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paulson@6121
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1 |
(**** FOL examples ****)
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paulson@6121
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2 |
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paulson@6121
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3 |
Pretty.setmargin 72; (*existing macros just allow this margin*)
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paulson@6121
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4 |
print_depth 0;
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paulson@6121
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5 |
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paulson@6121
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6 |
(*** Intuitionistic examples ***)
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paulson@6121
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7 |
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paulson@6121
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8 |
context IFOL.thy;
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paulson@6121
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9 |
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paulson@6121
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10 |
(*Quantifier example from Logic&Computation*)
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paulson@6121
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11 |
Goal "(EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))";
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paulson@6121
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12 |
by (resolve_tac [impI] 1);
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paulson@6121
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13 |
by (resolve_tac [allI] 1);
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paulson@6121
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14 |
by (resolve_tac [exI] 1);
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paulson@6121
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15 |
by (eresolve_tac [exE] 1);
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paulson@6121
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16 |
choplev 2;
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paulson@6121
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17 |
by (eresolve_tac [exE] 1);
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paulson@6121
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18 |
by (resolve_tac [exI] 1);
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paulson@6121
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19 |
by (eresolve_tac [allE] 1);
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paulson@6121
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20 |
by (assume_tac 1);
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paulson@6121
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21 |
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paulson@6121
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22 |
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paulson@6121
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23 |
(*Example of Dyckhoff's method*)
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paulson@6121
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24 |
Goalw [not_def] "~ ~ ((P-->Q) | (Q-->P))";
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paulson@6121
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25 |
by (resolve_tac [impI] 1);
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paulson@6121
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26 |
by (eresolve_tac [disj_impE] 1);
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paulson@6121
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27 |
by (eresolve_tac [imp_impE] 1);
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paulson@6121
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28 |
by (eresolve_tac [imp_impE] 1);
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paulson@6121
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29 |
by (REPEAT (eresolve_tac [FalseE] 2));
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paulson@6121
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30 |
by (assume_tac 1);
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paulson@6121
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31 |
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paulson@6121
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32 |
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paulson@6121
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33 |
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paulson@6121
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34 |
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paulson@6121
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35 |
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paulson@6121
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36 |
(*** Classical examples ***)
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paulson@6121
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37 |
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paulson@6121
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38 |
context FOL.thy;
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paulson@6121
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39 |
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paulson@6121
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40 |
Goal "EX y. ALL x. P(y)-->P(x)";
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paulson@6121
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41 |
by (resolve_tac [exCI] 1);
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paulson@6121
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42 |
by (resolve_tac [allI] 1);
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paulson@6121
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43 |
by (resolve_tac [impI] 1);
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paulson@6121
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44 |
by (eresolve_tac [allE] 1);
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paulson@6121
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45 |
prth (allI RSN (2,swap));
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paulson@6121
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46 |
by (eresolve_tac [it] 1);
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paulson@6121
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47 |
by (resolve_tac [impI] 1);
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paulson@6121
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48 |
by (eresolve_tac [notE] 1);
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paulson@6121
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49 |
by (assume_tac 1);
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paulson@6121
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50 |
Goal "EX y. ALL x. P(y)-->P(x)";
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paulson@6121
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51 |
by (Blast_tac 1);
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paulson@6121
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52 |
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paulson@6121
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53 |
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paulson@6121
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54 |
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paulson@6121
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55 |
- Goal "EX y. ALL x. P(y)-->P(x)";
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paulson@6121
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56 |
Level 0
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paulson@6121
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57 |
EX y. ALL x. P(y) --> P(x)
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paulson@6121
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58 |
1. EX y. ALL x. P(y) --> P(x)
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paulson@6121
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59 |
- by (resolve_tac [exCI] 1);
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paulson@6121
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Level 1
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paulson@6121
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61 |
EX y. ALL x. P(y) --> P(x)
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paulson@6121
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1. ALL y. ~(ALL x. P(y) --> P(x)) ==> ALL x. P(?a) --> P(x)
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paulson@6121
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- by (resolve_tac [allI] 1);
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paulson@6121
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64 |
Level 2
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paulson@6121
