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

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Proof step 3 from proof abs-gstore-crash (previous step)

# avol0 = # lpt, # lpt0 = # lpt, dom istore ∩ elems nx = ∅, dom gstore0 ∪ elems nx = dom gstore ∪ ginval, ginval < # lpt, awbuf-inv(avol0, bleb), flash-lpt-cons(lpt0, istore0, gstore0), synced(avol0), avol1 ↓ avol0, istore0 ⊑ istore, ∀ n. n < # lpt ∧ lpt0[n].flags = LP_GROUP_NODES ∧ lpt[n].flags = LP_GROUP_NODES ∧ ¬ n ∈ nx → lpt[n].size = lpt0[n].size, ∀ n. (n < # lpt ∧ lpt[n].flags = LP_INDEX_NODES ↔ n ∈ istore) ∧ (n ∈ istore → abs-leb(avol0[n], istore[n], lpt[n]) ∧ lpt[n].ref-size = istore[n].ref-size), ∀ n. (n < # lpt ∧ lpt[n].flags = LP_GROUP_NODES ↔ n ∈ gstore) ∧ (n ∈ gstore ∧ ¬ n ∈ ginval → abs-leb(avol0[n], gstore[n], lpt[n])) ∧ (n ∈ gstore → lpt[n].ref-size = gstore[n].ref-size), bleb = nobuffer ∨ bleb.leb < # lpt ∧ (nx ≠ [] ∧ bleb.leb = nx.last ∧ lpt[bleb.leb].flags = LP_GROUP_NODES ∨ lpt[bleb.leb].flags = LP_INDEX_NODES) ⊦ ( n < # lpt ∧ replay-lpt(lpt0.mapIndices(elems nx, λ lp. mklp(lp.ref-size, LEB_SIZE, LP_GROUP_NODES, lp.gcheapidx)), nx, # lpt)[n].flags = LP_GROUP_NODES ↔ n ∈ crash(gstore, gstore0, ginval, nx)) ∧ ( n ∈ gstore ∧ ¬ n ∈ ginval → abs-leb (avol1[n], crash(gstore, gstore0, ginval, nx)[n], replay-lpt(lpt0.mapIndices(elems nx, λ lp. mklp(lp.ref-size, LEB_SIZE, LP_GROUP_NODES, lp.gcheapidx)), nx, # lpt)[n])) ∧ ( n ∈ crash(gstore, gstore0, ginval, nx) → replay-lpt(lpt0.mapIndices(elems nx, λ lp. mklp(lp.ref-size, LEB_SIZE, LP_GROUP_NODES, lp.gcheapidx)), nx, # lpt)[n].ref-size = (n ∈ nx ∨ ¬ n ∈ ginval ⊃ groupblock(gstore[n].gndlist, gstore[n].ladrlist, (n ∈ gstore0 ⊃ gstore0[n].ref-size;0));gstore0[n]).ref-size)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 4:
# lpt0 = # lpt, # avol0 = # lpt, (n ∈ nx ∨ ¬ n ∈ ginval ⊃ groupblock(gstore[n].gndlist, gstore[n].ladrlist, (n ∈ gstore0 ⊃ gstore0[n].ref-size;0));gstore0[n]) = groupblock(gndx, lax, m), dom istore ∩ elems nx = ∅, dom gstore0 ∪ elems nx = dom gstore ∪ ginval, ginval < # lpt, awbuf-inv(avol0, bleb), flash-lpt-cons(lpt0, istore0, gstore0), synced(avol0), avol1 ↓ avol0, istore0 ⊑ istore, ∀ n. (n < # lpt ∧ lpt[n].flags = LP_GROUP_NODES ↔ n ∈ gstore) ∧ (n ∈ gstore ∧ ¬ n ∈ ginval → abs-leb(avol0[n], gstore[n], lpt[n])) ∧ (n ∈ gstore → lpt[n].ref-size = gstore[n].ref-size), ∀ n. (n < # lpt ∧ lpt[n].flags = LP_INDEX_NODES ↔ n ∈ istore) ∧ (n ∈ istore → abs-leb(avol0[n], istore[n], lpt[n]) ∧ lpt[n].ref-size = istore[n].ref-size), ∀ n. n < # lpt ∧ lpt0[n].flags = LP_GROUP_NODES ∧ lpt[n].flags = LP_GROUP_NODES ∧ ¬ n ∈ nx → lpt[n].size = lpt0[n].size, bleb = nobuffer ∨ bleb.leb < # lpt ∧ (nx ≠ [] ∧ bleb.leb = nx.last ∧ lpt[bleb.leb].flags = LP_GROUP_NODES ∨ lpt[bleb.leb].flags = LP_INDEX_NODES) ⊦ ( n < # lpt ∧ replay-lpt(lpt0.mapIndices(elems nx, λ lp. mklp(lp.ref-size, LEB_SIZE, LP_GROUP_NODES, lp.gcheapidx)), nx, # lpt)[n].flags = LP_GROUP_NODES ↔ n ∈ crash(gstore, gstore0, ginval, nx)) ∧ ( n ∈ gstore ∧ ¬ n ∈ ginval → abs-leb (avol1[n], crash(gstore, gstore0, ginval, nx)[n], replay-lpt(lpt0.mapIndices(elems nx, λ lp. mklp(lp.ref-size, LEB_SIZE, LP_GROUP_NODES, lp.gcheapidx)), nx, # lpt)[n])) ∧ ( n ∈ crash(gstore, gstore0, ginval, nx) → replay-lpt(lpt0.mapIndices(elems nx, λ lp. mklp(lp.ref-size, LEB_SIZE, LP_GROUP_NODES, lp.gcheapidx)), nx, # lpt)[n].ref-size = m)

Rule argument

spec: ablock, inst: group-store name: elim seq: ⊦ (gndx = gblock.gndlist ∧ lax = gblock.ladrlist) ∧ m = gblock.ref-size ↔ gblock = groupblock(gndx, lax, m) subst: [gblock, gndx, lax, m] → [(n ∈ nx ∨ ¬ n ∈ ginval ⊃ groupblock (gstore[n].gndlist, gstore[n].ladrlist, (n ∈ gstore0 ⊃ gstore0[n].ref-size;0)) ;gstore0[n] ), gndx, lax, m]

Additional information

Used simplifier rules: ⊦ (gndx = gblock.gndlist ∧ lax = gblock.ladrlist) ∧ m = gblock.ref-size ↔ gblock = groupblock(gndx, lax, m); (elim) Used terms: (n ∈ nx ∨ ¬ n ∈ ginval ⊃ groupblock (gstore[n].gndlist, gstore[n].ladrlist, (n ∈ gstore0 ⊃ gstore0[n].ref-size;0)) ;gstore0[n] ), gndx, lax, m