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-rw-r--r--hott/Equivalence.thy52
-rw-r--r--hott/Identity.thy147
-rw-r--r--hott/More_List.thy4
-rw-r--r--hott/More_Nat.thy6
-rw-r--r--hott/Nat.thy17
-rw-r--r--spartan/core/Spartan.thy186
-rw-r--r--spartan/core/context_tactical.ML152
-rw-r--r--spartan/core/elimination.ML6
-rw-r--r--spartan/core/eqsubst.ML36
-rw-r--r--spartan/core/lib.ML3
-rw-r--r--spartan/core/rewrite.ML26
-rw-r--r--spartan/core/tactics.ML285
-rw-r--r--spartan/core/typechecking.ML64
-rw-r--r--spartan/lib/List.thy13
-rw-r--r--spartan/lib/Maybe.thy7
-rw-r--r--spartan/lib/More_Types.thy30
16 files changed, 603 insertions, 431 deletions
diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy
index 66248a9..88adc8b 100644
--- a/hott/Equivalence.thy
+++ b/hott/Equivalence.thy
@@ -35,7 +35,7 @@ Lemma (derive) hsym:
"\<And>x. x: A \<Longrightarrow> B x: U i"
shows "H: f ~ g \<Longrightarrow> g ~ f"
unfolding homotopy_def
- apply intros
+ apply intro
apply (rule pathinv)
\<guillemotright> by (elim H)
\<guillemotright> by typechk
@@ -106,7 +106,8 @@ Lemma homotopy_id_left [typechk]:
Lemma homotopy_id_right [typechk]:
assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
shows "homotopy_refl A f: f \<circ> (id A) ~ f"
- unfolding homotopy_refl_def homotopy_def by reduce
+ unfolding homotopy_refl_def homotopy_def
+ by reduce
Lemma homotopy_funcomp_left:
assumes
@@ -165,11 +166,9 @@ Lemma (derive) id_qinv:
assumes "A: U i"
shows "qinv (id A)"
unfolding qinv_def
- apply intro defer
- apply intro defer
- apply htpy_id
- apply id_htpy
- done
+ proof intro
+ show "id A: A \<rightarrow> A" by typechk
+ qed (reduce, intro; refl)
Lemma (derive) quasiinv_qinv:
assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
@@ -272,7 +271,7 @@ Lemma (derive) biinv_imp_qinv:
also have ".. ~ (id A) \<circ> h" by (subst funcomp_assoc[symmetric]) (htpy_o, fact)
also have ".. ~ h" by reduce refl
finally have "g ~ h" by this
- then have "f \<circ> g ~ f \<circ> h" by o_htpy
+ then have "f \<circ> g ~ f \<circ> h" by (o_htpy, this)
also note \<open>H2:_\<close>
finally show "f \<circ> g ~ id B" by this
qed
@@ -313,10 +312,9 @@ Lemma (derive) equivalence_refl:
assumes "A: U i"
shows "A \<simeq> A"
unfolding equivalence_def
- apply intro defer
- apply (rule qinv_imp_biinv) defer
- apply (rule id_qinv)
- done
+ proof intro
+ show "biinv (id A)" by (rule qinv_imp_biinv) (rule id_qinv)
+ qed typechk
text \<open>
The following could perhaps be easier with transport (but then I think we need
@@ -340,11 +338,14 @@ 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> apply (rule funcompI) defer by assumption+ known
\<guillemotright> by (rule funcomp_biinv)
done
done
@@ -372,32 +373,31 @@ Lemma (derive) id_imp_equiv:
shows "<trans (id (U i)) p, ?isequiv>: A \<simeq> B"
unfolding equivalence_def
apply intros defer
-
- \<comment> \<open>Switch off auto-typechecking, which messes with universe levels\<close>
- supply [[auto_typechk=false]]
-
- \<guillemotright> apply (eq p, typechk)
+ \<guillemotright> apply (eq p)
\<^item> prems vars A B
apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric])
+ using [[solve_side_conds=1]]
apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric])
- apply (rule transport, rule U_in_U)
- apply (rule lift_universe_codomain, rule U_in_U)
- apply (typechk, rule U_in_U)
+ apply (rule transport, rule Ui_in_USi)
+ apply (rule lift_universe_codomain, rule Ui_in_USi)
+ apply (typechk, rule Ui_in_USi)
done
\<^item> prems vars A
+ using [[solve_side_conds=1]]
apply (subst transport_comp)
- \<^enum> by (rule U_in_U)
- \<^enum> by reduce (rule lift_U)
+ \<^enum> by (rule Ui_in_USi)
+ \<^enum> by reduce (rule in_USi_if_in_Ui)
\<^enum> by reduce (rule id_biinv)
done
done
\<guillemotright> \<comment> \<open>Similar proof as in the first subgoal above\<close>
apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric])
+ using [[solve_side_conds=1]]
apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric])
- apply (rule transport, rule U_in_U)
- apply (rule lift_universe_codomain, rule U_in_U)
- apply (typechk, rule U_in_U)
+ apply (rule transport, rule Ui_in_USi)
+ apply (rule lift_universe_codomain, rule Ui_in_USi)
+ apply (typechk, rule Ui_in_USi)
done
done
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
diff --git a/hott/More_List.thy b/hott/More_List.thy
index 460dc7b..1b8034f 100644
--- a/hott/More_List.thy
+++ b/hott/More_List.thy
@@ -10,8 +10,8 @@ section \<open>Length\<close>
definition [implicit]: "len \<equiv> ListRec ? Nat 0 (fn _ _ rec. suc rec)"
experiment begin
- Lemma "len [] \<equiv> ?n" by (subst comps)+
- Lemma "len [0, suc 0, suc (suc 0)] \<equiv> ?n" by (subst comps)+
+ Lemma "len [] \<equiv> ?n" by (subst comp)+
+ Lemma "len [0, suc 0, suc (suc 0)] \<equiv> ?n" by (subst comp)+
end
diff --git a/hott/More_Nat.thy b/hott/More_Nat.thy
index 8177170..59c8d54 100644
--- a/hott/More_Nat.thy
+++ b/hott/More_Nat.thy
@@ -15,15 +15,15 @@ Lemma (derive) eq: shows "Nat \<rightarrow> Nat \<rightarrow> U O"
apply intro focus vars n apply elim
\<guillemotright> by (rule TopF) \<comment> \<open>n \<equiv> 0\<close>
\<guillemotright> by (rule BotF) \<comment> \<open>n \<equiv> suc _\<close>
- \<guillemotright> by (rule U_in_U)
+ \<guillemotright> by (rule Ui_in_USi)
done
\<comment> \<open>m \<equiv> suc k\<close>
apply intro focus vars k k_eq n apply (elim n)
\<guillemotright> by (rule BotF) \<comment> \<open>n \<equiv> 0\<close>
\<guillemotright> prems vars l proof - show "k_eq l: U O" by typechk qed
- \<guillemotright> by (rule U_in_U)
+ \<guillemotright> by (rule Ui_in_USi)
done
- by (rule U_in_U)
+ by (rule Ui_in_USi)
done
text \<open>
diff --git a/hott/Nat.thy b/hott/Nat.thy
index 4abd315..177ec47 100644
--- a/hott/Nat.thy
+++ b/hott/Nat.thy
@@ -38,9 +38,10 @@ where
f n (NatInd (fn n. C n) c\<^sub>0 (fn k rec. f k rec) n)"
lemmas
- [intros] = NatF Nat_zero Nat_suc and
- [elims "?n"] = NatE and
- [comps] = Nat_comp_zero Nat_comp_suc
+ [form] = NatF and
+ [intro, intros] = Nat_zero Nat_suc and
+ [elim "?n"] = NatE and
+ [comp] = Nat_comp_zero Nat_comp_suc
abbreviation "NatRec C \<equiv> NatInd (fn _. C)"
@@ -66,12 +67,12 @@ lemma add_type [typechk]:
shows "m + n: Nat"
unfolding add_def by typechk
-lemma add_zero [comps]:
+lemma add_zero [comp]:
assumes "m: Nat"
shows "m + 0 \<equiv> m"
unfolding add_def by reduce
-lemma add_suc [comps]:
+lemma add_suc [comp]:
assumes "m: Nat" "n: Nat"
shows "m + suc n \<equiv> suc (m + n)"
unfolding add_def by reduce
@@ -127,17 +128,17 @@ lemma mul_type [typechk]:
shows "m * n: Nat"
unfolding mul_def by typechk
-lemma mul_zero [comps]:
+lemma mul_zero [comp]:
assumes "n: Nat"
shows "n * 0 \<equiv> 0"
unfolding mul_def by reduce
-lemma mul_one [comps]:
+lemma mul_one [comp]:
assumes "n: Nat"
shows "n * 1 \<equiv> n"
unfolding mul_def by reduce
-lemma mul_suc [comps]:
+lemma mul_suc [comp]:
assumes "m: Nat" "n: Nat"
shows "m * suc n \<equiv> m + m * n"
unfolding mul_def by reduce
diff --git a/spartan/core/Spartan.thy b/spartan/core/Spartan.thy
index 8fa6cfe..a4ad300 100644
--- a/spartan/core/Spartan.thy
+++ b/spartan/core/Spartan.thy
@@ -10,22 +10,29 @@ keywords
"focus" "\<guillemotright>" "\<^item>" "\<^enum>" "~" :: prf_script_goal % "proof" and
"congruence" "print_coercions" :: thy_decl and
"rhs" "derive" "prems" "vars" :: quasi_command
-
begin
section \<open>Prelude\<close>
+paragraph \<open>Set up the meta-logic just so.\<close>
+
+paragraph \<open>Notation.\<close>
+
declare [[eta_contract=false]]
+text \<open>
+ Rebind notation for meta-lambdas since we want to use `\<lambda>` for the object
+ lambdas. Meta-functions now use the binder `fn`.
