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Proof step 49 from proof program+leftmost (previous step)

∀ t0, r5, r, r0, r3, a, r6, t2, r1, t1, a0, P, H. # t0 < # branch(t3, a2, t4) → ( r5 = r ∧ r0 ≠ null ∧ r ≠ null ∧ (r0 => mknode(r3, a, r6) * tr(r3, t0) * tr(r6, t2) * tr(r1, t1) * r => mknode(r0, a0, r1) * P)(H) → 〈while r3 ≠ null do { r5 := r0 ; r0 := r3 ; r3 := H[r0] .left }〉〈a1 := H[r0] .val ; r3 := H[r0] .right ; H[r5] .left := r3〉 ((tr(r, branch((t0 = leaf ⊃ t2; branch(t0 .butleftmost, a, t2)), a0, t1)) * P * true)(H) ∧ a1 = ((t0 = leaf ⊃ a; t0 .leftmost)))), H[r3] = mknode(r8, a3, r9), r0 ≠ null, r ≠ null, (r0 => mknode(r3, a, r6) * tr(r3, branch(t3, a2, t4)) * tr(r6, t2) * tr(r1, t1) * r => mknode(r0, a0, r1) * P)(H) ⊦ 〈while r8 ≠ null do { r0 := r3 ; r3 := r8 ; r8 := H[r3] .left }〉〈a1 := H[r3] .val ; r8 := H[r3] .right ; H[r0] .left := r8〉 ( (tr(r, branch(branch((t3 = leaf ⊃ t4; branch(t3 .butleftmost, a2, t4)), a, t2), a0, t1)) * P * true)(H) ∧ a1 = ((t3 = leaf ⊃ a2; t3 .leftmost)))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 28:
⊦ tr(r, branch(t0, a, t1)) = _∃ λ r0, r1. r => mknode(r0, a, r1) * tr(r0, t0) * tr(r1, t1)
Proof step 48:
∀ t0, r5, r, r0, r3, a, r6, t2, r1, t1, a0, P, H. # t0 < # branch(t3, a2, t4) → ( r5 = r ∧ r0 ≠ null ∧ r ≠ null ∧ (r0 => mknode(r3, a, r6) * tr(r3, t0) * tr(r6, t2) * tr(r1, t1) * r => mknode(r0, a0, r1) * P)(H) → 〈while r3 ≠ null do { r5 := r0 ; r0 := r3 ; r3 := H[r0] .left }〉〈a1 := H[r0] .val ; r3 := H[r0] .right ; H[r5] .left := r3〉 ((tr(r, branch((t0 = leaf ⊃ t2; branch(t0 .butleftmost, a, t2)), a0, t1)) * P * true)(H) ∧ a1 = ((t0 = leaf ⊃ a; t0 .leftmost)))), H[r3] = mknode(r8, a3, r9), r0 ≠ null, r ≠ null, (r0 => mknode(r3, a, r6) * _∃ λ r0, r1. r3 => mknode(r0, a2, r1) * tr(r0, t3) * tr(r1, t4) * tr(r6, t2) * tr(r1, t1) * r => mknode(r0, a0, r1) * P)(H) ⊦ 〈while r8 ≠ null do { r0 := r3 ; r3 := r8 ; r8 := H[r3] .left }〉〈a1 := H[r3] .val ; r8 := H[r3] .right ; H[r0] .left := r8〉 ( (tr(r, branch(branch((t3 = leaf ⊃ t4; branch(t3 .butleftmost, a2, t4)), a, t2), a0, t1)) * P * true)(H) ∧ a1 = ((t3 = leaf ⊃ a2; t3 .leftmost)))

Rule argument

[x0lemmaarg: spec-name: linked-tree instance-name: lemma-name: tr-rec antecedent: succedent: tr(r, branch(t0, a, t1)) = _∃ λ r0, r1. r => mknode(r0, a, r1) * tr(r0, t0) * tr(r1, t1) substlist: [t1, a, t0, r] → [t4, a2, t3, r3] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((0, 0, 4, 0, 1, 1, 1, 0)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (t4, a2, t3, r3)]]