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Proof step 7 from proof encode (previous step)

buf0 = buf, # vtbl * (ENCODED_NAT_SIZE + ENCODED_VOLID_SIZE) + ENCODED_NAT_SIZE ≤ # buf ⊦ ⟪encode_nat#(# vtbl, 0; buf, n; err); n0 := n; vtbl0 := vtbl; while err = ESUCCESS ∧ vtbl ≠ ∅ do {/* (encode : invariant (err = ESUCCESS ∨ ⌊err⌋ ) ∧ # buf = # buf` ∧ ( err = ESUCCESS → (∃ buf0, buf1, buf2, vtbl0. buf = buf0 + buf1 + buf2 ∧ vtbl` = vtbl ∪ vtbl0 ∧ vtbl ⊥ vtbl0 ∧ # buf0 + # buf1 = n0 ∧ # buf2 ≥ # vtbl * (ENCODED_NAT_SIZE + ENCODED_VOLID_SIZE) ∧ is-encoded(# vtbl`, buf0) ∧ is-encoded-rec(vtbl0, buf1))) wfbound # vtbl) */; choose v with v ∈ vtbl in {encode_volid#(v, n0; buf, n; err); if err = ESUCCESS then {n0 := n0 + n; encode_nat#(vtbl[v], n0; buf, n; err); n0 := n0 + n}; vtbl := vtbl -- v}}⟫ ((err = ESUCCESS ∨ ⌊err⌋ ) ∧ # buf = # buf0 ∧ (err = ESUCCESS → is-encoded(vtbl0, buf)))

User applied rule insert program lemma

And obtained the following new goals

Proof step 8:
n1 + encoding-size(n) ≤ # buf, buf0 = buf ⊦ ⟪encode_nat#(n, n1; buf, n0; err)⟫ ( (err = ESUCCESS ∨ ⌊err⌋ ) ∧ # buf = # buf0 ∧ subarray(buf, 0, n1) = subarray(buf0, 0, n1) ∧ subarray(buf, n1 + encoding-size(n), # buf - n1 + encoding-size(n)) = subarray(buf0, n1 + encoding-size(n), # buf - n1 + encoding-size(n)) ∧ (err = ESUCCESS → is-encoded(n, subarray(buf, n1, encoding-size(n))) ∧ n0 = encoding-size(n)))
Proof step 9:
buf = buf, buf0 = buf, buf1 = buf, n2 = 0, n3 = # vtbl, # vtbl * (ENCODED_NAT_SIZE + ENCODED_VOLID_SIZE) + ENCODED_NAT_SIZE ≤ # buf, 0 + encoding-size(# vtbl) ≤ # buf ⊦ [encode_nat#(# vtbl, 0; buf, n; err)] ( (err = ESUCCESS ∨ ⌊err⌋ ) ∧ # buf = # buf1 ∧ subarray(buf, 0, n2) = subarray(buf1, 0, n2) ∧ subarray(buf, n2 + encoding-size(n3), # buf - n2 + encoding-size(n3)) = subarray(buf1, n2 + encoding-size(n3), # buf - n2 + encoding-size(n3)) ∧ (err = ESUCCESS → is-encoded(n3, subarray(buf, n2, encoding-size(n3))) ∧ n = encoding-size(n3)) → ⟪n0 := n; vtbl0 := vtbl; while err = ESUCCESS ∧ vtbl ≠ ∅ do {/* (encode : invariant (err = ESUCCESS ∨ ⌊err⌋ ) ∧ # buf = # buf` ∧ ( err = ESUCCESS → (∃ buf0, buf1, buf2, vtbl0. buf = buf0 + buf1 + buf2 ∧ vtbl` = vtbl ∪ vtbl0 ∧ vtbl ⊥ vtbl0 ∧ # buf0 + # buf1 = n0 ∧ # buf2 ≥ # vtbl * (ENCODED_NAT_SIZE + ENCODED_VOLID_SIZE) ∧ is-encoded(# vtbl`, buf0) ∧ is-encoded-rec(vtbl0, buf1))) wfbound # vtbl) */; choose v with v ∈ vtbl in {encode_volid#(v, n0; buf, n; err); if err = ESUCCESS then {n0 := n0 + n; encode_nat#(vtbl[v], n0; buf, n; err); n0 := n0 + n}; vtbl := vtbl -- v}}⟫ ((err = ESUCCESS ∨ ⌊err⌋ ) ∧ # buf = # buf0 ∧ (err = ESUCCESS → is-encoded(vtbl0, buf))))

Rule argument

spec: encoding, inst: encoding-nat name: encode-contract seq: n1 + encoding-size(n) ≤ # buf, buf0 = buf ⊦ ⟪encode_nat#(n, n1; buf, n0; err)⟫ ( (err = ESUCCESS ∨ ⌊err⌋ ) ∧ # buf = # buf0 ∧ subarray(buf, 0, n1) = subarray(buf0, 0, n1) ∧ subarray (buf, n1 + encoding-size(n), # buf - n1 + encoding-size(n)) = subarray (buf0, n1 + encoding-size(n), # buf - n1 + encoding-size(n)) ∧ ( err = ESUCCESS → is-encoded(n, subarray(buf, n1, encoding-size(n))) ∧ n0 = encoding-size(n))) subst: [n1, n, n0, buf, err, buf0] → [0, # vtbl, n, buf, err, buf] flags: <all paths>

Additional information

Used simplifier rules: ⊦ encoding-size(n) = ENCODED_NAT_SIZE; (s)% ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m ≤ n ↔ ¬ n < m; (ss)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ ENCODED_NAT_SIZE ≠ 0; (s)% ⊦ ¬ m + n < m; (s)