+\<close>
setup \<open>
let
val typ = Simple_Syntax.read_typ
fun mixfix (sy, ps, p) = Mixfix (Input.string sy, ps, p, Position.no_range)
in
Sign.del_syntax (Print_Mode.ASCII, true)
- [("_lambda", typ "pttrns \<Rightarrow> 'a \<Rightarrow> logic", mixfix ("(3fn _./ _)", [0, 3], 3))]
+ [("_lambda", typ "pttrns \<Rightarrow> 'a \<Rightarrow> logic", mixfix ("(3%_./ _)", [0, 3], 3))]
#> Sign.del_syntax Syntax.mode_default
[("_lambda", typ "pttrns \<Rightarrow> 'a \<Rightarrow> logic", mixfix ("(3\<lambda>_./ _)", [0, 3], 3))]
#> Sign.add_syntax Syntax.mode_default
@@ -38,11 +45,16 @@ translations "a $ b" \<rightharpoonup> "a (b)"
abbreviation (input) K where "K x \<equiv> fn _. x"
+paragraph \<open>
+ Deeply embed dependent type theory: meta-type of expressions, and typing
+ judgment.
+\<close>
+
typedecl o
judgment has_type :: \<open>o \<Rightarrow> o \<Rightarrow> prop\<close> ("(2_:/ _)" 999)
-text \<open>Type annotations:\<close>
+text \<open>Type annotations for type-checking and inference.\<close>
definition anno :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> ("'(_ : _')")
where "(a : A) \<equiv> a" if "a: A"
@@ -51,7 +63,9 @@ lemma anno: "a: A \<Longrightarrow> (a : A): A"
by (unfold anno_def)
-section \<open>Universes\<close>
+section \<open>Axioms\<close>
+
+subsection \<open>Universes\<close>
typedecl lvl \<comment> \<open>Universe levels\<close>
@@ -65,19 +79,19 @@ axiomatization
lt_trans: "i < j \<Longrightarrow> j < k \<Longrightarrow> i < k"
axiomatization U :: \<open>lvl \<Rightarrow> o\<close> where
- U_hierarchy: "i < j \<Longrightarrow> U i: U j" and
- U_cumulative: "A: U i \<Longrightarrow> i < j \<Longrightarrow> A: U j"
+ Ui_in_Uj: "i < j \<Longrightarrow> U i: U j" and
+ in_Uj_if_in_Ui: "A: U i \<Longrightarrow> i < j \<Longrightarrow> A: U j"
-lemma U_in_U:
+lemma Ui_in_USi:
"U i: U (S i)"
- by (rule U_hierarchy, rule lt_S)
+ by (rule Ui_in_Uj, rule lt_S)
-lemma lift_U:
+lemma in_USi_if_in_Ui:
"A: U i \<Longrightarrow> A: U (S i)"
- by (erule U_cumulative, rule lt_S)
+ by (erule in_Uj_if_in_Ui, rule lt_S)
-section \<open>\<Prod>-type\<close>
+subsection \<open>\<Prod>-type\<close>
axiomatization
Pi :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o\<close> and
@@ -100,9 +114,9 @@ translations
abbreviation Fn (infixr "\<rightarrow>" 40) where "A \<rightarrow> B \<equiv> \<Prod>_: A. B"
axiomatization where
- PiF: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; A: U i\<rbrakk> \<Longrightarrow> \<Prod>x: A. B x: U i" and
+ PiF: "\<lbrakk>A: U i; \<And>x. x: A \<Longrightarrow> B x: U i\<rbrakk> \<Longrightarrow> \<Prod>x: A. B x: U i" and
- PiI: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x: B x; A: U i\<rbrakk> \<Longrightarrow> \<lambda>x: A. b x: \<Prod>x: A. B x" and
+ PiI: "\<lbrakk>A: U i; \<And>x. x: A \<Longrightarrow> b x: B x\<rbrakk> \<Longrightarrow> \<lambda>x: A. b x: \<Prod>x: A. B x" and
PiE: "\<lbrakk>f: \<Prod>x: A. B x; a: A\<rbrakk> \<Longrightarrow> f `a: B a" and
@@ -113,14 +127,14 @@ axiomatization where
Pi_cong: "\<lbrakk>
\<And>x. x: A \<Longrightarrow> B x \<equiv> B' x;
A: U i;
- \<And>x. x: A \<Longrightarrow> B x: U i;
- \<And>x. x: A \<Longrightarrow> B' x: U i
+ \<And>x. x: A \<Longrightarrow> B x: U j;
+ \<And>x. x: A \<Longrightarrow> B' x: U j
\<rbrakk> \<Longrightarrow> \<Prod>x: A. B x \<equiv> \<Prod>x: A. B' x" and
lam_cong: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x \<equiv> c x; A: U i\<rbrakk> \<Longrightarrow> \<lambda>x: A. b x \<equiv> \<lambda>x: A. c x"
-section \<open>\<Sum>-type\<close>
+subsection \<open>\<Sum>-type\<close>
axiomatization
Sig :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o\<close> and
@@ -138,16 +152,16 @@ abbreviation "and" (infixl "\<and>" 60)
where "A \<and> B \<equiv> A \<times> B"
axiomatization where
- SigF: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; A: U i\<rbrakk> \<Longrightarrow> \<Sum>x: A. B x: U i" and
+ SigF: "\<lbrakk>A: U i; \<And>x. x: A \<Longrightarrow> B x: U i\<rbrakk> \<Longrightarrow> \<Sum>x: A. B x: U i" and
SigI: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; a: A; b: B a\<rbrakk> \<Longrightarrow> <a, b>: \<Sum>x: A. B x" and
SigE: "\<lbrakk>
p: \<Sum>x: A. B x;
A: U i;
- \<And>x. x : A \<Longrightarrow> B x: U i;
- \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x, y>;
- \<And>p. p: \<Sum>x: A. B x \<Longrightarrow> C p: U i
+ \<And>x. x : A \<Longrightarrow> B x: U j;
+ \<And>p. p: \<Sum>x: A. B x \<Longrightarrow> C p: U k;
+ \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x, y>
\<rbrakk> \<Longrightarrow> SigInd A (fn x. B x) (fn p. C p) f p: C p" and
Sig_comp: "\<lbrakk>
@@ -161,22 +175,23 @@ axiomatization where
Sig_cong: "\<lbrakk>
\<And>x. x: A \<Longrightarrow> B x \<equiv> B' x;
A: U i;
- \<And>x. x : A \<Longrightarrow> B x: U i;
- \<And>x. x : A \<Longrightarrow> B' x: U i
+ \<And>x. x : A \<Longrightarrow> B x: U j;
+ \<And>x. x : A \<Longrightarrow> B' x: U j
\<rbrakk> \<Longrightarrow> \<Sum>x: A. B x \<equiv> \<Sum>x: A. B' x"
section \<open>Infrastructure\<close>
ML_file \<open>lib.ML\<close>
-ML_file \<open>typechecking.ML\<close>
+ML_file \<open>context_tactical.ML\<close>
subsection \<open>Proof commands\<close>
+ML_file \<open>focus.ML\<close>
+
named_theorems typechk
ML_file \<open>goals.ML\<close>
-ML_file \<open>focus.ML\<close>
subsection \<open>Congruence automation\<close>
@@ -184,72 +199,75 @@ consts "rhs" :: \<open>'a\<close> ("..")
