Negation of a conjunction:
1. Proposition
(Ø(P Ù Q) ® (ØP Ú ØQ)) (hilb18)
1 | (ØØP ® P) | add proposition hilb6 |
2 | (ØØQ ® Q) | replace proposition variable P by Q in 1 |
3 | (ØØ(ØP Ú ØQ) ® (ØP Ú ØQ)) | replace proposition variable Q by (ØP Ú ØQ) in 2 |
4 | (Ø(P Ù Q) ® (ØP Ú ØQ)) | reverse abbreviation and in 3 at occurence 1 |
qed |
The reverse of a negation of a conjunction:
2. Proposition
((ØP Ú ØQ) ® Ø(P Ù Q)) (hilb19)
1 | (P ® ØØP) | add proposition hilb5 |
2 | (Q ® ØØQ) | replace proposition variable P by Q in 1 |
3 | ((ØP Ú ØQ) ® ØØ(ØP Ú ØQ)) | replace proposition variable Q by (ØP Ú ØQ) in 2 |
4 | ((ØP Ú ØQ) ® Ø(P Ù Q)) | reverse abbreviation and in 3 at occurence 1 |
qed |
Negation of a disjunction:
3. Proposition
(Ø(P Ú Q) ® (ØP Ù ØQ)) (hilb20)
1 | (P ® P) | add proposition hilb2 |
2 | (Q ® Q) | replace proposition variable P by Q in 1 |
3 | (Ø(P Ú Q) ® Ø(P Ú Q)) | replace proposition variable Q by Ø(P Ú Q) in 2 |
4 | (Ø(P Ú Q) ® Ø(ØØP Ú Q)) | elementary equivalence in 3 at 8 of hilb5 with hilb6 |
5 | (Ø(P Ú Q) ® Ø(ØØP Ú ØØQ)) | elementary equivalence in 4 at 11 of hilb5 with hilb6 |
6 | (Ø(P Ú Q) ® (ØP Ù ØQ)) | reverse abbreviation and in 5 at occurence 1 |
qed |
Reverse of a negation of a disjunction:
4. Proposition
((ØP Ù ØQ) ® Ø(P Ú Q)) (hilb21)
1 | (P ® P) | add proposition hilb2 |
2 | (Q ® Q) | replace proposition variable P by Q in 1 |
3 | (Ø(P Ú Q) ® Ø(P Ú Q)) | replace proposition variable Q by Ø(P Ú Q) in 2 |
4 | (Ø(ØØP Ú Q) ® Ø(P Ú Q)) | elementary equivalence in 3 at 4 of hilb5 with hilb6 |
5 | (Ø(ØØP Ú ØØQ) ® Ø(P Ú Q)) | elementary equivalence in 4 at 7 of hilb5 with hilb6 |
6 | ((ØP Ù ØQ) ® Ø(P Ú Q)) | reverse abbreviation and in 5 at occurence 1 |
qed |
The Conjunction is commutative:
5. Proposition
((P Ù Q) ® (Q Ù P)) (hilb22)
1 | (P ® P) | add proposition hilb2 |
2 | (Q ® Q) | replace proposition variable P by Q in 1 |
3 | ((P Ù Q) ® (P Ù Q)) | replace proposition variable Q by (P Ù Q) in 2 |
4 | ((P Ù Q) ® Ø(ØP Ú ØQ)) | use abbreviation and in 3 at occurence 2 |
5 | ((P Ù Q) ® Ø(ØQ Ú ØP)) | elementary equivalence in 4 at 1 of hilb9 with hilb10 |
6 | ((P Ù Q) ® (Q Ù P)) | reverse abbreviation and in 5 at occurence 1 |
qed |
A technical lemma that is simular to the previous one:
6. Proposition
((Q Ù P) ® (P Ù Q)) (hilb23)
1 | ((P Ù Q) ® (Q Ù P)) | add proposition hilb22 |
2 | ((P Ù A) ® (A Ù P)) | replace proposition variable Q by A in 1 |
3 | ((B Ù A) ® (A Ù B)) | replace proposition variable P by B in 2 |
4 | ((B Ù P) ® (P Ù B)) | replace proposition variable A by P in 3 |
5 | ((Q Ù P) ® (P Ù Q)) | replace proposition variable B by Q in 4 |
