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-rw-r--r--hott/Identity.thy147
1 files changed, 69 insertions, 78 deletions
diff --git a/hott/Identity.thy b/hott/Identity.thy
index dd2d6a0..1cb3946 100644
--- a/hott/Identity.thy
+++ b/hott/Identity.thy
@@ -20,7 +20,8 @@ syntax "_Id" :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close>
translations "a =\<^bsub>A\<^esub> b" \<rightleftharpoons> "CONST Id A a b"
axiomatization where
- IdF: "\<lbrakk>A: U i; a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^bsub>A\<^esub> b: U i" and
+ \<comment> \<open>Here `A: U i` comes last because A is often implicit\<close>
+ IdF: "\<lbrakk>a: A; b: A; A: U i\<rbrakk> \<Longrightarrow> a =\<^bsub>A\<^esub> b: U i" and
IdI: "a: A \<Longrightarrow> refl a: a =\<^bsub>A\<^esub> a" and
@@ -39,35 +40,36 @@ axiomatization where
\<rbrakk> \<Longrightarrow> IdInd A (fn x y p. C x y p) (fn x. f x) a a (refl a) \<equiv> f a"
lemmas
- [intros] = IdF IdI and
- [elims "?p" "?a" "?b"] = IdE and
- [comps] = Id_comp and
+ [form] = IdF and
+ [intro, intros] = IdI and
+ [elim "?p" "?a" "?b"] = IdE and
+ [comp] = Id_comp and
[refl] = IdI
section \<open>Path induction\<close>
-method_setup eq = \<open>
-Args.term >> (fn tm => fn ctxt => CONTEXT_METHOD (K (
- CONTEXT_SUBGOAL (fn (goal, i) =>
- let
- val facts = Proof_Context.facts_of ctxt
- val prems = Logic.strip_assums_hyp goal
- val template = \<^const>\<open>has_type\<close>
- $ tm
- $ (\<^const>\<open>Id\<close> $ Var (("*?A", 0), \<^typ>\<open>o\<close>)
- $ Var (("*?x", 0), \<^typ>\<open>o\<close>)
- $ Var (("*?y", 0), \<^typ>\<open>o\<close>))
- val types =
- map (Thm.prop_of o #1) (Facts.could_unify facts template)
- @ filter (fn prem => Term.could_unify (template, prem)) prems
- |> map Lib.type_of_typing
- in case types of
- (\<^const>\<open>Id\<close> $ _ $ x $ y)::_ =>
- elim_context_tac [tm, x, y] ctxt i
- | _ => Context_Tactic.CONTEXT_TACTIC no_tac
- end) 1)))
-\<close>
+method_setup eq =
+ \<open>Args.term >> (fn tm => K (CONTEXT_METHOD (
+ CHEADGOAL o SIDE_CONDS (
+ CONTEXT_SUBGOAL (fn (goal, i) => fn cst as (ctxt, st) =>
+ let
+ val facts = Proof_Context.facts_of ctxt
+ val prems = Logic.strip_assums_hyp goal
+ val template = \<^const>\<open>has_type\<close>
+ $ tm
+ $ (\<^const>\<open>Id\<close> $ Var (("*?A", 0), \<^typ>\<open>o\<close>)
+ $ Var (("*?x", 0), \<^typ>\<open>o\<close>)
+ $ Var (("*?y", 0), \<^typ>\<open>o\<close>))
+ val types =
+ map (Thm.prop_of o #1) (Facts.could_unify facts template)
+ @ filter (fn prem => Term.could_unify (template, prem)) prems
+ |> map Lib.type_of_typing
+ in case types of
+ (Const (\<^const_name>\<open>Id\<close>, _) $ _ $ x $ y) :: _ =>
+ elim_ctac [tm, x, y] i cst
+ | _ => no_ctac cst
+ end)))))\<close>
section \<open>Symmetry and transitivity\<close>
@@ -77,7 +79,7 @@ Lemma (derive) pathinv:
shows "y =\<^bsub>A\<^esub> x"
by (eq p) intro
-lemma pathinv_comp [comps]:
+lemma pathinv_comp [comp]:
assumes "x: A" "A: U i"
shows "pathinv A x x (refl x) \<equiv> refl x"
unfolding pathinv_def by reduce
@@ -95,7 +97,7 @@ Lemma (derive) pathcomp:
done
done
-lemma pathcomp_comp [comps]:
+lemma pathcomp_comp [comp]:
assumes "a: A" "A: U i"
shows "pathcomp A a a a (refl a) (refl a) \<equiv> refl a"
unfolding pathcomp_def by reduce
@@ -133,14 +135,14 @@ Lemma (derive) refl_pathcomp:
assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
shows "(refl x) \<bullet> p = p"