ML_file \<open>congruence.ML\<close>
+subsection \<open>Proof methods and type-checking/inference\<close>
+
+named_theorems form and intro and intros and comp
+\<comment> \<open>`intros` stores the introduction rule variants used by the
+ `intro` and `intros` methods.\<close>
+
ML_file \<open>elimination.ML\<close> \<comment> \<open>elimination rules\<close>
ML_file \<open>cases.ML\<close> \<comment> \<open>case reasoning rules\<close>
-subsection \<open>Methods\<close>
-
-named_theorems intros and comps
lemmas
- [intros] = PiF PiI SigF SigI and
- [elims "?f"] = PiE and
- [elims "?p"] = SigE and
- [comps] = beta Sig_comp and
+ [form] = PiF SigF and
+ [intro] = PiI SigI and
+ [intros] = PiI[rotated] SigI and
+ [elim "?f"] = PiE and
+ [elim "?p"] = SigE and
+ [comp] = beta Sig_comp and
[cong] = Pi_cong lam_cong Sig_cong
+ML_file \<open>typechecking.ML\<close>
ML_file \<open>tactics.ML\<close>
-method_setup assumptions =
- \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (
- CHANGED (TRYALL (assumptions_tac ctxt))))\<close>
+method_setup typechk =
+ \<open>Scan.succeed (K (CONTEXT_METHOD (
+ CHEADGOAL o Types.check_infer)))\<close>
method_setup known =
- \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (
- CHANGED (TRYALL (known_tac ctxt))))\<close>
+ \<open>Scan.succeed (K (CONTEXT_METHOD (
+ CHEADGOAL o Types.known_ctac)))\<close>
+
+method_setup rule =
+ \<open>Attrib.thms >> (fn ths => K (CONTEXT_METHOD (
+ CHEADGOAL o SIDE_CONDS (rule_ctac ths))))\<close>
+
+method_setup dest =
+ \<open>Scan.lift (Scan.option (Args.parens Parse.int))
+ -- Attrib.thms >> (fn (n_opt, ths) => K (CONTEXT_METHOD (
+ CHEADGOAL o SIDE_CONDS (dest_ctac n_opt ths))))\<close>
method_setup intro =
- \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (intro_tac ctxt)))\<close>
+ \<open>Scan.succeed (K (CONTEXT_METHOD (
+ CHEADGOAL o SIDE_CONDS (intro_ctac))))\<close>
method_setup intros =
- \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (intros_tac ctxt)))\<close>
+ \<open>Scan.succeed (K (CONTEXT_METHOD (
+ CHEADGOAL o SIDE_CONDS (CREPEAT o intro_ctac))))\<close>
method_setup elim =
- \<open>Scan.repeat Args.term >> (fn tms => fn ctxt =>
- CONTEXT_METHOD (K (elim_context_tac tms ctxt 1)))\<close>
+ \<open>Scan.repeat Args.term >> (fn tms => K (CONTEXT_METHOD (
+ CHEADGOAL o SIDE_CONDS (elim_ctac tms))))\<close>
method elims = elim+
method_setup cases =
- \<open>Args.term >> (fn tm => fn ctxt => SIMPLE_METHOD' (cases_tac tm ctxt))\<close>
-
-(*This could possibly use additional simplification hints via a simp: modifier*)
-method_setup typechk =
- \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (
- SIDE_CONDS (typechk_tac ctxt) ctxt))
- (* CHANGED (ALLGOALS (TRY o typechk_tac ctxt)))) *)\<close>
+ \<open>Args.term >> (fn tm => K (CONTEXT_METHOD (
+ CHEADGOAL o SIDE_CONDS (cases_ctac tm))))\<close>
-method_setup rule =
- \<open>Attrib.thms >> (fn ths => fn ctxt =>
- SIMPLE_METHOD (HEADGOAL (SIDE_CONDS (rule_tac ths ctxt) ctxt)))\<close>
-
-method_setup dest =
- \<open>Scan.lift (Scan.option (Args.parens Parse.int)) -- Attrib.thms
- >> (fn (opt_n, ths) => fn ctxt =>
- SIMPLE_METHOD (HEADGOAL (SIDE_CONDS (dest_tac opt_n ths ctxt) ctxt)))\<close>
subsection \<open>Reflexivity\<close>
named_theorems refl
method refl = (rule refl)
-subsection \<open>Trivial proofs modulo typechecking\<close>
+subsection \<open>Trivial proofs (modulo automatic discharge of side conditions)\<close>
method_setup this =
- \<open>Scan.succeed (fn ctxt => METHOD (
- EVERY o map (HEADGOAL o
- (fn ths => SIDE_CONDS (resolve_tac ctxt ths) ctxt) o
- single)
- ))\<close>
+ \<open>Scan.succeed (K (CONTEXT_METHOD (fn facts =>
+ CHEADGOAL (SIDE_CONDS
+ (CONTEXT_TACTIC' (fn ctxt => resolve_tac ctxt facts))
+ facts))))\<close>
subsection \<open>Rewriting\<close>
@@ -270,7 +288,7 @@ lemma eta_expand:
lemma rewr_imp:
assumes "PROP A \<equiv> PROP B"
shows "(PROP A \<Longrightarrow> PROP C) \<equiv> (PROP B \<Longrightarrow> PROP C)"
- apply (rule Pure.equal_intr_rule)
+ apply (Pure.rule Pure.equal_intr_rule)
apply (drule equal_elim_rule2[OF assms]; assumption)
apply (drule equal_elim_rule1[OF assms]; assumption)
done
@@ -288,9 +306,11 @@ ML_file \<open>~~/src/HOL/Library/cconv.ML\<close>
ML_file \<open>rewrite.ML\<close>
\<comment> \<open>\<open>reduce\<close> computes terms via judgmental equalities\<close>
-setup \<open>map_theory_simpset (fn ctxt => ctxt addSolver (mk_solver "" typechk_tac))\<close>
+setup \<open>map_theory_simpset (fn ctxt =>
+ ctxt addSolver (mk_solver "" (fn ctxt' =>
+ NO_CONTEXT_TACTIC ctxt' o Types.check_infer (Simplifier.prems_of ctxt'))))\<close>
-method reduce uses add = (simp add: comps add | subst comps)+
+method reduce uses add = changed \<open>((simp add: comp add | sub comp); typechk?)+\<close>
subsection \<open>Implicits\<close>
@@ -348,13 +368,19 @@ lemma eta_exp:
shows "f \<equiv> \<lambda>x: A. f x"
by (rule eta[symmetric])
+lemma refine_codomain:
+ assumes
+ "A: U i"
+ "f: \<Prod>x: A. B x"
+ "\<And>x. x: A \<Longrightarrow> f `x: C x"
+ shows "f: \<Prod>x: A. C x"
+ by (subst eta_exp)
+
lemma lift_universe_codomain:
assumes "A: U i" "f: A \<rightarrow> U j"
shows "f: A \<rightarrow> U (S j)"
- apply (sub eta_exp)
- apply known
- apply (Pure.rule intros; rule lift_U)
- done
+ using in_USi_if_in_Ui[of "f {}"]
+ by (rule refine_codomain)
subsection \<open>Function composition\<close>
@@ -376,7 +402,7 @@ lemma funcompI [typechk]:
"g \<circ>\<^bsub>A\<^esub> f: \<Prod>x: A. C (f x)"
unfolding funcomp_def by typechk
-lemma funcomp_assoc [comps]:
+lemma funcomp_assoc [comp]:
assumes
"f: A \<rightarrow> B"
"g: B \<rightarrow> C"
@@ -386,7 +412,7 @@ lemma funcomp_assoc [comps]:
"(h \<circ>\<^bsub>B\<^esub> g) \<circ>\<^bsub>A\<^esub> f \<equiv> h \<circ>\<^bsub>A\<^esub> g \<circ>\<^bsub>A\<^esub> f"
unfolding funcomp_def by reduce
-lemma funcomp_lambda_comp [comps]:
+lemma funcomp_lambda_comp [comp]:
assumes
"A: U i"
"\<And>x. x: A \<Longrightarrow> b x: B"
@@ -395,7 +421,7 @@ lemma funcomp_lambda_comp [comps]:
"(\<lambda>x: B. c x) \<circ>\<^bsub>A\<^esub> (\<lambda>x: A. b x) \<equiv> \<lambda>x: A. c (b x)"
unfolding funcomp_def by reduce
-lemma funcomp_apply_comp [comps]:
+lemma funcomp_apply_comp [comp]:
assumes
"f: A \<rightarrow> B" "g: \<Prod>x: B. C x"
"x: A"
