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869 Here \isa{wf\ r} expresses that the relation~\isa{r} is well-founded. |
869 Here \isa{wf\ r} expresses that the relation~\isa{r} is well-founded. |
870 |
870 |
871 Many familiar induction principles are instances of this rule. |
871 Many familiar induction principles are instances of this rule. |
872 For example, the predecessor relation on the natural numbers |
872 For example, the predecessor relation on the natural numbers |
873 is well-founded; induction over it is mathematical induction. |
873 is well-founded; induction over it is mathematical induction. |
874 The `tail of' relation on lists is well-founded; induction over |
874 The ``tail of'' relation on lists is well-founded; induction over |
875 it is structural induction. |
875 it is structural induction. |
876 |
876 |
877 Well-foundedness can be difficult to show. The various |
877 Well-foundedness can be difficult to show. The various |
878 formulations are all complicated. However, often a relation |
878 formulations are all complicated. However, often a relation |
879 is well-founded by construction. HOL provides |
879 is well-founded by construction. HOL provides |