[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 8 from proof prepost-gc-referenced-nodes (previous step)

# lax = # gndx, lax.to-adr(lnum) = ax, jvalid0 ↔ boolvar0, gstore2[lnum] = groupblock(gndx, lax, m), err = ESUCCESS, fi0 = ri, gstore = gstore1, gstore0 = gstore2, jro = ginval, ndx = [], nx = nx0, p_os = ns, pmaxino = maxino, inv(groupblock(gndx, lax, m)), inv(gstore2 \ ginval), ri-cons(ri, nodes(gstore2 ↓, ginval)), lnum ∈ gstore2 ⊦ ⟪while err = ESUCCESS ∧ ax ≠ [] do {/* (prepost-gc-referenced-nodes : invariant (err = ESUCCESS ∨ ⌊err⌋ ) ∧ ( err = ESUCCESS → (∃ ax0, gndx0. ax` = ax0 + ax ∧ gndx` = gndx0 + gndx ∧ # ax` = # gndx` ∧ # ax = # gndx ∧ ndx = nodes(gstore0 ↓, jro)[referenced-adrs(ax0, fi0)])) wfbound (err = ESUCCESS ⊃ # ax;0)) */; let jvalid = ?, adr0 = ? in {let adr = ax.head, gnd = gndx.head in let key = gnd.nd.key in {index_index_lookup#(key, fi0, fi, jro, gstore0, gstore, nx, pmaxino, p_os, jvalid0; jvalid, adr0; err); if err = ESUCCESS ∧ jvalid ∧ adr0 = adr then ndx := ndx + gnd.nd}; if err = ESUCCESS then {ax := ax.tail; gndx := gndx.tail}}}⟫ ((err = ESUCCESS ∨ ⌊err⌋ ) ∧ (err = ESUCCESS → ndx = nodes(gstore2 ↓, ginval)[referenced-adrs(lax.to-adr(lnum), ri)])), lnum ∈ ginval

Heuristic annotation applied rule annotation right

And obtained the following new goals

Proof step 9:
# lax = # gndx1, lax.to-adr(lnum) = ax2, jvalid0 ↔ boolvar0, gstore2[lnum] = groupblock(gndx1, lax, m), ax = ax1, err = err0, err = ESUCCESS, err1 = ESUCCESS, fi0 = ri, gstore = gstore1, gstore0 = gstore2, jro = ginval, ndx0 = [], nx = nx0, p_os = ns, pmaxino = maxino, inv(gstore2 \ ginval), inv(groupblock(gndx1, lax, m)), ri-cons(ri, nodes(gstore2 ↓, ginval)), lnum ∈ gstore2, err = ESUCCESS → (∃ ax0, gndx0. ax2 = ax0 + ax ∧ gndx1 = gndx0 + gndx ∧ # ax2 = # gndx1 ∧ # ax = # gndx ∧ ndx = nodes(gstore0 ↓, jro)[referenced-adrs(ax0, fi0)]), err = ESUCCESS ∨ ⌊err⌋ ⊦ ⟪let jvalid = ?, adr0 = ? in {let adr = ax.head, gnd = gndx.head in let key = gnd.nd.key in {index_index_lookup#(key, fi0, fi, jro, gstore0, gstore, nx, pmaxino, p_os, jvalid0; jvalid, adr0; err); if err = ESUCCESS ∧ jvalid ∧ adr0 = adr then ndx := ndx + gnd.nd}; if err = ESUCCESS then {ax := ax.tail; gndx := gndx.tail}}⟫ ( ( (err = ESUCCESS ∨ ⌊err⌋ ) ∧ ( err = ESUCCESS → (∃ ax0, gndx0. ax2 = ax0 + ax ∧ gndx1 = gndx0 + gndx ∧ # ax2 = # gndx1 ∧ # ax = # gndx ∧ ndx = nodes(gstore0 ↓, jro)[referenced-adrs(ax0, fi0)]))) ∧ (err = ESUCCESS ⊃ # ax;0) < (err0 = ESUCCESS ⊃ # ax1;0)), ax = [], lnum ∈ ginval

Rule argument

rule: invariant right arg: bound: (err = ESUCCESS ⊃ # ax;0) inv: (err = ESUCCESS ∨ ⌊err⌋ ) ∧ ( err = ESUCCESS → (∃ ax0, gndx0. ax` = ax0 + ax ∧ gndx` = gndx0 + gndx ∧ # ax` = # gndx` ∧ # ax = # gndx ∧ ndx = nodes(gstore0 ↓, jro)[referenced-adrs(ax0, fi0)])) fmapos: right 1 pos: right 1

Additional information

Used simplifier rules: ⊦ # lax.to-adr(n) = # lax; (s)% ⊦ #(gndx + gndx0) = # gndx + # gndx0; (s)%a: ⊦ (gndx + gndx0) + gndx1 = gndx + gndx0 + gndx1; from specification list[group-store]%fwd-inv-gblock-rem: ⊦ inv(gstore \ ginval) ∧ n ∈ gstore ∧ ¬ n ∈ ginval ∧ gstore[n] = gblock → inv(gblock); (forward) from specification apersistence-base% ⊦ # [] = 0; (s)% ⊦ # gndx = 0 ↔ gndx = []; (s)% ⊦ m + n = m + n0 ↔ n = n0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ ax + [] = ax; (s)%a: ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1; from specification list[address-list-set]% ⊦ ax = ax0 + ax ↔ ax0 = []; (s)% ⊦ gndx = gndx0 + gndx ↔ gndx0 = []; (s)% ⊦ inv(groupblock(gndx, lax, m0)) → (# gndx = # lax ↔ true); (ss)% ⊦ referenced-adrs([], ri) = []; (s)% ⊦ fs[[]] = []; (s) Used formulas: (err = ESUCCESS ∨ ⌊err⌋ ) ∧ ( err = ESUCCESS → (∃ ax0, gndx0. ax` = ax0 + ax ∧ gndx` = gndx0 + gndx ∧ # ax` = # gndx` ∧ # ax = # gndx ∧ ndx = nodes(gstore0 ↓, jro)[referenced-adrs(ax0, fi0)])), (err = ESUCCESS ⊃ # ax;0) = (err = ESUCCESS ⊃ # ax;0)