qed |
Reduction of a conjunction:
7. Proposition
((P Ù Q) ® P) (hilb24)
1 | (P ® (P Ú Q)) | add axiom axiom2 |
2 | (P ® (P Ú A)) | replace proposition variable Q by A in 1 |
3 | (B ® (B Ú A)) | replace proposition variable P by B in 2 |
4 | (B ® (B Ú ØQ)) | replace proposition variable A by ØQ in 3 |
5 | (ØP ® (ØP Ú ØQ)) | replace proposition variable B by ØP in 4 |
6 | ((P ® Q) ® (ØQ ® ØP)) | add proposition hilb7 |
7 | ((P ® A) ® (ØA ® ØP)) | replace proposition variable Q by A in 6 |
8 | ((B ® A) ® (ØA ® ØB)) | replace proposition variable P by B in 7 |
9 | ((B ® (ØP Ú ØQ)) ® (Ø(ØP Ú ØQ) ® ØB)) | replace proposition variable A by (ØP Ú ØQ) in 8 |
10 | ((ØP ® (ØP Ú ØQ)) ® (Ø(ØP Ú ØQ) ® ØØP)) | replace proposition variable B by ØP in 9 |
11 | (Ø(ØP Ú ØQ) ® ØØP) | modus ponens with 5, 10 |
12 | ((P Ù Q) ® ØØP) | reverse abbreviation and in 11 at occurence 1 |
13 | ((P Ù Q) ® P) | elementary equivalence in 12 at 1 of hilb6 with hilb5 |
qed |
Another form of a reduction of a conjunction:
8. Proposition
((P Ù Q) ® Q) (hilb25)
1 | ((P Ù Q) ® P) | add proposition hilb24 |
2 | ((P Ù A) ® P) | replace proposition variable Q by A in 1 |
3 | ((B Ù A) ® B) | replace proposition variable P by B in 2 |
4 | ((B Ù P) ® B) | replace proposition variable A by P in 3 |
5 | ((Q Ù P) ® Q) | replace proposition variable B by Q in 4 |
6 | ((P Ù Q) ® Q) | elementary equivalence in 5 at 1 of hilb22 with hilb23 |
qed |
The conjunction is associative too (first implication):
9. Proposition
((P Ù (Q Ù A)) ® ((P Ù Q) Ù A)) (hilb26)
1 | (((P Ú Q) Ú A) ® (P Ú (Q Ú A))) | add proposition hilb15 |
2 | ((P ® Q) ® (ØQ ® ØP)) | add proposition hilb7 |
3 | ((P ® A) ® (ØA ® ØP)) | replace proposition variable Q by A in 2 |
4 | ((B ® A) ® (ØA ® ØB)) | replace proposition variable P by B in 3 |
5 | ((B ® (P Ú (Q Ú A))) ® (Ø(P Ú (Q Ú A)) ® ØB)) | replace proposition variable A by (P Ú (Q Ú A)) in 4 |
6 | ((((P Ú Q) Ú A) ® (P Ú (Q Ú A))) ® (Ø(P Ú (Q Ú A)) ® Ø((P Ú Q) Ú A))) | replace proposition variable B by ((P Ú Q) Ú A) in 5 |
7 | (Ø(P Ú (Q Ú A)) ® Ø((P Ú Q) Ú A)) | modus ponens with 1, 6 |
8 | (Ø(P Ú ØØ(Q Ú A)) ® Ø((P Ú Q) Ú A)) | elementary equivalence in 7 at 5 of hilb5 with hilb6 |
9 | (Ø(P Ú ØØ(Q Ú A)) ® Ø(ØØ(P Ú Q) Ú A)) | elementary equivalence in 8 at 12 of hilb5 with hilb6 |
10 | (Ø(P Ú ØØ(Q Ú B)) ® Ø(ØØ(P Ú Q) Ú B)) | replace proposition variable A by B in 9 |
11 | (Ø(P Ú ØØ(C Ú B)) ® Ø(ØØ(P Ú C) Ú B)) | replace proposition variable Q by C in 10 |
12 | (Ø(D Ú ØØ(C Ú B)) ® Ø(ØØ(D Ú C) Ú B)) | replace proposition variable P by D in 11 |
