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Proof step 8 from proof induction-rev-list (previous step)

φ([]), ∀ gndx0, gnd. φ(gndx0) → φ(gndx0 + gnd '), ∀ gndx1, φ. # gndx1 < #(gndx0 + gnd ') → φ([]) ∧ (∀ gndx0, gnd. φ(gndx0) → φ(gndx0 + gnd ')) → φ(gndx1) ⊦ φ(gndx0 + gnd ')

User applied rule apply induction

And obtained the following new goals

Proof step 9:
φ([]), φ(gndx0), ∀ gndx0, gnd. φ(gndx0) → φ(gndx0 + gnd '), ∀ gndx1, φ. # gndx1 < #(gndx0 + gnd ') → φ([]) ∧ (∀ gndx0, gnd. φ(gndx0) → φ(gndx0 + gnd ')) → φ(gndx1) ⊦ φ(gndx0 + gnd ')

Rule argument

subst: [gndx1, φ] → [gndx0, φ]

Additional information

Used simplifier rules: ⊦ #(gndx + gndx0) = # gndx + # gndx0; (s)% ⊦ # gnd ' = 0 +1; (s)%a: ⊦ (gndx + gndx0) + gndx1 = gndx + gndx0 + gndx1; from specification list[group-store]% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ ¬ n < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: gndx0, φ