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-rw-r--r--spartan/theories/Equivalence.thy6
-rw-r--r--spartan/theories/Identity.thy140
2 files changed, 92 insertions, 54 deletions
diff --git a/spartan/theories/Equivalence.thy b/spartan/theories/Equivalence.thy
index 73f1e0d..3415249 100644
--- a/spartan/theories/Equivalence.thy
+++ b/spartan/theories/Equivalence.thy
@@ -86,7 +86,7 @@ Lemma (derive) commute_homotopy:
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]
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
focus vars x
apply reduce
\<comment> \<open>Here it would really be nice to have automation for transport and
@@ -366,7 +366,7 @@ Lemma
assumes
"A: U i" "B: U i" "p: A =\<^bsub>U i\<^esub> B"
shows "A \<simeq> B"
- by (equality \<open>p:_\<close>) (rule equivalence_refl)
+ by (eq p) (rule equivalence_refl)
text \<open>
The following proof is wordy because (1) the typechecker doesn't rewrite, and
@@ -383,7 +383,7 @@ Lemma (derive) id_imp_equiv:
\<comment> \<open>Switch off auto-typechecking, which messes with universe levels\<close>
supply [[auto_typechk=false]]
- \<guillemotright> apply (equality \<open>p:_\<close>)
+ \<guillemotright> apply (eq p, typechk)
\<^item> prems vars A B
apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric])
apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric])
diff --git a/spartan/theories/Identity.thy b/spartan/theories/Identity.thy
index 3f16f4c..1ab12ff 100644
--- a/spartan/theories/Identity.thy
+++ b/spartan/theories/Identity.thy
@@ -40,16 +40,33 @@ axiomatization where
lemmas
[intros] = IdF IdI and
- [elims "?p" (*"?x" "?y"*)] = IdE and
+ [elims "?p" "?a" "?b"] = IdE and
[comps] = Id_comp
-section \<open>Induction\<close>
-
-ML_file \<open>../lib/equality.ML\<close>
-
-method_setup equality = \<open>Scan.lift Parse.thm >> (fn (fact, _) => fn ctxt =>
- CONTEXT_METHOD (K (Equality.equality_context_tac fact ctxt)))\<close>
+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>
section \<open>Symmetry and transitivity\<close>
@@ -57,7 +74,7 @@ section \<open>Symmetry and transitivity\<close>
Lemma (derive) pathinv:
assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
shows "y =\<^bsub>A\<^esub> x"
- by (equality \<open>p:_\<close>) intro
+ by (eq p) intro
lemma pathinv_comp [comps]:
assumes "x: A" "A: U i"
@@ -70,9 +87,9 @@ Lemma (derive) pathcomp:
"p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z"
shows
"x =\<^bsub>A\<^esub> z"
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
focus prems vars x p
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
apply intro
done
done
@@ -82,19 +99,6 @@ lemma pathcomp_comp [comps]:
shows "pathcomp A a a a (refl a) (refl a) \<equiv> refl a"
unfolding pathcomp_def by reduce
-text \<open>Set up \<open>sym\<close> attribute for propositional equalities:\<close>
-
-ML \<open>
-structure Id_Sym_Attr = Sym_Attr (
- struct
- val name = "sym"
- val symmetry_rule = @{thm pathinv[rotated 3]}
- end
-)
-\<close>
-
-setup \<open>Id_Sym_Attr.setup\<close>
-
section \<open>Notation\<close>
@@ -113,42 +117,76 @@ translations
"p \<bullet> q" \<leftharpoondown> "CONST pathcomp A x y z p q"
-section \<open>Basic propositional equalities\<close>
+section \<open>Calculational reasoning\<close>
+
+consts rhs :: \<open>'a\<close> ("''''")
+
+ML \<open>
+local fun last_rhs ctxt =
+ let
+ val this_name = Name_Space.full_name (Proof_Context.naming_of ctxt)