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EX y. ALL x. P(y) --> P(x)
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paulson@6121
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1. !!x. ALL y. ~(ALL x. P(y) --> P(x)) ==> P(?a) --> P(x)
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paulson@6121
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- by (resolve_tac [impI] 1);
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paulson@6121
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Level 3
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paulson@6121
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EX y. ALL x. P(y) --> P(x)
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paulson@6121
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1. !!x. [| ALL y. ~(ALL x. P(y) --> P(x)); P(?a) |] ==> P(x)
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paulson@6121
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- by (eresolve_tac [allE] 1);
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paulson@6121
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Level 4
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paulson@6121
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73 |
EX y. ALL x. P(y) --> P(x)
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paulson@6121
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1. !!x. [| P(?a); ~(ALL xa. P(?y3(x)) --> P(xa)) |] ==> P(x)
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paulson@6121
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- prth (allI RSN (2,swap));
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paulson@6121
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[| ~(ALL x. ?P1(x)); !!x. ~?Q ==> ?P1(x) |] ==> ?Q
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paulson@6121
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- by (eresolve_tac [it] 1);
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paulson@6121
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Level 5
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paulson@6121
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EX y. ALL x. P(y) --> P(x)
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paulson@6121
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1. !!x xa. [| P(?a); ~P(x) |] ==> P(?y3(x)) --> P(xa)
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paulson@6121
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- by (resolve_tac [impI] 1);
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paulson@6121
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Level 6
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paulson@6121
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EX y. ALL x. P(y) --> P(x)
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paulson@6121
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1. !!x xa. [| P(?a); ~P(x); P(?y3(x)) |] ==> P(xa)
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paulson@6121
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- by (eresolve_tac [notE] 1);
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paulson@6121
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Level 7
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paulson@6121
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EX y. ALL x. P(y) --> P(x)
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paulson@6121
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1. !!x xa. [| P(?a); P(?y3(x)) |] ==> P(x)
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paulson@6121
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- by (assume_tac 1);
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paulson@6121
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Level 8
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paulson@6121
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EX y. ALL x. P(y) --> P(x)
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paulson@6121
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No subgoals!
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paulson@6121
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93 |
- Goal "EX y. ALL x. P(y)-->P(x)";
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paulson@6121
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94 |
Level 0
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paulson@6121
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EX y. ALL x. P(y) --> P(x)
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paulson@6121
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96 |
1. EX y. ALL x. P(y) --> P(x)
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paulson@6121
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- by (best_tac FOL_dup_cs 1);
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paulson@6121
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98 |
Level 1
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paulson@6121
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99 |
EX y. ALL x. P(y) --> P(x)
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paulson@6121
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100 |
No subgoals!
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paulson@6121
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101 |
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paulson@6121
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102 |
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paulson@6121
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103 |
(**** finally, the example FOL/ex/if.ML ****)
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paulson@6121
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104 |
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paulson@6121
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105 |
> val prems = goalw if_thy [if_def]
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paulson@6121
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# "[| P ==> Q; ~P ==> R |] ==> if(P,Q,R)";
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paulson@6121
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107 |
Level 0
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paulson@6121
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108 |
if(P,Q,R)
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paulson@6121
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109 |
1. P & Q | ~P & R
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paulson@6121
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110 |
> by (Classical.fast_tac (FOL_cs addIs prems) 1);
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paulson@6121
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111 |
Level 1
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paulson@6121
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112 |
if(P,Q,R)
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paulson@6121
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113 |
No subgoals!
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paulson@6121
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114 |
> val ifI = result();
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paulson@6121
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115 |
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paulson@6121
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116 |
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paulson@6121
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117 |
> val major::prems = goalw if_thy [if_def]
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paulson@6121
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118 |
# "[| if(P,Q,R); [| P; Q |] ==> S; [| ~P; R |] ==> S |] ==> S";
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paulson@6121
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119 |
Level 0
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paulson@6121
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120 |
S
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paulson@6121
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121 |
1. S
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paulson@6121
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122 |
> by (cut_facts_tac [major] 1);
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paulson@6121
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123 |
Level 1
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paulson@6121
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124 |
S
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paulson@6121
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125 |
1. P & Q | ~P & R ==> S
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paulson@6121
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126 |
> by (Classical.fast_tac (FOL_cs addIs prems) 1);
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paulson@6121
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127 |
Level 2
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paulson@6121
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128 |
S
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paulson@6121
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129 |
No subgoals!
