doc-src/Logics/CTT-eg.txt

fix to spacing in switches, for Vampire under SML/NJ

(**** CTT examples -- process using Doc/tout CTT-eg.txt  ****)

Pretty.setmargin 72;  (*existing macros just allow this margin*)
print_depth 0;


(*** Type inference, from CTT/ex/typechk.ML ***)

Goal "lam n. rec(n, 0, %x y. x) : ?A";
by (resolve_tac [ProdI] 1);
by (eresolve_tac [NE] 2);
by (resolve_tac [NI0] 2);
by (assume_tac 2);
by (resolve_tac [NF] 1);



(*** Logical reasoning, from CTT/ex/elim.ML ***)

val prems = Goal
    "[| A type;  \
\       !!x. x:A ==> B(x) type;       \
\       !!x. x:A ==> C(x) type;       \
\       p: SUM x:A. B(x) + C(x)       \
\    |] ==>  ?a : (SUM x:A. B(x)) + (SUM x:A. C(x))";
by (resolve_tac (prems RL [SumE]) 1);
by (eresolve_tac [PlusE] 1);
by (resolve_tac [PlusI_inl] 1);
by (resolve_tac [SumI] 1);
by (assume_tac 1);
by (assume_tac 1);
by (typechk_tac prems);
by (pc_tac prems 1);


(*** Currying, from CTT/ex/elim.ML ***)

val prems = Goal
    "[| A type; !!x. x:A ==> B(x) type;                         \
\               !!z. z: (SUM x:A. B(x)) ==> C(z) type           \
\    |] ==> ?a : PROD f: (PROD z : (SUM x:A . B(x)) . C(z)).    \
\                     (PROD x:A . PROD y:B(x) . C(<x,y>))";
by (intr_tac prems);
by (eresolve_tac [ProdE] 1);
by (intr_tac prems);


(*** Axiom of Choice ***)

val prems = Goal   
    "[| A type;  !!x. x:A ==> B(x) type;  \
\       !!x y.[| x:A;  y:B(x) |] ==> C(x,y) type  \
\    |] ==> ?a : PROD h: (PROD x:A. SUM y:B(x). C(x,y)).    \
\                        (SUM f: (PROD x:A. B(x)). PROD x:A. C(x, f`x))";
by (intr_tac prems);
by (eresolve_tac [ProdE RS SumE_fst] 1);
by (assume_tac 1);
by (resolve_tac [replace_type] 1);
by (resolve_tac [subst_eqtyparg] 1);
by (resolve_tac [ProdC] 1);
by (typechk_tac (SumE_fst::prems));
by (eresolve_tac [ProdE RS SumE_snd] 1);
by (typechk_tac prems);





STOP_HERE;