+ (Binding.name Auto_Bind.thisN)
+ val this = #thms (the (Proof_Context.lookup_fact ctxt this_name))
+ handle Option => []
+ val rhs = case map Thm.prop_of this of
+ [\<^const>\<open>has_type\<close> $ _ $ (\<^const>\<open>Id\<close> $ _ $ _ $ y)] => y
+ | _ => Term.dummy
+ in
+ map_aterms (fn t => case t of Const (\<^const_name>\<open>rhs\<close>, _) => rhs | _ => t)
+ end
+in val _ = Context.>>
+ (Syntax_Phases.term_check 5 "" (fn ctxt => map (last_rhs ctxt)))
+end
+\<close>
+
+text \<open>Set up \<open>sym\<close> attribute for propositional equalities:\<close>
-(*TODO: Better integration of equality type directly into reasoning...*)
+ML \<open>
+structure Id_Sym_Attr = Sym_Attr (
+ struct
+ val name = "sym"
+ val symmetry_rule = @{thm pathinv[rotated 3]}
+ end
+)
+\<close>
+
+setup \<open>Id_Sym_Attr.setup\<close>
+
+
+section \<open>Basic propositional equalities\<close>
Lemma (derive) pathcomp_left_refl:
assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
shows "(refl x) \<bullet> p = p"
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
apply (reduce; intros)
done
Lemma (derive) pathcomp_right_refl:
assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
shows "p \<bullet> (refl y) = p"
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
apply (reduce; intros)
done
Lemma (derive) pathcomp_left_inv:
assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
shows "p\<inverse> \<bullet> p = refl y"
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
apply (reduce; intros)
done
Lemma (derive) pathcomp_right_inv:
assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
shows "p \<bullet> p\<inverse> = refl x"
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
apply (reduce; intros)
done
Lemma (derive) pathinv_pathinv:
assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
shows "p\<inverse>\<inverse> = p"
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
apply (reduce; intros)
done
@@ -157,11 +195,11 @@ Lemma (derive) pathcomp_assoc:
"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"
shows "p \<bullet> (q \<bullet> r) = p \<bullet> q \<bullet> r"
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
focus prems vars x p
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
focus prems vars x p
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
apply (reduce; intros)
done
done
@@ -177,7 +215,7 @@ Lemma (derive) ap:
"f: A \<rightarrow> B"
"p: x =\<^bsub>A\<^esub> y"
shows "f x = f y"
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
apply intro
done
@@ -199,9 +237,9 @@ Lemma (derive) ap_pathcomp:
"p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z"
shows
"f[p \<bullet> q] = f[p] \<bullet> f[q]"
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
focus prems vars x p
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
apply (reduce; intro)
done
done
@@ -213,7 +251,7 @@ Lemma (derive) ap_pathinv:
"f: A \<rightarrow> B"
"p: x =\<^bsub>A\<^esub> y"
shows "f[p\<inverse>] = f[p]\<inverse>"
- by (equality \<open>p:_\<close>) (reduce; intro)
+ by (eq p) (reduce; intro)
text \<open>The next two proofs currently use some low-level rewriting.\<close>
@@ -224,7 +262,7 @@ Lemma (derive) ap_funcomp:
"f: A \<rightarrow> B" "g: B \<rightarrow> C"
"p: x =\<^bsub>A\<^esub> y"
shows "(g \<circ> f)[p] = g[f[p]]"
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
apply (simp only: funcomp_apply_comp[symmetric])
apply (reduce; intro)
done
