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Proof step 47 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)))), r ≠ null, r0 ≠ null, ((r3 => mknode(r8, a2, r4) * tr(r8, t3) * tr(r4, t4)) * r => mknode(r0, a0, r1) * tr(r1, t1) * tr(r6, t2) * r0 => mknode(r3, a, r6) * 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 apply induction

And obtained the following new goals

Proof step 46:
〈P := P * r => mknode(r0, a0, r1) * tr(r1, t1), r3 := r8, r0 := r3, r := r0, r5 := r0〉〈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((t3 = leaf ⊃ t4; branch(t3 .butleftmost, a2, t4)), a, t2)) * P * true)(H) ∧ a1 = ((t3 = leaf ⊃ a2; t3 .leftmost))), ∀ 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)))), r ≠ null, r0 ≠ null, ((r3 => mknode(r8, a2, r4) * tr(r8, t3) * tr(r4, t4)) * r => mknode(r0, a0, r1) * tr(r1, t1) * tr(r6, t2) * r0 => mknode(r3, a, r6) * 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

[substlist-arg: [t0, r5, r, r0, r3, a, r6, t2, r1, t1, a0, P, H] → [t3, r0, r0, r3, r8, a2, r4, t4, r6, t2, a, P * r => mknode(r0, a0, r1) * tr(r1, t1), H] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ m + n = n + m], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ ¬ m + n < m], [CSIMPSEQ : ⊦ m < n + 1 ↔ ¬ n < m], [CSIMPSEQ : ⊦ m +1 = m + 1], [CSIMPSEQ : ⊦ # branch(t, a, t0) = (# t + # t0) +1], [CSIMPSEQ : ⊦ (P0 * P1) * P2 = P0 * P1 * P2], [CSIMPSEQ : ⊦ P0 * P1 = P1 * P0], [CSIMPSEQ : ⊦ null => nd = ⌜false⌝ ], [CSIMPSEQ : ⊦ ⌜false⌝ = false], [CSIMPSEQ : ⊦ false * P = false], [CSIMPSEQ : ⊦ ¬ false(H)])], [EXTRATERMS : (t3, r0, r0, r3, r8, a2, r4, t4, r6, t2, a, P * r => mknode(r0, a0, r1) * tr(r1, t1), H)]]