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-rw-r--r--hott/Equivalence.thy23
-rw-r--r--hott/Identity.thy82
2 files changed, 55 insertions, 50 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
diff --git a/hott/Identity.thy b/hott/Identity.thy
index 29ce26a..728537c 100644
--- a/hott/Identity.thy
+++ b/hott/Identity.thy
@@ -135,14 +135,14 @@ lemmas
section \<open>Basic propositional equalities\<close>
Lemma (derive) refl_pathcomp:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
+ assumes "A: U i" "x: A" "y: A" "p: x = y"
shows "(refl x) \<bullet> p = p"
apply (eq p)
apply (reduce; intro)
done
Lemma (derive) pathcomp_refl:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
+ assumes "A: U i" "x: A" "y: A" "p: x = y"
shows "p \<bullet> (refl y) = p"
apply (eq p)
apply (reduce; intro)
@@ -166,24 +166,24 @@ Lemma ru_refl [comp]:
unfolding pathcomp_refl_def by reduce
Lemma (derive) inv_pathcomp:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
+ assumes "A: U i" "x: A" "y: A" "p: x = y"
shows "p\<inverse> \<bullet> p = refl y"
by (eq p) (reduce; intro)
Lemma (derive) pathcomp_inv:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
+ assumes "A: U i" "x: A" "y: A" "p: x = y"
shows "p \<bullet> p\<inverse> = refl x"
by (eq p) (reduce; intro)
Lemma (derive) pathinv_pathinv:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
+ assumes "A: U i" "x: A" "y: A" "p: x = y"
shows "p\<inverse>\<inverse> = p"
by (eq p) (reduce; intro)
Lemma (derive) pathcomp_assoc:
assumes
"A: U i" "x: A" "y: A" "z: A" "w: A"
- "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z" "r: z =\<^bsub>A\<^esub> w"
+ "p: x = y" "q: y = z" "r: z = w"
shows "p \<bullet> (q \<bullet> r) = p \<bullet> q \<bullet> r"
apply (eq p)
focus prems vars x p
@@ -203,7 +203,7 @@ Lemma (derive) ap:
"A: U i" "B: U i"
"x: A" "y: A"
"f: A \<rightarrow> B"
- "p: x =\<^bsub>A\<^esub> y"
+ "p: x = y"
shows "f x = f y"
by (eq p) intro
@@ -222,7 +222,7 @@ Lemma (derive) ap_pathcomp:
"A: U i" "B: U i"
"x: A" "y: A" "z: A"
"f: A \<rightarrow> B"
- "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z"
+ "p: x = y" "q: y = z"
shows
"f[p \<bullet> q] = f[p] \<bullet> f[q]"
apply (eq p)
@@ -237,7 +237,7 @@ Lemma (derive) ap_pathinv:
"A: U i" "B: U i"
"x: A" "y: A"
"f: A \<rightarrow> B"
- "p: x =\<^bsub>A\<^esub> y"
+ "p: x = y"
shows "f[p\<inverse>] = f[p]\<inverse>"
by (eq p) (reduce; intro)
@@ -248,7 +248,7 @@ Lemma (derive) ap_funcomp:
"A: U i" "B: U i" "C: U i"
"x: A" "y: A"
"f: A \<rightarrow> B" "g: B \<rightarrow> C"
- "p: x =\<^bsub>A\<^esub> y"
+ "p: x = y"
shows "(g \<circ> f)[p] = g[f[p]]"
apply (eq p)
\<guillemotright> by reduce
@@ -256,7 +256,7 @@ Lemma (derive) ap_funcomp:
done
Lemma (derive) ap_id:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
+ assumes "A: U i" "x: A" "y: A" "p: x = y"
shows "(id A)[p] = p"
apply (eq p)
\<guillemotright> by reduce
@@ -271,7 +271,7 @@ Lemma (derive) transport:
