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Proof step 1 from proof projstate-rec-addboth-one (conclusion) (back to tree)

pa = rren(ra) ⊦ rstep(rs, ra, rs0) ∧ lren(la) = rren(ra) ∧ lstep(ls, la, ls0) → πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs0) ', pa ') = rs ' + πrs(lstep, rstep, lren, rren, (ls0, rs0) ', [])

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof projstate-rec-addboth-one (previous step) (back to tree)

lren(la) = rren(ra), pa = rren(ra), πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs0) ', pa ') ≠ rs ' + πrs(lstep, rstep, lren, rren, (ls0, rs0) ', []), lstep(ls, la, ls0), rstep(rs, ra, rs0) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:projstate-rec-addboth, seq:pa = lren(la) ⊦ # psl = # pal ∧ rstep(rs, ra, rs0) ∧ rren(ra) = lren(la) ∧ lstep(ls, la, ls0) → πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs0) ' + psl, pa ' + pal) = rs ' + πrs(lstep, rstep, lren, rren, (ls0, rs0) ' + psl, pal), subst:[pa, lstep, ra, ls, rs, ls0, pal, rs0, la, lren, rstep, rren, psl] → [rren(ra), lstep, ra, ls, rs, ls0, [], rs0, la, lren, rstep, rren, []]; ['ls, 'ra, 'la, 'pa, 'rs] → ['ls, 'ra, 'la, 'pa, 'rs]

Additional information

Used simplifier rules: projstate-base: ⊦ πrs(lstep, rstep, lren, rren, (ls, rs) ', []) = rs '; ⊦ # [] = 0; (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: rren(ra), [], rs, ra, [], rstep, lren, rren, la, rs0, lstep, ls, ls0

Proof step 3 from proof projstate-rec-addboth-one (previous step) (back to tree)

pa = lren(la) ⊦ # psl = # pal ∧ rstep(rs, ra, rs0) ∧ rren(ra) = lren(la) ∧ lstep(ls, la, ls0) → πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs0) ' + psl, pa ' + pal) = rs ' + πrs(lstep, rstep, lren, rren, (ls0, rs0) ' + psl, pal)

Lemma projstate-rec-addboth

Proof Tree for projstate-rec-addboth-one (View conclusion)

The following image shows the proof tree of sequent calculus. The proved goal is at the bottom. You can click on the nodes to view additional rules. See here for details on colors in a proof tree.

the proof tree