diff options
Diffstat (limited to '')
| -rw-r--r-- | hott/Equivalence.thy | 4 | ||||
| -rw-r--r-- | hott/Identity.thy | 77 | ||||
| -rw-r--r-- | hott/List_HoTT.thy | 2 |
3 files changed, 38 insertions, 45 deletions
diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy index 9fe11a7..745bc67 100644 --- a/hott/Equivalence.thy +++ b/hott/Equivalence.thy @@ -400,8 +400,8 @@ Lemma by (eq p) (rule equivalence_refl) text \<open> - The following proof is wordy because (1) the typechecker doesn't normalize, - and (2) we don't yet have universe level inference. + The following proof is wordy because (1) typechecker normalization is still + rudimentary, and (2) we don't yet have universe level inference. \<close> Lemma (def) equiv_if_equal: diff --git a/hott/Identity.thy b/hott/Identity.thy index caab2e3..0531b74 100644 --- a/hott/Identity.thy +++ b/hott/Identity.thy @@ -6,7 +6,7 @@ text \<open> \<close> theory Identity -imports Spartan +imports MLTT begin @@ -109,7 +109,7 @@ method pathcomp for p q :: o = rule pathcomp[where ?p=p and ?q=q] section \<open>Notation\<close> definition Id_i (infix "=" 110) - where [implicit]: "Id_i x y \<equiv> x =\<^bsub>{}\<^esub> y" + where [implicit]: "x = y \<equiv> x =\<^bsub>{}\<^esub> y" definition pathinv_i ("_\<inverse>" [1000]) where [implicit]: "pathinv_i p \<equiv> pathinv {} {} {} p" @@ -532,47 +532,40 @@ method right_whisker = (rule right_whisker) section \<open>Horizontal path-composition\<close> -locale horiz_pathcomposable = -\<comment> \<open>Conditions under which horizontal path-composition is defined.\<close> -fixes - i A a b c p q r s -assumes [type]: - "A: U i" "a: A" "b: A" "c: A" - "p: a =\<^bsub>A\<^esub> b" "q: a =\<^bsub>A\<^esub> b" - "r: b =\<^bsub>A\<^esub> c" "s: b =\<^bsub>A\<^esub> c" +locale horiz_pathcomposable = \<comment> \<open>Conditions under which horizontal path-composition is defined.\<close> +fixes i A a b c p q r s +assumes [type]: "A: U i" "a: A" "b: A" "c: A" + "p: a =\<^bsub>A\<^esub> b" "q: a =\<^bsub>A\<^esub> b" "r: b =\<^bsub>A\<^esub> c" "s: b =\<^bsub>A\<^esub> c" begin - -Lemma (def) horiz_pathcomp: - assumes "\<alpha>: p = q" "\<beta>: r = s" - shows "p \<bullet> r = q \<bullet> s" -proof (rule pathcomp) - show "p \<bullet> r = q \<bullet> r" by right_whisker fact - show ".. = q \<bullet> s" by left_whisker fact -qed typechk - -text \<open>A second horizontal composition:\<close> - -Lemma (def) horiz_pathcomp': - assumes "\<alpha>: p = q" "\<beta>: r = s" - shows "p \<bullet> r = q \<bullet> s" -proof (rule pathcomp) - show "p \<bullet> r = p \<bullet> s" by left_whisker fact - show ".. = q \<bullet> s" by right_whisker fact -qed typechk - -notation horiz_pathcomp (infix "\<star>" 121) -notation horiz_pathcomp' (infix "\<star>''" 121) - -Lemma (def) horiz_pathcomp_eq_horiz_pathcomp': - assumes "\<alpha>: p = q" "\<beta>: r = s" - shows "\<alpha> \<star> \<beta> = \<alpha> \<star>' \<beta>" - unfolding horiz_pathcomp_def horiz_pathcomp'_def - apply (eq \<alpha>, eq \<beta>) - focus vars p apply (eq p) - focus vars a _ _ _ r by (eq r) (compute, refl) - done - done - + Lemma (def) horiz_pathcomp: + assumes "\<alpha>: p = q" "\<beta>: r = s" shows "p \<bullet> r = q \<bullet> s" + proof (rule pathcomp) + show "p \<bullet> r = q \<bullet> r" by right_whisker fact + show ".. = q \<bullet> s" by left_whisker fact + qed typechk + + Lemma (def) horiz_pathcomp': + assumes "\<alpha>: p = q" "\<beta>: r = s" shows "p \<bullet> r = q \<bullet> s" + proof (rule pathcomp) + show "p \<bullet> r = p \<bullet> s" by left_whisker fact + show ".. = q \<bullet> s" by right_whisker fact + qed typechk + + notation horiz_pathcomp (infix "\<star>" 121) + notation horiz_pathcomp' (infix "\<star>''" 121) + + Lemma (def) horiz_pathcomp_eq_horiz_pathcomp': + assumes "\<alpha>: p = q" "\<beta>: r = s" shows "\<alpha> \<star> \<beta> = \<alpha> \<star>' \<beta>" + unfolding horiz_pathcomp_def horiz_pathcomp'_def + proof (eq \<alpha>, eq \<beta>) + fix p q assuming "p: a = b" "q: b = c" + show "refl p \<bullet>\<^sub>r q \<bullet> (p \<bullet>\<^sub>l refl q) = p \<bullet>\<^sub>l refl q \<bullet> (refl p \<bullet>\<^sub>r q)" + proof (eq p) + fix a r assuming "a: A" "r: a = c" + show "refl (refl a) \<bullet>\<^sub>r r \<bullet> (refl a \<bullet>\<^sub>l refl r) = refl a \<bullet>\<^sub>l refl r \<bullet> (refl (refl a) \<bullet>\<^sub>r r)" + by (eq r) (compute, refl) + qed + qed end diff --git a/hott/List_HoTT.thy b/hott/List_HoTT.thy index 9bd1517..d866f59 100644 --- a/hott/List_HoTT.thy +++ b/hott/List_HoTT.thy @@ -1,6 +1,6 @@ theory List_HoTT imports - Spartan.List + MLTT.List Nat begin |