@@ -417,22 +443,22 @@ abbreviation id where "id A \<equiv> \<lambda>x: A. x"
lemma
id_type[typechk]: "A: U i \<Longrightarrow> id A: A \<rightarrow> A" and
- id_comp [comps]: "x: A \<Longrightarrow> (id A) x \<equiv> x" \<comment> \<open>for the occasional manual rewrite\<close>
- by reduce
+ id_comp [comp]: "x: A \<Longrightarrow> (id A) x \<equiv> x" \<comment> \<open>for the occasional manual rewrite\<close>
+ by reduce+
-lemma id_left [comps]:
+lemma id_left [comp]:
assumes "f: A \<rightarrow> B" "A: U i" "B: U i"
shows "(id B) \<circ>\<^bsub>A\<^esub> f \<equiv> f"
by (subst eta_exp[of f]) (reduce, rule eta)
-lemma id_right [comps]:
+lemma id_right [comp]:
assumes "f: A \<rightarrow> B" "A: U i" "B: U i"
shows "f \<circ>\<^bsub>A\<^esub> (id A) \<equiv> f"
by (subst eta_exp[of f]) (reduce, rule eta)
lemma id_U [typechk]:
"id (U i): U i \<rightarrow> U i"
- by typechk (fact U_in_U)
+ by typechk (rule Ui_in_USi) (*FIXME: Add annotation rule to typechecker*)
section \<open>Pairs\<close>
@@ -445,7 +471,7 @@ lemma fst_type [typechk]:
shows "fst A B: (\<Sum>x: A. B x) \<rightarrow> A"
unfolding fst_def by typechk
-lemma fst_comp [comps]:
+lemma fst_comp [comp]:
assumes
"a: A"
"b: B a"
@@ -459,10 +485,10 @@ lemma snd_type [typechk]:
shows "snd A B: \<Prod>p: \<Sum>x: A. B x. B (fst A B p)"
unfolding snd_def by typechk reduce
-lemma snd_comp [comps]:
+lemma snd_comp [comp]:
assumes "a: A" "b: B a" "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i"
shows "snd A B <a, b> \<equiv> b"
- unfolding snd_def by reduce
+ unfolding snd_def by reduce+
subsection \<open>Notation\<close>
@@ -483,7 +509,7 @@ Lemma fst [typechk]:
"p: \<Sum>x: A. B x"
"A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i"
shows "fst p: A"
- by typechk
+ by typechk
Lemma snd [typechk]:
assumes
diff --git a/spartan/core/context_tactical.ML b/spartan/core/context_tactical.ML
new file mode 100644
index 0000000..a5159f6
--- /dev/null
+++ b/spartan/core/context_tactical.ML
@@ -0,0 +1,152 @@
+(* Title: context_tactical.ML
+ Author: Joshua Chen
+
+More context tactics, and context tactic combinators.
+*)
+
+infix 1 CTHEN CTHEN' CTHEN_ALL_NEW CTHEN_ALL_NEW_FWD
+infix 0 CORELSE CAPPEND CORELSE' CAPPEND'
+
+structure Context_Tactical:
+sig
+
+type context_tactic' = int -> context_tactic
+val CONTEXT_TACTIC': (Proof.context -> int -> tactic) -> context_tactic'
+val all_ctac: context_tactic
+val no_ctac: context_tactic
+val print_ctac: (Proof.context -> string) -> context_tactic
+val CTHEN: context_tactic * context_tactic -> context_tactic
+val CORELSE: context_tactic * context_tactic -> context_tactic
+val CAPPEND: context_tactic * context_tactic -> context_tactic
+val CTHEN': context_tactic' * context_tactic' -> context_tactic'
+val CORELSE': context_tactic' * context_tactic' -> context_tactic'
+val CAPPEND': context_tactic' * context_tactic' -> context_tactic'
+val CTRY: context_tactic -> context_tactic
+val CREPEAT: context_tactic -> context_tactic
+val CFILTER: (context_state -> bool) -> context_tactic -> context_tactic
+val CCHANGED: context_tactic -> context_tactic
+val CTHEN_ALL_NEW: context_tactic' * context_tactic' -> context_tactic'
+val CREPEAT_IN_RANGE: int -> int -> context_tactic' -> context_tactic
+val CREPEAT_ALL_NEW: context_tactic' -> context_tactic'
+val CTHEN_ALL_NEW_FWD: context_tactic' * context_tactic' -> context_tactic'
+val CREPEAT_ALL_NEW_FWD: context_tactic' -> context_tactic'
+val CHEADGOAL: context_tactic' -> context_tactic
+val CALLGOALS: context_tactic' -> context_tactic
+val CSOMEGOAL: context_tactic' -> context_tactic
+val CRANGE: context_tactic' list -> context_tactic'
+
+end = struct
+
+type context_tactic' = int -> context_tactic
+
+fun CONTEXT_TACTIC' tac i (ctxt, st) = TACTIC_CONTEXT ctxt ((tac ctxt i) st)
+
+val all_ctac = Seq.make_results o Seq.single
+val no_ctac = K Seq.empty
+fun print_ctac f (ctxt, st) = CONTEXT_TACTIC (print_tac ctxt (f ctxt)) (ctxt, st)
+
+fun (ctac1 CTHEN ctac2) cst = Seq.maps_results ctac2 (ctac1 cst)
+
+fun (ctac1 CORELSE ctac2) cst =
+ (case Seq.pull (ctac1 cst) of
+ NONE => ctac2 cst
+ | some => Seq.make (fn () => some))
+
+fun (ctac1 CAPPEND ctac2) cst =
+ Seq.append (ctac1 cst) (Seq.make (fn () => Seq.pull (ctac2 cst)))
+
+fun (ctac1 CTHEN' ctac2) x = ctac1 x CTHEN ctac2 x
+fun (ctac1 CORELSE' ctac2) x = ctac1 x CORELSE ctac2 x
+fun (ctac1 CAPPEND' ctac2) x = ctac1 x CAPPEND ctac2 x
+
+fun CTRY ctac = ctac CORELSE all_ctac
+
+fun CREPEAT ctac =
+ let
+ fun rep qs cst =
+ (case Seq.pull (Seq.filter_results (ctac cst)) of
+ NONE => SOME (cst, Seq.make (fn () => repq qs))
+ | SOME (cst', q) => rep (q :: qs) cst')
+ and repq [] = NONE
+ | repq (q :: qs) =
+ (case Seq.pull q of
+ NONE => repq qs
+ | SOME (cst, q) => rep (q :: qs) cst);
+ in fn cst => Seq.make_results (Seq.make (fn () => rep [] cst)) end
+
+fun CFILTER pred ctac cst =
+ ctac cst
+ |> Seq.filter_results
+ |> Seq.filter pred
+ |> Seq.make_results
+
+(*Only accept next states where the subgoals have changed*)
+fun CCHANGED ctac (cst as (_, st)) =
+ CFILTER (fn (_, st') => not (Thm.eq_thm (st, st'))) ctac cst
+
+local
+ fun op THEN (f, g) x = Seq.maps_results g (f x)
+
+ fun INTERVAL f i j x =
+ if i > j then Seq.make_results (Seq.single x)
+ else op THEN (f j, INTERVAL f i (j - 1)) x
+
+ (*By Peter Lammich: apply tactic to subgoals in interval in a forward manner,
+ skipping over emerging subgoals*)
+ fun INTERVAL_FWD ctac l u (cst as (_, st)) = cst |>
+ (if l > u then all_ctac
+ else (ctac l CTHEN (fn cst' as (_, st') =>
+ let val ofs = Thm.nprems_of st' - Thm.nprems_of st in
+ if ofs < ~1
+ then raise THM (
+ "INTERVAL_FWD: tactic solved more than one goal", ~1, [st, st'])
+ else INTERVAL_FWD ctac (l + 1 + ofs) (u + ofs) cst'
+ end)))
+in
+
+fun (ctac1 CTHEN_ALL_NEW ctac2) i (cst as (_, st)) =
+ cst |> (ctac1 i CTHEN (fn cst' as (_, st') =>
+ INTERVAL ctac2 i (i + Thm.nprems_of st' - Thm.nprems_of st) cst'))
+
+(*By Peter Lammich: apply ctac2 to all subgoals emerging from ctac1, in forward
+ manner*)
+fun (ctac1 CTHEN_ALL_NEW_FWD ctac2) i (cst as (_, st)) =
+ cst |> (ctac1 i CTHEN (fn cst' as (_, st') =>
+ INTERVAL_FWD ctac2 i (i + Thm.nprems_of st' - Thm.nprems_of st) cst'))
+
+(*Repeatedly apply ctac to the i-th until the k-th-from-last subgoals
+ (i.e. leave the last k subgoals alone), until no more changes appear in the