apply (eq p)
- apply (reduce; intros)
+ apply (reduce; intro)
done
Lemma (derive) pathcomp_refl:
assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
shows "p \<bullet> (refl y) = p"
apply (eq p)
- apply (reduce; intros)
+ apply (reduce; intro)
done
definition [implicit]: "ru p \<equiv> pathcomp_refl ? ? ? p"
@@ -150,36 +152,30 @@ translations
"CONST ru p" \<leftharpoondown> "CONST pathcomp_refl A x y p"
"CONST lu p" \<leftharpoondown> "CONST refl_pathcomp A x y p"
-Lemma lu_refl [comps]:
+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 [comps]:
+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"
shows "p\<inverse> \<bullet> p = refl y"
- apply (eq p)
- apply (reduce; intros)
- done
+ by (eq p) (reduce; intro)
Lemma (derive) pathcomp_inv:
assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
shows "p \<bullet> p\<inverse> = refl x"
- apply (eq p)
- apply (reduce; intros)
- done
+ by (eq p) (reduce; intro)
Lemma (derive) pathinv_pathinv:
assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
shows "p\<inverse>\<inverse> = p"
- apply (eq p)
- apply (reduce; intros)
- done
+ by (eq p) (reduce; intro)
Lemma (derive) pathcomp_assoc:
assumes
@@ -191,7 +187,7 @@ Lemma (derive) pathcomp_assoc:
apply (eq p)
focus prems vars x p
apply (eq p)
- apply (reduce; intros)
+ apply (reduce; intro)
done
done
done
@@ -206,16 +202,14 @@ Lemma (derive) ap:
"f: A \<rightarrow> B"
"p: x =\<^bsub>A\<^esub> y"
shows "f x = f y"
- apply (eq p)
- apply intro
- done
+ by (eq p) intro
definition ap_i ("_[_]" [1000, 0])
where [implicit]: "ap_i f p \<equiv> ap ? ? ? ? f p"
translations "f[p]" \<leftharpoondown> "CONST ap A B x y f p"
-Lemma ap_refl [comps]:
+Lemma ap_refl [comp]:
assumes "f: A \<rightarrow> B" "x: A" "A: U i" "B: U i"
shows "f[refl x] \<equiv> refl (f x)"
unfolding ap_def by reduce
@@ -254,17 +248,16 @@ Lemma (derive) ap_funcomp:
"p: x =\<^bsub>A\<^esub> y"
shows "(g \<circ> f)[p] = g[f[p]]"
apply (eq p)
- apply (simp only: funcomp_apply_comp[symmetric])
- apply (reduce; intro)
+ \<guillemotright> by reduce
+ \<guillemotright> by reduce intro
done
Lemma (derive) ap_id:
assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
shows "(id A)[p] = p"
apply (eq p)
- apply (rewrite at "\<hole> = _" id_comp[symmetric])
- apply (rewrite at "_ = \<hole>" id_comp[symmetric])
- apply (reduce; intros)
+ \<guillemotright> by reduce
+ \<guillemotright> by reduce intro
done
@@ -284,7 +277,7 @@ definition transport_i ("trans")
translations "trans P p" \<leftharpoondown> "CONST transport A P x y p"
-Lemma transport_comp [comps]:
+Lemma transport_comp [comp]:
assumes
"a: A"
"A: U i"
@@ -302,6 +295,7 @@ Lemma use_transport:
"A: U i"
"\<And>x. x: A \<Longrightarrow> P x: U i"
shows "trans P p\<inverse> u: P y"
+ unfolding transport_def pathinv_def
by typechk
method transport uses eq = (rule use_transport[OF eq])
@@ -322,7 +316,7 @@ Lemma (derive) transport_right_inv:
"x: A" "y: A"
"p: x =\<^bsub>A\<^esub> y"
shows "(trans P p) \<circ> (trans P p\<inverse>) = id (P y)"
- by (eq p) (reduce; intros)
+ by (eq p) (reduce; intro)
Lemma (derive) transport_pathcomp:
assumes
@@ -335,7 +329,7 @@ Lemma (derive) transport_pathcomp:
apply (eq p)
focus prems vars x p
apply (eq p)
- apply (reduce; intros)
+ apply (reduce; intro)
done
done
@@ -348,7 +342,7 @@ Lemma (derive) transport_compose_typefam:
"p: x =\<^bsub>A\<^esub> y"
"u: P (f x)"