"A: U i"
"\<And>x. x: A \<Longrightarrow> P x: U i"
"x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
+ "p: x = y"
shows "P x \<rightarrow> P y"
by (eq p) intro
@@ -308,7 +308,7 @@ Lemma (derive) transport_left_inv:
"A: U i"
"\<And>x. x: A \<Longrightarrow> P x: U i"
"x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
+ "p: x = y"
shows "(trans P p\<inverse>) \<circ> (trans P p) = id (P x)"
by (eq p) (reduce; refl)
@@ -317,7 +317,7 @@ Lemma (derive) transport_right_inv:
"A: U i"
"\<And>x. x: A \<Longrightarrow> P x: U i"
"x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
+ "p: x = y"
shows "(trans P p) \<circ> (trans P p\<inverse>) = id (P y)"
by (eq p) (reduce; intro)
@@ -327,7 +327,7 @@ Lemma (derive) transport_pathcomp:
"\<And>x. x: A \<Longrightarrow> P x: U i"
"x: A" "y: A" "z: A"
"u: P x"
- "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z"
+ "p: x = y" "q: y = z"
shows "trans P q (trans P p u) = trans P (p \<bullet> q) u"
apply (eq p)
focus prems vars x p
@@ -342,7 +342,7 @@ Lemma (derive) transport_compose_typefam:
"\<And>x. x: B \<Longrightarrow> P x: U i"
"f: A \<rightarrow> B"
"x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
+ "p: x = y"
"u: P (f x)"
shows "trans (fn x. P (f x)) p u = trans P f[p] u"
by (eq p) (reduce; intro)
@@ -355,7 +355,7 @@ Lemma (derive) transport_function_family:
"f: \<Prod>x: A. P x \<rightarrow> Q x"
"x: A" "y: A"
"u: P x"
- "p: x =\<^bsub>A\<^esub> y"
+ "p: x = y"
shows "trans Q p ((f x) u) = (f y) (trans P p u)"
by (eq p) (reduce; intro)
@@ -363,7 +363,7 @@ Lemma (derive) transport_const:
assumes
"A: U i" "B: U i"
"x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
+ "p: x = y"
shows "\<Prod>b: B. trans (fn _. B) p b = b"
by (intro, eq p) (reduce; intro)
@@ -384,7 +384,7 @@ Lemma (derive) pathlift:
"A: U i"
"\<And>x. x: A \<Longrightarrow> P x: U i"
"x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
+ "p: x = y"
"u: P x"
shows "<x, u> = <y, trans P p u>"
by (eq p) (reduce; intro)
@@ -409,7 +409,7 @@ Lemma (derive) pathlift_fst:
"\<And>x. x: A \<Longrightarrow> P x: U i"
"x: A" "y: A"
"u: P x"
- "p: x =\<^bsub>A\<^esub> y"
+ "p: x = y"
shows "fst[lift P p u] = p"
apply (eq p)
\<guillemotright> by reduce
@@ -425,7 +425,7 @@ Lemma (derive) apd:
"\<And>x. x: A \<Longrightarrow> P x: U i"
"f: \<Prod>x: A. P x"
"x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
+ "p: x = y"
shows "trans P p (f x) = f y"
by (eq p) (reduce; intro)
@@ -448,7 +448,7 @@ Lemma (derive) apd_ap:
"A: U i" "B: U i"
"f: A \<rightarrow> B"
"x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
+ "p: x = y"
shows "apd f p = trans_const B p (f x) \<bullet> f[p]"
by (eq p) (reduce; intro)
@@ -457,10 +457,9 @@ section \<open>Whiskering\<close>
Lemma (derive) right_whisker:
assumes "A: U i" "a: A" "b: A" "c: A"
- shows "\<lbrakk>p: a = b; q: a = b; r: b = c; \<alpha>: p =\<^bsub>a = b\<^esub> q\<rbrakk> \<Longrightarrow> p \<bullet> r = q \<bullet> r"
