# Theory CTT

```(*  Title:      CTT/CTT.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

theory CTT
imports Pure
begin

section ‹Constructive Type Theory: axiomatic basis›

ML_file ‹~~/src/Provers/typedsimp.ML›
setup Pure_Thy.old_appl_syntax_setup

typedecl i
typedecl t
typedecl o

consts
― ‹Types›
F         :: "t"
T         :: "t"          ― ‹‹F› is empty, ‹T› contains one element›
contr     :: "i⇒i"
tt        :: "i"
― ‹Natural numbers›
N         :: "t"
succ      :: "i⇒i"
rec       :: "[i, i, [i,i]⇒i] ⇒ i"
― ‹Unions›
inl       :: "i⇒i"
inr       :: "i⇒i"
"when"    :: "[i, i⇒i, i⇒i]⇒i"
― ‹General Sum and Binary Product›
Sum       :: "[t, i⇒t]⇒t"
fst       :: "i⇒i"
snd       :: "i⇒i"
split     :: "[i, [i,i]⇒i] ⇒i"
― ‹General Product and Function Space›
Prod      :: "[t, i⇒t]⇒t"
― ‹Types›
Plus      :: "[t,t]⇒t"           (infixr "+" 40)
― ‹Equality type›
Eq        :: "[t,i,i]⇒t"
eq        :: "i"
― ‹Judgements›
Type      :: "t ⇒ prop"          ("(_ type)" [10] 5)
Eqtype    :: "[t,t]⇒prop"        ("(_ =/ _)" [10,10] 5)
Elem      :: "[i, t]⇒prop"       ("(_ /: _)" [10,10] 5)
Eqelem    :: "[i,i,t]⇒prop"      ("(_ =/ _ :/ _)" [10,10,10] 5)
Reduce    :: "[i,i]⇒prop"        ("Reduce[_,_]")

― ‹Types›

― ‹Functions›
lambda    :: "(i ⇒ i) ⇒ i"      (binder "❙λ" 10)
app       :: "[i,i]⇒i"           (infixl "`" 60)
― ‹Natural numbers›
Zero      :: "i"                  ("0")
― ‹Pairing›
pair      :: "[i,i]⇒i"           ("(1<_,/_>)")

syntax
"_PROD"   :: "[idt,t,t]⇒t"       ("(3∏_:_./ _)" 10)
"_SUM"    :: "[idt,t,t]⇒t"       ("(3∑_:_./ _)" 10)
translations
"∏x:A. B" ⇌ "CONST Prod(A, λx. B)"
"∑x:A. B" ⇌ "CONST Sum(A, λx. B)"

abbreviation Arrow :: "[t,t]⇒t"  (infixr "⟶" 30)
where "A ⟶ B ≡ ∏_:A. B"

abbreviation Times :: "[t,t]⇒t"  (infixr "×" 50)
where "A × B ≡ ∑_:A. B"

text ‹
Reduction: a weaker notion than equality;  a hack for simplification.
‹Reduce[a,b]› means either that ‹a = b : A› for some ‹A› or else
that ‹a› and ‹b› are textually identical.

Does not verify ‹a:A›!  Sound because only ‹trans_red› uses a ‹Reduce›
premise. No new theorems can be proved about the standard judgements.
