Metatheory - University of Cambridge

6. Completeness 47
5 A want to use ∨E with line 1
6 ¬B want to use ∨E with line 2
7 ⊥ ⊥I 5, 3
8 ¬C want to use ∨E with line 2
9 ⊥ ⊥I 5, 3
i B want to use ∨E with line 1
i + 1 ¬B want to use ∨E with line 2
i + 2 ⊥ ⊥I i, i + 1
i + 3 ¬C want to use ∨E with line 2
i + 4 ⊥ ⊥I 4, i + 3
Recall that our aim is to end up with a proof of ‘⊥’ on no assumptions. So the
most we can hope for, from each assumption, is that it generates ‘⊥’. When
an assumption does generate ‘⊥’, we will have no further use for it, except in a
potential application of ∨E. So, at this point, we can start to close some of the
gaps in our proof-skeleton, and put some flesh on its bones by adding in line
numbers. I have done this in the above example. I have also added in some
dummy line numbers, lines i through to i + 4, for ease of reference.)
Given what I have just said, we must next apply ∨E as often as we can:
5 A want to use ∨E with line 1
6 ¬B want to use ∨E with line 2
7 ⊥ ⊥I 5, 3
8 ¬C want to use ∨E with line 2
9 ⊥ ⊥I 5, 3
10 ⊥ ∨E 2, 6–7, 8–9
11 B want to use ∨E with line 1
12 ¬B want to use ∨E with line 2
13 ⊥ ⊥I 10, 11
14 ¬C want to use ∨E with line 2
15 ⊥ ⊥I 4, 13
16 ⊥ ∨E 2, 12–13, 14–15