# Theory CTT

Up to index of Isabelle/CTT

theory CTT
imports Pure
`(*  Title:      CTT/CTT.thy    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory    Copyright   1993  University of Cambridge*)header {* Constructive Type Theory *}theory CTTimports PurebeginML_file "~~/src/Provers/typedsimp.ML"setup Pure_Thy.old_appl_syntax_setuptypedecl itypedecl ttypedecl oconsts  (*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 "lam " 10)  app       :: "[i,i]=>i"           (infixl "`" 60)  (*Natural numbers*)  Zero      :: "i"                  ("0")  (*Pairing*)  pair      :: "[i,i]=>i"           ("(1<_,/_>)")syntax  "_PROD"   :: "[idt,t,t]=>t"       ("(3PROD _:_./ _)" 10)  "_SUM"    :: "[idt,t,t]=>t"       ("(3SUM _:_./ _)" 10)translations  "PROD x:A. B" == "CONST Prod(A, %x. B)"  "SUM x:A. B"  == "CONST Sum(A, %x. B)"abbreviation  Arrow     :: "[t,t]=>t"  (infixr "-->" 30) where  "A --> B == PROD _:A. B"abbreviation  Times     :: "[t,t]=>t"  (infixr "*" 50) where  "A * B == SUM _:A. B"notation (xsymbols)  lambda  (binder "λλ" 10) and  Elem  ("(_ /∈ _)" [10,10] 5) and  Eqelem  ("(2_ =/ _ ∈/ _)" [10,10,10] 5) and  Arrow  (infixr "-->" 30) and  Times  (infixr "×" 50)notation (HTML output)  lambda  (binder "λλ" 10) and  Elem  ("(_ /∈ _)" [10,10] 5) and  Eqelem  ("(2_ =/ _ ∈/ _)" [10,10,10] 5) and  Times  (infixr "×" 50)syntax (xsymbols)  "_PROD"   :: "[idt,t,t] => t"     ("(3Π _∈_./ _)"    10)  "_SUM"    :: "[idt,t,t] => t"     ("(3Σ _∈_./ _)" 10)syntax (HTML output)  "_PROD"   :: "[idt,t,t] => t"     ("(3Π _∈_./ _)"    10)  "_SUM"    :: "[idt,t,t] => t"     ("(3Σ _∈_./ _)" 10)axioms  (*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.*)  refl_red: "Reduce[a,a]"  red_if_equal: "a = b : A ==> Reduce[a,b]"  trans_red: "[| a = b : A;  Reduce[b,c] |] ==> a = c : A"  (*Reflexivity*)  refl_type: "A type ==> A = A"  refl_elem: "a : A ==> a = a : A"  (*Symmetry*)  sym_type:  "A = B ==> B = A"  sym_elem:  "a = b : A ==> b = a : A"  (*Transitivity*)  trans_type:   "[| A = B;  B = C |] ==> A = C"  trans_elem:   "[| a = b : A;  b = c : A |] ==> a = c : A"  equal_types:  "[| a : A;  A = B |] ==> a : B"  equal_typesL: "[| a = b : A;  A = B |] ==> a = b : B"  (*Substitution*)  subst_type:   "[| a : A;  !!z. z:A ==> B(z) type |] ==> B(a) type"  subst_typeL:  "[| a = c : A;  !!z. z:A ==> B(z) = D(z) |] ==> B(a) = D(c)"  subst_elem:   "[| a : A;  !!z. z:A ==> b(z):B(z) |] ==> b(a):B(a)"  subst_elemL:    "[| a=c : A;  !!z. z:A ==> b(z)=d(z) : B(z) |] ==> b(a)=d(c) : B(a)"  (*The type N -- natural numbers*)  NF: "N type"  NI0: "0 : N"  NI_succ: "a : N ==> succ(a) : N"  NI_succL:  "a = b : N ==> succ(a) = succ(b) : N"  NE:   "[| p: N;  a: C(0);  !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |]   ==> rec(p, a, %u v. b(u,v)) : C(p)"  NEL:   "[| 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, %u v. b(u,v)) = rec(q,c,d) : C(p)"  NC0:   "[| a: C(0);  !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |]   ==> rec(0, a, %u v. b(u,v)) = a : C(0)"  NC_succ:   "[| p: N;  a: C(0);       !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |] ==>   rec(succ(p), a, %u v. b(u,v)) = b(p, rec(p, a, %u v. b(u,v))) : C(succ(p))"  (*The fourth Peano axiom.  