›
axiomatization
where
refl_red: "⋀a. Reduce[a,a]" and
red_if_equal: "⋀a b A. a = b : A ⟹ Reduce[a,b]" and
trans_red: "⋀a b c A. ⟦a = b : A; Reduce[b,c]⟧ ⟹ a = c : A" and

― ‹Reflexivity›

refl_type: "⋀A. A type ⟹ A = A" and
refl_elem: "⋀a A. a : A ⟹ a = a : A" and

― ‹Symmetry›

sym_type:  "⋀A B. A = B ⟹ B = A" and
sym_elem:  "⋀a b A. a = b : A ⟹ b = a : A" and

― ‹Transitivity›

trans_type:   "⋀A B C. ⟦A = B; B = C⟧ ⟹ A = C" and
trans_elem:   "⋀a b c A. ⟦a = b : A; b = c : A⟧ ⟹ a = c : A" and

equal_types:  "⋀a A B. ⟦a : A; A = B⟧ ⟹ a : B" and
equal_typesL: "⋀a b A B. ⟦a = b : A; A = B⟧ ⟹ a = b : B" and

― ‹Substitution›

subst_type:   "⋀a A B. ⟦a : A; ⋀z. z:A ⟹ B(z) type⟧ ⟹ B(a) type" and
subst_typeL:  "⋀a c A B D. ⟦a = c : A; ⋀z. z:A ⟹ B(z) = D(z)⟧ ⟹ B(a) = D(c)" and

subst_elem:   "⋀a b A B. ⟦a : A; ⋀z. z:A ⟹ b(z):B(z)⟧ ⟹ b(a):B(a)" and
subst_elemL:
"⋀a b c d A B. ⟦a = c : A; ⋀z. z:A ⟹ b(z)=d(z) : B(z)⟧ ⟹ b(a)=d(c) : B(a)" and

― ‹The type ‹N› -- natural numbers›

NF: "N type" and
NI0: "0 : N" and
NI_succ: "⋀a. a : N ⟹ succ(a) : N" and
NI_succL:  "⋀a b. a = b : N ⟹ succ(a) = succ(b) : N" and

NE:
"⋀p a b C. ⟦p: N; a: C(0); ⋀u v. ⟦u: N; v: C(u)⟧ ⟹ b(u,v): C(succ(u))⟧
⟹ rec(p, a, λu v. b(u,v)) : C(p)" and

NEL:
"⋀p q a b c d C. ⟦p = q : N; a = c : C(0);
⋀u v. ⟦u: N; v: C(u)⟧ ⟹ b(u,v) = d(u,v): C(succ(u))⟧
⟹ rec(p, a, λu v. b(u,v)) = rec(q,c,d) : C(p)" and

NC0:
"⋀a b C. ⟦a: C(0); ⋀u v. ⟦u: N; v: C(u)⟧ ⟹ b(u,v): C(succ(u))⟧
⟹ rec(0, a, λu v. b(u,v)) = a : C(0)" and

NC_succ:
"⋀p a b C. ⟦p: N;  a: C(0); ⋀u v. ⟦u: N; v: C(u)⟧ ⟹ b(u,v): C(succ(u))⟧ ⟹
rec(succ(p), a, λu v. b(u,v)) = b(p, rec(p, a, λu v. b(u,v))) : C(succ(p))" and

― ‹The fourth Peano axiom.  See page 91 of Martin-Löf's book.›
zero_ne_succ: "⋀a. ⟦a: N; 0 = succ(a) : N⟧ ⟹ 0: F" and

― ‹The Product of a family of types›

ProdF: "⋀A B. ⟦A type; ⋀x. x:A ⟹ B(x) type⟧ ⟹ ∏x:A. B(x) type" and

ProdFL:
"⋀A B C D. ⟦A = C; ⋀x. x:A ⟹ B(x) = D(x)⟧ ⟹ ∏x:A. B(x) = ∏x:C. D(x)" and

ProdI:
"⋀b A B. ⟦A type; ⋀x. x:A ⟹ b(x):B(x)⟧ ⟹ ❙λx. b(x) : ∏x:A. B(x)" and

ProdIL: "⋀b c A B. ⟦A type; ⋀x. x:A ⟹ b(x) = c(x) : B(x)⟧ ⟹
❙λx. b(x) = ❙λx. c(x) : ∏x:A. B(x)" and

ProdE:  "⋀p a A B. ⟦p : ∏x:A. B(x); a : A⟧ ⟹ p`a : B(a)" and
ProdEL: "⋀p q a b A B. ⟦p = q: ∏x:A. B(x); a = b : A⟧ ⟹ p`a = q`b : B(a)" and

ProdC: "⋀a b A B. ⟦a : A; ⋀x. x:A ⟹ b(x) : B(x)⟧ ⟹ (❙λx. b(x)) ` a = b(a) : B(a)" and

ProdC2: "⋀p A B. p : ∏x:A. B(x) ⟹ (❙λx. p`x) = p : ∏x:A. B(x)" and

