(* Title: CTT/CTT.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
*)
theory CTT
imports Pure
begin
section \<open>Constructive Type Theory: axiomatic basis\<close>
ML_file "~~/src/Provers/typedsimp.ML"
setup Pure_Thy.old_appl_syntax_setup
typedecl i
typedecl t
typedecl o
consts
\<comment> \<open>Types\<close>
F :: "t"
T :: "t" \<comment> \<open>\<open>F\<close> is empty, \<open>T\<close> contains one element\<close>
contr :: "i\<Rightarrow>i"
tt :: "i"
\<comment> \<open>Natural numbers\<close>
N :: "t"
succ :: "i\<Rightarrow>i"
rec :: "[i, i, [i,i]\<Rightarrow>i] \<Rightarrow> i"
\<comment> \<open>Unions\<close>
inl :: "i\<Rightarrow>i"
inr :: "i\<Rightarrow>i"
"when" :: "[i, i\<Rightarrow>i, i\<Rightarrow>i]\<Rightarrow>i"
\<comment> \<open>General Sum and Binary Product\<close>
Sum :: "[t, i\<Rightarrow>t]\<Rightarrow>t"
fst :: "i\<Rightarrow>i"
snd :: "i\<Rightarrow>i"
split :: "[i, [i,i]\<Rightarrow>i] \<Rightarrow>i"
\<comment> \<open>General Product and Function Space\<close>
Prod :: "[t, i\<Rightarrow>t]\<Rightarrow>t"
\<comment> \<open>Types\<close>
Plus :: "[t,t]\<Rightarrow>t" (infixr "+" 40)
\<comment> \<open>Equality type\<close>
Eq :: "[t,i,i]\<Rightarrow>t"
eq :: "i"
\<comment> \<open>Judgements\<close>
Type :: "t \<Rightarrow> prop" ("(_ type)" [10] 5)
Eqtype :: "[t,t]\<Rightarrow>prop" ("(_ =/ _)" [10,10] 5)
Elem :: "[i, t]\<Rightarrow>prop" ("(_ /: _)" [10,10] 5)
Eqelem :: "[i,i,t]\<Rightarrow>prop" ("(_ =/ _ :/ _)" [10,10,10] 5)
Reduce :: "[i,i]\<Rightarrow>prop" ("Reduce[_,_]")
\<comment> \<open>Types\<close>
\<comment> \<open>Functions\<close>
lambda :: "(i \<Rightarrow> i) \<Rightarrow> i" (binder "\<^bold>\<lambda>" 10)
app :: "[i,i]\<Rightarrow>i" (infixl "`" 60)
\<comment> \<open>Natural numbers\<close>
Zero :: "i" ("0")
\<comment> \<open>Pairing\<close>
pair :: "[i,i]\<Rightarrow>i" ("(1<_,/_>)")
syntax
"_PROD" :: "[idt,t,t]\<Rightarrow>t" ("(3\<Prod>_:_./ _)" 10)
"_SUM" :: "[idt,t,t]\<Rightarrow>t" ("(3\<Sum>_:_./ _)" 10)
translations
"\<Prod>x:A. B" \<rightleftharpoons> "CONST Prod(A, \<lambda>x. B)"
"\<Sum>x:A. B" \<rightleftharpoons> "CONST Sum(A, \<lambda>x. B)"
abbreviation Arrow :: "[t,t]\<Rightarrow>t" (infixr "\<longrightarrow>" 30)
where "A \<longrightarrow> B \<equiv> \<Prod>_:A. B"
abbreviation Times :: "[t,t]\<Rightarrow>t" (infixr "\<times>" 50)
where "A \<times> B \<equiv> \<Sum>_:A. B"
text \<open>
Reduction: a weaker notion than equality; a hack for simplification.
\<open>Reduce[a,b]\<close> means either that \<open>a = b : A\<close> for some \<open>A\<close> or else
that \<open>a\<close> and \<open>b\<close> are textually identical.
Does not verify \<open>a:A\<close>! Sound because only \<open>trans_red\<close> uses a \<open>Reduce\<close>
premise. No new theorems can be proved about the standard judgements.
