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c0a5a79 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 | # AlienLogic: the public demo world
The proof format, the standard rules, and the reference manual are those of the evaluation world (`explorationbench/sandboxes/logic/engine.py`). The demo world adds three side conditions; the evaluation world adds 24 of its own, and no demo condition is one of them.
## Side conditions
| Id | Name | Condition |
|---|---|---|
| X01 | `AND_E_LEFT_ONLY` | AND_E2 is disabled: from AND(φ, ψ) only the left conjunct φ can be extracted, with AND_E1. |
| X02 | `IMPL_E_MAJOR_FIRST` | IMPL_E must cite the implication line first and its antecedent second: [IMPL_E i, j] where line i is IMPL(φ, ψ) and line j is φ. |
| X03 | `BOT_E_TOP_LEVEL_ONLY` | BOT_E is refused inside an assumption box; it is only legal at indent 0. |
## Fixed worked examples
Three accepted proofs and two refused ones, with the verdict the checker returns. A refusal carries only an opaque reason id.
**S01.** Modus Ponens, implication cited first. Verdict: ACCEPTED.
```
PROOF
premises: IMPL(p, q), p
goal: q
1. IMPL(p, q) [premise]
2. p [premise]
3. q [IMPL_E 1, 2]
END
```
**S02.** Left projection of a conjunction. Verdict: ACCEPTED.
```
PROOF
premises: AND(p, q)
goal: p
1. AND(p, q) [premise]
2. p [AND_E1 1]
END
```
**S03.** Explosion at the top level. Verdict: ACCEPTED.
```
PROOF
premises: p, NOT(p)
goal: q
1. p [premise]
2. NOT(p) [premise]
3. BOT [NOT_E 1, 2]
4. q [BOT_E 3]
END
```
**R01.** 用 AND_E2 取出右侧合取项。 Verdict: ALIEN_AXIOM_DISABLED / X01.
```
PROOF
premises: AND(p, q)
goal: q
1. AND(p, q) [premise]
2. q [AND_E2 1]
END
```
**R02.** IMPL_E 先引用前件，再引用蕴含式。 Verdict: ALIEN_SIDE_CONDITION / X02.
```
PROOF
premises: p, IMPL(p, q)
goal: q
1. p [premise]
2. IMPL(p, q) [premise]
3. q [IMPL_E 1, 2]
END
```
## Held-out theorems
### P01 · sanity
Premises: `a, b`. Goal: `AND(a, b)`. Built around: none.
Reference proof:
```
PROOF
premises: a, b
goal: AND(a, b)
1. a [premise]
2. b [premise]
3. AND(a, b) [AND_I 1, 2]
END
```
### P02 · single_rule
Premises: `r, IMPL(r, t)`. Goal: `t`. Built around: IMPL_E_MAJOR_FIRST.
Reference proof:
```
PROOF
premises: r, IMPL(r, t)
goal: t
1. r [premise]
2. IMPL(r, t) [premise]
3. t [IMPL_E 2, 1]
END
```
The standard-calculus proof below is refused here (X02):
```
PROOF
premises: r, IMPL(r, t)
goal: t
1. r [premise]
2. IMPL(r, t) [premise]
3. t [IMPL_E 1, 2]
END
```
### P03 · single_rule
Premises: `NOT(p)`. Goal: `IMPL(p, q)`. Built around: BOT_E_TOP_LEVEL_ONLY.
Reference proof:
```
PROOF
premises: NOT(p)
goal: IMPL(p, q)
1. NOT(p) [premise]
2. | p [assume]
3. | | NOT(q) [assume]
4. | | BOT [NOT_E 2, 1]
5. | NOT(NOT(q)) [NOT_I 3-4]
6. | q [DNE 5]
7. IMPL(p, q) [IMPL_I 2-6]
END
```
The standard-calculus proof below is refused here (X03):
```
PROOF
premises: NOT(p)
goal: IMPL(p, q)
1. NOT(p) [premise]
2. | p [assume]
3. | BOT [NOT_E 2, 1]
4. | q [BOT_E 3]
5. IMPL(p, q) [IMPL_I 2-4]
END
```
### P04 · two_rule
Premises: `AND(s, t), IMPL(s, t)`. Goal: `t`. Built around: AND_E_LEFT_ONLY, IMPL_E_MAJOR_FIRST.
Reference proof:
```
PROOF
premises: AND(s, t), IMPL(s, t)
goal: t
1. AND(s, t) [premise]
2. IMPL(s, t) [premise]
3. s [AND_E1 1]
4. t [IMPL_E 2, 3]
END
```
The standard-calculus proof below is refused here (X01):
```
PROOF
premises: AND(s, t), IMPL(s, t)
goal: t
1. AND(s, t) [premise]
2. IMPL(s, t) [premise]
3. t [AND_E2 1]
END
```
### U01 · unprovable
Premises: `AND(p, q)`. Goal: `q`. Built around: AND_E_LEFT_ONLY.
Unprovable. The correct answer is `ANSWER: UNPROVABLE`. Certificate: with AND_E2 gone every remaining rule is sound when AND(φ, ψ) is read as φ, and under that reading the assignment {'p': True, 'q': False} makes the premises true and the goal false.
A naive attempt the checker refuses:
```
PROOF
premises: AND(p, q)
goal: q
1. AND(p, q) [premise]
2. q [AND_E2 1]
END
```
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