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\DOC IMP_TRANS
\TYPE {IMP_TRANS : thm -> thm -> thm}
\SYNOPSIS
Implements the transitivity of implication.
\KEYWORDS
rule, implication, transitivity.
\DESCRIBE
When applied to theorems {A1 |- t1 ==> t2} and {A2 |- t2 ==> t3},
the inference rule {IMP_TRANS} returns the theorem {A1 u A2 |- t1 ==> t3}.
{
A1 |- t1 ==> t2 A2 |- t2 ==> t3
----------------------------------- IMP_TRANS
A1 u A2 |- t1 ==> t3
}
\FAILURE
Fails unless the theorems are both implicative, with the consequent of the
first being the same as the antecedent of the second (up to alpha-conversion).
\EXAMPLE
{
# let th1 = TAUT `p /\ q /\ r ==> p /\ q`
and th2 = TAUT `p /\ q ==> p`;;
val th1 : thm = |- p /\ q /\ r ==> p /\ q
val th2 : thm = |- p /\ q ==> p
# IMP_TRANS th1 th2;;
val it : thm = |- p /\ q /\ r ==> p
}
\SEEALSO
IMP_ANTISYM_RULE, SYM, TRANS.
\ENDDOC
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