| # AlienLogic: the public demo world |
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| 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. |
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| ## Side conditions |
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| | 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 |
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| 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. |
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|
| ``` |
| 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 |
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| Premises: `a, b`. Goal: `AND(a, b)`. Built around: none. |
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| Reference proof: |
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| ``` |
| 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 |
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| Premises: `r, IMPL(r, t)`. Goal: `t`. Built around: IMPL_E_MAJOR_FIRST. |
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| Reference proof: |
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| ``` |
| 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 |
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| Premises: `NOT(p)`. Goal: `IMPL(p, q)`. Built around: BOT_E_TOP_LEVEL_ONLY. |
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| Reference proof: |
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| ``` |
| 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): |
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|
| ``` |
| 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 |
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| Premises: `AND(s, t), IMPL(s, t)`. Goal: `t`. Built around: AND_E_LEFT_ONLY, IMPL_E_MAJOR_FIRST. |
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| 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 |
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| Premises: `AND(p, q)`. Goal: `q`. Built around: AND_E_LEFT_ONLY. |
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| 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. |
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| A naive attempt the checker refuses: |
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| ``` |
| PROOF |
| premises: AND(p, q) |
| goal: q |
| 1. AND(p, q) [premise] |
| 2. q [AND_E2 1] |
| END |
| ``` |
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