\<close>
axiomatization
where
refl_red: "\<And>a. Reduce[a,a]" and
red_if_equal: "\<And>a b A. a = b : A \<Longrightarrow> Reduce[a,b]" and
trans_red: "\<And>a b c A. \<lbrakk>a = b : A; Reduce[b,c]\<rbrakk> \<Longrightarrow> a = c : A" and
\<comment> \<open>Reflexivity\<close>
refl_type: "\<And>A. A type \<Longrightarrow> A = A" and
refl_elem: "\<And>a A. a : A \<Longrightarrow> a = a : A" and
\<comment> \<open>Symmetry\<close>
sym_type: "\<And>A B. A = B \<Longrightarrow> B = A" and
sym_elem: "\<And>a b A. a = b : A \<Longrightarrow> b = a : A" and
\<comment> \<open>Transitivity\<close>
trans_type: "\<And>A B C. \<lbrakk>A = B; B = C\<rbrakk> \<Longrightarrow> A = C" and
trans_elem: "\<And>a b c A. \<lbrakk>a = b : A; b = c : A\<rbrakk> \<Longrightarrow> a = c : A" and
equal_types: "\<And>a A B. \<lbrakk>a : A; A = B\<rbrakk> \<Longrightarrow> a : B" and
equal_typesL: "\<And>a b A B. \<lbrakk>a = b : A; A = B\<rbrakk> \<Longrightarrow> a = b : B" and
\<comment> \<open>Substitution\<close>
subst_type: "\<And>a A B. \<lbrakk>a : A; \<And>z. z:A \<Longrightarrow> B(z) type\<rbrakk> \<Longrightarrow> B(a) type" and
subst_typeL: "\<And>a c A B D. \<lbrakk>a = c : A; \<And>z. z:A \<Longrightarrow> B(z) = D(z)\<rbrakk> \<Longrightarrow> B(a) = D(c)" and
subst_elem: "\<And>a b A B. \<lbrakk>a : A; \<And>z. z:A \<Longrightarrow> b(z):B(z)\<rbrakk> \<Longrightarrow> b(a):B(a)" and
subst_elemL:
"\<And>a b c d A B. \<lbrakk>a = c : A; \<And>z. z:A \<Longrightarrow> b(z)=d(z) : B(z)\<rbrakk> \<Longrightarrow> b(a)=d(c) : B(a)" and
\<comment> \<open>The type \<open>N\<close> -- natural numbers\<close>
NF: "N type" and
NI0: "0 : N" and
NI_succ: "\<And>a. a : N \<Longrightarrow> succ(a) : N" and
NI_succL: "\<And>a b. a = b : N \<Longrightarrow> succ(a) = succ(b) : N" and
NE:
"\<And>p a b C. \<lbrakk>p: N; a: C(0); \<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v): C(succ(u))\<rbrakk>
\<Longrightarrow> rec(p, a, \<lambda>u v. b(u,v)) : C(p)" and
NEL:
"\<And>p q a b c d C. \<lbrakk>p = q : N; a = c : C(0);
\<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v) = d(u,v): C(succ(u))\<rbrakk>
\<Longrightarrow> rec(p, a, \<lambda>u v. b(u,v)) = rec(q,c,d) : C(p)" and
NC0:
"\<And>a b C. \<lbrakk>a: C(0); \<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v): C(succ(u))\<rbrakk>
\<Longrightarrow> rec(0, a, \<lambda>u v. b(u,v)) = a : C(0)" and
NC_succ:
"\<And>p a b C. \<lbrakk>p: N; a: C(0); \<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v): C(succ(u))\<rbrakk> \<Longrightarrow>
rec(succ(p), a, \<lambda>u v. b(u,v)) = b(p, rec(p, a, \<lambda>u v. b(u,v))) : C(succ(p))" and
\<comment> \<open>The fourth Peano axiom. See page 91 of Martin-Löf's book.\<close>
zero_ne_succ: "\<And>a. \<lbrakk>a: N; 0 = succ(a) : N\<rbrakk> \<Longrightarrow> 0: F" and
\<comment> \<open>The Product of a family of types\<close>
ProdF: "\<And>A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> B(x) type\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x) type" and
ProdFL:
"\<And>A B C D. \<lbrakk>A = C; \<And>x. x:A \<Longrightarrow> B(x) = D(x)\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x) = \<Prod>x:C. D(x)" and
ProdI:
"\<And>b A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> b(x):B(x)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b(x) : \<Prod>x:A. B(x)" and
ProdIL: "\<And>b c A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> b(x) = c(x) : B(x)\<rbrakk> \<Longrightarrow>
\<^bold>\<lambda>x. b(x) = \<^bold>\<lambda>x. c(x) : \<Prod>x:A. B(x)" and
ProdE: "\<And>p a A B. \<lbrakk>p : \<Prod>x:A. B(x); a : A\<rbrakk> \<Longrightarrow> p`a : B(a)" and
ProdEL: "\<And>p q a b A B. \<lbrakk>p = q: \<Prod>x:A. B(x); a = b : A\<rbrakk> \<Longrightarrow> p`a = q`b : B(a)" and
ProdC: "\<And>a b A B. \<lbrakk>a : A; \<And>x. x:A \<Longrightarrow> b(x) : B(x)\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. b(x)) ` a = b(a) : B(a)" and
ProdC2: "\<And>p A B. p : \<Prod>x:A. B(x) \<Longrightarrow> (\<^bold>\<lambda>x. p`x) = p : \<Prod>x:A. B(x)" and
\<comment> \<open>The Sum of a family of types\<close>
SumF: "\<And>A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> B(x) type\<rbrakk> \<Longrightarrow> \<Sum>x:A. B(x) type" and
SumFL: "\<And>A B C D. \<lbrakk>A = C; \<And>x. x:A \<Longrightarrow> B(x) = D(x)\<rbrakk> \<Longrightarrow> \<Sum>x:A. B(x) = \<Sum>x:C. D(x)" and
SumI: "\<And>a b A B. \<lbrakk>a : A; b : B(a)\<rbrakk> \<Longrightarrow> <a,b> : \<Sum>x:A. B(x)" and
SumIL: "\<And>a b c d A B. \<lbrakk> a = c : A; b = d : B(a)\<rbrakk> \<Longrightarrow> <a,b> = <c,d> : \<Sum>x:A. B(x)" and
