# The proof · Every link checked by a solver

> Eighty-eight checks by a solver, each with the result expected.

Gavin Robert McGowen · interstellarblends.com. A study of the texts, not legal advice.

Canonical: https://clause5loop.com/proof/

## The scalpel

- Logic by machine: the Z3 solver, 88 checks.
- Each conclusion follows. Each fails when one premise is dropped. Each rival is tested for consistency with the words.
- All 88 come back as they should.

## In plain words

A logic program checked every step: 88 checks, each with the expected result.

## The question

**How was the logic checked?**

With the Z3 solver: every conclusion follows from its premises and fails when any one is dropped, and every rival construction is tested for consistency with the words. All 88 checks give the expected result.

## The study, as written

Every link in the study is checked by a solver, Z3, and so is every claim that a rival construction exists. The solver runs three kinds of check. A conclusion follows when its premises, joined to the conclusion's denial, cannot all be true. A control drops one premise at a time and confirms the conclusion then fails, so no premise listed is idle. A rival is consistent with the words when the solver finds them true together, and that is how the claim of completeness is tested. All 88 checks give the result expected of them.

### What each conclusion needs

| Conclusion | What it needs (routes joined by *or*) |
| --- | --- |
| AC1 | (A4, AU4) |
| AC2 | (AC1, AU5, AU7, A3) |
| AC3 | (A1, A2, AU2) |
| AC4 | (AU9, AU10, W-orig, F-orig) |
| AC4-refuted | (N4c, W-orig, F-orig) |
| AC5 | (AC3, AU11, A8) |
| BC1 | (B6, B7) |
| BC2 | (BU5, F-cpi) |
| BC3 | (BC2, BU6, BU1, B8) |
| binary | (A4, AU4, AU5, AU7) *or* (AU12) |
| BC4 | (BU2, F-debt) *or* (BU3, BU8) |
| BC5 | (BC3) *or* (BC4) *or* (BU7) |

