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Proof step 1 from proof pmon-pia (conclusion) (back to tree)

⊦ pmon(frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))) (λ psl, pal. πra(lstep, rstep, lren, rren, psl, pal))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof pmon-pia (previous step) (back to tree)

¬ pmon(frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))) (λ psl, pal. πra(lstep, rstep, lren, rren, psl, pal)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: contfun-lim, name:montrl-relative-def, seq:⊦ pmon(trlp)(trf) ↔ (∀ sl1, al1, sl2, al2. # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 ∧ al1 ⊑ al2 ∧ sl1 ⊑ sl2 ∧ trlp(sl1, al1) ∧ trlp(sl2, al2) → trf(sl1, al1) ⊑ trf(sl2, al2)), subst:[trlp, trf] → [frag* (IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))), λ psl, pal. πra(lstep, rstep, lren, rren, psl, pal)]; ['s, 'a, 'b] → [('ls, 'rs), 'pa, 'ra], <all paths>

Additional information

Used terms: frag* (IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))), λ psl, pal. πra(lstep, rstep, lren, rren, psl, pal)

Proof step 3 from proof pmon-pia (previous step) (back to tree)

⊦ pmon(trlp)(trf) ↔ (∀ sl1, al1, sl2, al2. # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 ∧ al1 ⊑ al2 ∧ sl1 ⊑ sl2 ∧ trlp(sl1, al1) ∧ trlp(sl2, al2) → trf(sl1, al1) ⊑ trf(sl2, al2))

A spec-lemma

Proof step 4 from proof pmon-pia (previous step) (back to tree)

∃ sl1, al1, sl2, al2. ¬ ( # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 ∧ al1 ⊑ al2 ∧ sl1 ⊑ sl2 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(sl1, al1) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(sl2, al2) → πra(lstep, rstep, lren, rren, sl1, al1) ⊑ πra(lstep, rstep, lren, rren, sl2, al2)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 5 from proof pmon-pia (previous step) (back to tree)

# psl = # pal + 1, # psl0 = # pal0 + 1, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl0, pal0), pal ⊑ pal0, psl ⊑ psl0, ¬ πra(lstep, rstep, lren, rren, psl, pal) ⊑ πra(lstep, rstep, lren, rren, psl0, pal0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last, name:prefix, seq: ⊦ x ⊑ y ↔ (∃ z. x + z = y), subst:[x, y] → [pal, pal0]; ['a] → ['pa], <all paths>

Additional information

Used terms: pal, pal0

Proof step 6 from proof pmon-pia (previous step) (back to tree)

⊦ x ⊑ y ↔ (∃ z. x + z = y)

A spec-lemma

Proof step 7 from proof pmon-pia (previous step) (back to tree)

# psl0 = # pal0 + 1, # psl = # pal + 1, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl0, pal0), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal), psl ⊑ psl0, ¬ πra(lstep, rstep, lren, rren, psl, pal) ⊑ πra(lstep, rstep, lren, rren, psl0, pal0), ∃ z. pal + z = pal0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ #(x + y) = # x + # y; (s)

Proof step 8 from proof pmon-pia (previous step) (back to tree)

# psl = # pal + 1, # psl0 = (# pal + # pal1) + 1, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl0, pal + pal1), psl ⊑ psl0, ¬ πra(lstep, rstep, lren, rren, psl, pal) ⊑ πra(lstep, rstep, lren, rren, psl0, pal + pal1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last, name:prefix, seq: ⊦ x ⊑ y ↔ (∃ z. x + z = y), subst:[x, y] → [psl, psl0]; ['a] → [('ls, 'rs)], <all paths>

Additional information

Used terms: psl, psl0

Proof step 9 from proof pmon-pia (previous step) (back to tree)

⊦ x ⊑ y ↔ (∃ z. x + z = y)

A spec-lemma

Proof step 10 from proof pmon-pia (previous step) (back to tree)

