NOR is not Associative/Proof 1

Theorem

$p \downarrow \paren {q \downarrow r} \not \vdash \paren {p \downarrow q} \downarrow r$

Proof

By the tableau method of natural deduction:

$\neg \paren {p \downarrow \paren {q \downarrow r} \implies \paren {p \downarrow q} \downarrow r} $
Line	Pool	Formula	Rule	Depends upon	Notes
1	1	$\neg p \land r$	Assumption	(None)
2	1	$\neg p$	Rule of Simplification: $\land \EE_1$	1
3	1	$r$	Rule of Simplification: $\land \EE_2$	1
4	1	$q \lor r$	Rule of Addition: $\lor \II_2$	3
5	1	$\neg \neg \paren {q \lor r}$	Double Negation Introduction: $\neg \neg \II$	4
6	1	$\neg \paren {q \downarrow r}$	Sequent Introduction	5	Definition of Logical NOR
7	1	$\neg p \land \neg \paren {q \downarrow r}$	Rule of Conjunction: $\land \II$	2, 6
8	1	$\neg \paren {p \lor \paren {q \downarrow r} }$	Sequent Introduction	7	De Morgan's Laws: Conjunction of Negations
9	1	$p \downarrow \paren {q \downarrow r}$	Sequent Introduction	8	Definition of Logical NOR
10	1	$\paren {p \downarrow q} \lor r$	Rule of Addition: $\lor \II_2$	3
11	1	$\neg \neg \paren {\paren {p \downarrow q} \lor r}$	Double Negation Introduction: $\neg \neg \II$	10
12	1	$\neg \paren {\paren {p \downarrow q} \downarrow r}$	Sequent Introduction	11	Definition of Logical NOR
13	1	$\neg \neg \paren {p \downarrow \paren {q \downarrow r} }$	Double Negation Introduction: $\neg \neg \II$	9
14	1	$\paren {\neg \neg \paren {p \downarrow \paren {q \downarrow r} } } \land \paren {\neg \paren {\paren {p \downarrow q} \downarrow r} }$	Rule of Conjunction: $\land \II$	13, 12
15	1	$\neg \paren {\neg \paren {p \downarrow \paren {q \downarrow r} } \lor \paren {\paren {p \downarrow q} \downarrow r} }$	Sequent Introduction	14	De Morgan's Laws: Conjunction of Negations
16	1	$\neg \paren {p \downarrow \paren {q \downarrow r} \implies \paren {p \downarrow q} \downarrow r}$	Sequent Introduction	15	Rule of Material Implication

Taking $p = \bot$ and $r = \top$, we have $\vdash \neg p \land r$, discharging the last assumption.

Hence the result.

$\blacksquare$