proof - Proving non-existence of an infinite inductive value in Coq -


suppose have simple inductive type:

inductive ind : set :=     | ind0 : ind     | ind1 : ind -> ind.

and i'd prove values can't exist. specifically, there can't non-well-founded values: ~exists i, = ind1 i.

 

i've looked around bit on internet , came nothing. able write:

fixpoint depth (i : ind) : nat :=     match     | ind0 => 0     | ind1 i2 => 1 + depth i2     end.  goal ~exists i, = ind1 i. proof.     intro. inversion_clear h.     remember (depth x) d.     induction d.         rewrite h0 in heqd; simpl in heqd. discriminate.         rewrite h0 in heqd; simpl in heqd. injection heqd. assumption. qed.

which works, seems really ugly , non-general.

i don't think there general method proving such goals. experience, has not seemed such problem, however. in case, there simpler proof, using congruence tactic:

inductive ind : type := | ind0 : ind | ind1 : ind -> ind.  goal ~exists i, = ind1 i. proof.   intros [x hx]. induction x; congruence. qed.

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