+ goal state.*)
+fun CREPEAT_IN_RANGE i k ctac =
+ let fun interval_ctac (cst as (_, st)) =
+ INTERVAL_FWD ctac i (Thm.nprems_of st - k) cst
+ in CREPEAT (CCHANGED interval_ctac) end
+
+end
+
+fun CREPEAT_ALL_NEW ctac =
+ ctac CTHEN_ALL_NEW (CTRY o (fn i => CREPEAT_ALL_NEW ctac i))
+
+fun CREPEAT_ALL_NEW_FWD ctac =
+ ctac CTHEN_ALL_NEW_FWD (CTRY o (fn i => CREPEAT_ALL_NEW_FWD ctac i))
+
+fun CHEADGOAL ctac = ctac 1
+
+fun CALLGOALS ctac (cst as (_, st)) =
+ let
+ fun doall 0 = all_ctac
+ | doall n = ctac n CTHEN doall (n - 1);
+ in doall (Thm.nprems_of st) cst end
+
+fun CSOMEGOAL ctac (cst as (_, st)) =
+ let
+ fun find 0 = no_ctac
+ | find n = ctac n CORELSE find (n - 1);
+ in find (Thm.nprems_of st) cst end
+
+fun CRANGE [] _ = all_ctac
+ | CRANGE (ctac :: ctacs) i = CRANGE ctacs (i + 1) CTHEN ctac i
+
+end
+
+open Context_Tactical
diff --git a/spartan/core/elimination.ML b/spartan/core/elimination.ML
index 11b3af9..cd4e414 100644
--- a/spartan/core/elimination.ML
+++ b/spartan/core/elimination.ML
@@ -34,13 +34,13 @@ fun register_rule tms rl =
in Rules.map (Termtab.update (hd, (rl, map (#1 o dest_Var) tms))) end
-(* [elims] attribute *)
+(* [elim] attribute *)
val _ = Theory.setup (
- Attrib.setup \<^binding>\<open>elims\<close>
+ Attrib.setup \<^binding>\<open>elim\<close>
(Scan.repeat Args.term_pattern >>
(Thm.declaration_attribute o register_rule))
""
- #> Global_Theory.add_thms_dynamic (\<^binding>\<open>elims\<close>,
+ #> Global_Theory.add_thms_dynamic (\<^binding>\<open>elim\<close>,
fn context => (map #1 (rules (Context.proof_of context))))
)
diff --git a/spartan/core/eqsubst.ML b/spartan/core/eqsubst.ML
index ea6f098..e7ecf63 100644
--- a/spartan/core/eqsubst.ML
+++ b/spartan/core/eqsubst.ML
@@ -416,19 +416,27 @@ fun eqsubst_asm_tac ctxt occs thms i st =
should be done to an assumption, false = apply to the conclusion of
the goal) as well as the theorems to use *)
val _ =
- Theory.setup
- (Method.setup \<^binding>\<open>sub\<close>
- (Scan.lift (Args.mode "asm" -- Scan.optional (Args.parens (Scan.repeat Parse.nat)) [0]) --
- Attrib.thms >> (fn ((asm, occs), inthms) => fn ctxt =>
- SIMPLE_METHOD' ((if asm then eqsubst_asm_tac else eqsubst_tac) ctxt occs inthms)))
- "single-step substitution"
- #>
- (Method.setup \<^binding>\<open>subst\<close>
- (Scan.lift (Args.mode "asm" -- Scan.optional (Args.parens (Scan.repeat Parse.nat)) [0]) --
- Attrib.thms >> (fn ((asm, occs), inthms) => fn ctxt =>
- SIMPLE_METHOD' (SIDE_CONDS
- ((if asm then eqsubst_asm_tac else eqsubst_tac) ctxt occs inthms)
- ctxt)))
- "single-step substitution with auto-typechecking"))
+ let
+ val parser =
+ Scan.lift (Args.mode "asm"
+ -- Scan.optional (Args.parens (Scan.repeat Parse.nat)) [0])
+ -- Attrib.thms
+ fun eqsubst_asm_ctac occs inthms =
+ CONTEXT_TACTIC' (fn ctxt => eqsubst_asm_tac ctxt occs inthms)
+ fun eqsubst_ctac occs inthms =
+ CONTEXT_TACTIC' (fn ctxt => eqsubst_tac ctxt occs inthms)
+ in
+ Theory.setup (
+ Method.setup \<^binding>\<open>sub\<close>
+ (parser >> (fn ((asm, occs), inthms) => fn ctxt => SIMPLE_METHOD' (
+ (if asm then eqsubst_asm_tac else eqsubst_tac) ctxt occs inthms)))
+ "single-step substitution" #>
+ Method.setup \<^binding>\<open>subst\<close>
+ (parser >> (fn ((asm, occs), inthms) => K (CONTEXT_METHOD (
+ CHEADGOAL o SIDE_CONDS
+ ((if asm then eqsubst_asm_ctac else eqsubst_ctac) occs inthms)))))
+ "single-step substitution with automatic discharge of side conditions"
+ )
+ end
end;
diff --git a/spartan/core/lib.ML b/spartan/core/lib.ML
index 615f601..7b93a08 100644
--- a/spartan/core/lib.ML
+++ b/spartan/core/lib.ML
@@ -7,6 +7,7 @@ val maxint: int list -> int
(*Terms*)
val is_rigid: term -> bool
+val no_vars: term -> bool
val dest_eq: term -> term * term
val mk_Var: string -> int -> typ -> term
val lambda_var: term -> term -> term
@@ -50,6 +51,8 @@ val maxint = max (op >)
val is_rigid = not o is_Var o head_of
+val no_vars = not o exists_subterm is_Var
+
fun dest_eq (Const (\<^const_name>\<open>Pure.eq\<close>, _) $ t $ def) = (t, def)
| dest_eq _ = error "dest_eq"
diff --git a/spartan/core/rewrite.ML b/spartan/core/rewrite.ML
index f9c5d8e..eba0e81 100644
--- a/spartan/core/rewrite.ML
+++ b/spartan/core/rewrite.ML
@@ -15,8 +15,8 @@ patterns but has diverged significantly during its development.
We also allow introduction of identifiers for bound variables,
which can then be used to match arbitrary subterms inside abstractions.
-This code is slightly modified from the original at HOL/Library/rewrite.ML,
-to incorporate auto-typechecking for type theory.
+This code has been slightly modified from the original at HOL/Library/rewrite.ML
+to incorporate automatic discharge of type-theoretic side conditions.
*)
infix 1 then_pconv;
@@ -448,18 +448,18 @@ val _ =
val subst_parser =
let val scan = raw_pattern -- to_parser -- Parse.thms1
in context_lift scan prep_args end
+
+ fun rewrite_export_ctac inputs inthms =
+ CONTEXT_TACTIC' (fn ctxt => rewrite_export_tac ctxt inputs inthms)
in
Method.setup \<^binding>\<open>rewr\<close> (subst_parser >>
- (fn (pattern, inthms, (to, pat_ctxt)) => fn orig_ctxt =>
- SIMPLE_METHOD'
- (rewrite_export_tac orig_ctxt ((pattern, to), SOME pat_ctxt) inthms)))
- "single-step rewriting, allowing subterm selection via patterns"
- #>
- (Method.setup \<^binding>\<open>rewrite\<close> (subst_parser >>
- (fn (pattern, inthms, (to, pat_ctxt)) => fn orig_ctxt =>
- SIMPLE_METHOD' (SIDE_CONDS
- (rewrite_export_tac orig_ctxt ((pattern, to), SOME pat_ctxt) inthms)
- orig_ctxt)))
- "single-step rewriting with auto-typechecking")
+ (fn (pattern, inthms, (to, pat_ctxt)) => fn orig_ctxt => SIMPLE_METHOD'
+ (rewrite_export_tac orig_ctxt ((pattern, to), SOME pat_ctxt) inthms)))
+ "single-step rewriting, allowing subterm selection via patterns" #>
+ Method.setup \<^binding>\<open>rewrite\<close> (subst_parser >>
+ (fn (pattern, inthms, (to, pat_ctxt)) => K (CONTEXT_METHOD (
+ CHEADGOAL o SIDE_CONDS
+ (rewrite_export_ctac ((pattern, to), SOME pat_ctxt) inthms)))))
+ "single-step rewriting with auto-typechecking"
end
end
diff --git a/spartan/core/tactics.ML b/spartan/core/tactics.ML
index 0c71665..3922ae0 100644
--- a/spartan/core/tactics.ML
+++ b/spartan/core/tactics.ML
@@ -7,109 +7,66 @@ General tactics for dependent type theory.
