lean2/tests/lean/run/tree_subterm_pred.lean

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import logic
open eq.ops
inductive tree (A : Type) :=
| leaf : A → tree A
| node : tree A → tree A → tree A
namespace tree
inductive direct_subterm {A : Type} : tree A → tree A → Prop :=
| node_l : Π (l r : tree A), direct_subterm l (node l r)
| node_r : Π (l r : tree A), direct_subterm r (node l r)
definition direct_subterm.wf {A : Type} : well_founded (@direct_subterm A) :=
well_founded.intro (λ t : tree A,
tree.rec_on t
(λ (a : A), acc.intro (leaf a) (λ (s : tree A) (H : direct_subterm s (leaf a)),
have gen : ∀ r : tree A, direct_subterm s r → r = leaf a → acc direct_subterm s, from
λ r H, direct_subterm.rec_on H (λ l r e, tree.no_confusion e) (λ l r e, tree.no_confusion e),
gen (leaf a) H rfl))
(λ (l r : tree A) (ihl : acc direct_subterm l) (ihr : acc direct_subterm r),
acc.intro (node l r) (λ (s : tree A) (H : direct_subterm s (node l r)),
have gen : ∀ n₁ : tree A, direct_subterm s n₁ → node l r = n₁ → acc direct_subterm s, from
λ n₁ H, direct_subterm.rec_on H
(λ (l' r' : tree A) (Heq : node l r = node l' r'), tree.no_confusion Heq (λ leq req, eq.rec_on leq ihl))
(λ (l' r' : tree A) (Heq : node l r = node l' r'), tree.no_confusion Heq (λ leq req, eq.rec_on req ihr)),
gen (node l r) H rfl)))
definition direct_subterm.wf₂ {A : Type} : well_founded (@direct_subterm A) :=
begin
constructor, intro t, induction t,
repeat (constructor; intro y hlt; cases hlt; repeat assumption)
end
definition subterm {A : Type} : tree A → tree A → Prop := tc (@direct_subterm A)
definition subterm.wf {A : Type} : well_founded (@subterm A) :=
tc.wf (@direct_subterm.wf A)
open nat
example : subterm (leaf (2:nat)) (node (leaf 1) (leaf 2)) :=
!tc.base !direct_subterm.node_r
example : subterm (leaf (2:nat)) (node (node (leaf 1) (leaf 2)) (leaf 3)) :=
have s₁ : subterm (leaf 2) (node (leaf 1) (leaf 2)), from
!tc.base !direct_subterm.node_r,
have s₂ : subterm (node (leaf 1) (leaf 2)) (node (node (leaf 1) (leaf 2)) (leaf 3)), from
!tc.base !direct_subterm.node_l,
!tc.trans s₁ s₂
end tree