― ‹The Sum of a family of types›

SumF:  "⋀A B. ⟦A type; ⋀x. x:A ⟹ B(x) type⟧ ⟹ ∑x:A. B(x) type" and
SumFL: "⋀A B C D. ⟦A = C; ⋀x. x:A ⟹ B(x) = D(x)⟧ ⟹ ∑x:A. B(x) = ∑x:C. D(x)" and

SumI:  "⋀a b A B. ⟦a : A; b : B(a)⟧ ⟹ <a,b> : ∑x:A. B(x)" and
SumIL: "⋀a b c d A B. ⟦ a = c : A; b = d : B(a)⟧ ⟹ <a,b> = <c,d> : ∑x:A. B(x)" and

SumE: "⋀p c A B C. ⟦p: ∑x:A. B(x); ⋀x y. ⟦x:A; y:B(x)⟧ ⟹ c(x,y): C(<x,y>)⟧
⟹ split(p, λx y. c(x,y)) : C(p)" and

SumEL: "⋀p q c d A B C. ⟦p = q : ∑x:A. B(x);
⋀x y. ⟦x:A; y:B(x)⟧ ⟹ c(x,y)=d(x,y): C(<x,y>)⟧
⟹ split(p, λx y. c(x,y)) = split(q, λx y. d(x,y)) : C(p)" and

SumC: "⋀a b c A B C. ⟦a: A;  b: B(a); ⋀x y. ⟦x:A; y:B(x)⟧ ⟹ c(x,y): C(<x,y>)⟧
⟹ split(<a,b>, λx y. c(x,y)) = c(a,b) : C(<a,b>)" and

fst_def:   "⋀a. fst(a) ≡ split(a, λx y. x)" and
snd_def:   "⋀a. snd(a) ≡ split(a, λx y. y)" and

― ‹The sum of two types›

PlusF: "⋀A B. ⟦A type; B type⟧ ⟹ A+B type" and
PlusFL: "⋀A B C D. ⟦A = C; B = D⟧ ⟹ A+B = C+D" and

PlusI_inl: "⋀a A B. ⟦a : A; B type⟧ ⟹ inl(a) : A+B" and
PlusI_inlL: "⋀a c A B. ⟦a = c : A; B type⟧ ⟹ inl(a) = inl(c) : A+B" and

PlusI_inr: "⋀b A B. ⟦A type; b : B⟧ ⟹ inr(b) : A+B" and
PlusI_inrL: "⋀b d A B. ⟦A type; b = d : B⟧ ⟹ inr(b) = inr(d) : A+B" and

PlusE:
"⋀p c d A B C. ⟦p: A+B;
⋀x. x:A ⟹ c(x): C(inl(x));
⋀y. y:B ⟹ d(y): C(inr(y)) ⟧ ⟹ when(p, λx. c(x), λy. d(y)) : C(p)" and

PlusEL:
"⋀p q c d e f A B C. ⟦p = q : A+B;
⋀x. x: A ⟹ c(x) = e(x) : C(inl(x));
⋀y. y: B ⟹ d(y) = f(y) : C(inr(y))⟧
⟹ when(p, λx. c(x), λy. d(y)) = when(q, λx. e(x), λy. f(y)) : C(p)" and

PlusC_inl:
"⋀a c d A B C. ⟦a: A;
⋀x. x:A ⟹ c(x): C(inl(x));
⋀y. y:B ⟹ d(y): C(inr(y)) ⟧
⟹ when(inl(a), λx. c(x), λy. d(y)) = c(a) : C(inl(a))" and

PlusC_inr:
"⋀b c d A B C. ⟦b: B;
⋀x. x:A ⟹ c(x): C(inl(x));
⋀y. y:B ⟹ d(y): C(inr(y))⟧
⟹ when(inr(b), λx. c(x), λy. d(y)) = d(b) : C(inr(b))" and

― ‹The type ‹Eq››

EqF: "⋀a b A. ⟦A type; a : A; b : A⟧ ⟹ Eq(A,a,b) type" and
EqFL: "⋀a b c d A B. ⟦A = B; a = c : A; b = d : A⟧ ⟹ Eq(A,a,b) = Eq(B,c,d)" and
EqI: "⋀a b A. a = b : A ⟹ eq : Eq(A,a,b)" and
EqE: "⋀p a b A. p : Eq(A,a,b) ⟹ a = b : A" and

― ‹By equality of types, can prove ‹C(p)› from ‹C(eq)›, an elimination rule›
EqC: "⋀p a b A. p : Eq(A,a,b) ⟹ p = eq : Eq(A,a,b)" and

― ‹The type ‹F››

FF: "F type" and
FE: "⋀p C. ⟦p: F; C type⟧ ⟹ contr(p) : C" and
FEL: "⋀p q C. ⟦p = q : F; C type⟧ ⟹ contr(p) = contr(q) : C" and

― ‹The type T›
― ‹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 ‹C(x)› type ‹x:T›.
