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Proof step 17 from proof wflex (previous step)

∀ a, b1. (∀ a1, b0. a1 ≪1 a ∨ a1 = a ∧ b0 ≪2 b1 → IHPQ(a1, b0)) → IHPQ(a, b1), ∀ a1. a1 ≪1 a0 → (∀ IHPQ0, b1. (∀ a2, b2. (∀ a3, b0. a3 ≪1 a2 ∨ a3 = a2 ∧ b0 ≪2 b2 → IHPQ0(a3, b0)) → IHPQ0(a2, b2)) → IHPQ0(a1, b1)) ⊦ IHPQ(a0, b0)

User applied rule induction

And obtained the following new goals

Proof step 18:
∀ b. (∀ b0. b0 ≪2 b → IHQ(b0)) → IHQ(b) ⊦ IHQ(b)
Proof step 19:
⊦ ∀ b. (∀ b0. b0 ≪2 b → (∀ IHPQ, a0, b1. b0 = b1 ∧ (∀ a, b1. (∀ a1, b0. a1 ≪1 a ∨ a1 = a ∧ b0 ≪2 b1 → IHPQ(a1, b0)) → IHPQ(a, b1)) ∧ (∀ a1. a1 ≪1 a0 → (∀ IHPQ0, b1. (∀ a2, b2. (∀ a3, b0. a3 ≪1 a2 ∨ a3 = a2 ∧ b0 ≪2 b2 → IHPQ0(a3, b0)) → IHPQ0(a2, b2)) → IHPQ0(a1, b1))) → IHPQ(a0, b1))) → (∀ IHPQ, a0, b0. b = b0 ∧ (∀ a, b1. (∀ a1, b0. a1 ≪1 a ∨ a1 = a ∧ b0 ≪2 b1 → IHPQ(a1, b0)) → IHPQ(a, b1)) ∧ (∀ a1. a1 ≪1 a0 → (∀ IHPQ0, b1. (∀ a2, b2. (∀ a3, b0. a3 ≪1 a2 ∨ a3 = a2 ∧ b0 ≪2 b2 → IHPQ0(a3, b0)) → IHPQ0(a2, b2)) → IHPQ0(a1, b1))) → IHPQ(a0, b0))

Rule argument

spec: wellfounded, inst: lexical-param1 name: wellfounded seq: ∀ b. (∀ b0. b0 ≪2 b → IHQ(b0)) → IHQ(b) ⊦ IHQ(b) subst: [IHQ, b] → [λ b. ∀ IHPQ, a0, b0. b = b0 ∧ (∀ a, b1. (∀ a1, b0. a1 ≪1 a ∨ a1 = a ∧ b0 ≪2 b1 → IHPQ(a1, b0)) → IHPQ(a, b1)) ∧ (∀ a1. a1 ≪1 a0 → (∀ IHPQ0, b1. (∀ a2, b2. (∀ a3, b0. a3 ≪1 a2 ∨ a3 = a2 ∧ b0 ≪2 b2 → IHPQ0(a3, b0)) → IHPQ0(a2, b2)) → IHPQ0(a1, b1))) → IHPQ(a0, b0), b0]

Additional information

Used terms: λ b. ∀ IHPQ, a0, b0. b = b0 ∧ (∀ a, b1. (∀ a1, b0. a1 ≪1 a ∨ a1 = a ∧ b0 ≪2 b1 → IHPQ(a1, b0)) → IHPQ(a, b1)) ∧ (∀ a1. a1 ≪1 a0 → (∀ IHPQ0, b1. (∀ a2, b2. (∀ a3, b0. a3 ≪1 a2 ∨ a3 = a2 ∧ b0 ≪2 b2 → IHPQ0(a3, b0)) → IHPQ0(a2, b2)) → IHPQ0(a1, b1))) → IHPQ(a0, b0), b0