- \<comment> \<open>TODO: In the above we need to annotate the type of \<alpha> with the type `a = b`
- in order for the `eq` method to work correctly. This should be fixed with a
- pre-proof elaborator.\<close>
+ and "p: a = b" "q: a = b" "r: b = c"
+ and "\<alpha>: p = q"
+ shows "p \<bullet> r = q \<bullet> r"
apply (eq r)
focus prems vars x s t
proof -
@@ -473,7 +472,9 @@ Lemma (derive) right_whisker:
Lemma (derive) left_whisker:
assumes "A: U i" "a: A" "b: A" "c: A"
- shows "\<lbrakk>p: b = c; q: b = c; r: a = b; \<alpha>: p =\<^bsub>b = c\<^esub> q\<rbrakk> \<Longrightarrow> r \<bullet> p = r \<bullet> q"
+ and "p: b = c" "q: b = c" "r: a = b"
+ and "\<alpha>: p = q"
+ shows "r \<bullet> p = r \<bullet> q"
apply (eq r)
focus prems prms vars x s t
proof -
@@ -495,15 +496,13 @@ translations
"r \<bullet>\<^sub>l\<^bsub>c\<^esub> \<alpha>" \<leftharpoondown> "CONST left_whisker A a b c p q r \<alpha>"
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>"
+ assumes "A: U i" "a: A" "b: A" "p: a = b" "q: a = b" "\<alpha>: p = q"
+ shows "\<alpha> \<bullet>\<^sub>r\<^bsub>a\<^esub> (refl b) \<equiv> ru p \<bullet> \<alpha> \<bullet> (ru q)\<inverse>"
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>"
+ assumes "A: U i" "a: A" "b: A" "p: a = b" "q: a = b" "\<alpha>: p = q"
+ shows "(refl a) \<bullet>\<^sub>l\<^bsub>b\<^esub> \<alpha> \<equiv> (lu p) \<bullet> \<alpha> \<bullet> (lu q)\<inverse>"
unfolding left_whisker_def by reduce
method left_whisker = (rule left_whisker)
@@ -524,20 +523,22 @@ begin
Lemma (derive) horiz_pathcomp:
notes assums
- shows "\<lbrakk>\<alpha>: p = q; \<beta>: r = s\<rbrakk> \<Longrightarrow> ?prf \<alpha> \<beta>: p \<bullet> r = q \<bullet> s"
+ assumes "\<alpha>: p = q" "\<beta>: r = s"
+ shows "p \<bullet> r = q \<bullet> s"
proof (rule pathcomp)
- show "\<alpha>: p = q \<Longrightarrow> p \<bullet> r = q \<bullet> r" by right_whisker
- show "\<beta>: r = s \<Longrightarrow> .. = q \<bullet> s" by left_whisker
+ 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 (derive) horiz_pathcomp':
notes assums
- shows "\<lbrakk>\<alpha>: p = q; \<beta>: r = s\<rbrakk> \<Longrightarrow> ?prf \<alpha> \<beta>: p \<bullet> r = q \<bullet> s"
+ assumes "\<alpha>: p = q" "\<beta>: r = s"
+ shows "p \<bullet> r = q \<bullet> s"
proof (rule pathcomp)
- show "\<beta>: r = s \<Longrightarrow> p \<bullet> r = p \<bullet> s" by left_whisker
- show "\<alpha>: p = q \<Longrightarrow> .. = q \<bullet> s" by right_whisker
+ 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)
@@ -545,7 +546,8 @@ notation horiz_pathcomp' (infix "\<star>''" 121)
Lemma (derive) horiz_pathcomp_eq_horiz_pathcomp':
notes assums
- shows "\<lbrakk>\<alpha>: p = q; \<beta>: r = s\<rbrakk> \<Longrightarrow> \<alpha> \<star> \<beta> = \<alpha> \<star>' \<beta>"
+ 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)