Also computation can be derived from elimination.›

TF: "T type" and
TI: "tt : T" and
TE: "⋀p c C. ⟦p : T; c : C(tt)⟧ ⟹ c : C(p)" and
TEL: "⋀p q c d C. ⟦p = q : T; c = d : C(tt)⟧ ⟹ c = d : C(p)" and
TC: "⋀p. p : T ⟹ p = tt : T"

subsection "Tactics and derived rules for Constructive Type Theory"

text ‹Formation rules.›
lemmas form_rls = NF ProdF SumF PlusF EqF FF TF
and formL_rls = ProdFL SumFL PlusFL EqFL

text ‹
Introduction rules. OMITTED:
▪ ‹EqI›, because its premise is an ‹eqelem›, not an ‹elem›.
›
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 ‹
Elimination rules. OMITTED:
▪ ‹EqE›, because its conclusion is an ‹eqelem›, not an ‹elem›
▪ ‹TE›, because it does not involve a constructor.
›
lemmas elim_rls = NE ProdE SumE PlusE FE
and elimL_rls = NEL ProdEL SumEL PlusEL FEL

text ‹OMITTED: ‹eqC› are ‹TC› because they make rewriting loop: ‹p = un = un = …››
lemmas comp_rls = NC0 NC_succ ProdC SumC PlusC_inl PlusC_inr

text ‹Rules with conclusion ‹a:A›, an elem judgement.›
lemmas element_rls = intr_rls elim_rls

text ‹Definitions are (meta)equality axioms.›
lemmas basic_defs = fst_def snd_def

text ‹Compare with standard version: ‹B› is applied to UNSIMPLIFIED expression!›
lemma SumIL2: "⟦c = a : A; d = b : B(a)⟧ ⟹ <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 ‹
Exploit ‹p:Prod(A,B)› to create the assumption ‹z:B(a)›.
A more natural form of product elimination.
›
lemma subst_prodE:
assumes "p: Prod(A,B)"
and "a: A"
and "⋀z. z: B(a) ⟹ c(z): C(z)"
shows "c(p`a): C(p`a)"
by (rule assms ProdE)+

subsection ‹Tactics for type checking›

ML ‹
local

fun is_rigid_elem \<^Const_>‹Elem for a _› = not(is_Var (head_of a))
| is_rigid_elem \<^Const_>‹Eqelem for a _ _› = not(is_Var (head_of a))
| is_rigid_elem \<^Const_>‹Type for 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
›

text ‹
For simplification: type formation and checking,
but no equalities between terms.