> val prems = Goal
#     "[| A type;  \
# \       !!x. x:A ==> B(x) type;       \
# \       !!x. x:A ==> C(x) type;       \
# \       p: SUM x:A. B(x) + C(x)       \
# \    |] ==>  ?a : (SUM x:A. B(x)) + (SUM x:A. C(x))";
Level 0
?a : (SUM x:A. B(x)) + (SUM x:A. C(x))
 1. ?a : (SUM x:A. B(x)) + (SUM x:A. C(x))
> by (resolve_tac (prems RL [SumE]) 1);
Level 1
split(p,?c1) : (SUM x:A. B(x)) + (SUM x:A. C(x))
 1. !!x y.
       [| x : A; y : B(x) + C(x) |] ==>
       ?c1(x,y) : (SUM x:A. B(x)) + (SUM x:A. C(x))
> by (eresolve_tac [PlusE] 1);
Level 2
split(p,%x y. when(y,?c2(x,y),?d2(x,y)))
: (SUM x:A. B(x)) + (SUM x:A. C(x))
 1. !!x y xa.
       [| x : A; xa : B(x) |] ==>
       ?c2(x,y,xa) : (SUM x:A. B(x)) + (SUM x:A. C(x))
 2. !!x y ya.
       [| x : A; ya : C(x) |] ==>
       ?d2(x,y,ya) : (SUM x:A. B(x)) + (SUM x:A. C(x))
> by (resolve_tac [PlusI_inl] 1);
Level 3
split(p,%x y. when(y,%xa. inl(?a3(x,y,xa)),?d2(x,y)))
: (SUM x:A. B(x)) + (SUM x:A. C(x))
 1. !!x y xa. [| x : A; xa : B(x) |] ==> ?a3(x,y,xa) : SUM x:A. B(x)
 2. !!x y xa. [| x : A; xa : B(x) |] ==> SUM x:A. C(x) type
 3. !!x y ya.
       [| x : A; ya : C(x) |] ==>
       ?d2(x,y,ya) : (SUM x:A. B(x)) + (SUM x:A. C(x))
> by (resolve_tac [SumI] 1);
Level 4
split(p,%x y. when(y,%xa. inl(<?a4(x,y,xa),?b4(x,y,xa)>),?d2(x,y)))
: (SUM x:A. B(x)) + (SUM x:A. C(x))
 1. !!x y xa. [| x : A; xa : B(x) |] ==> ?a4(x,y,xa) : A
 2. !!x y xa. [| x : A; xa : B(x) |] ==> ?b4(x,y,xa) : B(?a4(x,y,xa))
 3. !!x y xa. [| x : A; xa : B(x) |] ==> SUM x:A. C(x) type
 4. !!x y ya.
       [| x : A; ya : C(x) |] ==>
       ?d2(x,y,ya) : (SUM x:A. B(x)) + (SUM x:A. C(x))
> by (assume_tac 1);
Level 5
split(p,%x y. when(y,%xa. inl(<x,?b4(x,y,xa)>),?d2(x,y)))
: (SUM x:A. B(x)) + (SUM x:A. C(x))
 1. !!x y xa. [| x : A; xa : B(x) |] ==> ?b4(x,y,xa) : B(x)
 2. !!x y xa. [| x : A; xa : B(x) |] ==> SUM x:A. C(x) type
 3. !!x y ya.
       [| x : A; ya : C(x) |] ==>
       ?d2(x,y,ya) : (SUM x:A. B(x)) + (SUM x:A. C(x))
> by (assume_tac 1);
Level 6
split(p,%x y. when(y,%xa. inl(<x,xa>),?d2(x,y)))
: (SUM x:A. B(x)) + (SUM x:A. C(x))
 1. !!x y xa. [| x : A; xa : B(x) |] ==> SUM x:A. C(x) type
 2. !!x y ya.
       [| x : A; ya : C(x) |] ==>
       ?d2(x,y,ya) : (SUM x:A. B(x)) + (SUM x:A. C(x))
> by (typechk_tac prems);
Level 7
split(p,%x y. when(y,%xa. inl(<x,xa>),?d2(x,y)))
: (SUM x:A. B(x)) + (SUM x:A. C(x))
 1. !!x y ya.
       [| x : A; ya : C(x) |] ==>
       ?d2(x,y,ya) : (SUM x:A. B(x)) + (SUM x:A. C(x))
> by (pc_tac prems 1);
Level 8
split(p,%x y. when(y,%xa. inl(<x,xa>),%y. inr(<x,y>)))
: (SUM x:A. B(x)) + (SUM x:A. C(x))
No subgoals!




> val prems = Goal
#     "[| A type; !!x. x:A ==> B(x) type; \
# \               !!z. z: (SUM x:A. B(x)) ==> C(z) type |] \
# \    ==> ?a : (PROD z : (SUM x:A . B(x)) . C(z)) \
# \         --> (PROD x:A . PROD y:B(x) . C(<x,y>))";
Level 0
?a : (PROD z:SUM x:A. B(x). C(z)) --> (PROD x:A. PROD y:B(x). C(<x,y>))
 1. ?a : (PROD z:SUM x:A. B(x). C(z)) -->
         (PROD x:A. PROD y:B(x). C(<x,y>))
> by (intr_tac prems);
Level 1
lam x xa xb. ?b7(x,xa,xb)
: (PROD z:SUM x:A. B(x). C(z)) --> (PROD x:A. PROD y:B(x). C(<x,y>))
 1. !!uu x y.
       [| uu : PROD z:SUM x:A. B(x). C(z); x : A; y : B(x) |] ==>
       ?b7(uu,x,y) : C(<x,y>)
> by (eresolve_tac [ProdE] 1);
Level 2
lam x xa xb. x ` <xa,xb>
: (PROD z:SUM x:A. B(x). C(z)) --> (PROD x:A. PROD y:B(x). C(<x,y>))
 1. !!uu x y. [| x : A; y : B(x) |] ==> <x,y> : SUM x:A. B(x)
> by (intr_tac prems);
Level 3
lam x xa xb. x ` <xa,xb>
: (PROD z:SUM x:A. B(x). C(z)) --> (PROD x:A. PROD y:B(x). C(<x,y>))
No subgoals!