# psl0 = (# pal + # pal1) + 1, # psl = # pal + 1, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl0, pal + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal), ¬ πra(lstep, rstep, lren, rren, psl, pal) ⊑ πra(lstep, rstep, lren, rren, psl0, pal + pal1), ∃ z. psl + z = psl0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ #(x + y) = # x + # y; (s)

Proof step 11 from proof pmon-pia (previous step) (back to tree)

# psl = # pal + 1, # psl + # psl1 = (# pal + # pal1) + 1, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal + pal1), ¬ πra(lstep, rstep, lren, rren, psl, pal) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal + pal1) ⊦

User applied rule induction

And obtained the following new goals

Rule argument

indvar: # pal + # pal1 wfpred: <

Additional information

Used simplifier rules: <; (wellfounded) Used terms: # pal + # pal1

Proof step 12 from proof pmon-pia (previous step) (back to tree)

# psl = # pal + 1, # psl + # psl1 = (# pal + # pal1) + 1, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal + pal1), ¬ πra(lstep, rstep, lren, rren, psl, pal) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < # pal + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: pal

Additional information

Used simplifier rules: List('a) freely generated by [], +;;

Proof step 13 from proof pmon-pia (previous step) (back to tree)

# psl = # pal + 1, # psl + # psl1 = (# pal + # pal1) + 1, pal = [], frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal), ¬ πra(lstep, rstep, lren, rren, psl, pal) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < # pal + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ [] + x = x; (s) ⊦ frag*(IOAut(start, step, ext))(sl, []) ↔ sl = sl.head '; (s) ⊦ x + [] = x; (s) ⊦ a + x = a ' + x; (s) ⊦ # x = 1 ↔ x = x.head + []; (as) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ # [] = 0; (s)

Proof step 14 from proof pmon-pia (previous step) (back to tree)

# psl + # psl1 = 1 + # pal1, psl = psl.head ', frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal1), ¬ πra(lstep, rstep, lren, rren, psl, []) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < # [] + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-list-c, seq:x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y, subst:[x, a, y] → [psl, ps, psl0]; ['a] → [('ls, 'rs)]

Additional information

Used simplifier rules: ⊦ a ' ≠ []; (s) x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y; (elim) Used terms: psl, ps, psl0

Proof step 15 from proof pmon-pia (previous step) (back to tree)

# psl + # psl1 = 1 + # pal1, psl = ps ', psl = ps + psl0, psl ≠ [], frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal1), ¬ πra(lstep, rstep, lren, rren, psl, []) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < # [] + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ [] + x = x; (s) ⊦ x + [] = x; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ n + 0 = n; (s) ⊦ # [] = 0; (s) ⊦ a + x = a ' + x; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ m + n = m + n0 ↔ n = n0; (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s) ⊦ a ' = a0 ' + x ↔ a = a0 ∧ x = []; (ws)

Proof step 16 from proof pmon-pia (previous step) (back to tree)

# psl1 = # pal1, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(ps ' + psl1, pal1), ¬ πra(lstep, rstep, lren, rren, ps ', []) ⊑ πra(lstep, rstep, lren, rren, ps ' + psl1, pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < # [] + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

User applied rule constructor cut

And closed the goal!

Rule argument

var: ps

Additional information

Used simplifier rules: projaction-base: ⊦ πra(lstep, rstep, lren, rren, (ls, rs) ', []) = []; ⊦ [] ⊑ x; (s)

Proof step 17 from proof pmon-pia (previous step) (back to tree)

# psl = # pal + 1, # psl + # psl1 = (# pal + # pal1) + 1, pal = pa + pal3, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal), ¬ πra(lstep, rstep, lren, rren, psl, pal) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < # pal + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a + x = a ' + x; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 18 from proof pmon-pia (previous step) (back to tree)

# psl = 2 + # pal3, # psl + # psl1 = 2 + # pal3 + # pal1, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pa ' + pal3), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, (pa ' + pal3) + pal1), ¬ πra(lstep, rstep, lren, rren, psl, pa ' + pal3) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, (pa ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < #(pa + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: psl