›
lemmas routine_rls = form_rls formL_rls refl_type element_rls

ML ‹
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}))))
›

method_setup form = ‹Scan.succeed (fn ctxt => SIMPLE_METHOD (form_tac ctxt))›
method_setup typechk = ‹Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (typechk_tac ctxt ths))›
method_setup intr = ‹Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (intr_tac ctxt ths))›
method_setup equal = ‹Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (equal_tac ctxt ths))›

subsection ‹Simplification›

text ‹To simplify the type in a goal.›
lemma replace_type: "⟦B = A; a : A⟧ ⟹ a : B"
apply (rule equal_types)
apply (rule_tac [2] sym_type)
apply assumption+
done

text ‹Simplify the parameter of a unary type operator.›
lemma subst_eqtyparg:
assumes 1: "a=c : A"
and 2: "⋀z. z:A ⟹ 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 ‹Simplification rules for Constructive Type Theory.›
lemmas reduction_rls = comp_rls [THEN trans_elem]

ML ‹
(*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) *)
resolve_tac ctxt @{thms subst_prodE} i  THEN  assume_tac ctxt i  THEN  assume_tac ctxt i

(*Finds P⟶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)
›

method_setup eqintr = ‹Scan.succeed (SIMPLE_METHOD o eqintr_tac)›
method_setup NE = ‹
Scan.lift Parse.embedded_inner_syntax >> (fn s => fn ctxt => SIMPLE_METHOD' (NE_tac ctxt s))
›
method_setup pc = ‹Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (pc_tac ctxt ths))›

ML_file ‹rew.ML›
method_setup rew = ‹Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (rew_tac ctxt ths))›
method_setup hyp_rew = ‹Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (hyp_rew_tac ctxt ths))›

subsection ‹The elimination rules for fst/snd›

lemma SumE_fst: "p : Sum(A,B) ⟹ fst(p) : A"
apply (unfold basic_defs)
apply (erule SumE)
apply assumption
done

text ‹The first premise must be ‹p:Sum(A,B)›!!.›
lemma SumE_snd:
assumes major: "p: Sum(A,B)"
and "A type"
and "⋀x. x:A ⟹ 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 ‹The two-element type (booleans and conditionals)›

definition Bool :: "t"
where "Bool ≡ T+T"

definition true :: "i"
where "true ≡ inl(tt)"

definition false :: "i"
where "false ≡ inr(tt)"

definition cond :: "[i,i,i]⇒i"
where "cond(a,b,c) ≡ when(a, λ_. b, λ_. c)"

lemmas bool_defs = Bool_def true_def false_def cond_def

subsection ‹Derivation of rules for the type ‹Bool››

text ‹Formation rule.›
lemma boolF: "Bool type"
unfolding bool_defs by typechk

text ‹Introduction rules for ‹true›, ‹false›.›

lemma boolI_true: "true : Bool"
unfolding bool_defs by typechk

lemma boolI_false: "false : Bool"
unfolding bool_defs by typechk

text ‹Elimination rule: typing of ‹cond›.›
lemma boolE: "⟦p:Bool; a : C(true); b : C(false)⟧ ⟹ cond(p,a,b) : C(p)"
unfolding bool_defs
apply (typechk; erule TE)
apply typechk
done

lemma boolEL: "⟦p = q : Bool; a = c : C(true); b = d : C(false)⟧
⟹ 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 ‹Computation rules for ‹true›, ‹false›.›

lemma boolC_true: "⟦a : C(true); b : C(false)⟧ ⟹ 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: "⟦a : C(true); b : C(false)⟧ ⟹ cond(false,a,b) = b : C(false)"