structure Tactics:
sig
-val assumptions_tac: Proof.context -> int -> tactic
-val known_tac: Proof.context -> int -> tactic
-val typechk_tac: Proof.context -> int -> tactic
-val auto_typechk: bool Config.T
-val SIDE_CONDS: (int -> tactic) -> Proof.context -> int -> tactic
-val rule_tac: thm list -> Proof.context -> int -> tactic
-val dest_tac: int option -> thm list -> Proof.context -> int -> tactic
-val intro_tac: Proof.context -> int -> tactic
-val intros_tac: Proof.context -> int -> tactic
-val elim_context_tac: term list -> Proof.context -> int -> context_tactic
-val cases_tac: term -> Proof.context -> int -> tactic
+val solve_side_conds: int Config.T
+val SIDE_CONDS: context_tactic' -> thm list -> context_tactic'
+val rule_ctac: thm list -> context_tactic'
+val dest_ctac: int option -> thm list -> context_tactic'
+val intro_ctac: context_tactic'
+val elim_ctac: term list -> context_tactic'
+val cases_ctac: term -> context_tactic'
end = struct
-(*An assumption tactic that only solves typing goals with rigid terms and
- judgmental equalities without schematic variables*)
-fun assumptions_tac ctxt = SUBGOAL (fn (goal, i) =>
- let
- val concl = Logic.strip_assums_concl goal
- in
- if
- Lib.is_typing concl andalso Lib.is_rigid (Lib.term_of_typing concl)
- orelse not ((exists_subterm is_Var) concl)
- then assume_tac ctxt i
- else no_tac
- end)
-
-(*Solves typing goals with rigid term by resolving with context facts and
- simplifier premises, or arbitrary goals by *non-unifying* assumption*)
-fun known_tac ctxt = SUBGOAL (fn (goal, i) =>
- let
- val concl = Logic.strip_assums_concl goal
- in
- ((if Lib.is_typing concl andalso Lib.is_rigid (Lib.term_of_typing concl)
- then
- let val ths = map fst (Facts.props (Proof_Context.facts_of ctxt))
- in resolve_tac ctxt (ths @ Simplifier.prems_of ctxt) end
- else K no_tac)
- ORELSE' assumptions_tac ctxt) i
- end)
-
-(*Typechecking: try to solve goals of the form "a: A" where a is rigid*)
-fun typechk_tac ctxt =
- let
- val tac = SUBGOAL (fn (goal, i) =>
- if Lib.rigid_typing_concl goal
- then
- let val net = Tactic.build_net
- ((Named_Theorems.get ctxt \<^named_theorems>\<open>typechk\<close>)
- @(Named_Theorems.get ctxt \<^named_theorems>\<open>intros\<close>)
- @(map #1 (Elim.rules ctxt)))
- in (resolve_from_net_tac ctxt net) i end
- else no_tac)
- in
- REPEAT_ALL_NEW (known_tac ctxt ORELSE' tac)
- end
-fun typechk_context_tac (ctxt, st) =
- let
-
- in
- ()
- end
+(* Side conditions *)
+val solve_side_conds =
+ Attrib.setup_config_int \<^binding>\<open>solve_side_conds\<close> (K 2)
-(*Many methods try to automatically discharge side conditions by typechecking.
- Switch this flag off to discharge by non-unifying assumption instead.*)
-val auto_typechk = Attrib.setup_config_bool \<^binding>\<open>auto_typechk\<close> (K true)
+fun SIDE_CONDS ctac facts i (cst as (ctxt, st)) = cst |> (ctac i CTHEN
+ (case Config.get ctxt solve_side_conds of
+ 1 => CALLGOALS (CTRY o Types.known_ctac facts)
+ | 2 => CREPEAT_IN_RANGE i (Thm.nprems_of st - i)
+ (CTRY o CREPEAT_ALL_NEW_FWD (Types.check_infer facts))
+ | _ => all_ctac))
-fun side_cond_tac ctxt = CHANGED o REPEAT o
- (if Config.get ctxt auto_typechk then typechk_tac ctxt else known_tac ctxt)
-(*Combinator runs tactic and tries to discharge all new typing side conditions*)
-fun SIDE_CONDS tac ctxt = tac THEN_ALL_NEW (TRY o side_cond_tac ctxt)
+(* rule, dest, intro *)
local
-fun mk_rules _ ths [] = ths
- | mk_rules n ths ths' =
- let val ths'' = foldr1 (op @)
- (map (fn th => [rotate_prems n (th RS @{thm PiE})] handle THM _ => []) ths')
- in
- mk_rules n (ths @ ths') ths''
- end
+ fun mk_rules _ ths [] = ths
+ | mk_rules n ths ths' =
+ let val ths'' = foldr1 (op @)
+ (map
+ (fn th => [rotate_prems n (th RS @{thm PiE})] handle THM _ => [])
+ ths')
+ in
+ mk_rules n (ths @ ths') ths''
+ end
in
-(*Resolves with given rules, discharging as many side conditions as possible*)
-fun rule_tac ths ctxt = resolve_tac ctxt (mk_rules 0 [] ths)
+(*Resolves with given rules*)
+fun rule_ctac ths i (ctxt, st) =
+ TACTIC_CONTEXT ctxt (resolve_tac ctxt (mk_rules 0 [] ths) i st)
(*Attempts destruct-resolution with the n-th premise of the given rules*)
-fun dest_tac opt_n ths ctxt = dresolve_tac ctxt
- (mk_rules (case opt_n of NONE => 0 | SOME 0 => 0 | SOME n => n-1) [] ths)
+fun dest_ctac opt_n ths i (ctxt, st) =
+ TACTIC_CONTEXT ctxt (dresolve_tac ctxt
+ (mk_rules (case opt_n of NONE => 0 | SOME 0 => 0 | SOME n => n-1) [] ths)
+ i st)
end
(*Applies some introduction rule*)
-fun intro_tac ctxt = SUBGOAL (fn (_, i) => SIDE_CONDS
- (resolve_tac ctxt (Named_Theorems.get ctxt \<^named_theorems>\<open>intros\<close>)) ctxt i)
+fun intro_ctac i (ctxt, st) = TACTIC_CONTEXT ctxt (resolve_tac ctxt
+ (Named_Theorems.get ctxt \<^named_theorems>\<open>intros\<close>) i st)
-fun intros_tac ctxt = SUBGOAL (fn (_, i) =>
- (CHANGED o REPEAT o CHANGED o intro_tac ctxt) i)
(* Induction/elimination *)
-(*Pushes a context/goal premise typing t:T into a \<Prod>-type*)
+(*Pushes a known typing t:T into a \<Prod>-type.
