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92 lines
3.1 KiB
92 lines
3.1 KiB
First Conflict in Step 6: \\
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\begin{conflictgraph}
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\node[base node] (notA) {$\lnot a$};
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\node[base node] (notB) [below of=notA] {$\lnot b$};
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\node[base node] (C) [right of=notB] {$c$};
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\node[base node] (notE) [above right of=C] {$\lnot e$};
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\node[base node] (D) [above right of=notE] {$d$};
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\node[base node] (notD) [below right of=notE] {$\lnot d$};
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\node[base node] (bot) [below right of=D] {$\bot$};
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\path[]
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(notA) edge [] node {$6$} (notE)
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(notB) edge [] node {$2$} (C)
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(C) edge [] node {$6$} (notE)
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(notE) edge [] node {$4$} (D)
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(notE) edge [] node {$5$} (notD)
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(notD) edge [] node {} (bot)
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(D) edge [] node {} (bot);
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\end{conflictgraph}
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\begin{prooftree}
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\AxiomC{$4. \; \lnot d \lor e$}
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\AxiomC{$5. \; d \lor e$}
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\BinaryInfC{$e$}
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\AxiomC{$6. \; a \lor \lnot c \lor \lnot e$}
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\BinaryInfC{$a \lor \lnot c$}
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\AxiomC{$2. \; b \lor c$}
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\BinaryInfC{$ a \lor b$}
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\end{prooftree}
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\vspace{1cm}
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Second Conflict in Step 11: \\
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\begin{conflictgraph}
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\node[base node] (notA) {$\lnot a$};
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\node[base node] (B) [right of=notA] {$b$};
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\node[base node] (notD) [right of=B] {$\lnot d$};
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\node[base node] (E) [right of=notD] {$e$};
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\node[base node] (C) [below of=E] {$c$};
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\node[base node] (notE) [right of=C] {$\lnot e$};
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\node[base node] (bot) [above right of=notE] {$\bot$};
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\path[]
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(notA) edge [] node {$8$} (B)
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(B) edge [] node {$3$} (notD)
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(notD) edge [] node {$5$} (E)
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(notD) edge [] node {$7$} (C)
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(C) edge [] node {$6$} (notE)
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(notE) edge [] node {} (bot)
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(E) edge [] node {} (bot);
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\end{conflictgraph}
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\begin{prooftree}
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\AxiomC{$6. \; a \lor \lnot c \lor \lnot e$}
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\AxiomC{$7. \; \lnot b \lor c \lor d$}
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\BinaryInfC{$a \lor \lnot b \lor d \lor \lnot e$}
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\AxiomC{$5. \; d \lor e$}
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\BinaryInfC{$a \lor \lnot b \lor d$}
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\AxiomC{$3. \; \lnot b \lor \lnot d$}
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\BinaryInfC{$ a \lor \lnot b$}
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\AxiomC{$8. \; a \lor b$}
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\BinaryInfC{$a$}
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\end{prooftree}
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\vspace{1cm}
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Second Conflict in Step 16: \\
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\begin{conflictgraph}
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\node (0) {};
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\node[base node] (A) [right of=0] {$a$};
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\node[base node] (notC) [right of=A] {$\lnot c$};
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\node[base node] (B) [right of=notC] {$b$};
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\node[base node] (notD) [right of=B] {$\lnot d$};
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\node[base node] (D) [below of=notD] {$d$};
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\node[base node] (bot) [above right of=D] {$\bot$};
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\path[]
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(0) edge [] node {$9$} (A)
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(A) edge [] node {$1$} (notC)
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(notC) edge [] node {$2$} (B)
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(B) edge [] node {$3$} (D)
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(B) edge [] node {$7$} (notD)
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(notD) edge [] node {} (bot)
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(D) edge [] node {} (bot);
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\end{conflictgraph}
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\begin{prooftree}
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\AxiomC{$7. \; \lnot b \lor c \lor d$}
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\AxiomC{$3. \; \lnot b \lor \lnot d$}
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\BinaryInfC{$\lnot b \lor c$}
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\AxiomC{$2. \; b \lor c$}
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\BinaryInfC{$c$}
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\AxiomC{$1. \; \lnot a \lor \lnot c$}
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\BinaryInfC{$\lnot a$}
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\AxiomC{$9. \; a$}
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\BinaryInfC{$\bot$}
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\end{prooftree}
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