unfolding bool_defs
apply (rule comp_rls)
apply typechk
apply (erule_tac [!] TE)
apply typechk
done

section ‹Elementary arithmetic›

subsection ‹Arithmetic operators and their definitions›

definition add :: "[i,i]⇒i"   (infixr "#+" 65)
where "a#+b ≡ rec(a, b, λu v. succ(v))"

definition diff :: "[i,i]⇒i"   (infixr "-" 65)
where "a-b ≡ rec(b, a, λu v. rec(v, 0, λx y. x))"

definition absdiff :: "[i,i]⇒i"   (infixr "|-|" 65)
where "a|-|b ≡ (a-b) #+ (b-a)"

definition mult :: "[i,i]⇒i"   (infixr "#*" 70)
where "a#*b ≡ rec(a, 0, λu v. b #+ v)"

definition mod :: "[i,i]⇒i"   (infixr "mod" 70)
where "a mod b ≡ rec(a, 0, λu v. rec(succ(v) |-| b, 0, λx y. succ(v)))"

definition div :: "[i,i]⇒i"   (infixr "div" 70)
where "a div b ≡ rec(a, 0, λu v. rec(succ(u) mod b, succ(v), λx y. v))"

lemmas arith_defs = add_def diff_def absdiff_def mult_def mod_def div_def

text ‹Typing of ‹add›: short and long versions.›

lemma add_typing: "⟦a:N; b:N⟧ ⟹ a #+ b : N"
unfolding arith_defs by typechk

lemma add_typingL: "⟦a = c:N; b = d:N⟧ ⟹ a #+ b = c #+ d : N"
unfolding arith_defs by equal

text ‹Computation for ‹add›: 0 and successor cases.›

lemma addC0: "b:N ⟹ 0 #+ b = b : N"
unfolding arith_defs by rew

lemma addC_succ: "⟦a:N; b:N⟧ ⟹ succ(a) #+ b = succ(a #+ b) : N"
unfolding arith_defs by rew

subsubsection ‹Multiplication›

text ‹Typing of ‹mult›: short and long versions.›

lemma mult_typing: "⟦a:N; b:N⟧ ⟹ a #* b : N"

lemma mult_typingL: "⟦a = c:N; b = d:N⟧ ⟹ a #* b = c #* d : N"

text ‹Computation for ‹mult›: 0 and successor cases.›

lemma multC0: "b:N ⟹ 0 #* b = 0 : N"
unfolding arith_defs by rew

lemma multC_succ: "⟦a:N; b:N⟧ ⟹ succ(a) #* b = b #+ (a #* b) : N"
unfolding arith_defs by rew

subsubsection ‹Difference›

text ‹Typing of difference.›

lemma diff_typing: "⟦a:N; b:N⟧ ⟹ a - b : N"
unfolding arith_defs by typechk

lemma diff_typingL: "⟦a = c:N; b = d:N⟧ ⟹ a - b = c - d : N"
unfolding arith_defs by equal

text ‹Computation for difference: 0 and successor cases.›

lemma diffC0: "a:N ⟹ a - 0 = a : N"
unfolding arith_defs by rew

text ‹Note: ‹rec(a, 0, λz w.z)› is ‹pred(a).››

lemma diff_0_eq_0: "b:N ⟹ 0 - b = 0 : N"
unfolding arith_defs
apply (NE b)
apply hyp_rew
done

text ‹
Essential to simplify FIRST!!  (Else we get a critical pair)
‹succ(a) - succ(b)› rewrites to ‹pred(succ(a) - b)›.
›
lemma diff_succ_succ: "⟦a:N; b:N⟧ ⟹ succ(a) - succ(b) = a - b : N"
unfolding arith_defs
apply hyp_rew
apply (NE b)
apply hyp_rew
done

subsection ‹Simplification›

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 =
multC0 multC_succ
diffC0 diff_0_eq_0 diff_succ_succ

ML ‹
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))
›

method_setup arith_rew = ‹
Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (arith_rew_tac ctxt ths))
›

method_setup hyp_arith_rew = ‹
Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (hyp_arith_rew_tac ctxt ths))
›

lemma add_assoc: "⟦a:N; b:N; c:N⟧ ⟹ (a #+ b) #+ c = a #+ (b #+ c) : N"
apply (NE a)
apply hyp_arith_rew
done

text ‹Commutative law for addition.  Can be proved using three inductions.
