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Proof step 3 from proof valid-group-nodes-shorter-prefix (previous step)

valid-group-nodes(prefixn(n0, gndx)) < valid-group-nodes(prefixn(n1, gndx)), valid-group-nodes(prefixn(n0, gndx)) < # prefixn(n1, gndx), n1 ≤ n0 ⊦ gndx = [], n1 = 0

User applied rule induction

And obtained the following new goals

Proof step 4:
φ([]) ∧ (∀ gndx, gnd. φ(gndx) → φ(gndx + gnd)) ⊦ φ(gndx)
Proof step 5:
⊦ (∀ n0, gndx, n1. [] = gndx ∧ valid-group-nodes(prefixn(n0, gndx)) < valid-group-nodes(prefixn(n1, gndx)) ∧ valid-group-nodes(prefixn(n0, gndx)) < # prefixn(n1, gndx) ∧ n1 ≤ n0 → gndx = [] ∨ n1 = 0) ∧ (∀ gndx, gnd. (∀ n0, gndx1, n1. gndx = gndx1 ∧ valid-group-nodes(prefixn(n0, gndx1)) < valid-group-nodes(prefixn(n1, gndx1)) ∧ valid-group-nodes(prefixn(n0, gndx1)) < # prefixn(n1, gndx1) ∧ n1 ≤ n0 → gndx1 = [] ∨ n1 = 0) → (∀ n0, gndx1, n1. gndx + gnd = gndx1 ∧ valid-group-nodes(prefixn(n0, gndx1)) < valid-group-nodes(prefixn(n1, gndx1)) ∧ valid-group-nodes(prefixn(n0, gndx1)) < # prefixn(n1, gndx1) ∧ n1 ≤ n0 → gndx1 = [] ∨ n1 = 0))

Rule argument

spec: <toplevel> name: induction-rev-list seq: φ([]) ∧ (∀ gndx, gnd. φ(gndx) → φ(gndx + gnd)) ⊦ φ(gndx) subst: [φ, gndx] → [λ gndx0. ∀ n0, gndx, n1. gndx0 = gndx ∧ valid-group-nodes(prefixn(n0, gndx)) < valid-group-nodes(prefixn(n1, gndx)) ∧ valid-group-nodes(prefixn(n0, gndx)) < # prefixn(n1, gndx) ∧ n1 ≤ n0 → gndx = [] ∨ n1 = 0, gndx]

Additional information

Used terms: λ gndx0. ∀ n0, gndx, n1. gndx0 = gndx ∧ valid-group-nodes(prefixn(n0, gndx)) < valid-group-nodes(prefixn(n1, gndx)) ∧ valid-group-nodes(prefixn(n0, gndx)) < # prefixn(n1, gndx) ∧ n1 ≤ n0 → gndx = [] ∨ n1 = 0, gndx