SumE: "\<And>p c A B C. \<lbrakk>p: \<Sum>x:A. B(x); \<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> c(x,y): C(<x,y>)\<rbrakk>
\<Longrightarrow> split(p, \<lambda>x y. c(x,y)) : C(p)" and
SumEL: "\<And>p q c d A B C. \<lbrakk>p = q : \<Sum>x:A. B(x);
\<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> c(x,y)=d(x,y): C(<x,y>)\<rbrakk>
\<Longrightarrow> split(p, \<lambda>x y. c(x,y)) = split(q, \<lambda>x y. d(x,y)) : C(p)" and
SumC: "\<And>a b c A B C. \<lbrakk>a: A; b: B(a); \<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> c(x,y): C(<x,y>)\<rbrakk>
\<Longrightarrow> split(<a,b>, \<lambda>x y. c(x,y)) = c(a,b) : C(<a,b>)" and
fst_def: "\<And>a. fst(a) \<equiv> split(a, \<lambda>x y. x)" and
snd_def: "\<And>a. snd(a) \<equiv> split(a, \<lambda>x y. y)" and
\<comment> \<open>The sum of two types\<close>
PlusF: "\<And>A B. \<lbrakk>A type; B type\<rbrakk> \<Longrightarrow> A+B type" and
PlusFL: "\<And>A B C D. \<lbrakk>A = C; B = D\<rbrakk> \<Longrightarrow> A+B = C+D" and
PlusI_inl: "\<And>a A B. \<lbrakk>a : A; B type\<rbrakk> \<Longrightarrow> inl(a) : A+B" and
PlusI_inlL: "\<And>a c A B. \<lbrakk>a = c : A; B type\<rbrakk> \<Longrightarrow> inl(a) = inl(c) : A+B" and
PlusI_inr: "\<And>b A B. \<lbrakk>A type; b : B\<rbrakk> \<Longrightarrow> inr(b) : A+B" and
PlusI_inrL: "\<And>b d A B. \<lbrakk>A type; b = d : B\<rbrakk> \<Longrightarrow> inr(b) = inr(d) : A+B" and
PlusE:
"\<And>p c d A B C. \<lbrakk>p: A+B;
\<And>x. x:A \<Longrightarrow> c(x): C(inl(x));
\<And>y. y:B \<Longrightarrow> d(y): C(inr(y)) \<rbrakk> \<Longrightarrow> when(p, \<lambda>x. c(x), \<lambda>y. d(y)) : C(p)" and
PlusEL:
"\<And>p q c d e f A B C. \<lbrakk>p = q : A+B;
\<And>x. x: A \<Longrightarrow> c(x) = e(x) : C(inl(x));
\<And>y. y: B \<Longrightarrow> d(y) = f(y) : C(inr(y))\<rbrakk>
\<Longrightarrow> when(p, \<lambda>x. c(x), \<lambda>y. d(y)) = when(q, \<lambda>x. e(x), \<lambda>y. f(y)) : C(p)" and
PlusC_inl:
"\<And>a c d A B C. \<lbrakk>a: A;
\<And>x. x:A \<Longrightarrow> c(x): C(inl(x));
\<And>y. y:B \<Longrightarrow> d(y): C(inr(y)) \<rbrakk>
\<Longrightarrow> when(inl(a), \<lambda>x. c(x), \<lambda>y. d(y)) = c(a) : C(inl(a))" and
PlusC_inr:
"\<And>b c d A B C. \<lbrakk>b: B;
\<And>x. x:A \<Longrightarrow> c(x): C(inl(x));
\<And>y. y:B \<Longrightarrow> d(y): C(inr(y))\<rbrakk>
\<Longrightarrow> when(inr(b), \<lambda>x. c(x), \<lambda>y. d(y)) = d(b) : C(inr(b))" and
\<comment> \<open>The type \<open>Eq\<close>\<close>
EqF: "\<And>a b A. \<lbrakk>A type; a : A; b : A\<rbrakk> \<Longrightarrow> Eq(A,a,b) type" and
EqFL: "\<And>a b c d A B. \<lbrakk>A = B; a = c : A; b = d : A\<rbrakk> \<Longrightarrow> Eq(A,a,b) = Eq(B,c,d)" and
EqI: "\<And>a b A. a = b : A \<Longrightarrow> eq : Eq(A,a,b)" and
EqE: "\<And>p a b A. p : Eq(A,a,b) \<Longrightarrow> a = b : A" and
\<comment> \<open>By equality of types, can prove \<open>C(p)\<close> from \<open>C(eq)\<close>, an elimination rule\<close>
EqC: "\<And>p a b A. p : Eq(A,a,b) \<Longrightarrow> p = eq : Eq(A,a,b)" and
\<comment> \<open>The type \<open>F\<close>\<close>
FF: "F type" and
FE: "\<And>p C. \<lbrakk>p: F; C type\<rbrakk> \<Longrightarrow> contr(p) : C" and
FEL: "\<And>p q C. \<lbrakk>p = q : F; C type\<rbrakk> \<Longrightarrow> contr(p) = contr(q) : C" and
\<comment> \<open>The type T\<close>
\<comment> \<open>
Martin-Löf's book (page 68) discusses elimination and computation.
Elimination can be derived by computation and equality of types,
but with an extra premise \<open>C(x)\<close> type \<open>x:T\<close>.
Also computation can be derived from elimination.
\<close>
TF: "T type" and
TI: "tt : T" and
TE: "\<And>p c C. \<lbrakk>p : T; c : C(tt)\<rbrakk> \<Longrightarrow> c : C(p)" and
TEL: "\<And>p q c d C. \<lbrakk>p = q : T; c = d : C(tt)\<rbrakk> \<Longrightarrow> c = d : C(p)" and
TC: "\<And>p. p : T \<Longrightarrow> p = tt : T"
subsection "Tactics and derived rules for Constructive Type Theory"
text \<open>Formation rules.\<close>
lemmas form_rls = NF ProdF SumF PlusF EqF FF TF
and formL_rls = ProdFL SumFL PlusFL EqFL
text \<open>
Introduction rules. OMITTED:
\<^item> \<open>EqI\<close>, because its premise is an \<open>eqelem\<close>, not an \<open>elem\<close>.
\<close>
lemmas intr_rls = NI0 NI_succ ProdI SumI PlusI_inl PlusI_inr TI
and intrL_rls = NI_succL ProdIL SumIL PlusI_inlL PlusI_inrL
text \<open>
Elimination rules. OMITTED:
\<^item> \<open>EqE\<close>, because its conclusion is an \<open>eqelem\<close>, not an \<open>elem\<close>
\<^item> \<open>TE\<close>, because it does not involve a constructor.