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ n + 0 = n; (s) ⊦ # [] = 0; (s) ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic

Proof step 19 from proof pmon-pia (previous step) (back to tree)

# psl = 2 + # pal3, # psl + # psl1 = 2 + # pal3 + # pal1, psl = ps + psl0, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, (pa ' + pal3) + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pa ' + pal3), ¬ πra(lstep, rstep, lren, rren, psl, pa ' + pal3) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, (pa ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < #(pa + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a + x = a ' + x; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic n < n0 ⊦ m + (* n) = m0 + (* n0) ↔ m = m0 + (n0 - n); (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 20 from proof pmon-pia (previous step) (back to tree)

# psl0 = # pal3 + 1, # psl0 + # psl1 = # pal3 + # pal1 + 1, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(ps ' + psl0, pa ' + pal3), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ps ' + psl0) + psl1, (pa ' + pal3) + pal1), ¬ πra(lstep, rstep, lren, rren, ps ' + psl0, pa ' + pal3) ⊑ πra(lstep, rstep, lren, rren, (ps ' + psl0) + psl1, (pa ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < #(pa + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: ps

Additional information

Proof step 21 from proof pmon-pia (previous step) (back to tree)

# psl0 = # pal3 + 1, # psl0 + # psl1 = # pal3 + # pal1 + 1, ps = (ls, rs), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ps ' + psl0) + psl1, (pa ' + pal3) + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(ps ' + psl0, pa ' + pal3), ¬ πra(lstep, rstep, lren, rren, ps ' + psl0, pa ' + pal3) ⊑ πra(lstep, rstep, lren, rren, (ps ' + psl0) + psl1, (pa ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < #(pa + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 22 from proof pmon-pia (previous step) (back to tree)

# psl0 = # pal3 + 1, # psl0 + # psl1 = # pal3 + # pal1 + 1, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls, rs) ' + psl0, pa ' + pal3), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls, rs) ' + psl0) + psl1, (pa ' + pal3) + pal1), ¬ πra(lstep, rstep, lren, rren, (ls, rs) ' + psl0, pa ' + pal3) ⊑ πra(lstep, rstep, lren, rren, ((ls, rs) ' + psl0) + psl1, (pa ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < #(pa + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:finfrag-rec, seq:⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al), subst:[a, al, step, s, start, ext, sl] → [pa, pal3 + pal1, Π(lstep, rstep, lren, rren, lext, rext), (ls, rs), Π(lstart, rstart), Π(lext, rext, lren, rren), psl0 + psl1]; ['s, 'a, 'e] → [('ls, 'rs), 'pa, 'e], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: pal3 + pal1, (ls, rs), Π(lext, rext, lren, rren), psl0 + psl1, pa, Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext)

Proof step 23 from proof pmon-pia (previous step) (back to tree)

⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al)

A spec-lemma

Proof step 24 from proof pmon-pia (previous step) (back to tree)

# psl0 = # pal3 + 1, # psl0 + # psl1 = # pal3 + # pal1 + 1, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl0 + psl1, pal3 + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls, rs) ' + psl0, pa ' + pal3), Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (psl0 + psl1).head), ¬ πra(lstep, rstep, lren, rren, (ls, rs) ' + psl0, pa ' + pal3) ⊑ πra(lstep, rstep, lren, rren, ((ls, rs) ' + psl0) + psl1, (pa ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < #(pa + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: psl0

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ n + 0 = n; (s) ⊦ # [] = 0; (s) ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws) ⊦ # x = 0 ↔ x = []; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic

Proof step 25 from proof pmon-pia (previous step) (back to tree)

# psl0 = # pal3 + 1, # psl0 + # psl1 = # pal3 + # pal1 + 1, psl0 = ps + psl2, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls, rs) ' + psl0, pa ' + pal3), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl0 + psl1, pal3 + pal1), Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (psl0 + psl1).head), ¬ πra(lstep, rstep, lren, rren, (ls, rs) ' + psl0, pa ' + pal3) ⊑ πra(lstep, rstep, lren, rren, ((ls, rs) ' + psl0) + psl1, (pa ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < #(pa + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + n = m + n0 ↔ n = n0; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a + x = a ' + x; (s) ⊦ (a ' + x).head = a; (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 26 from proof pmon-pia (previous step) (back to tree)