@@ -232,7 +270,7 @@ Lemma (derive) ap_funcomp:
Lemma (derive) ap_id:
assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
shows "(id A)[p] = p"
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
apply (rewrite at "\<hole> = _" id_comp[symmetric])
apply (rewrite at "_ = \<hole>" id_comp[symmetric])
apply (reduce; intros)
@@ -248,7 +286,7 @@ Lemma (derive) transport:
"x: A" "y: A"
"p: x =\<^bsub>A\<^esub> y"
shows "P x \<rightarrow> P y"
- by (equality \<open>p:_\<close>) intro
+ by (eq p) intro
definition transport_i ("trans")
where [implicit]: "trans P p \<equiv> transport ? P ? ? p"
@@ -282,7 +320,7 @@ Lemma (derive) transport_left_inv:
"x: A" "y: A"
"p: x =\<^bsub>A\<^esub> y"
shows "(trans P p\<inverse>) \<circ> (trans P p) = id (P x)"
- by (equality \<open>p:_\<close>) (reduce; intro)
+ by (eq p) (reduce; intro)
Lemma (derive) transport_right_inv:
assumes
@@ -291,7 +329,7 @@ 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 (equality \<open>p:_\<close>) (reduce; intros)
+ by (eq p) (reduce; intros)
Lemma (derive) transport_pathcomp:
assumes
@@ -301,9 +339,9 @@ Lemma (derive) transport_pathcomp:
"u: P x"
"p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z"
shows "trans P q (trans P p u) = trans P (p \<bullet> q) u"
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
focus prems vars x p
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
apply (reduce; intros)
done
done
@@ -317,7 +355,7 @@ Lemma (derive) transport_compose_typefam:
"p: x =\<^bsub>A\<^esub> y"
"u: P (f x)"
shows "trans (\<lambda>x. P (f x)) p u = trans P f[p] u"
- by (equality \<open>p:_\<close>) (reduce; intros)
+ by (eq p) (reduce; intros)
Lemma (derive) transport_function_family:
assumes
@@ -329,7 +367,7 @@ 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 (equality \<open>p:_\<close>) (reduce; intros)
+ by (eq p) (reduce; intros)
Lemma (derive) transport_const:
assumes
@@ -337,7 +375,7 @@ Lemma (derive) transport_const:
"x: A" "y: A"
"p: x =\<^bsub>A\<^esub> y"
shows "\<Prod>b: B. trans (\<lambda>_. B) p b = b"
- by (intro, equality \<open>p:_\<close>) (reduce; intro)
+ by (intro, eq p) (reduce; intro)
definition transport_const_i ("trans'_const")
where [implicit]: "trans_const B p \<equiv> transport_const ? B ? ? p"
@@ -359,7 +397,7 @@ Lemma (derive) pathlift:
"p: x =\<^bsub>A\<^esub> y"
"u: P x"
shows "<x, u> = <y, trans P p u>"
- by (equality \<open>p:_\<close>) (reduce; intros)
+ by (eq p) (reduce; intros)
definition pathlift_i ("lift")
where [implicit]: "lift P p u \<equiv> pathlift ? P ? ? p u"
@@ -383,7 +421,7 @@ Lemma (derive) pathlift_fst:
"u: P x"
"p: x =\<^bsub>A\<^esub> y"
shows "fst[lift P p u] = p"
- apply (equality \<open>p:_\<close>)
+ apply (eq p)
text \<open>Some rewriting needed here:\<close>
\<guillemotright> vars x y
(*Here an automatic reordering tactic would be helpful*)
@@ -405,7 +443,7 @@ Lemma (derive) apd:
"x: A" "y: A"
"p: x =\<^bsub>A\<^esub> y"
shows "trans P p (f x) = f y"
- by (equality \<open>p:_\<close>) (reduce; intros; typechk)
+ by (eq p) (reduce; intros; typechk)
definition apd_i ("apd")
where [implicit]: "apd f p \<equiv> Identity.apd ? ? f ? ? p"
@@ -428,7 +466,7 @@ Lemma (derive) apd_ap:
"x: A" "y: A"
"p: x =\<^bsub>A\<^esub> y"
shows "apd f p = trans_const B p (f x) \<bullet> f[p]"
- by (equality \<open>p:_\<close>) (reduce; intro)
+ by (eq p) (reduce; intro)
end
generated by cgit v1.2.3 (git 2.25.1) at 2024-12-23 01:22:49 +0000