+ This tactic is well-behaved only when t is sufficiently well specified
+ (otherwise there might be multiple possible judgments t:T that unify, and
+ which is chosen is undefined).*)
fun internalize_fact_tac t =
Subgoal.FOCUS_PARAMS (fn {context = ctxt, concl = raw_concl, ...} =>
let
@@ -123,18 +80,10 @@ fun internalize_fact_tac t =
in
HEADGOAL (resolve_tac ctxt [resolvent])
(*known_tac infers the correct type T inferred by unification*)
- THEN SOMEGOAL (known_tac ctxt)
+ THEN SOMEGOAL (NO_CONTEXT_TACTIC ctxt o Types.known_ctac [])
end)
-(*Premises that have already been pushed into the \<Prod>-type*)
-structure Inserts = Proof_Data (
- type T = term Item_Net.T
- val init = K (Item_Net.init Term.aconv_untyped single)
-)
-
-local
-
-fun elim_core_tac tms types ctxt = SUBGOAL (K (
+fun elim_core_tac tms types ctxt =
let
val rule_insts = map ((Elim.lookup_rule ctxt) o Term.head_of) types
val rules = flat (map
@@ -144,84 +93,86 @@ fun elim_core_tac tms types ctxt = SUBGOAL (K (
(idxnames ~~ map (Thm.cterm_of ctxt) tms) rl])
rule_insts)
in
- HEADGOAL (resolve_tac ctxt rules)
- THEN RANGE (replicate (length tms) (typechk_tac ctxt)) 1
- end handle Option => no_tac))
+ resolve_tac ctxt rules
+ THEN' RANGE (replicate (length tms) (NO_CONTEXT_TACTIC ctxt o Types.check_infer []))
+ end handle Option => K no_tac
-in
-
-fun elim_context_tac tms ctxt = case tms of
- [] => CONTEXT_SUBGOAL (K (Context_Tactic.CONTEXT_TACTIC (HEADGOAL (
- SIDE_CONDS (eresolve_tac ctxt (map #1 (Elim.rules ctxt))) ctxt))))
- | major::_ => CONTEXT_SUBGOAL (fn (goal, _) =>
- let
- val facts = Proof_Context.facts_of ctxt
- val prems = Logic.strip_assums_hyp goal
- val template = Lib.typing_of_term major
- 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
- [] => Context_Tactic.CONTEXT_TACTIC no_tac
- | _ =>
- let
- val inserts = map (Thm.prop_of o fst) (Facts.props facts) @ prems
- |> filter Lib.is_typing
- |> map Lib.dest_typing
- |> filter_out (fn (t, _) =>
- Term.aconv (t, major) orelse Item_Net.member (Inserts.get ctxt) t)
- |> map (fn (t, T) => ((t, T), Lib.subterm_count_distinct tms T))
- |> filter (fn (_, i) => i > 0)
- (*`t1: T1` comes before `t2: T2` if T1 contains t2 as subterm.
- If they are incomparable, then order by decreasing
- `subterm_count [p, x, y] T`*)
- |> sort (fn (((t1, _), i), ((_, T2), j)) =>
- Lib.cond_order (Lib.subterm_order T2 t1) (int_ord (j, i)))
- |> map (#1 o #1)
- val record_inserts = Inserts.map (fold Item_Net.update inserts)
- val tac =
- (*Push premises having a subterm in `tms` into a \<Prod>*)
- fold (fn t => fn tac =>
- tac THEN HEADGOAL (internalize_fact_tac t ctxt))
- inserts all_tac
- (*Apply elimination rule*)
- THEN (HEADGOAL (
- elim_core_tac tms types ctxt
- (*Pull pushed premises back out*)
- THEN_ALL_NEW (SUBGOAL (fn (_, i) =>
- REPEAT_DETERM_N (length inserts)
- (resolve_tac ctxt @{thms PiI} i)))
- ))
- (*Side conditions*)
- THEN ALLGOALS (TRY o side_cond_tac ctxt)
- in
- fn (ctxt, st) => Context_Tactic.TACTIC_CONTEXT
- (record_inserts ctxt) (tac st)
- end
- end)
+(*Premises that have already been pushed into the \<Prod>-type*)
+structure Inserts = Proof_Data (
+ type T = term Item_Net.T
+ val init = K (Item_Net.init Term.aconv_untyped single)
+)
-fun cases_tac tm ctxt = SUBGOAL (fn (goal, i) =>
- let
- val facts = Proof_Context.facts_of ctxt
- val prems = Logic.strip_assums_hyp goal
- val template = Lib.typing_of_term tm
- 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
- val res = (case types of
- [typ] => Drule.infer_instantiate' ctxt [SOME (Thm.cterm_of ctxt tm)]
- (the (Case.lookup_rule ctxt (Term.head_of typ)))
- | [] => raise Option
- | _ => raise error (Syntax.string_of_term ctxt tm ^ "not uniquely typed"))
- handle Option => error ("no case rule known for "
- ^ (Syntax.string_of_term ctxt tm))
- in
- SIDE_CONDS (resolve_tac ctxt [res]) ctxt i
- end)
+fun elim_ctac tms =
+ case tms of
+ [] => CONTEXT_TACTIC' (fn ctxt => eresolve_tac ctxt (map #1 (Elim.rules ctxt)))
+ | major :: _ => CONTEXT_SUBGOAL (fn (goal, _) => fn cst as (ctxt, st) =>
+ let
+ val facts = Proof_Context.facts_of ctxt
+ val prems = Logic.strip_assums_hyp goal
+ val template = Lib.typing_of_term major
+ 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
+ [] => no_ctac cst
+ | _ =>
+ let
+ val inserts = map (Thm.prop_of o fst) (Facts.props facts) @ prems
+ |> filter Lib.is_typing
+ |> map Lib.dest_typing
+ |> filter_out (fn (t, _) =>
+ Term.aconv (t, major) orelse Item_Net.member (Inserts.get ctxt) t)
+ |> map (fn (t, T) => ((t, T), Lib.subterm_count_distinct tms T))
+ |> filter (fn (_, i) => i > 0)
+ (*`t1: T1` comes before `t2: T2` if T1 contains t2 as subterm.
+ If they are incomparable, then order by decreasing
+ `subterm_count [p, x, y] T`*)
+ |> sort (fn (((t1, _), i), ((_, T2), j)) =>
+ Lib.cond_order (Lib.subterm_order T2 t1) (int_ord (j, i)))
+ |> map (#1 o #1)
+ val record_inserts = Inserts.map (fold Item_Net.update inserts)
+ val tac =
+ (*Push premises having a subterm in `tms` into a \<Prod>*)
+ fold (fn t => fn tac =>
+ tac THEN HEADGOAL (internalize_fact_tac t ctxt))
+ inserts all_tac
+ (*Apply elimination rule*)
+ THEN HEADGOAL (
+ elim_core_tac tms types ctxt
+ (*Pull pushed premises back out*)
+ THEN_ALL_NEW (SUBGOAL (fn (_, i) =>
+ REPEAT_DETERM_N (length inserts)
+ (resolve_tac ctxt @{thms PiI[rotated]} i))))
+ in
+ TACTIC_CONTEXT (record_inserts ctxt) (tac st)
+ end
+ end)
+
+fun cases_ctac tm =
+ let fun tac ctxt =
+ SUBGOAL (fn (goal, i) =>
+ let
+ val facts = Proof_Context.facts_of ctxt
+ val prems = Logic.strip_assums_hyp goal
+ val template = Lib.typing_of_term tm
+ 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
+ val res = (case types of
+ [typ] => Drule.infer_instantiate' ctxt [SOME (Thm.cterm_of ctxt tm)]
+ (the (Case.lookup_rule ctxt (Term.head_of typ)))
+ | [] => raise Option
+ | _ => raise error (Syntax.string_of_term ctxt tm ^ "not uniquely typed"))
+ handle Option =>
+ error ("no case rule known for " ^ (Syntax.string_of_term ctxt tm))
+ in
+ resolve_tac ctxt [res] i
+ end)
+ in CONTEXT_TACTIC' tac end
-end
end
diff --git a/spartan/core/typechecking.ML b/spartan/core/typechecking.ML
index 437a2dc..946ecd6 100644
--- a/spartan/core/typechecking.ML
+++ b/spartan/core/typechecking.ML
@@ -1,7 +1,7 @@
(* Title: typechecking.ML
Author: Joshua Chen
-Type information and typechecking infrastructure.
+Type information and type-checking infrastructure.
