Thanks, John! We need to get rid of the new_axiom biz at some point,
I think, but our code sure looks nice. Thanks again! I have two miz3
comments for Freek and you:
1) I was wrong about comments: you can use
::
for comments inside a proof, just as in Mizar. That's almost
explained in Freek's pdf, but it ought to be clearer. BUT (and I
think this is a bug) you can't have ` in the comment.
2) Here's a miz3 problem I had that I couldn't solve, and the error
message doesn't help me. There's only one error message, so I'll just
send the miz3 Exception: Mizar_error, which contains this:
cases by X1, X2;
:: #2
:: 2: inference time-out
Mizar thought that was fine. So I rewrote the code to say first
cases by X1;
and then
cases by X2;
and it didn't help. So I don't think it's a cases problem. I think
there's some other error I made which isn't getting reported.
Exception:
Mizar_error
(`;
let a b x be point;
assume ~(a = b) [H1];
assume ~(a = x) [H2];
assume on_line(x,a,b) [H3];
thus ! c .
on_line(c,a,b) ==> on_line(c,a,x)
proof
let c be point;
assume on_line(c,a,b) [H4];
Between (a,b,x) \/ Between (b,x,a) \/ Between (x,a,b) by H3, Line_DEF;
Between (a,b,x) \/ Between (a,x,b) \/ Between (x,a,b) [X1]
by -, Bsymmetry_THM;
Between (a,b,c) \/ Between (b,c,a) \/ Between (c,a,b) by H4, Line_DEF;
Between (a,b,c) \/ Between (a,c,b) \/ Between (c,a,b) [X2]
by -, Bsymmetry_THM;
x = b \/ b = c \/ (~(x = b) /\ ~(b = c));
cases by -;
suppose x = b [Case1];
on_line(c,a,x) by -, H4;
qed by -;
suppose b = c [Case3];
Between (a,c,x) \/ Between (c,x,a) \/ Between (x,a,c) by -, H3,
Line_DEF;
Between (x,c,a) \/ Between (a,x,c) \/ Between (c,a,x) by -,
Bsymmetry_THM;
on_line(c,a,x) by -, H2, Line_DEF;
qed by -;
suppose ~(x = b) /\ ~(b = c) [Case2];
Between (a,x,c) \/ Between (a,c,x) \/ Between (x,a,c)
proof
cases by X1, X2;
:: #2
:: 2: inference time-out
suppose Between (a,b,x) /\ Between (a,b,c) [X3];
Between (b,x,c) \/ Between (b,c,x) by -, H1, Gupta_THM;
is_ordered (a,b,x,c) \/ is_ordered (a,b,c,x) by -, Case2, X3,
BTransitivityOrdered_THM;
qed by -, is_ordered_DEF;
suppose Between (a,b,x) /\ Between (a,c,b);
is_ordered (a,c,b,x) by -, B123and134Ordered_THM;
qed by -, is_ordered_DEF;
suppose Between (a,b,x) /\ Between (c,a,b);
is_ordered (c,a,b,x) by -, H1, BTransitivityOrdered_THM;
Between (c,a,x) by -, is_ordered_DEF;
qed by -, Bsymmetry_THM;
suppose Between (a,x,b) /\ Between (a,b,c);
is_ordered (a,x,b,c) by -, B123and134Ordered_THM;
qed by -, is_ordered_DEF;
suppose Between (a,x,b) /\ Between (a,c,b) [X4];
consider m such that
Between (b,a,m) /\ a,m === a,b [X5] by -, A4;
~(a = m) [X6] by H1, X5, EquivSymmetric, A3;
Between (m,a,b) by X5, Bsymmetry_THM;
Between (m,a,c) /\ Between (m,a,x) by -, X4, B124and234then123_THM;
Between (a,c,x) \/ Between (a,x,c) by -, X6, Gupta_THM;
qed by -;
suppose Between (a,x,b) /\ Between (c,a,b);
Between (c,a,x) by -, B124and234then123_THM;
qed by -, Bsymmetry_THM;
suppose Between (x,a,b) /\ Between (a,b,c);
is_ordered (x,a,b,c) by -, H1, BTransitivityOrdered_THM;
qed by -, is_ordered_DEF;
suppose Between (x,a,b) /\ Between (a,c,b);
qed by -, B124and234then123_THM;
suppose Between (x,a,b) /\ Between (c,a,b);
Between (b,a,x) /\ Between (b,a,c) by -, Bsymmetry_THM;
Between (a,x,c) \/ Between (a,c,x) by -, H1, Gupta_THM;
qed by -;
end;
Between (a,x,c) \/ Between (x,c,a) \/ Between (c,a,x) by -,
Bsymmetry_THM;
qed by -, H2, Line_DEF;
end
;`,
(0, 1, 0)).
#
--
Best,
Bill
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