> val prems = Goal
#     "[| A type;  !!x. x:A ==> B(x) type;  \
# \       !!x y.[| x:A;  y:B(x) |] ==> C(x,y) type  \
# \    |] ==> ?a :    (PROD x:A. SUM y:B(x). C(x,y))    \
# \               --> (SUM f: (PROD x:A. B(x)). PROD x:A. C(x, f`x))";
Level 0
?a : (PROD x:A. SUM y:B(x). C(x,y)) -->
     (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))
 1. ?a : (PROD x:A. SUM y:B(x). C(x,y)) -->
         (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))
> by (intr_tac prems);
Level 1
lam x. <lam xa. ?b7(x,xa),lam xa. ?b8(x,xa)>
: (PROD x:A. SUM y:B(x). C(x,y)) -->
  (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))
 1. !!uu x.
       [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>
       ?b7(uu,x) : B(x)
 2. !!uu x.
       [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>
       ?b8(uu,x) : C(x,(lam x. ?b7(uu,x)) ` x)
> by (eresolve_tac [ProdE RS SumE_fst] 1);
Level 2
lam x. <lam xa. fst(x ` xa),lam xa. ?b8(x,xa)>
: (PROD x:A. SUM y:B(x). C(x,y)) -->
  (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))
 1. !!uu x. x : A ==> x : A
 2. !!uu x.
       [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>
       ?b8(uu,x) : C(x,(lam x. fst(uu ` x)) ` x)
> by (assume_tac 1);
Level 3
lam x. <lam xa. fst(x ` xa),lam xa. ?b8(x,xa)>
: (PROD x:A. SUM y:B(x). C(x,y)) -->
  (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))
 1. !!uu x.
       [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>
       ?b8(uu,x) : C(x,(lam x. fst(uu ` x)) ` x)
> by (resolve_tac [replace_type] 1);
Level 4
lam x. <lam xa. fst(x ` xa),lam xa. ?b8(x,xa)>
: (PROD x:A. SUM y:B(x). C(x,y)) -->
  (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))
 1. !!uu x.
       [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>
       C(x,(lam x. fst(uu ` x)) ` x) = ?A13(uu,x)
 2. !!uu x.
       [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>
       ?b8(uu,x) : ?A13(uu,x)
> by (resolve_tac [subst_eqtyparg] 1);
Level 5
lam x. <lam xa. fst(x ` xa),lam xa. ?b8(x,xa)>
: (PROD x:A. SUM y:B(x). C(x,y)) -->
  (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))
 1. !!uu x.
       [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>
       (lam x. fst(uu ` x)) ` x = ?c14(uu,x) : ?A14(uu,x)
 2. !!uu x z.
       [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A;
          z : ?A14(uu,x) |] ==>
       C(x,z) type
 3. !!uu x.
       [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>
       ?b8(uu,x) : C(x,?c14(uu,x))
> by (resolve_tac [ProdC] 1);
Level 6
lam x. <lam xa. fst(x ` xa),lam xa. ?b8(x,xa)>
: (PROD x:A. SUM y:B(x). C(x,y)) -->
  (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))
 1. !!uu x.
       [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==> x : ?A15(uu,x)
 2. !!uu x xa.
       [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A;
          xa : ?A15(uu,x) |] ==>
       fst(uu ` xa) : ?B15(uu,x,xa)
 3. !!uu x z.
       [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A;
          z : ?B15(uu,x,x) |] ==>
       C(x,z) type
 4. !!uu x.
       [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>
       ?b8(uu,x) : C(x,fst(uu ` x))
> by (typechk_tac (SumE_fst::prems));
Level 7
lam x. <lam xa. fst(x ` xa),lam xa. ?b8(x,xa)>
: (PROD x:A. SUM y:B(x). C(x,y)) -->
  (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))
 1. !!uu x.
       [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>
       ?b8(uu,x) : C(x,fst(uu ` x))
> by (eresolve_tac [ProdE RS SumE_snd] 1);
Level 8
lam x. <lam xa. fst(x ` xa),lam xa. snd(x ` xa)>
: (PROD x:A. SUM y:B(x). C(x,y)) -->
  (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))
 1. !!uu x. x : A ==> x : A
 2. !!uu x. x : A ==> B(x) type
 3. !!uu x xa. [| x : A; xa : B(x) |] ==> C(x,xa) type
> by (typechk_tac prems);
Level 9
lam x. <lam xa. fst(x ` xa),lam xa. snd(x ` xa)>
: (PROD x:A. SUM y:B(x). C(x,y)) -->
  (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))
No subgoals!