Must simplify after first induction!  Orientation of rewrites is delicate.›
lemma add_commute: "⟦a:N; b:N⟧ ⟹ 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 ‹Multiplication›

text ‹Right annihilation in product.›
lemma mult_0_right: "a:N ⟹ a #* 0 = 0 : N"
apply (NE a)
apply hyp_arith_rew
done

text ‹Right successor law for multiplication.›
lemma mult_succ_right: "⟦a:N; b:N⟧ ⟹ a #* succ(b) = a #+ (a #* b) : N"
apply (NE a)
done

text ‹Commutative law for multiplication.›
lemma mult_commute: "⟦a:N; b:N⟧ ⟹ a #* b = b #* a : N"
apply (NE a)
apply (hyp_arith_rew mult_0_right mult_succ_right)
done

lemma add_mult_distrib: "⟦a:N; b:N; c:N⟧ ⟹ (a #+ b) #* c = (a #* c) #+ (b #* c) : N"
apply (NE a)
done

text ‹Associative law for multiplication.›
lemma mult_assoc: "⟦a:N; b:N; c:N⟧ ⟹ (a #* b) #* c = a #* (b #* c) : N"
apply (NE a)
done

subsection ‹Difference›

text ‹
Difference on natural numbers, without negative numbers
▪ ‹a - b = 0›  iff  ‹a ≤ b›
▪ ‹a - b = succ(c)› iff ‹a > b›
›

lemma diff_self_eq_0: "a:N ⟹ a - a = 0 : N"
apply (NE a)
apply hyp_arith_rew
done

lemma add_0_right: "⟦c : N; 0 : N; c : N⟧ ⟹ c #+ 0 = c : N"

text ‹
Addition is the inverse of subtraction: if ‹b ≤ x› then ‹b #+ (x - b) = x›.
An example of induction over a quantified formula (a product).
Uses rewriting with a quantified, implicative inductive hypothesis.
›
"b:N ⟹ ?a : ∏x:N. Eq(N, b-x, 0) ⟶ Eq(N, b #+ (x-b), x)"
apply (NE b)
― ‹strip one "universal quantifier" but not the "implication"›
apply (rule_tac [3] intr_rls)
― ‹case analysis on ‹x› in ‹succ(u) ≤ x ⟶ succ(u) #+ (x - succ(u)) = x››
prefer 4
apply (NE x)
apply assumption
― ‹Prepare for simplification of types -- the antecedent ‹succ(u) ≤ x››
apply (rule_tac [2] replace_type)
apply (rule_tac [1] replace_type)
apply arith_rew
― ‹Solves first 0 goal, simplifies others.  Two sugbgoals remain.
Both follow by rewriting, (2) using quantified induction hyp.›
apply intr ― ‹strips remaining ‹∏›s›
apply assumption
done

text ‹
Version of above with premise ‹b - a = 0› i.e. ‹a ≥ b›.
Using @{thm ProdE} does not work -- for ‹?B(?a)› is ambiguous.
the use of ‹THEN› below instantiates Vars in @{thm ProdE} automatically.
›
lemma add_diff_inverse: "⟦a:N; b:N; b - a = 0 : N⟧ ⟹ b #+ (a-b) = a : N"
apply (rule EqE)
apply (rule add_diff_inverse_lemma [THEN ProdE, THEN ProdE])
apply (assumption | rule EqI)+
done

subsection ‹Absolute difference›

text ‹Typing of absolute difference: short and long versions.›

lemma absdiff_typing: "⟦a:N; b:N⟧ ⟹ a |-| b : N"
unfolding arith_defs by typechk

lemma absdiff_typingL: "⟦a = c:N; b = d:N⟧ ⟹ a |-| b = c |-| d : N"
unfolding arith_defs by equal

lemma absdiff_self_eq_0: "a:N ⟹ a |-| a = 0 : N"
unfolding absdiff_def by (arith_rew diff_self_eq_0)

lemma absdiffC0: "a:N ⟹ 0 |-| a = a : N"
unfolding absdiff_def by hyp_arith_rew

lemma absdiff_succ_succ: "⟦a:N; b:N⟧ ⟹ succ(a) |-| succ(b)  =  a |-| b : N"
unfolding absdiff_def by hyp_arith_rew

text ‹Note how easy using commutative laws can be?  ...not always...›
lemma absdiff_commute: "⟦a:N; b:N⟧ ⟹ a |-| b = b |-| a : N"
unfolding absdiff_def
apply (typechk diff_typing)
done

text ‹If ‹a + b = 0› then ‹a = 0›. Surprisingly tedious.›
schematic_goal add_eq0_lemma: "⟦a:N; b:N⟧ ⟹ ?c : Eq(N,a#+b,0) ⟶ Eq(N,a,0)"
apply (NE a)
apply (rule_tac [3] replace_type)
apply arith_rew
apply intr  ― ‹strips remaining ‹∏›s›
apply (rule_tac [2] zero_ne_succ [THEN FE])
apply (erule_tac [3] EqE [THEN sym_elem])
done

text ‹
Version of above with the premise ‹a + b = 0›.