# psl2 = # pal3, # psl1 = # pal1, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls, rs) ' + ps ' + psl2, pa ' + pal3), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ps ' + psl2) + psl1, pal3 + pal1), Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, ps), ¬ πra(lstep, rstep, lren, rren, (ls, rs) ' + ps ' + psl2, pa ' + pal3) ⊑ πra(lstep, rstep, lren, rren, (ls, rs) ' + (ps ' + psl2) + psl1, (pa ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < #(pa + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: ps

Additional information

Proof step 27 from proof pmon-pia (previous step) (back to tree)

# psl1 = # pal1, # psl2 = # pal3, ps = (ls0, rs0), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ps ' + psl2) + psl1, pal3 + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls, rs) ' + ps ' + psl2, pa ' + pal3), Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, ps), ¬ πra(lstep, rstep, lren, rren, (ls, rs) ' + ps ' + psl2, pa ' + pal3) ⊑ πra(lstep, rstep, lren, rren, (ls, rs) ' + (ps ' + psl2) + psl1, (pa ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < #(pa + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 28 from proof pmon-pia (previous step) (back to tree)

# psl2 = # pal3, # psl1 = # pal1, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls, rs) ' + (ls0, rs0) ' + psl2, pa ' + pal3), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls0, rs0) ' + psl2) + psl1, pal3 + pal1), Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (ls0, rs0)), ¬ πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs0) ' + psl2, pa ' + pal3) ⊑ πra(lstep, rstep, lren, rren, (ls, rs) ' + ((ls0, rs0) ' + psl2) + psl1, (pa ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < #(pa + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:finfrag-rec, seq:⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al), subst:[a, al, step, s, start, ext, sl] → [pa, pal3, Π(lstep, rstep, lren, rren, lext, rext), (ls, rs), Π(lstart, rstart), Π(lext, rext, lren, rren), (ls0, rs0) ' + psl2]; ['s, 'a, 'e] → [('ls, 'rs), 'pa, 'e], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: pal3, (ls, rs), Π(lext, rext, lren, rren), (ls0, rs0) ' + psl2, pa, Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext)

Proof step 29 from proof pmon-pia (previous step) (back to tree)

⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al)

A spec-lemma

Proof step 30 from proof pmon-pia (previous step) (back to tree)

# psl1 = # pal1, # psl2 = # pal3, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls0, rs0) ' + psl2) + psl1, pal3 + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs0) ' + psl2, pal3), Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (ls0, rs0)), Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, ((ls0, rs0) ' + psl2).head), ¬ πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs0) ' + psl2, pa ' + pal3) ⊑ πra(lstep, rstep, lren, rren, (ls, rs) ' + ((ls0, rs0) ' + psl2) + psl1, (pa ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < #(pa + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (a ' + x).head = a; (s)

Proof step 31 from proof pmon-pia (previous step) (back to tree)

# psl2 = # pal3, # psl1 = # pal1, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs0) ' + psl2, pal3), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls0, rs0) ' + psl2) + psl1, pal3 + pal1), Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (ls0, rs0)), ¬ πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs0) ' + psl2, pa ' + pal3) ⊑ πra(lstep, rstep, lren, rren, (ls, rs) ' + ((ls0, rs0) ' + psl2) + psl1, (pa ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < #(pa + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: product, name:lrstep-def, seq:⊦ Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (ls0, rs0)) ↔ rs = rs0 ∧ (∃ la. lstep(ls, la, ls0) ∧ pa = lren(la) ∧ ¬ cod(rren)(pa)) ∨ ls = ls0 ∧ (∃ ra. rstep(rs, ra, rs0) ∧ pa = rren(ra) ∧ ¬ cod(lren)(pa)) ∨ (∃ la, ra. lstep(ls, la, ls0) ∧ lext(la) ≠ None ∧ pa = lren(la) ∧ rstep(rs, ra, rs0) ∧ rext(ra) ≠ None ∧ pa = rren(ra)), subst:[pa, ls0, rext, lstep, rs0, ls, rs, rren, lren, lext, rstep] → [pa, ls0, rext, lstep, rs0, ls, rs, rren, lren, lext, rstep]; ['ls, 'ra, 'la, 'pa, 'rs, 'e] → ['ls, 'ra, 'la, 'pa, 'rs, 'e], <all paths>