\<close>
lemmas elim_rls = NE ProdE SumE PlusE FE
and elimL_rls = NEL ProdEL SumEL PlusEL FEL
text \<open>OMITTED: \<open>eqC\<close> are \<open>TC\<close> because they make rewriting loop: \<open>p = un = un = \<dots>\<close>\<close>
lemmas comp_rls = NC0 NC_succ ProdC SumC PlusC_inl PlusC_inr
text \<open>Rules with conclusion \<open>a:A\<close>, an elem judgement.\<close>
lemmas element_rls = intr_rls elim_rls
text \<open>Definitions are (meta)equality axioms.\<close>
lemmas basic_defs = fst_def snd_def
text \<open>Compare with standard version: \<open>B\<close> is applied to UNSIMPLIFIED expression!\<close>
lemma SumIL2: "\<lbrakk>c = a : A; d = b : B(a)\<rbrakk> \<Longrightarrow> <c,d> = <a,b> : Sum(A,B)"
by (rule sym_elem) (rule SumIL; rule sym_elem)
lemmas intrL2_rls = NI_succL ProdIL SumIL2 PlusI_inlL PlusI_inrL
text \<open>
Exploit \<open>p:Prod(A,B)\<close> to create the assumption \<open>z:B(a)\<close>.
A more natural form of product elimination.
\<close>
lemma subst_prodE:
assumes "p: Prod(A,B)"
and "a: A"
and "\<And>z. z: B(a) \<Longrightarrow> c(z): C(z)"
shows "c(p`a): C(p`a)"
by (rule assms ProdE)+
subsection \<open>Tactics for type checking\<close>
ML \<open>
local
fun is_rigid_elem (Const(@{const_name Elem},_) $ a $ _) = not(is_Var (head_of a))
| is_rigid_elem (Const(@{const_name Eqelem},_) $ a $ _ $ _) = not(is_Var (head_of a))
| is_rigid_elem (Const(@{const_name Type},_) $ a) = not(is_Var (head_of a))
| is_rigid_elem _ = false
in
(*Try solving a:A or a=b:A by assumption provided a is rigid!*)
fun test_assume_tac ctxt = SUBGOAL (fn (prem, i) =>
if is_rigid_elem (Logic.strip_assums_concl prem)
then assume_tac ctxt i else no_tac)
fun ASSUME ctxt tf i = test_assume_tac ctxt i ORELSE tf i
end
\<close>
text \<open>
For simplification: type formation and checking,
but no equalities between terms.
\<close>
lemmas routine_rls = form_rls formL_rls refl_type element_rls
ML \<open>
fun routine_tac rls ctxt prems =
ASSUME ctxt (filt_resolve_from_net_tac ctxt 4 (Tactic.build_net (prems @ rls)));
(*Solve all subgoals "A type" using formation rules. *)
val form_net = Tactic.build_net @{thms form_rls};
fun form_tac ctxt =
REPEAT_FIRST (ASSUME ctxt (filt_resolve_from_net_tac ctxt 1 form_net));
(*Type checking: solve a:A (a rigid, A flexible) by intro and elim rules. *)
fun typechk_tac ctxt thms =
let val tac =
filt_resolve_from_net_tac ctxt 3
(Tactic.build_net (thms @ @{thms form_rls} @ @{thms element_rls}))
in REPEAT_FIRST (ASSUME ctxt tac) end
(*Solve a:A (a flexible, A rigid) by introduction rules.
Cannot use stringtrees (filt_resolve_tac) since
goals like ?a:SUM(A,B) have a trivial head-string *)
fun intr_tac ctxt thms =
let val tac =
filt_resolve_from_net_tac ctxt 1
(Tactic.build_net (thms @ @{thms form_rls} @ @{thms intr_rls}))
in REPEAT_FIRST (ASSUME ctxt tac) end
(*Equality proving: solve a=b:A (where a is rigid) by long rules. *)
fun equal_tac ctxt thms =
REPEAT_FIRST
(ASSUME ctxt
(filt_resolve_from_net_tac ctxt 3
(Tactic.build_net (thms @ @{thms form_rls element_rls intrL_rls elimL_rls refl_elem}))))
\<close>
method_setup form = \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (form_tac ctxt))\<close>
method_setup typechk = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (typechk_tac ctxt ths))\<close>
method_setup intr = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (intr_tac ctxt ths))\<close>
method_setup equal = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (equal_tac ctxt ths))\<close>
subsection \<open>Simplification\<close>
text \<open>To simplify the type in a goal.\<close>
lemma replace_type: "\<lbrakk>B = A; a : A\<rbrakk> \<Longrightarrow> a : B"
apply (rule equal_types)
apply (rule_tac [2] sym_type)
apply assumption+
done
text \<open>Simplify the parameter of a unary type operator.\<close>
lemma subst_eqtyparg:
assumes 1: "a=c : A"
and 2: "\<And>z. z:A \<Longrightarrow> B(z) type"
shows "B(a) = B(c)"
apply (rule subst_typeL)
apply (rule_tac [2] refl_type)
apply (rule 1)
apply (erule 2)
done
text \<open>Simplification rules for Constructive Type Theory.\<close>
lemmas reduction_rls = comp_rls [THEN trans_elem]
ML \<open>
(*Converts each goal "e : Eq(A,a,b)" into "a=b:A" for simplification.
Uses other intro rules to avoid changing flexible goals.*)
val eqintr_net = Tactic.build_net @{thms EqI intr_rls}
fun eqintr_tac ctxt =
REPEAT_FIRST (ASSUME ctxt (filt_resolve_from_net_tac ctxt 1 eqintr_net))
(** Tactics that instantiate CTT-rules.
Vars in the given terms will be incremented!
