ACL2-PC_colon__colon_EQUIV.html -- ACL2 Version 6.3

### ACL2-PC::EQUIV

(primitive) `` attempt an equality (or congruence-based) substitution

    Major Section:  PROOF-CHECKER-COMMANDS

    Examples:
    (equiv (* x y) 3) -- replace (* x y) by 3 everywhere inside the
                         current subterm, if their equality is among the
                         top-level hypotheses or the governors
    (equiv x t iff)   -- replace x by t everywhere inside the current
                         subterm, where only propositional equivalence
                         needs to be maintained at each occurrence of x

    General form:
    (equiv old new &optional relation)

Substitute new for old everywhere inside the current subterm, provided that either (relation old new) or (relation new old) is among the top-level hypotheses or the governors (possibly by way of backchaining and/or refinement; see below). If relation is `nil` or is not supplied, then it defaults to `equal`. See also the command `=`, which is much more flexible. Note that this command fails if no substitution is actually made.

Remark: No substitution takes place inside explicit values. So for example, the instruction `(equiv 3 x)` will cause `3` to be replaced by `x` if the current subterm is, say, `(* 3 y)`, but not if the current subterm is `(* 4 y)` even though `4 = (1+ 3)`.

The following remarks are quite technical and mostly describe a certain weak form of ``backchaining'' that has been implemented for `equiv` in order to support the `=` command. In fact neither the term `(relation old new)` nor the term `(relation new old)` needs to be explicitly among the current ``assumptions'', i.e., the top-level hypothesis or the governors. Rather, there need only be such an assumption that ``tells us'' `(r old new)` or `(r new old)`, for some equivalence relation `r` that refines `relation`. Here, ``tells us'' means that either one of the indicated terms is among those assumptions, or else there is an assumption that is an implication whose conclusion is one of the indicated terms and whose hypotheses (gathered up by appropriately flattening the first argument of the `implies` term) are all among the current assumptions.