Additional information

Used terms: pa, rs, rstep, lren, rren, lext, rs0, rext, lstep, ls, ls0

Proof step 32 from proof pmon-pia (previous step) (back to tree)

⊦ Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (ls0, rs0)) ↔ rs = rs0 ∧ (∃ la. lstep(ls, la, ls0) ∧ pa = lren(la) ∧ ¬ cod(rren)(pa)) ∨ ls = ls0 ∧ (∃ ra. rstep(rs, ra, rs0) ∧ pa = rren(ra) ∧ ¬ cod(lren)(pa)) ∨ (∃ la, ra. lstep(ls, la, ls0) ∧ lext(la) ≠ None ∧ pa = lren(la) ∧ rstep(rs, ra, rs0) ∧ rext(ra) ≠ None ∧ pa = rren(ra))

A spec-lemma

Proof step 33 from proof pmon-pia (previous step) (back to tree)

# psl1 = # pal1, # psl2 = # pal3, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls0, rs0) ' + psl2) + psl1, pal3 + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs0) ' + psl2, pal3), ¬ πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs0) ' + psl2, pa ' + pal3) ⊑ πra(lstep, rstep, lren, rren, (ls, rs) ' + ((ls0, rs0) ' + psl2) + psl1, (pa ' + pal3) + pal1), rs = rs0 ∧ (∃ la. lstep(ls, la, ls0) ∧ pa = lren(la) ∧ ¬ cod(rren)(pa)) ∨ ls = ls0 ∧ (∃ ra. rstep(rs, ra, rs0) ∧ pa = rren(ra) ∧ ¬ cod(lren)(pa)) ∨ (∃ la, ra. lstep(ls, la, ls0) ∧ lext(la) ≠ None ∧ pa = lren(la) ∧ rstep(rs, ra, rs0) ∧ rext(ra) ≠ None ∧ pa = rren(ra)), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < #(pa + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 34 from proof pmon-pia (previous step) (back to tree)

# psl2 = # pal3, # psl1 = # pal1, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs0) ' + psl2, pal3), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls0, rs0) ' + psl2) + psl1, pal3 + pal1), ¬ πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs0) ' + psl2, pa ' + pal3) ⊑ πra(lstep, rstep, lren, rren, (ls, rs) ' + ((ls0, rs0) ' + psl2) + psl1, (pa ' + pal3) + pal1), rs = rs0 ∧ lstep(ls, la, ls0) ∧ pa = lren(la) ∧ ¬ cod(rren)(lren(la)) ∨ ls = ls0 ∧ rstep(rs, ra, rs0) ∧ pa = rren(ra) ∧ ¬ cod(lren)(rren(ra)) ∨ lstep(ls, la0, ls0) ∧ lext(la0) ≠ None ∧ rren(ra0) = lren(la0) ∧ rstep(rs, ra0, rs0) ∧ rext(ra0) ≠ None ∧ pa = rren(ra0), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < #(pa + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 6

Additional information

Proof step 35 from proof pmon-pia (previous step) (back to tree)

