:
assumes "A: U i" "x: A"
shows "ru (refl x) \<equiv> refl (refl x)"
- unfolding pathcomp_refl_def by reduce+
+ unfolding pathcomp_refl_def by reduce
Lemma (derive) inv_pathcomp:
assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
@@ -374,7 +377,7 @@ Lemma transport_const_comp [comp]:
"x: A" "b: B"
"A: U i" "B: U i"
shows "trans_const B (refl x) b\<equiv> refl b"
- unfolding transport_const_def by reduce+
+ unfolding transport_const_def by reduce
Lemma (derive) pathlift:
assumes
@@ -398,7 +401,7 @@ Lemma pathlift_comp [comp]:
"\<And>x. x: A \<Longrightarrow> P x: U i"
"A: U i"
shows "lift P (refl x) u \<equiv> refl <x, u>"
- unfolding pathlift_def by reduce+
+ unfolding pathlift_def by reduce
Lemma (derive) pathlift_fst:
assumes
@@ -438,7 +441,7 @@ Lemma dependent_map_comp [comp]:
"A: U i"
"\<And>x. x: A \<Longrightarrow> P x: U i"
shows "apd f (refl x) \<equiv> refl (f x)"
- unfolding apd_def by reduce+
+ unfolding apd_def by reduce
Lemma (derive) apd_ap:
assumes
@@ -495,13 +498,13 @@ Lemma whisker_refl [comp]:
assumes "A: U i" "a: A" "b: A"
shows "\<lbrakk>p: a = b; q: a = b; \<alpha>: p =\<^bsub>a = b\<^esub> q\<rbrakk> \<Longrightarrow>
\<alpha> \<bullet>\<^sub>r\<^bsub>a\<^esub> (refl b) \<equiv> ru p \<bullet> \<alpha> \<bullet> (ru q)\<inverse>"
- unfolding right_whisker_def by reduce+
+ unfolding right_whisker_def by reduce
Lemma refl_whisker [comp]:
assumes "A: U i" "a: A" "b: A"
shows "\<lbrakk>p: a = b; q: a = b; \<alpha>: p = q\<rbrakk> \<Longrightarrow>
(refl a) \<bullet>\<^sub>l\<^bsub>b\<^esub> \<alpha> \<equiv> (lu p) \<bullet> \<alpha> \<bullet> (lu q)\<inverse>"
- unfolding left_whisker_def by reduce+
+ unfolding left_whisker_def by reduce
method left_whisker = (rule left_whisker)
method right_whisker = (rule right_whisker)