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| 1 | +definition: forall I$ J$ (filled2(I$,J$) <-> |
| 2 | + place(I$,J$,2) or place(I$-1,J$,2) or place(I$-2,J$,2)). |
| 3 | + |
| 4 | +definition: forall I$ J$ (filled3(I$,J$) <-> |
| 5 | + place(I$,J$,3) or place(I$,J$-1,3) or place(I$,J$-2,3)). |
| 6 | + |
| 7 | +lemma(forward): |
| 8 | +filled2(I$,J$) -> h(J$,I$) or h(J$,I$-1) or h(J$,I$-2). |
| 9 | + |
| 10 | +lemma(forward): |
| 11 | +filled2(I$,J$) -> covered(J$,I$). |
| 12 | + |
| 13 | +lemma(forward): |
| 14 | +filled3(I$,J$) -> v(J$,I$) or v(J$-1,I$) or v(J$-2,I$). |
| 15 | + |
| 16 | +lemma(forward): |
| 17 | +filled3(I$,J$) -> covered(J$,I$). |
| 18 | + |
| 19 | +lemma(forward): |
| 20 | +filled2(I$,J$) -> h(J$,I$) or h(J$,I$-1) or h(J$,I$-2). |
| 21 | + |
| 22 | +lemma(forward): |
| 23 | +filled2(I$,J$) -> covered(J$,I$). |
| 24 | + |
| 25 | +lemma(forward): |
| 26 | + filled(I$,J$) -> |
| 27 | + place(I$,J$,1) or filled2(I$,J$) or filled3(I$,J$). |
| 28 | + |
| 29 | +lemma(forward): |
| 30 | + square(I$,J$) -> |
| 31 | + place(I$,J$,1) or filled2(I$,J$) or filled3(I$,J$). |
| 32 | + |
| 33 | +lemma(forward): |
| 34 | + square(I$,J$) -> place(I$,J$,1) or covered(J$,I$). |
| 35 | + |
| 36 | + |
| 37 | + |
| 38 | +lemma(backward): |
| 39 | + place(V1, V2, V3) -> |
| 40 | + V3 = 1 and square(V2, V1) and not covered(V2, V1) or |
| 41 | + V3 = 2 and h(V2, V1) or |
| 42 | + V3 = 3 and v(V2, V1). |
| 43 | + |
| 44 | +lemma(backward): |
| 45 | + not(h(R,C) and v(R,C)). |
| 46 | + |
| 47 | +lemma(backward): |
| 48 | + not(h(R$,C$) and h(R$,C$+I$) and 1 <= I$ <= 2). |
| 49 | + |
| 50 | +lemma(backward): |
| 51 | + not(h(R$,C$) and v(R$,C$+I$) and 0 <= I$ <= 2). |
| 52 | + |
| 53 | +lemma(backward): |
| 54 | + not(v(R$,C$) and h(R$+I$,C$) and 0 <= I$ <= 2). |
| 55 | + |
| 56 | +lemma(backward): |
| 57 | + not(v(R$,C$) and h(R$+I$,C$-J$) and 0 <= J$ <= 2 and 1 <= I$ <= 2). |
| 58 | + |
| 59 | +lemma(backward): |
| 60 | + square(I$,J$) -> place(I$,J$,1) or filled2(I$,J$) or filled3(I$,J$). |
| 61 | + |
| 62 | +lemma(backward): filled2(I$,J$) -> filled(I$,J$). |
| 63 | + |
| 64 | +lemma(backward): filled3(I$,J$) -> filled(I$,J$). |
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