# psl1 = # pal1, # psl2 = # pal3, pa = lren(la), rs = rs0, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls0, rs0) ' + psl2) + psl1, pal3 + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs0) ' + psl2, pal3), lstep(ls, la, ls0), ¬ cod(rren)(lren(la)), ¬ πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs0) ' + psl2, pa ' + pal3) ⊑ πra(lstep, rstep, lren, rren, (ls, rs) ' + ((ls0, rs0) ' + psl2) + psl1, (pa ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < #(pa + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: projaction-rec-skipL: ⊦ # psl = # pal ∧ lstep(ls, la, ls0) ∧ ¬ cod(rren)(lren(la)) → πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psl, lren(la) ' + pal) = πra(lstep, rstep, lren, rren, (ls0, rs) ' + psl, pal); a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 36 from proof pmon-pia (previous step) (back to tree)

# psl2 = # pal3, # psl1 = # pal1, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs0) ' + psl2, pal3), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls0, rs0) ' + psl2) + psl1, pal3 + pal1), lstep(ls, la, ls0), ¬ cod(rren)(lren(la)), ¬ πra(lstep, rstep, lren, rren, (ls0, rs0) ' + psl2, pal3) ⊑ πra(lstep, rstep, lren, rren, (ls, rs0) ' + ((ls0, rs0) ' + psl2) + psl1, (lren(la) ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren0, rren, lext, rext. # pal0 + # pal2 < #(lren(la) + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren0, rren, lext, rext), Π(lext, rext, lren0, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren0, rren, lext, rext), Π(lext, rext, lren0, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren0, rren, psl, pal0) ⊑ πra(lstep, rstep, lren0, rren, psl + psl1, pal0 + pal2) → false ⊦

User applied rule apply induction

And obtained the following new goals

Rule argument

subst: [pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren0, rren, lext, rext] → [pal3, pal1, (ls0, rs0) ' + psl2, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext]

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s) ⊦ #(a ' + x) = (# x)+1; (s) ⊦ m +1 = m + 1; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m + n < m + n0 ↔ n < n0; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ ¬ n < n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: pal3, pal1, (ls0, rs0) ' + psl2, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext

Proof step 37 from proof pmon-pia (previous step) (back to tree)

# psl1 = # pal1, # psl2 = # pal3, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls0, rs0) ' + psl2) + psl1, pal3 + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs0) ' + psl2, pal3), lstep(ls, la, ls0), ¬ cod(rren)(lren(la)), ¬ πra(lstep, rstep, lren, rren, (ls0, rs0) ' + psl2, pal3) ⊑ πra(lstep, rstep, lren, rren, (ls, rs0) ' + ((ls0, rs0) ' + psl2) + psl1, (lren(la) ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren0, rren, lext, rext. # pal0 + # pal2 < #(lren(la) + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren0, rren, lext, rext), Π(lext, rext, lren0, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren0, rren, lext, rext), Π(lext, rext, lren0, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren0, rren, psl, pal0) ⊑ πra(lstep, rstep, lren0, rren, psl + psl1, pal0 + pal2) → false ⊦ #((ls0, rs0) ' + psl2) = # pal3 + 1 ∧ #((ls0, rs0) ' + psl2) + # psl1 = (# pal3 + # pal1) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs0) ' + psl2, pal3) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls0, rs0) ' + psl2) + psl1, pal3 + pal1) ∧ ¬ πra(lstep, rstep, lren, rren, (ls0, rs0) ' + psl2, pal3) ⊑ πra(lstep, rstep, lren, rren, ((ls0, rs0) ' + psl2) + psl1, pal3 + pal1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + n = m + n0 ↔ n = n0; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 38 from proof pmon-pia (previous step) (back to tree)