See page 91 of Martin-Lof's book*)  zero_ne_succ:    "[| a: N;  0 = succ(a) : N |] ==> 0: F"  (*The Product of a family of types*)  ProdF:  "[| A type; !!x. x:A ==> B(x) type |] ==> PROD x:A. B(x) type"  ProdFL:   "[| A = C;  !!x. x:A ==> B(x) = D(x) |] ==>   PROD x:A. B(x) = PROD x:C. D(x)"  ProdI:   "[| A type;  !!x. x:A ==> b(x):B(x)|] ==> lam x. b(x) : PROD x:A. B(x)"  ProdIL:   "[| A type;  !!x. x:A ==> b(x) = c(x) : B(x)|] ==>   lam x. b(x) = lam x. c(x) : PROD x:A. B(x)"  ProdE:  "[| p : PROD x:A. B(x);  a : A |] ==> p`a : B(a)"  ProdEL: "[| p=q: PROD x:A. B(x);  a=b : A |] ==> p`a = q`b : B(a)"  ProdC:   "[| a : A;  !!x. x:A ==> b(x) : B(x)|] ==>   (lam x. b(x)) ` a = b(a) : B(a)"  ProdC2:   "p : PROD x:A. B(x) ==> (lam x. p`x) = p : PROD x:A. B(x)"  (*The Sum of a family of types*)  SumF:  "[| A type;  !!x. x:A ==> B(x) type |] ==> SUM x:A. B(x) type"  SumFL:    "[| A = C;  !!x. x:A ==> B(x) = D(x) |] ==> SUM x:A. B(x) = SUM x:C. D(x)"  SumI:  "[| a : A;  b : B(a) |] ==> <a,b> : SUM x:A. B(x)"  SumIL: "[| a=c:A;  b=d:B(a) |] ==> <a,b> = <c,d> : SUM x:A. B(x)"  SumE:    "[| p: SUM x:A. B(x);  !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |]    ==> split(p, %x y. c(x,y)) : C(p)"  SumEL:    "[| p=q : SUM x:A. B(x);       !!x y. [| x:A; y:B(x) |] ==> c(x,y)=d(x,y): C(<x,y>)|]    ==> split(p, %x y. c(x,y)) = split(q, % x y. d(x,y)) : C(p)"  SumC:    "[| a: A;  b: B(a);  !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |]    ==> split(<a,b>, %x y. c(x,y)) = c(a,b) : C(<a,b>)"  fst_def:   "fst(a) == split(a, %x y. x)"  snd_def:   "snd(a) == split(a, %x y. y)"  (*The sum of two types*)  PlusF:   "[| A type;  B type |] ==> A+B type"  PlusFL:  "[| A = C;  B = D |] ==> A+B = C+D"  PlusI_inl:   "[| a : A;  B type |] ==> inl(a) : A+B"  PlusI_inlL: "[| a = c : A;  B type |] ==> inl(a) = inl(c) : A+B"  PlusI_inr:   "[| A type;  b : B |] ==> inr(b) : A+B"  PlusI_inrL: "[| A type;  b = d : B |] ==> inr(b) = inr(d) : A+B"  PlusE:    "[| p: A+B;  !!x. x:A ==> c(x): C(inl(x));                !!y. y:B ==> d(y): C(inr(y)) |]    ==> when(p, %x. c(x), %y. d(y)) : C(p)"  PlusEL:    "[| 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, %x. c(x), %y. d(y)) = when(q, %x. e(x), %y. f(y)) : C(p)"  PlusC_inl:    "[| a: A;  !!x. x:A ==> c(x): C(inl(x));              !!y. y:B ==> d(y): C(inr(y)) |]    ==> when(inl(a), %x. c(x), %y. d(y)) = c(a) : C(inl(a))"  PlusC_inr:    "[| b: B;  !!x. x:A ==> c(x): C(inl(x));              !!y. y:B ==> d(y): C(inr(y)) |]    ==> when(inr(b), %x. c(x), %y. d(y)) = d(b) : C(inr(b))"  (*The type Eq*)  EqF:    "[| A type;  a : A;  b : A |] ==> Eq(A,a,b) type"  EqFL: "[| A=B;  a=c: A;  b=d : A |] ==> Eq(A,a,b) = Eq(B,c,d)"  EqI: "a = b : A ==> eq : Eq(A,a,b)"  EqE: "p : Eq(A,a,b) ==> a = b : A"  (*By equality of types, can prove C(p) from C(eq), an elimination rule*)  EqC: "p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)"  (*The type F*)  FF: "F type"  FE: "[| p: F;  C type |] ==> contr(p) : C"  FEL:  "[| p = q : F;  C type |] ==> contr(p) = contr(q) : C"  (*The type T     Martin-Lof'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"  TI: "tt : T"  TE: "[| p : T;  c : C(tt) |] ==> c : C(p)"  TEL: "[| p = q : T;  c = d : C(tt) |] ==> c = d : C(p)"  TC: "p : T ==> p = tt : T"subsection "Tactics and derived rules for Constructive Type Theory"(*Formation rules*)lemmas form_rls = NF ProdF SumF PlusF EqF FF TF  and formL_rls = ProdFL SumFL PlusFL EqFL(*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(*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(*OMITTED: eqC are TC because they make rewriting loop: p = un = un = ... *)lemmas comp_rls = NC0 NC_succ ProdC SumC PlusC_inl PlusC_inr(*rules with conclusion a:A, an