The (rtac EqE i) lets them apply to equality judgements. **)
fun NE_tac ctxt sp i =
TRY (resolve_tac ctxt @{thms EqE} i) THEN
Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm NE} i
fun SumE_tac ctxt sp i =
TRY (resolve_tac ctxt @{thms EqE} i) THEN
Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm SumE} i
fun PlusE_tac ctxt sp i =
TRY (resolve_tac ctxt @{thms EqE} i) THEN
Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm PlusE} i
(** Predicate logic reasoning, WITH THINNING!! Procedures adapted from NJ. **)
(*Finds f:Prod(A,B) and a:A in the assumptions, concludes there is z:B(a) *)
fun add_mp_tac ctxt i =
resolve_tac ctxt @{thms subst_prodE} i THEN assume_tac ctxt i THEN assume_tac ctxt i
(*Finds P\<longrightarrow>Q and P in the assumptions, replaces implication by Q *)
fun mp_tac ctxt i = eresolve_tac ctxt @{thms subst_prodE} i THEN assume_tac ctxt i
(*"safe" when regarded as predicate calculus rules*)
val safe_brls = sort (make_ord lessb)
[ (true, @{thm FE}), (true,asm_rl),
(false, @{thm ProdI}), (true, @{thm SumE}), (true, @{thm PlusE}) ]
val unsafe_brls =
[ (false, @{thm PlusI_inl}), (false, @{thm PlusI_inr}), (false, @{thm SumI}),
(true, @{thm subst_prodE}) ]
(*0 subgoals vs 1 or more*)
val (safe0_brls, safep_brls) =
List.partition (curry (op =) 0 o subgoals_of_brl) safe_brls
fun safestep_tac ctxt thms i =
form_tac ctxt ORELSE
resolve_tac ctxt thms i ORELSE
biresolve_tac ctxt safe0_brls i ORELSE mp_tac ctxt i ORELSE
DETERM (biresolve_tac ctxt safep_brls i)
fun safe_tac ctxt thms i = DEPTH_SOLVE_1 (safestep_tac ctxt thms i)
fun step_tac ctxt thms = safestep_tac ctxt thms ORELSE' biresolve_tac ctxt unsafe_brls
(*Fails unless it solves the goal!*)
fun pc_tac ctxt thms = DEPTH_SOLVE_1 o (step_tac ctxt thms)
\<close>
method_setup eqintr = \<open>Scan.succeed (SIMPLE_METHOD o eqintr_tac)\<close>
method_setup NE = \<open>
Scan.lift Args.embedded_inner_syntax >> (fn s => fn ctxt => SIMPLE_METHOD' (NE_tac ctxt s))
\<close>
method_setup pc = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (pc_tac ctxt ths))\<close>
method_setup add_mp = \<open>Scan.succeed (SIMPLE_METHOD' o add_mp_tac)\<close>
ML_file "rew.ML"
method_setup rew = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (rew_tac ctxt ths))\<close>
method_setup hyp_rew = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (hyp_rew_tac ctxt ths))\<close>
subsection \<open>The elimination rules for fst/snd\<close>
lemma SumE_fst: "p : Sum(A,B) \<Longrightarrow> fst(p) : A"
apply (unfold basic_defs)
apply (erule SumE)
apply assumption
done
text \<open>The first premise must be \<open>p:Sum(A,B)\<close>!!.\<close>
lemma SumE_snd:
assumes major: "p: Sum(A,B)"
and "A type"
and "\<And>x. x:A \<Longrightarrow> B(x) type"
shows "snd(p) : B(fst(p))"
apply (unfold basic_defs)
apply (rule major [THEN SumE])
apply (rule SumC [THEN subst_eqtyparg, THEN replace_type])
apply (typechk assms)
done
section \<open>The two-element type (booleans and conditionals)\<close>
definition Bool :: "t"
where "Bool \<equiv> T+T"
definition true :: "i"
where "true \<equiv> inl(tt)"
definition false :: "i"
where "false \<equiv> inr(tt)"
definition cond :: "[i,i,i]\<Rightarrow>i"
where "cond(a,b,c) \<equiv> when(a, \<lambda>_. b, \<lambda>_. c)"
lemmas bool_defs = Bool_def true_def false_def cond_def
subsection \<open>Derivation of rules for the type \<open>Bool\<close>\<close>
text \<open>Formation rule.\<close>
lemma boolF: "Bool type"
unfolding bool_defs by typechk
text \<open>Introduction rules for \<open>true\<close>, \<open>false\<close>.\<close>
lemma boolI_true: "true : Bool"
unfolding bool_defs by typechk
lemma boolI_false: "false : Bool"
unfolding bool_defs by typechk
text \<open>Elimination rule: typing of \<open>cond\<close>.\<close>
lemma boolE: "\<lbrakk>p:Bool; a : C(true); b : C(false)\<rbrakk> \<Longrightarrow> cond(p,a,b) : C(p)"
unfolding bool_defs
apply (typechk; erule TE)
apply typechk
done
lemma boolEL: "\<lbrakk>p = q : Bool; a = c : C(true); b = d : C(false)\<rbrakk>
\<Longrightarrow> cond(p,a,b) = cond(q,c,d) : C(p)"
unfolding bool_defs
apply (rule PlusEL)
apply (erule asm_rl refl_elem [THEN TEL])+
done
text \<open>Computation rules for \<open>true\<close>, \<open>false\<close>.\<close>
lemma boolC_true: "\<lbrakk>a : C(true); b : C(false)\<rbrakk> \<Longrightarrow> cond(true,a,b) = a : C(true)"
unfolding bool_defs
apply (rule comp_rls)
apply typechk
apply (erule_tac [!] TE)
apply typechk
done
lemma boolC_false: "\<lbrakk>a : C(true); b : C(false)\<rbrakk> \<Longrightarrow> cond(false,a,b) = b : C(false)"
unfolding bool_defs
apply (rule comp_rls)
apply typechk
apply (erule_tac [!] TE)
apply typechk
done
section \<open>Elementary arithmetic\<close>