13 | (Ø(D Ú ØØ(C Ú ØA)) ® Ø(ØØ(D Ú C) Ú ØA)) | replace proposition variable B by ØA in 12 |
14 | (Ø(D Ú ØØ(ØQ Ú ØA)) ® Ø(ØØ(D Ú ØQ) Ú ØA)) | replace proposition variable C by ØQ in 13 |
15 | (Ø(ØP Ú ØØ(ØQ Ú ØA)) ® Ø(ØØ(ØP Ú ØQ) Ú ØA)) | replace proposition variable D by ØP in 14 |
16 | ((P Ù Ø(ØQ Ú ØA)) ® Ø(ØØ(ØP Ú ØQ) Ú ØA)) | reverse abbreviation and in 15 at occurence 1 |
17 | ((P Ù (Q Ù A)) ® Ø(ØØ(ØP Ú ØQ) Ú ØA)) | reverse abbreviation and in 16 at occurence 1 |
18 | ((P Ù (Q Ù A)) ® (Ø(ØP Ú ØQ) Ù A)) | reverse abbreviation and in 17 at occurence 1 |
19 | ((P Ù (Q Ù A)) ® ((P Ù Q) Ù A)) | reverse abbreviation and in 18 at occurence 1 |
qed |
The conjunction is associative (second implication):
10. Proposition
(((P Ù Q) Ù A) ® (P Ù (Q Ù A))) (hilb27)
1 | ((P Ú (Q Ú A)) ® ((P Ú Q) Ú A)) | add proposition hilb14 |
2 | ((P ® Q) ® (ØQ ® ØP)) | add proposition hilb7 |
3 | ((P ® A) ® (ØA ® ØP)) | replace proposition variable Q by A in 2 |
4 | ((B ® A) ® (ØA ® ØB)) | replace proposition variable P by B in 3 |
5 | ((B ® ((P Ú Q) Ú A)) ® (Ø((P Ú Q) Ú A) ® ØB)) | replace proposition variable A by ((P Ú Q) Ú A) in 4 |
6 | (((P Ú (Q Ú A)) ® ((P Ú Q) Ú A)) ® (Ø((P Ú Q) Ú A) ® Ø(P Ú (Q Ú A)))) | replace proposition variable B by (P Ú (Q Ú A)) in 5 |
7 | (Ø((P Ú Q) Ú A) ® Ø(P Ú (Q Ú A))) | modus ponens with 1, 6 |
8 | (Ø(ØØ(P Ú Q) Ú A) ® Ø(P Ú (Q Ú A))) | elementary equivalence in 7 at 4 of hilb5 with hilb6 |
9 | (Ø(ØØ(P Ú Q) Ú A) ® Ø(P Ú ØØ(Q Ú A))) | elementary equivalence in 8 at 13 of hilb5 with hilb6 |
10 | (Ø(ØØ(P Ú Q) Ú B) ® Ø(P Ú ØØ(Q Ú B))) | replace proposition variable A by B in 9 |
11 | (Ø(ØØ(P Ú C) Ú B) ® Ø(P Ú ØØ(C Ú B))) | replace proposition variable Q by C in 10 |
12 | (Ø(ØØ(D Ú C) Ú B) ® Ø(D Ú ØØ(C Ú B))) | replace proposition variable P by D in 11 |
13 | (Ø(ØØ(D Ú C) Ú ØA) ® Ø(D Ú ØØ(C Ú ØA))) | replace proposition variable B by ØA in 12 |
14 | (Ø(ØØ(D Ú ØQ) Ú ØA) ® Ø(D Ú ØØ(ØQ Ú ØA))) | replace proposition variable C by ØQ in 13 |
15 | (Ø(ØØ(ØP Ú ØQ) Ú ØA) ® Ø(ØP Ú ØØ(ØQ Ú ØA))) | replace proposition variable D by ØP in 14 |
16 | ((Ø(ØP Ú ØQ) Ù A) ® Ø(ØP Ú ØØ(ØQ Ú ØA))) | reverse abbreviation and in 15 at occurence 1 |
17 | (((P Ù Q) Ù A) ® Ø(ØP Ú ØØ(ØQ Ú ØA))) | reverse abbreviation and in 16 at occurence 1 |
18 | (((P Ù Q) Ù A) ® (P Ù Ø(ØQ Ú ØA))) | reverse abbreviation and in 17 at occurence 1 |
19 | (((P Ù Q) Ù A) ® (P Ù (Q Ù A))) | reverse abbreviation and in 18 at occurence 1 |
qed |
Form for the conjunction rule:
11. Proposition
(P ® (Q ® (P Ù Q))) (hilb28)
1 | (P Ú ØP) | add proposition hilb4 |
2 | ((ØP Ú ØQ) Ú Ø(ØP Ú ØQ)) | replace proposition variable P by (ØP Ú ØQ) in 1 |