elem judgement*)lemmas element_rls = intr_rls elim_rls(*Definitions are (meta)equality axioms*)lemmas basic_defs = fst_def snd_def(*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)"apply (rule sym_elem)apply (rule SumIL)apply (rule_tac [!] sym_elem)apply assumption+donelemmas intrL2_rls = NI_succL ProdIL SumIL2 PlusI_inlL PlusI_inrL(*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)"apply (rule assms ProdE)+donesubsection {* Tactics for type checking *}ML {*localfun is_rigid_elem (Const("CTT.Elem",_) \$ a \$ _) = not(is_Var (head_of a))  | is_rigid_elem (Const("CTT.Eqelem",_) \$ a \$ _ \$ _) = not(is_Var (head_of a))  | is_rigid_elem (Const("CTT.Type",_) \$ a) = not(is_Var (head_of a))  | is_rigid_elem _ = falsein(*Try solving a:A or a=b:A by assumption provided a is rigid!*)val test_assume_tac = SUBGOAL(fn (prem,i) =>    if is_rigid_elem (Logic.strip_assums_concl prem)    then  assume_tac i  else  no_tac)fun ASSUME tf i = test_assume_tac i  ORELSE  tf iend;*}(*For simplification: type formation and checking,  but no equalities between terms*)lemmas routine_rls = form_rls formL_rls refl_type element_rlsML {*local  val equal_rls = @{thms form_rls} @ @{thms element_rls} @ @{thms intrL_rls} @    @{thms elimL_rls} @ @{thms refl_elem}infun routine_tac rls prems = ASSUME (filt_resolve_tac (prems @ rls) 4);(*Solve all subgoals "A type" using formation rules. *)val form_tac = REPEAT_FIRST (ASSUME (filt_resolve_tac @{thms form_rls} 1));(*Type checking: solve a:A (a rigid, A flexible) by intro and elim rules. *)fun typechk_tac thms =  let val tac = filt_resolve_tac (thms @ @{thms form_rls} @ @{thms element_rls}) 3  in  REPEAT_FIRST (ASSUME 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 thms =  let val tac = filt_resolve_tac(thms @ @{thms form_rls} @ @{thms intr_rls}) 1  in  REPEAT_FIRST (ASSUME tac)  end(*Equality proving: solve a=b:A (where a is rigid) by long rules. *)fun equal_tac thms =  REPEAT_FIRST (ASSUME (filt_resolve_tac (thms @ equal_rls) 3))end*}subsection {* Simplification *}(*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(*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(*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_tac = REPEAT_FIRST (ASSUME (filt_resolve_tac (@{thm EqI} :: @{thms intr_rls}) 1))(** 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 (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm NE} ifun SumE_tac ctxt sp i =  TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm SumE} ifun PlusE_tac ctxt sp i =  TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), 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 i =    rtac @{thm subst_prodE} i  THEN  assume_tac i  THEN  assume_tac i(*Finds P-->Q and P in the assumptions, replaces implication by Q *)fun mp_tac i = etac @{thm subst_prodE} i  THEN  assume_tac 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_brlsfun safestep_tac thms i =    form_tac  ORELSE    resolve_tac thms i  ORELSE    biresolve_tac safe0_brls i  ORELSE  mp_tac i  ORELSE    DETERM (biresolve_tac safep_brls i)fun safe_tac thms i = DEPTH_SOLVE_1 (safestep_tac thms i)fun step_tac thms = safestep_tac thms  ORELSE'  biresolve_tac unsafe_brls(*Fails unless it solves the goal!*)fun pc_tac thms = DEPTH_SOLVE_1 o (step_tac thms)*}ML_file "rew.ML"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 assumptiondone(*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 (tactic {* typechk_tac @{thms assms} *})  doneend`