*)
structure Types: sig
@@ -11,11 +11,12 @@ val types: Proof.context -> term -> thm list
val put_type: thm -> Proof.context -> Proof.context
val put_types: thm list -> Proof.context -> Proof.context
-val check: Proof.context -> thm -> int -> tactic
-val infer: Proof.context -> thm -> int -> tactic
+val known_ctac: thm list -> int -> context_tactic
+val check_infer: thm list -> int -> context_tactic
end = struct
+
(* Context data *)
structure Data = Generic_Data (
@@ -32,22 +33,55 @@ fun put_type typing = Context.proof_map (Data.map (Item_Net.update typing))
fun put_types typings = foldr1 (op o) (map put_type typings)
-(* Checking and inference *)
-
+(* Context tactics for type-checking *)
+
+(*Solves goals without metavariables and type inference problems by resolving
+ with facts or assumption from inline premises.*)
+fun known_ctac facts = CONTEXT_SUBGOAL (fn (goal, i) => fn (ctxt, st) =>
+ TACTIC_CONTEXT ctxt
+ let val concl = Logic.strip_assums_concl goal in
+ if Lib.no_vars concl orelse
+ (Lib.is_typing concl andalso Lib.no_vars (Lib.term_of_typing concl))
+ then
+ let val ths = facts
+ (*FIXME: Shouldn't pull nameless facts directly from context*)
+ @ map fst (Facts.props (Proof_Context.facts_of ctxt))
+ in (resolve_tac ctxt ths i ORELSE assume_tac ctxt i) st end
+ else Seq.empty
+ end)
+
+(*Simple bidirectional typing tactic, with some nondeterminism from backtracking
+ search over arbitrary `typechk` rules. The current implementation does not
+ perform any normalization.*)
local
-
-fun checkable prop = Lib.is_typing prop
- andalso not (exists_subterm is_Var (Lib.type_of_typing prop))
-
+ fun check_infer_step facts i (ctxt, st) =
+ let
+ val tac = SUBGOAL (fn (goal, i) =>
+ if Lib.rigid_typing_concl goal
+ then
+ let val net = Tactic.build_net (facts
+ (*MAYBE FIXME: Remove `typechk` from this list, and instead perform
+ definitional unfolding to (w?)hnf.*)
+ @(Named_Theorems.get ctxt \<^named_theorems>\<open>typechk\<close>)
+ @(Named_Theorems.get ctxt \<^named_theorems>\<open>form\<close>)
+ @(Named_Theorems.get ctxt \<^named_theorems>\<open>intro\<close>)
+ @(map #1 (Elim.rules ctxt)))
+ in (resolve_from_net_tac ctxt net) i end
+ else no_tac)
+ val ctxt' = ctxt
+ in
+ TACTIC_CONTEXT ctxt' (tac i st)
+ end
in
-fun check ctxt rule = Subgoal.FOCUS_PREMS (
- fn {context = goal_ctxt, prems, concl, ...} => no_tac) ctxt
-
-fun infer ctxt rule = Subgoal.FOCUS_PREMS (
- fn {context = goal_ctxt, prems, concl, ...} => no_tac) ctxt
+fun check_infer facts i (cst as (_, st)) =
+ let
+ val ctac = known_ctac facts CORELSE' check_infer_step facts
+ in
+ cst |> (ctac i CTHEN
+ CREPEAT_IN_RANGE i (Thm.nprems_of st - i) (CTRY o CREPEAT_ALL_NEW_FWD ctac))
+ end
end
-
end
diff --git a/spartan/lib/List.thy b/spartan/lib/List.thy
index a755859..be86b63 100644
--- a/spartan/lib/List.thy
+++ b/spartan/lib/List.thy
@@ -44,9 +44,10 @@ where
f x xs (ListInd A (fn xs. C xs) c\<^sub>0 (fn x xs rec. f x xs rec) xs)"
lemmas
- [intros] = ListF List_nil List_cons and
- [elims "?xs"] = ListE and
- [comps] = List_comp_nil List_comp_cons
+ [form] = ListF and
+ [intro, intros] = List_nil List_cons and
+ [elim "?xs"] = ListE and
+ [comp] = List_comp_nil List_comp_cons
abbreviation "ListRec A C \<equiv> ListInd A (fn _. C)"
@@ -110,7 +111,7 @@ Lemma head_type [typechk]:
shows "head xs: Maybe A"
unfolding head_def by typechk
-Lemma head_of_cons [comps]:
+Lemma head_of_cons [comp]:
assumes "A: U i" "x: A" "xs: List A"
shows "head (x # xs) \<equiv> some x"
unfolding head_def by reduce
@@ -120,7 +121,7 @@ Lemma tail_type [typechk]:
shows "tail xs: List A"
unfolding tail_def by typechk
-Lemma tail_of_cons [comps]:
+Lemma tail_of_cons [comp]:
assumes "A: U i" "x: A" "xs: List A"
shows "tail (x # xs) \<equiv> xs"
unfolding tail_def by reduce
@@ -181,7 +182,7 @@ Lemma rev_type [typechk]:
shows "rev xs: List A"
unfolding rev_def by typechk
-Lemma rev_nil [comps]:
+Lemma rev_nil [comp]:
assumes "A: U i"
shows "rev (nil A) \<equiv> nil A"
unfolding rev_def by reduce
diff --git a/spartan/lib/Maybe.thy b/spartan/lib/Maybe.thy
index d821920..0ce534c 100644
--- a/spartan/lib/Maybe.thy
+++ b/spartan/lib/Maybe.thy
@@ -54,9 +54,10 @@ Lemma Maybe_comp_some:
unfolding MaybeInd_def some_def by (reduce add: Maybe_def)
lemmas
- [intros] = MaybeF Maybe_none Maybe_some and
- [comps] = Maybe_comp_none Maybe_comp_some and
- MaybeE [elims "?m"] = MaybeInd[rotated 4]
+ [form] = MaybeF and
+ [intro, intros] = Maybe_none Maybe_some and
+ [comp] = Maybe_comp_none Maybe_comp_some and
+ MaybeE [elim "?m"] = MaybeInd[rotated 4]
lemmas
Maybe_cases [cases] = MaybeE
diff --git a/spartan/lib/More_Types.thy b/spartan/lib/More_Types.thy
index 0d7096f..55e6554 100644
--- a/spartan/lib/More_Types.thy
+++ b/spartan/lib/More_Types.thy
@@ -16,9 +16,9 @@ notation Sum (infixl "\<or>" 50)
axiomatization where
SumF: "\<lbrakk>A: U i; B: U i\<rbrakk> \<Longrightarrow> A \<or> B: U i" and
- Sum_inl: "\<lbrakk>a: A; B: U i\<rbrakk> \<Longrightarrow> inl A B a: A \<or> B" and
+ Sum_inl: "\<lbrakk>B: U i; a: A\<rbrakk> \<Longrightarrow> inl A B a: A \<or> B" and
- Sum_inr: "\<lbrakk>b: B; A: U i\<rbrakk> \<Longrightarrow> inr A B b: A \<or> B" and
+ Sum_inr: "\<lbrakk>A: U i; b: B\<rbrakk> \<Longrightarrow> inr A B b: A \<or> B" and
SumE: "\<lbrakk>
s: A \<or> B;
@@ -42,9 +42,11 @@ axiomatization where
\<rbrakk> \<Longrightarrow> SumInd A B (fn s. C s) (fn a. c a) (fn b. d b) (inr A B b) \<equiv> d b"
lemmas
- [intros] = SumF Sum_inl Sum_inr and
- [elims ?s] = SumE and
- [comps] = Sum_comp_inl Sum_comp_inr
+ [form] = SumF and
+ [intro] = Sum_inl Sum_inr and
+ [intros] = Sum_inl[rotated] Sum_inr[rotated] and
+ [elim ?s] = SumE and
+ [comp] = Sum_comp_inl Sum_comp_inr
method left = rule Sum_inl
method right = rule Sum_inr
@@ -76,10 +78,11 @@ and
BotE: "\<lbrakk>x: \<bottom>; \<And>x. x: \<bottom> \<Longrightarrow> C x: U i\<rbrakk> \<Longrightarrow> BotInd (fn x. C x) x: C x"
lemmas
- [intros] = TopF TopI BotF and
- [elims ?a] = TopE and
- [elims ?x] = BotE and
- [comps] = Top_comp
+ [form] = TopF BotF and
+ [intro, intros] = TopI and
+ [elim ?a] = TopE and
+ [elim ?x] = BotE and
+ [comp] = Top_comp
section \<open>Booleans\<close>
@@ -125,9 +128,10 @@ Lemma if_false:
by reduce
lemmas
- [intros] = BoolF Bool_true Bool_false and
- [comps] = if_true if_false and
- [elims ?x] = ifelse
+ [form] = BoolF and
+ [intro, intros] = Bool_true Bool_false and
+ [comp] = if_true if_false and
+ [elim ?x] = ifelse
lemmas
BoolE = ifelse
@@ -136,7 +140,7 @@ subsection \<open>Notation\<close>
definition ifelse_i ("if _ then _ else _")
where [implicit]: "if x then a else b \<equiv> ifelse ? x a b"
-no_translations "if x then a else b" \<leftharpoondown> "CONST ifelse C x a b"
+translations "if x then a else b" \<leftharpoondown> "CONST ifelse C x a b"
subsection \<open>Logical connectives\<close>
generated by cgit v1.2.3 (git 2.25.1) at 2024-12-22 11:30:16 +0000