3 | (((P Ú Q) Ú A) ® (P Ú (Q Ú A))) | add proposition hilb15 |
4 | (((P Ú Q) Ú B) ® (P Ú (Q Ú B))) | replace proposition variable A by B in 3 |
5 | (((P Ú C) Ú B) ® (P Ú (C Ú B))) | replace proposition variable Q by C in 4 |
6 | (((D Ú C) Ú B) ® (D Ú (C Ú B))) | replace proposition variable P by D in 5 |
7 | (((D Ú C) Ú Ø(ØP Ú ØQ)) ® (D Ú (C Ú Ø(ØP Ú ØQ)))) | replace proposition variable B by Ø(ØP Ú ØQ) in 6 |
8 | (((D Ú ØQ) Ú Ø(ØP Ú ØQ)) ® (D Ú (ØQ Ú Ø(ØP Ú ØQ)))) | replace proposition variable C by ØQ in 7 |
9 | (((ØP Ú ØQ) Ú Ø(ØP Ú ØQ)) ® (ØP Ú (ØQ Ú Ø(ØP Ú ØQ)))) | replace proposition variable D by ØP in 8 |
10 | (ØP Ú (ØQ Ú Ø(ØP Ú ØQ))) | modus ponens with 2, 9 |
11 | (P ® (ØQ Ú Ø(ØP Ú ØQ))) | reverse abbreviation impl in 10 at occurence 1 |
12 | (P ® (Q ® Ø(ØP Ú ØQ))) | reverse abbreviation impl in 11 at occurence 1 |
13 | (P ® (Q ® (P Ù Q))) | reverse abbreviation and in 12 at occurence 1 |
qed |
Preconditions could be put together in a conjunction (first direction):
12. Proposition
((P ® (Q ® A)) ® ((P Ù Q) ® A)) (hilb29)
1 | (P ® P) | add proposition hilb2 |
2 | (Q ® Q) | replace proposition variable P by Q in 1 |
3 | ((P ® (Q ® A)) ® (P ® (Q ® A))) | replace proposition variable Q by (P ® (Q ® A)) in 2 |
4 | ((P ® (Q ® A)) ® (ØP Ú (Q ® A))) | use abbreviation impl in 3 at occurence 4 |
5 | ((P ® (Q ® A)) ® (ØP Ú (ØQ Ú A))) | use abbreviation impl in 4 at occurence 4 |
6 | ((P ® (Q ® A)) ® ((ØP Ú ØQ) Ú A)) | elementary equivalence in 5 at 1 of hilb14 with hilb15 |
7 | ((P ® (Q ® A)) ® (ØØ(ØP Ú ØQ) Ú A)) | elementary equivalence in 6 at 8 of hilb5 with hilb6 |
8 | ((P ® (Q ® A)) ® (Ø(ØP Ú ØQ) ® A)) | reverse abbreviation impl in 7 at occurence 1 |
9 | ((P ® (Q ® A)) ® ((P Ù Q) ® A)) | reverse abbreviation and in 8 at occurence 1 |
qed |
Preconditions could be put together in a conjunction (second direction):
13. Proposition
(((P Ù Q) ® A) ® (P ® (Q ® A))) (hilb30)
1 | (P ® P) | add proposition hilb2 |
2 | (Q ® Q) | replace proposition variable P by Q in 1 |
3 | ((P ® (Q ® A)) ® (P ® (Q ® A))) | replace proposition variable Q by (P ® (Q ® A)) in 2 |
4 | ((ØP Ú (Q ® A)) ® (P ® (Q ® A))) | use abbreviation impl in 3 at occurence 2 |
5 | ((ØP Ú (ØQ Ú A)) ® (P ® (Q ® A))) | use abbreviation impl in 4 at occurence 2 |
6 | (((ØP Ú ØQ) Ú A) ® (P ® (Q ® A))) | elementary equivalence in 5 at 1 of hilb14 with hilb15 |
7 | ((ØØ(ØP Ú ØQ) Ú A) ® (P ® (Q ® A))) | elementary equivalence in 6 at 3 of hilb5 with hilb6 |
8 | ((Ø(ØP Ú ØQ) ® A) ® (P ® (Q ® A))) | reverse abbreviation impl in 7 at occurence 1 |