### The checks, one by one

|  | Check | Result | Expected |
| --- | --- | --- | --- |
| AL1 | route 1: the conclusion follows from A4, AU4 | true | true |
| AL1 | route 1: control, without A4 | false | false |
| AL1 | route 1: control, without AU4 | false | false |
| AL1 | A5 (a purely nominal increase is no accession) follows from AU4 and the definition | true | true |
| AL1 | with the rival measure (an accession counted in dollars), the same part is income | true | true |
| AL1 | A4 alone, with the definition, decides neither way (not income) | false | false |
| AL1 | A4 alone, with the definition, decides neither way (income) | false | false |
| AL2 | route 1: the conclusion follows from AC1, AU5, AU7, A3 | true | true |
| AL2 | route 1: control, without AC1 | false | false |
| AL2 | route 1: control, without AU5 | false | false |
| AL2 | route 1: control, without AU7 | false | false |
| AL2 | route 1: control, without A3 | false | false |
| AL2 | K-A4 (an excise on the sale, valid without the Amendment) is consistent with AC1 and AU5 when AU7 is not entered | true | true |
| AL2 | K-A4 contradicts AC1 and AU5 once AU7 is entered | false | false |
| AL2 | Q5, a mixed gain: with AC1, the real part r taxable and the inflation part not (K-A5) | true | true |
| AL2 | Q5: without AU5, "deficiency = zero" on the whole gain does not follow from AC1 | false | false |
| AL3 | route 1: the conclusion follows from A1, A2, AU2 | true | true |
| AL3 | route 1: control, without A1 | false | false |
| AL3 | route 1: control, without A2 | false | false |
| AL3 | route 1: control, without AU2 | false | false |
| AL3 | without AU2, the conclusion stands with an actor that is not the House acting by a Bill (stated by 12 U.S.C. § 263) | true | true |
| AL4 | route 1: the conclusion follows from AU9, AU10, W-orig, F-orig | true | true |
| AL4 | route 1: control, without AU9 | false | false |
| AL4 | route 1: control, without AU10 | false | false |
| AL4 | route 1: control, without W-orig | false | false |
| AL4 | route 1: control, without F-orig | false | false |
| AN4 | route 1: the conclusion follows from N4c, W-orig, F-orig | true | true |
| AN4 | route 1: control, without N4c | false | false |
| AN4 | route 1: control, without W-orig | false | false |
| AN4 | route 1: control, without F-orig | false | false |
| AN4 | AU10 contradicts the canon's definition of surplusage, given the clause, the facts and AU9 | false | false |
| AN4 | control: without AU10, the canon, the clause, the facts and AU9 are consistent | true | true |
| AN4 | AU9 contradicts the 1828 meaning of revenue (the produce of the tax that levies it), given who levies | false | false |
| AN4 | control: without the 1828 meaning, AU9 is consistent with who levies | true | true |
| AN4 | the clause as written does not entail AU8 (that whatever raises revenue must be a House Bill) | false | false |
| AN4 | with AU8 entered, the facts and AU9 contradict it: AU8 would make the monetary action a violation | false | false |
| AN4 | the third construction (the law raises the increment) is consistent with the clause as written | true | true |
| AL5 | route 1: the conclusion follows from AC3, AU11, A8 | true | true |
| AL5 | route 1: control, without AC3 | false | false |
| AL5 | route 1: control, without AU11 | false | false |
| AL5 | route 1: control, without A8 | false | false |
| AL5 | without AU11, a statute can authorize it (the reading against: Hampton, Consumers' Research) | true | true |
| AL6 | route 1: the conclusion follows from A4, AU4, AU5, AU7 | true | true |
| AL6 | route 1: control, without A4 | false | false |
| AL6 | route 1: control, without AU4 | false | false |
| AL6 | route 1: control, without AU5 | false | false |
| AL6 | route 1: control, without AU7 | false | false |
| AL6 | route 2: the conclusion follows from AU12 | true | true |
| AL6 | route 2: control, without AU12 | false | false |
| AL6 | A9 tested: the third construction K-A3 is consistent with the words entered as written (A4 granted) | true | true |
| AL6 | with AU4, AU5 and AU7 entered, K-A3 is excluded | false | false |
| AL6 | with AU12 entered, K-A3 is excluded | false | false |
| AL6 | control: without AU4 (and without AU12), K-A3 survives | true | true |
| AL6 | control: without AU5 (and without AU12), K-A3 survives | true | true |
| AL6 | control: without AU7 (and without AU12), K-A3 survives | true | true |
| AL6 | the words permit K-A1 (enforce) | true | true |
| AL6 | the words permit K-A2 (double extraction) | true | true |
| AL6 | the words permit K-A3 (the unit of account) | true | true |
| AL6 | the words permit K-A5 (the real part only: the whole gain is not the inflation part) | true | true |
| BL1 | route 1: the conclusion follows from B6, B7 | true | true |
| BL1 | route 1: control, without B6 | false | false |
| BL1 | route 1: control, without B7 | false | false |
| BL2 | route 1: the conclusion follows from BU5, F-cpi | true | true |
| BL2 | route 1: control, without BU5 | false | false |
| BL2 | route 1: control, without F-cpi | false | false |
| BL3 | route 1: the conclusion follows from BC2, BU6, BU1, B8 | true | true |
| BL3 | route 1: control, without BC2 | false | false |
| BL3 | route 1: control, without BU6 | false | false |
| BL3 | route 1: control, without BU1 | false | false |
| BL3 | route 1: control, without B8 | false | false |
| BL3 | K-B2 (inflation is not a tax) is consistent with the clause, B8 and the transfer | true | true |
| BL4 | route 1: the conclusion follows from BU2, F-debt | true | true |
| BL4 | route 1: control, without BU2 | false | false |
| BL4 | route 1: control, without F-debt | false | false |
| BL4 | route 2: the conclusion follows from BU3, BU8 | true | true |
| BL4 | route 2: control, without BU3 | false | false |
| BL4 | route 2: control, without BU8 | false | false |
| BL4 | BU2 contradicts the borrowing clause ("To borrow Money"), given that the debt is met by borrowing | false | false |
| BL4 | control: without BU2, the borrowing clause and the facts are consistent | true | true |
| BL5 | route 1: the conclusion follows from BC3 | true | true |
| BL5 | route 1: control, without BC3 | false | false |
| BL5 | route 2: the conclusion follows from BC4 | true | true |
| BL5 | route 2: control, without BC4 | false | false |
| BL5 | route 3: the conclusion follows from BU7 | true | true |
| BL5 | route 3: control, without BU7 | false | false |
| BL6 | the words (borrowing granted; no new Bill; the transfer; the House's consent to the framework) permit K-B2: inflation not a tax, no bypass | true | true |
| BL6 | with K-B2 the practice is authorized | true | true |
| BL6 | with BU6 and BU1 entered, K-B2 is excluded | false | false |