Metamath Proof Explorer

Theorem a1ii

Description: (_Note_: This inference rule and the previous one, idi , will normally never appear in a completed proof.)

This is a technical inference to assist proof development. It provides a temporary way to add an independent subproof to a proof under development, for later assignment to a normal proof step.

The Metamath (Metamath-exe) program Proof Assistant requires proofs to be developed backwards from the conclusion with no gaps, and it has no mechanism that lets the user work on isolated subproofs. This inference provides a workaround for this limitation. It can be inserted at any point in a proof to allow an independent subproof to be developed on the side, for later use as part of the final proof.

Instructions:

  1. Assign this inference to any unknown step in the proof. Typically, the last unknown step is the most convenient, since MM-PA> ASSIGN LAST can be used. This step will be replicated in hypothesis a1ii.1, from where the development of the main proof can continue.
  2. Develop the independent subproof backwards from hypothesis a1ii.2. If desired, use a MM-PA> LET STEP command to pre-assign the conclusion of the independent subproof to a1ii.2.
  3. After the independent subproof is complete, use MM-PA> IMPROVE ALL to assign it automatically to an unknown step in the main proof that matches it.
  4. After the entire proof is complete, use MM-PA> MINIMIZE_WITH * to clean up (discard) all a1ii references automatically.

This can also be used to apply subproofs from other theorems. In step 2, simply assign the theorem to a1ii.2, and run MM-PA> EXPAND <theorem> to "import" a subproof from another theorem.

This inference was originally designed to assist importing partially completed Proof Worksheets from the mmj2 Proof Assistant GUI, but it can also be useful on its own. Interestingly, no axioms are required for its proof. It is the inference associated with a1i . (Contributed by NM, 7-Feb-2006) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses a1ii.1 
|- ph
a1ii.2 
|- ps
Assertion a1ii 
|- ph

Proof

Step Hyp Ref Expression
1 a1ii.1 
 |-  ph
2 a1ii.2 
 |-  ps