9 | (((P Ù Q) ® A) ® (P ® (Q ® A))) | reverse abbreviation and in 8 at occurence 1 |
qed |
Absorbtion of a conjunction (first direction):
14. Proposition
((P Ù P) ® P) (hilb31)
1 | ((P Ù Q) ® P) | add proposition hilb24 |
2 | ((P Ù P) ® P) | replace proposition variable Q by P in 1 |
qed |
Absorbtion of a conjunction (second direction):
15. Proposition
(P ® (P Ù P)) (hilb32)
1 | ((P Ú P) ® P) | add proposition hilb11 |
2 | ((P ® Q) ® (ØQ ® ØP)) | add proposition hilb7 |
3 | ((P ® A) ® (ØA ® ØP)) | replace proposition variable Q by A in 2 |
4 | ((B ® A) ® (ØA ® ØB)) | replace proposition variable P by B in 3 |
5 | ((B ® P) ® (ØP ® ØB)) | replace proposition variable A by P in 4 |
6 | (((P Ú P) ® P) ® (ØP ® Ø(P Ú P))) | replace proposition variable B by (P Ú P) in 5 |
7 | (ØP ® Ø(P Ú P)) | modus ponens with 1, 6 |
8 | (ØQ ® Ø(Q Ú Q)) | replace proposition variable P by Q in 7 |
9 | (ØØP ® Ø(ØP Ú ØP)) | replace proposition variable Q by ØP in 8 |
10 | (P ® Ø(ØP Ú ØP)) | elementary equivalence in 9 at 1 of hilb6 with hilb5 |
11 | (P ® (P Ù P)) | reverse abbreviation and in 10 at occurence 1 |
qed |
Absorbtion of identical preconditions (first direction):
16. Proposition
((P ® (P ® Q)) ® (P ® Q)) (hilb33)
1 | ((P ® (Q ® A)) ® ((P Ù Q) ® A)) | add proposition hilb29 |
2 | ((P ® (Q ® B)) ® ((P Ù Q) ® B)) | replace proposition variable A by B in 1 |
3 | ((P ® (C ® B)) ® ((P Ù C) ® B)) | replace proposition variable Q by C in 2 |
4 | ((D ® (C ® B)) ® ((D Ù C) ® B)) | replace proposition variable P by D in 3 |
5 | ((D ® (C ® Q)) ® ((D Ù C) ® Q)) | replace proposition variable B by Q in 4 |
6 | ((D ® (P ® Q)) ® ((D Ù P) ® Q)) | replace proposition variable C by P in 5 |
7 | ((P ® (P ® Q)) ® ((P Ù P) ® Q)) | replace proposition variable D by P in 6 |
8 | ((P ® (P ® Q)) ® (P ® Q)) | elementary equivalence in 7 at 1 of hilb31 with hilb32 |
qed |
Absorbtion of identical preconditions (second direction):
17. Proposition
((P ® Q) ® (P ® (P ® Q))) (hilb34)
1 | (((P Ù Q) ® A) ® (P ® (Q ® A))) | add proposition hilb30 |
2 | (((P Ù Q) ® B) ® (P ® (Q ® B))) | replace proposition variable A by B in 1 |
3 | (((P Ù C) ® B) ® (P ® (C ® B))) | replace proposition variable Q by C in 2 |
4 | (((D Ù C) ® B) ® (D ® (C ® B))) | replace proposition variable P by D in 3 |
5 | (((D Ù C) ® Q) ® (D ® (C ® Q))) | replace proposition variable B by Q in 4 |
6 | (((D Ù P) ® Q) ® (D ® (P ® Q))) | replace proposition variable C by P in 5 |
7 | (((P Ù P) ® Q) ® (P ® (P ® Q))) | replace proposition variable D by P in 6 |
8 | ((P ® Q) ® (P ® (P ® Q))) | elementary equivalence in 7 at 1 of hilb31 with hilb32 |
qed |