Again, resolution instantiates variables in @{thm ProdE}.
›
lemma add_eq0: "⟦a:N; b:N; a #+ b = 0 : N⟧ ⟹ a = 0 : N"
apply (rule EqE)
apply (rule_tac [3] EqI)
apply typechk
done

text ‹Here is a lemma to infer ‹a - b = 0› and ‹b - a = 0› from ‹a |-| b = 0›, below.›
schematic_goal absdiff_eq0_lem:
"⟦a:N; b:N; a |-| b = 0 : N⟧ ⟹ ?a : Eq(N, a-b, 0) × Eq(N, b-a, 0)"
apply (unfold absdiff_def)
apply intr
apply eqintr
apply (rule_tac [6] add_commute [THEN trans_elem])
apply (typechk diff_typing)
done

text ‹If ‹a |-| b = 0› then ‹a = b›
proof: ‹a - b = 0› and ‹b - a = 0›, so ‹b = a + (b - a) = a + 0 = a›.
›
lemma absdiff_eq0: "⟦a |-| b = 0 : N; a:N; b:N⟧ ⟹ 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)
done

subsection ‹Remainder and Quotient›

text ‹Typing of remainder: short and long versions.›

lemma mod_typing: "⟦a:N; b:N⟧ ⟹ a mod b : N"
unfolding mod_def by (typechk absdiff_typing)

lemma mod_typingL: "⟦a = c:N; b = d:N⟧ ⟹ a mod b = c mod d : N"
unfolding mod_def by (equal absdiff_typingL)

text ‹Computation for ‹mod›: 0 and successor cases.›

lemma modC0: "b:N ⟹ 0 mod b = 0 : N"
unfolding mod_def by (rew absdiff_typing)

lemma modC_succ: "⟦a:N; b:N⟧ ⟹
succ(a) mod b = rec(succ(a mod b) |-| b, 0, λx y. succ(a mod b)) : N"
unfolding mod_def by (rew absdiff_typing)

text ‹Typing of quotient: short and long versions.›

lemma div_typing: "⟦a:N; b:N⟧ ⟹ a div b : N"
unfolding div_def by (typechk absdiff_typing mod_typing)

lemma div_typingL: "⟦a = c:N; b = d:N⟧ ⟹ 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 ‹Computation for quotient: 0 and successor cases.›

lemma divC0: "b:N ⟹ 0 div b = 0 : N"
unfolding div_def by (rew mod_typing absdiff_typing)

lemma divC_succ: "⟦a:N; b:N⟧ ⟹
succ(a) div b = rec(succ(a) mod b, succ(a div b), λx y. a div b) : N"
unfolding div_def by (rew mod_typing)

text ‹Version of above with same condition as the ‹mod› one.›
lemma divC_succ2: "⟦a:N; b:N⟧ ⟹
succ(a) div b =rec(succ(a mod b) |-| b, succ(a div b), λx 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 ‹For case analysis on whether a number is 0 or a successor.›
lemma iszero_decidable: "a:N ⟹ rec(a, inl(eq), λka kb. inr(<ka, eq>)) :
Eq(N,a,0) + (∑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 ‹Main Result. Holds when ‹b› is 0 since ‹a mod 0 = a› and ‹a div 0 = 0›.›
lemma mod_div_equality: "⟦a:N; b:N⟧ ⟹ 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)
― ‹case analysis on ‹succ(u mod b) |-| b››
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)
― ‹Replace one occurrence of ‹b› by ‹succ(u mod b)›. Clumsy!›
apply (erule EqE [THEN absdiff_eq0, THEN sym_elem])
apply (rule_tac [3] refl_elem)
apply (hyp_arith_rew div_typing_rls)
done

end
```