subsection \<open>Arithmetic operators and their definitions\<close>
definition add :: "[i,i]\<Rightarrow>i" (infixr "#+" 65)
where "a#+b \<equiv> rec(a, b, \<lambda>u v. succ(v))"
definition diff :: "[i,i]\<Rightarrow>i" (infixr "-" 65)
where "a-b \<equiv> rec(b, a, \<lambda>u v. rec(v, 0, \<lambda>x y. x))"
definition absdiff :: "[i,i]\<Rightarrow>i" (infixr "|-|" 65)
where "a|-|b \<equiv> (a-b) #+ (b-a)"
definition mult :: "[i,i]\<Rightarrow>i" (infixr "#*" 70)
where "a#*b \<equiv> rec(a, 0, \<lambda>u v. b #+ v)"
definition mod :: "[i,i]\<Rightarrow>i" (infixr "mod" 70)
where "a mod b \<equiv> rec(a, 0, \<lambda>u v. rec(succ(v) |-| b, 0, \<lambda>x y. succ(v)))"
definition div :: "[i,i]\<Rightarrow>i" (infixr "div" 70)
where "a div b \<equiv> rec(a, 0, \<lambda>u v. rec(succ(u) mod b, succ(v), \<lambda>x y. v))"
lemmas arith_defs = add_def diff_def absdiff_def mult_def mod_def div_def
subsection \<open>Proofs about elementary arithmetic: addition, multiplication, etc.\<close>
subsubsection \<open>Addition\<close>
text \<open>Typing of \<open>add\<close>: short and long versions.\<close>
lemma add_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #+ b : N"
unfolding arith_defs by typechk
lemma add_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a #+ b = c #+ d : N"
unfolding arith_defs by equal
text \<open>Computation for \<open>add\<close>: 0 and successor cases.\<close>
lemma addC0: "b:N \<Longrightarrow> 0 #+ b = b : N"
unfolding arith_defs by rew
lemma addC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) #+ b = succ(a #+ b) : N"
unfolding arith_defs by rew
subsubsection \<open>Multiplication\<close>
text \<open>Typing of \<open>mult\<close>: short and long versions.\<close>
lemma mult_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #* b : N"
unfolding arith_defs by (typechk add_typing)
lemma mult_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a #* b = c #* d : N"
unfolding arith_defs by (equal add_typingL)
text \<open>Computation for \<open>mult\<close>: 0 and successor cases.\<close>
lemma multC0: "b:N \<Longrightarrow> 0 #* b = 0 : N"
unfolding arith_defs by rew
lemma multC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) #* b = b #+ (a #* b) : N"
unfolding arith_defs by rew
subsubsection \<open>Difference\<close>
text \<open>Typing of difference.\<close>
lemma diff_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a - b : N"
unfolding arith_defs by typechk
lemma diff_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a - b = c - d : N"
unfolding arith_defs by equal
text \<open>Computation for difference: 0 and successor cases.\<close>
lemma diffC0: "a:N \<Longrightarrow> a - 0 = a : N"
unfolding arith_defs by rew
text \<open>Note: \<open>rec(a, 0, \<lambda>z w.z)\<close> is \<open>pred(a).\<close>\<close>
lemma diff_0_eq_0: "b:N \<Longrightarrow> 0 - b = 0 : N"
unfolding arith_defs
apply (NE b)
apply hyp_rew
done
text \<open>
Essential to simplify FIRST!! (Else we get a critical pair)
\<open>succ(a) - succ(b)\<close> rewrites to \<open>pred(succ(a) - b)\<close>.
\<close>
lemma diff_succ_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) - succ(b) = a - b : N"
unfolding arith_defs
apply hyp_rew
apply (NE b)
apply hyp_rew
done
subsection \<open>Simplification\<close>
lemmas arith_typing_rls = add_typing mult_typing diff_typing
and arith_congr_rls = add_typingL mult_typingL diff_typingL
lemmas congr_rls = arith_congr_rls intrL2_rls elimL_rls
lemmas arithC_rls =
addC0 addC_succ
multC0 multC_succ
diffC0 diff_0_eq_0 diff_succ_succ
ML \<open>
structure Arith_simp = TSimpFun(
val refl = @{thm refl_elem}
val sym = @{thm sym_elem}
val trans = @{thm trans_elem}
val refl_red = @{thm refl_red}
val trans_red = @{thm trans_red}
val red_if_equal = @{thm red_if_equal}
val default_rls = @{thms arithC_rls comp_rls}
val routine_tac = routine_tac @{thms arith_typing_rls routine_rls}
)
fun arith_rew_tac ctxt prems =
make_rew_tac ctxt (Arith_simp.norm_tac ctxt (@{thms congr_rls}, prems))
fun hyp_arith_rew_tac ctxt prems =
make_rew_tac ctxt
(Arith_simp.cond_norm_tac ctxt (prove_cond_tac ctxt, @{thms congr_rls}, prems))
\<close>
method_setup arith_rew = \<open>
Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (arith_rew_tac ctxt ths))
\<close>
method_setup hyp_arith_rew = \<open>
Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (hyp_arith_rew_tac ctxt ths))
\<close>
subsection \<open>Addition\<close>
text \<open>Associative law for addition.\<close>
lemma add_assoc: "\<lbrakk>a:N; b:N; c:N\<rbrakk> \<Longrightarrow> (a #+ b) #+ c = a #+ (b #+ c) : N"
apply (NE a)
apply hyp_arith_rew
done
text \<open>Commutative law for addition. Can be proved using three inductions.
