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Mirrors > Home > ILE Home > Th. List > df-sb | Unicode version |
Description: Define proper
substitution. Remark 9.1 in [Megill] p. 447 (p.
15 of the
preprint). For our notation, we use ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
Our notation was introduced in Haskell B. Curry's Foundations of
Mathematical Logic (1977), p. 316 and is frequently used in textbooks
of
lambda calculus and combinatory logic. This notation improves the common
but ambiguous notation, " In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 1721, sbcom2 1863 and sbid2v 1872).
Note that our definition is valid even when
When
In classical logic, another possible definition is
There are no restrictions on any of the variables, including what
variables may occur in wff |
Ref | Expression |
---|---|
df-sb |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph |
. . 3
![]() ![]() | |
2 | vx |
. . 3
![]() ![]() | |
3 | vy |
. . 3
![]() ![]() | |
4 | 1, 2, 3 | wsb 1645 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | 2, 3 | weq 1392 |
. . . 4
![]() ![]() ![]() ![]() |
6 | 5, 1 | wi 4 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | 5, 1 | wa 97 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | 7, 2 | wex 1381 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | 6, 8 | wa 97 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | 4, 9 | wb 98 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
This definition is referenced by: sbimi 1647 sb1 1649 sb2 1650 sbequ1 1651 sbequ2 1652 drsb1 1680 spsbim 1724 sbequ8 1727 sbidm 1731 sb6 1766 hbsbv 1817 |
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