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paulson@6121
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130 |
> val ifE = result();
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paulson@6121
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131 |
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paulson@6121
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132 |
> goal if_thy "if(P, if(Q,A,B), if(Q,C,D)) <-> if(Q, if(P,A,C), if(P,B,D))";
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paulson@6121
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133 |
Level 0
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paulson@6121
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134 |
if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
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135 |
1. if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
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136 |
> by (resolve_tac [iffI] 1);
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paulson@6121
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137 |
Level 1
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paulson@6121
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138 |
if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
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139 |
1. if(P,if(Q,A,B),if(Q,C,D)) ==> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
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140 |
2. if(Q,if(P,A,C),if(P,B,D)) ==> if(P,if(Q,A,B),if(Q,C,D))
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paulson@6121
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141 |
> by (eresolve_tac [ifE] 1);
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paulson@6121
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142 |
Level 2
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paulson@6121
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143 |
if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
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144 |
1. [| P; if(Q,A,B) |] ==> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
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145 |
2. [| ~P; if(Q,C,D) |] ==> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
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146 |
3. if(Q,if(P,A,C),if(P,B,D)) ==> if(P,if(Q,A,B),if(Q,C,D))
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paulson@6121
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147 |
> by (eresolve_tac [ifE] 1);
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paulson@6121
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148 |
Level 3
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paulson@6121
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149 |
if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
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150 |
1. [| P; Q; A |] ==> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
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151 |
2. [| P; ~Q; B |] ==> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
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152 |
3. [| ~P; if(Q,C,D) |] ==> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
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153 |
4. if(Q,if(P,A,C),if(P,B,D)) ==> if(P,if(Q,A,B),if(Q,C,D))
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paulson@6121
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154 |
> by (resolve_tac [ifI] 1);
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paulson@6121
|
155 |
Level 4
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paulson@6121
|
156 |
if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
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157 |
1. [| P; Q; A; Q |] ==> if(P,A,C)
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paulson@6121
|
158 |
2. [| P; Q; A; ~Q |] ==> if(P,B,D)
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paulson@6121
|
159 |
3. [| P; ~Q; B |] ==> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
|
160 |
4. [| ~P; if(Q,C,D) |] ==> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
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161 |
5. if(Q,if(P,A,C),if(P,B,D)) ==> if(P,if(Q,A,B),if(Q,C,D))
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paulson@6121
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162 |
> by (resolve_tac [ifI] 1);
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paulson@6121
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163 |
Level 5
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paulson@6121
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164 |
if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
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165 |
1. [| P; Q; A; Q; P |] ==> A
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paulson@6121
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166 |
2. [| P; Q; A; Q; ~P |] ==> C
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paulson@6121
|
167 |
3. [| P; Q; A; ~Q |] ==> if(P,B,D)
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paulson@6121
|
168 |
4. [| P; ~Q; B |] ==> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
|
169 |
5. [| ~P; if(Q,C,D) |] ==> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
|
170 |
6. if(Q,if(P,A,C),if(P,B,D)) ==> if(P,if(Q,A,B),if(Q,C,D))
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paulson@6121
|
171 |
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paulson@6121
|
172 |
> choplev 0;
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paulson@6121
|
173 |
Level 0
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paulson@6121
|
174 |
if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
|
175 |
1. if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
|
176 |
> val if_cs = FOL_cs addSIs [ifI] addSEs[ifE];
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paulson@6121
|
177 |
> by (Classical.fast_tac if_cs 1);
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paulson@6121
|
178 |
Level 1
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paulson@6121
|
179 |
if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D))
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paulson@6121
|
180 |
No subgoals!
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paulson@6121
|
181 |
> val if_commute = result();
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paulson@6121
|
182 |
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paulson@6121
|
183 |
> goal if_thy "if(if(P,Q,R), A, B) <-> if(P, if(Q,A,B), if(R,A,B))";
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paulson@6121
|
184 |
Level 0
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paulson@6121
|
185 |
if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B))
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paulson@6121
|
186 |
1. if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B))
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paulson@6121
|
187 |
> by (Classical.fast_tac if_cs 1);
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paulson@6121
|
188 |
Level 1
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paulson@6121
|
189 |
if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B))
|
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paulson@6121
|
190 |
No subgoals!