Must simplify after first induction! Orientation of rewrites is delicate.\<close>
lemma add_commute: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #+ b = b #+ a : N"
apply (NE a)
apply hyp_arith_rew
apply (rule sym_elem)
prefer 2
apply (NE b)
prefer 4
apply (NE b)
apply hyp_arith_rew
done
subsection \<open>Multiplication\<close>
text \<open>Right annihilation in product.\<close>
lemma mult_0_right: "a:N \<Longrightarrow> a #* 0 = 0 : N"
apply (NE a)
apply hyp_arith_rew
done
text \<open>Right successor law for multiplication.\<close>
lemma mult_succ_right: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #* succ(b) = a #+ (a #* b) : N"
apply (NE a)
apply (hyp_arith_rew add_assoc [THEN sym_elem])
apply (assumption | rule add_commute mult_typingL add_typingL intrL_rls refl_elem)+
done
text \<open>Commutative law for multiplication.\<close>
lemma mult_commute: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #* b = b #* a : N"
apply (NE a)
apply (hyp_arith_rew mult_0_right mult_succ_right)
done
text \<open>Addition distributes over multiplication.\<close>
lemma add_mult_distrib: "\<lbrakk>a:N; b:N; c:N\<rbrakk> \<Longrightarrow> (a #+ b) #* c = (a #* c) #+ (b #* c) : N"
apply (NE a)
apply (hyp_arith_rew add_assoc [THEN sym_elem])
done
text \<open>Associative law for multiplication.\<close>
lemma mult_assoc: "\<lbrakk>a:N; b:N; c:N\<rbrakk> \<Longrightarrow> (a #* b) #* c = a #* (b #* c) : N"
apply (NE a)
apply (hyp_arith_rew add_mult_distrib)
done
subsection \<open>Difference\<close>
text \<open>
Difference on natural numbers, without negative numbers
\<^item> \<open>a - b = 0\<close> iff \<open>a \<le> b\<close>
\<^item> \<open>a - b = succ(c)\<close> iff \<open>a > b\<close>
\<close>
lemma diff_self_eq_0: "a:N \<Longrightarrow> a - a = 0 : N"
apply (NE a)
apply hyp_arith_rew
done
lemma add_0_right: "\<lbrakk>c : N; 0 : N; c : N\<rbrakk> \<Longrightarrow> c #+ 0 = c : N"
by (rule addC0 [THEN [3] add_commute [THEN trans_elem]])
text \<open>
Addition is the inverse of subtraction: if \<open>b \<le> x\<close> then \<open>b #+ (x - b) = x\<close>.
An example of induction over a quantified formula (a product).
Uses rewriting with a quantified, implicative inductive hypothesis.
\<close>
schematic_goal add_diff_inverse_lemma:
"b:N \<Longrightarrow> ?a : \<Prod>x:N. Eq(N, b-x, 0) \<longrightarrow> Eq(N, b #+ (x-b), x)"
apply (NE b)
\<comment> \<open>strip one "universal quantifier" but not the "implication"\<close>
apply (rule_tac [3] intr_rls)
\<comment> \<open>case analysis on \<open>x\<close> in \<open>succ(u) \<le> x \<longrightarrow> succ(u) #+ (x - succ(u)) = x\<close>\<close>
prefer 4
apply (NE x)
apply assumption
\<comment> \<open>Prepare for simplification of types -- the antecedent \<open>succ(u) \<le> x\<close>\<close>
apply (rule_tac [2] replace_type)
apply (rule_tac [1] replace_type)
apply arith_rew
\<comment> \<open>Solves first 0 goal, simplifies others. Two sugbgoals remain.
Both follow by rewriting, (2) using quantified induction hyp.\<close>
apply intr \<comment> \<open>strips remaining \<open>\<Prod>\<close>s\<close>
apply (hyp_arith_rew add_0_right)
apply assumption
done
text \<open>
Version of above with premise \<open>b - a = 0\<close> i.e. \<open>a \<ge> b\<close>.
Using @{thm ProdE} does not work -- for \<open>?B(?a)\<close> is ambiguous.
Instead, @{thm add_diff_inverse_lemma} states the desired induction scheme;
the use of \<open>THEN\<close> below instantiates Vars in @{thm ProdE} automatically.
\<close>
lemma add_diff_inverse: "\<lbrakk>a:N; b:N; b - a = 0 : N\<rbrakk> \<Longrightarrow> b #+ (a-b) = a : N"
apply (rule EqE)
apply (rule add_diff_inverse_lemma [THEN ProdE, THEN ProdE])
apply (assumption | rule EqI)+
done
subsection \<open>Absolute difference\<close>
text \<open>Typing of absolute difference: short and long versions.\<close>
lemma absdiff_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a |-| b : N"
unfolding arith_defs by typechk
lemma absdiff_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a |-| b = c |-| d : N"
unfolding arith_defs by equal
lemma absdiff_self_eq_0: "a:N \<Longrightarrow> a |-| a = 0 : N"
unfolding absdiff_def by (arith_rew diff_self_eq_0)
lemma absdiffC0: "a:N \<Longrightarrow> 0 |-| a = a : N"
unfolding absdiff_def by hyp_arith_rew
lemma absdiff_succ_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) |-| succ(b) = a |-| b : N"
unfolding absdiff_def by hyp_arith_rew
text \<open>Note how easy using commutative laws can be? ...not always...\<close>
lemma absdiff_commute: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a |-| b = b |-| a : N"
unfolding absdiff_def
apply (rule add_commute)
apply (typechk diff_typing)
done
text \<open>If \<open>a + b = 0\<close> then \<open>a = 0\<close>. Surprisingly tedious.\<close>
schematic_goal add_eq0_lemma: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> ?c : Eq(N,a#+b,0) \<longrightarrow> Eq(N,a,0)"
apply (NE a)
apply (rule_tac [3] replace_type)
apply arith_rew
apply intr \<comment> \<open>strips remaining \<open>\<Prod>\<close>s\<close>
apply (rule_tac [2] zero_ne_succ [THEN FE])
apply (erule_tac [3] EqE [THEN sym_elem])
apply (typechk add_typing)
done
text \<open>
Version of above with the premise \<open>a + b = 0\<close>.
