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Proof step 2 from proof valid-group-nodes-drop-superfluous-prefixn (previous step)

valid-group-nodes(gndx) ≤ n ⊦ valid-group-nodes(prefixn(n, gndx)) = valid-group-nodes(gndx)

User applied rule induction

And obtained the following new goals

Proof step 3:
φ([]) ∧ (∀ gndx, gnd. φ(gndx) → φ(gndx + gnd)) ⊦ φ(gndx)
Proof step 4:
⊦ (∀ gndx, n. [] = gndx ∧ valid-group-nodes(gndx) ≤ n → valid-group-nodes(prefixn(n, gndx)) = valid-group-nodes(gndx)) ∧ (∀ gndx, gnd. (∀ gndx1, n. gndx = gndx1 ∧ valid-group-nodes(gndx1) ≤ n → valid-group-nodes(prefixn(n, gndx1)) = valid-group-nodes(gndx1)) → (∀ gndx1, n. gndx + gnd = gndx1 ∧ valid-group-nodes(gndx1) ≤ n → valid-group-nodes(prefixn(n, gndx1)) = valid-group-nodes(gndx1)))

Rule argument

spec: <toplevel> name: induction-rev-list seq: φ([]) ∧ (∀ gndx, gnd. φ(gndx) → φ(gndx + gnd)) ⊦ φ(gndx) subst: [φ, gndx] → [λ gndx0. ∀ gndx, n. gndx0 = gndx ∧ valid-group-nodes(gndx) ≤ n → valid-group-nodes(prefixn(n, gndx)) = valid-group-nodes(gndx), gndx]

Additional information

Used terms: λ gndx0. ∀ gndx, n. gndx0 = gndx ∧ valid-group-nodes(gndx) ≤ n → valid-group-nodes(prefixn(n, gndx)) = valid-group-nodes(gndx), gndx