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paulson@6121
|
191 |
> val nested_ifs = result();
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paulson@6121
|
192 |
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paulson@6121
|
193 |
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paulson@6121
|
194 |
> choplev 0;
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paulson@6121
|
195 |
Level 0
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paulson@6121
|
196 |
if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B))
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paulson@6121
|
197 |
1. if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B))
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paulson@6121
|
198 |
> by (rewrite_goals_tac [if_def]);
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paulson@6121
|
199 |
Level 1
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paulson@6121
|
200 |
if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B))
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paulson@6121
|
201 |
1. (P & Q | ~P & R) & A | ~(P & Q | ~P & R) & B <->
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paulson@6121
|
202 |
P & (Q & A | ~Q & B) | ~P & (R & A | ~R & B)
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paulson@6121
|
203 |
> by (Classical.fast_tac FOL_cs 1);
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paulson@6121
|
204 |
Level 2
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paulson@6121
|
205 |
if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B))
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paulson@6121
|
206 |
No subgoals!
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paulson@6121
|
207 |
|
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paulson@6121
|
208 |
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paulson@6121
|
209 |
> goal if_thy "if(if(P,Q,R), A, B) <-> if(P, if(Q,A,B), if(R,B,A))";
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paulson@6121
|
210 |
Level 0
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paulson@6121
|
211 |
if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A))
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paulson@6121
|
212 |
1. if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A))
|
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paulson@6121
|
213 |
> by (REPEAT (Classical.step_tac if_cs 1));
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paulson@6121
|
214 |
Level 1
|
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paulson@6121
|
215 |
if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A))
|
|
paulson@6121
|
216 |
1. [| A; ~P; R; ~P; R |] ==> B
|
|
paulson@6121
|
217 |
2. [| B; ~P; ~R; ~P; ~R |] ==> A
|
|
paulson@6121
|
218 |
3. [| ~P; R; B; ~P; R |] ==> A
|
|
paulson@6121
|
219 |
4. [| ~P; ~R; A; ~B; ~P |] ==> R
|
|
paulson@6121
|
220 |
|
|
paulson@6121
|
221 |
> choplev 0;
|
|
paulson@6121
|
222 |
Level 0
|
|
paulson@6121
|
223 |
if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A))
|
|
paulson@6121
|
224 |
1. if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A))
|
|
paulson@6121
|
225 |
> by (rewrite_goals_tac [if_def]);
|
|
paulson@6121
|
226 |
Level 1
|
|
paulson@6121
|
227 |
if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A))
|
|
paulson@6121
|
228 |
1. (P & Q | ~P & R) & A | ~(P & Q | ~P & R) & B <->
|
|
paulson@6121
|
229 |
P & (Q & A | ~Q & B) | ~P & (R & B | ~R & A)
|
|
paulson@6121
|
230 |
> by (Classical.fast_tac FOL_cs 1);
|
|
paulson@6121
|
231 |
by: tactic failed
|
|
paulson@6121
|
232 |
Exception- ERROR raised
|
|
paulson@6121
|
233 |
Exception failure raised
|
|
paulson@6121
|
234 |
|
|
paulson@6121
|
235 |
> by (REPEAT (Classical.step_tac FOL_cs 1));
|
|
paulson@6121
|
236 |
Level 2
|
|
paulson@6121
|
237 |
if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A))
|
|
paulson@6121
|
238 |
1. [| A; ~P; R; ~P; R; ~False |] ==> B
|
|
paulson@6121
|
239 |
2. [| A; ~P; R; R; ~False; ~B; ~B |] ==> Q
|
|
paulson@6121
|
240 |
3. [| B; ~P; ~R; ~P; ~A |] ==> R
|
|
paulson@6121
|
241 |
4. [| B; ~P; ~R; ~Q; ~A |] ==> R
|
|
paulson@6121
|
242 |
5. [| B; ~R; ~P; ~A; ~R; Q; ~False |] ==> A
|
|
paulson@6121
|
243 |
6. [| ~P; R; B; ~P; R; ~False |] ==> A
|
|
paulson@6121
|
244 |
7. [| ~P; ~R; A; ~B; ~R |] ==> P
|
|
paulson@6121
|
245 |
8. [| ~P; ~R; A; ~B; ~R |] ==> Q
|