Again, resolution instantiates variables in @{thm ProdE}.
\<close>
lemma add_eq0: "\<lbrakk>a:N; b:N; a #+ b = 0 : N\<rbrakk> \<Longrightarrow> a = 0 : N"
apply (rule EqE)
apply (rule add_eq0_lemma [THEN ProdE])
apply (rule_tac [3] EqI)
apply typechk
done
text \<open>Here is a lemma to infer \<open>a - b = 0\<close> and \<open>b - a = 0\<close> from \<open>a |-| b = 0\<close>, below.\<close>
schematic_goal absdiff_eq0_lem:
"\<lbrakk>a:N; b:N; a |-| b = 0 : N\<rbrakk> \<Longrightarrow> ?a : Eq(N, a-b, 0) \<times> Eq(N, b-a, 0)"
apply (unfold absdiff_def)
apply intr
apply eqintr
apply (rule_tac [2] add_eq0)
apply (rule add_eq0)
apply (rule_tac [6] add_commute [THEN trans_elem])
apply (typechk diff_typing)
done
text \<open>If \<open>a |-| b = 0\<close> then \<open>a = b\<close>
proof: \<open>a - b = 0\<close> and \<open>b - a = 0\<close>, so \<open>b = a + (b - a) = a + 0 = a\<close>.
\<close>
lemma absdiff_eq0: "\<lbrakk>a |-| b = 0 : N; a:N; b:N\<rbrakk> \<Longrightarrow> a = b : N"
apply (rule EqE)
apply (rule absdiff_eq0_lem [THEN SumE])
apply eqintr
apply (rule add_diff_inverse [THEN sym_elem, THEN trans_elem])
apply (erule_tac [3] EqE)
apply (hyp_arith_rew add_0_right)
done
subsection \<open>Remainder and Quotient\<close>
text \<open>Typing of remainder: short and long versions.\<close>
lemma mod_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a mod b : N"
unfolding mod_def by (typechk absdiff_typing)
lemma mod_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a mod b = c mod d : N"
unfolding mod_def by (equal absdiff_typingL)
text \<open>Computation for \<open>mod\<close>: 0 and successor cases.\<close>
lemma modC0: "b:N \<Longrightarrow> 0 mod b = 0 : N"
unfolding mod_def by (rew absdiff_typing)
lemma modC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow>
succ(a) mod b = rec(succ(a mod b) |-| b, 0, \<lambda>x y. succ(a mod b)) : N"
unfolding mod_def by (rew absdiff_typing)
text \<open>Typing of quotient: short and long versions.\<close>
lemma div_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a div b : N"
unfolding div_def by (typechk absdiff_typing mod_typing)
lemma div_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a div b = c div d : N"
unfolding div_def by (equal absdiff_typingL mod_typingL)
lemmas div_typing_rls = mod_typing div_typing absdiff_typing
text \<open>Computation for quotient: 0 and successor cases.\<close>
lemma divC0: "b:N \<Longrightarrow> 0 div b = 0 : N"
unfolding div_def by (rew mod_typing absdiff_typing)
lemma divC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow>
succ(a) div b = rec(succ(a) mod b, succ(a div b), \<lambda>x y. a div b) : N"
unfolding div_def by (rew mod_typing)
text \<open>Version of above with same condition as the \<open>mod\<close> one.\<close>
lemma divC_succ2: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow>
succ(a) div b =rec(succ(a mod b) |-| b, succ(a div b), \<lambda>x y. a div b) : N"
apply (rule divC_succ [THEN trans_elem])
apply (rew div_typing_rls modC_succ)
apply (NE "succ (a mod b) |-|b")
apply (rew mod_typing div_typing absdiff_typing)
done
text \<open>For case analysis on whether a number is 0 or a successor.\<close>
lemma iszero_decidable: "a:N \<Longrightarrow> rec(a, inl(eq), \<lambda>ka kb. inr(<ka, eq>)) :
Eq(N,a,0) + (\<Sum>x:N. Eq(N,a, succ(x)))"
apply (NE a)
apply (rule_tac [3] PlusI_inr)
apply (rule_tac [2] PlusI_inl)
apply eqintr
apply equal
done
text \<open>Main Result. Holds when \<open>b\<close> is 0 since \<open>a mod 0 = a\<close> and \<open>a div 0 = 0\<close>.\<close>
lemma mod_div_equality: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a mod b #+ (a div b) #* b = a : N"
apply (NE a)
apply (arith_rew div_typing_rls modC0 modC_succ divC0 divC_succ2)
apply (rule EqE)
\<comment> \<open>case analysis on \<open>succ(u mod b) |-| b\<close>\<close>
apply (rule_tac a1 = "succ (u mod b) |-| b" in iszero_decidable [THEN PlusE])
apply (erule_tac [3] SumE)
apply (hyp_arith_rew div_typing_rls modC0 modC_succ divC0 divC_succ2)
\<comment> \<open>Replace one occurrence of \<open>b\<close> by \<open>succ(u mod b)\<close>. Clumsy!\<close>
apply (rule add_typingL [THEN trans_elem])
apply (erule EqE [THEN absdiff_eq0, THEN sym_elem])
apply (rule_tac [3] refl_elem)
apply (hyp_arith_rew div_typing_rls)
done
end