shows "trans (fn x. P (f x)) p u = trans P f[p] u"
- by (eq p) (reduce; intros)
+ by (eq p) (reduce; intro)
Lemma (derive) transport_function_family:
assumes
@@ -360,7 +354,7 @@ Lemma (derive) transport_function_family:
"u: P x"
"p: x =\<^bsub>A\<^esub> y"
shows "trans Q p ((f x) u) = (f y) (trans P p u)"
- by (eq p) (reduce; intros)
+ by (eq p) (reduce; intro)
Lemma (derive) transport_const:
assumes
@@ -375,12 +369,12 @@ definition transport_const_i ("trans'_const")
translations "trans_const B p" \<leftharpoondown> "CONST transport_const A B x y p"
-Lemma transport_const_comp [comps]:
+Lemma transport_const_comp [comp]:
assumes
"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
@@ -390,21 +384,21 @@ Lemma (derive) pathlift:
"p: x =\<^bsub>A\<^esub> y"
"u: P x"
shows "<x, u> = <y, trans P p u>"
- by (eq p) (reduce; intros)
+ by (eq p) (reduce; intro)
definition pathlift_i ("lift")
where [implicit]: "lift P p u \<equiv> pathlift ? P ? ? p u"
translations "lift P p u" \<leftharpoondown> "CONST pathlift A P x y p u"
-Lemma pathlift_comp [comps]:
+Lemma pathlift_comp [comp]:
assumes
"u: P x"
"x: A"
"\<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
@@ -415,13 +409,7 @@ Lemma (derive) pathlift_fst:
"p: x =\<^bsub>A\<^esub> y"
shows "fst[lift P p u] = p"
apply (eq p)
- text \<open>Some rewriting needed here:\<close>
- \<guillemotright> vars x y
- (*Here an automatic reordering tactic would be helpful*)
- apply (rewrite at x in "x = y" fst_comp[symmetric])
- prefer 4
- apply (rewrite at y in "_ = y" fst_comp[symmetric])
- done
+ \<guillemotright> by reduce
\<guillemotright> by reduce intro
done
@@ -436,21 +424,21 @@ Lemma (derive) apd:
"x: A" "y: A"
"p: x =\<^bsub>A\<^esub> y"
shows "trans P p (f x) = f y"
- by (eq p) (reduce; intros; typechk)
+ by (eq p) (reduce; intro)
definition apd_i ("apd")
where [implicit]: "apd f p \<equiv> Identity.apd ? ? f ? ? p"
translations "apd f p" \<leftharpoondown> "CONST Identity.apd A P f x y p"
-Lemma dependent_map_comp [comps]:
+Lemma dependent_map_comp [comp]:
assumes
"f: \<Prod>x: A. P x"
"x: A"
"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
@@ -467,6 +455,9 @@ 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>
apply (eq r)
focus prems vars x s t
proof -
@@ -500,17 +491,17 @@ translations
"\<alpha> \<bullet>\<^sub>r\<^bsub>a\<^esub> r" \<leftharpoondown> "CONST right_whisker A a b c p q r \<alpha>"
"r \<bullet>\<^sub>l\<^bsub>c\<^esub> \<alpha>" \<leftharpoondown> "CONST left_whisker A a b c p q r \<alpha>"
-Lemma whisker_refl [comps]:
+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 [comps]:
+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)
@@ -632,12 +623,12 @@ Lemma (derive) eckmann_hilton:
proof -
have "\<alpha> \<bullet> \<beta> = \<alpha> \<star> \<beta>"
by (rule pathcomp_eq_horiz_pathcomp)
- also have [simplified comps]: ".. = \<alpha> \<star>' \<beta>"
+ also have [simplified comp]: ".. = \<alpha> \<star>' \<beta>"
by (transport eq: \<Omega>2.horiz_pathcomp_eq_horiz_pathcomp') refl
also have ".. = \<beta> \<bullet> \<alpha>"
by (rule pathcomp_eq_horiz_pathcomp')
finally show "\<alpha> \<bullet> \<beta> = \<beta> \<bullet> \<alpha>"
- by this (reduce add: \<Omega>2.horiz_pathcomp_def \<Omega>2.horiz_pathcomp'_def)
+ by this (reduce add: \<Omega>2.horiz_pathcomp_def \<Omega>2.horiz_pathcomp'_def)+
qed
end
generated by cgit v1.2.3 (git 2.25.1) at 2024-05-17 09:55:13 +0000