# psl1 = # pal1, # psl2 = # pal3, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls0, rs0) ' + psl2) + psl1, pal3 + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs0) ' + psl2, pal3), lstep(ls, la, ls0), πra(lstep, rstep, lren, rren, (ls0, rs0) ' + psl2, pal3) ⊑ πra(lstep, rstep, lren, rren, ((ls0, rs0) ' + psl2) + psl1, pal3 + pal1), ¬ cod(rren)(lren(la)), ¬ πra(lstep, rstep, lren, rren, (ls0, rs0) ' + psl2, pal3) ⊑ πra(lstep, rstep, lren, rren, (ls, rs0) ' + ((ls0, rs0) ' + psl2) + psl1, (lren(la) ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren0, rren, lext, rext. # pal0 + # pal2 < #(lren(la) + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren0, rren, lext, rext), Π(lext, rext, lren0, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren0, rren, lext, rext), Π(lext, rext, lren0, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren0, rren, psl, pal0) ⊑ πra(lstep, rstep, lren0, rren, psl + psl1, pal0 + pal2) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:projaction-rec-skipL, seq:⊦ # psl = # pal ∧ lstep(ls, la, ls0) ∧ ¬ cod(rren)(lren(la)) → πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psl, lren(la) ' + pal) = πra(lstep, rstep, lren, rren, (ls0, rs) ' + psl, pal), subst:[lstep, la, ls, rs, ls0, pal, lren, rstep, rren, psl] → [lstep, la, ls, rs0, ls0, pal3 + pal1, lren, rstep, rren, psl2 + psl1]; ['ls, 'ra, 'la, 'pa, 'rs] → ['ls, 'ra, 'la, 'pa, 'rs], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ #(x + y) = # x + # y; (s) Used terms: psl2 + psl1, rs0, rstep, pal3 + pal1, lren, rren, la, lstep, ls, ls0

Proof step 39 from proof pmon-pia (previous step) (back to tree)

⊦ # psl = # pal ∧ lstep(ls, la, ls0) ∧ ¬ cod(rren)(lren(la)) → πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psl, lren(la) ' + pal) = πra(lstep, rstep, lren, rren, (ls0, rs) ' + psl, pal)

Lemma projaction-rec-skipL

Proof step 40 from proof pmon-pia (previous step) (back to tree)

# psl1 = # pal1, # psl2 = # pal3, ls = ls0, pa = rren(ra), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls0, rs0) ' + psl2) + psl1, pal3 + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs0) ' + psl2, pal3), rstep(rs, ra, rs0), ¬ cod(lren)(rren(ra)), ¬ πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs0) ' + psl2, pa ' + pal3) ⊑ πra(lstep, rstep, lren, rren, (ls, rs) ' + ((ls0, rs0) ' + psl2) + psl1, (pa ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < #(pa + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: projaction-rec-addR: ⊦ # psl = # pal ∧ rstep(rs, ra, rs0) ∧ ¬ cod(lren)(rren(ra)) → πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls, rs0) ' + psl, rren(ra) ' + pal) = ra + πra(lstep, rstep, lren, rren, (ls, rs0) + psl, pal); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a + x = a ' + x; (s)

Proof step 41 from proof pmon-pia (previous step) (back to tree)

# psl2 = # pal3, # psl1 = # pal1, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs0) ' + psl2, pal3), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls0, rs0) ' + psl2) + psl1, pal3 + pal1), rstep(rs, ra, rs0), ¬ cod(lren)(rren(ra)), ¬ ra ' + πra(lstep, rstep, lren, rren, (ls0, rs0) ' + psl2, pal3) ⊑ πra(lstep, rstep, lren, rren, (ls0, rs) ' + ((ls0, rs0) ' + psl2) + psl1, (rren(ra) ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren0, lext, rext. # pal0 + # pal2 < #(rren(ra) + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren0, lext, rext), Π(lext, rext, lren, rren0)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren0, lext, rext), Π(lext, rext, lren, rren0)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren0, psl, pal0) ⊑ πra(lstep, rstep, lren, rren0, psl + psl1, pal0 + pal2) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:projaction-rec-addR, seq:⊦ # psl = # pal ∧ rstep(rs, ra, rs0) ∧ ¬ cod(lren)(rren(ra)) → πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls, rs0) ' + psl, rren(ra) ' + pal) = ra + πra(lstep, rstep, lren, rren, (ls, rs0) + psl, pal), subst:[pal, lstep, rs0, ra, ls, rs, lren, rstep, rren, psl] → [pal3 + pal1, lstep, rs0, ra, ls0, rs, lren, rstep, rren, psl2 + psl1]; ['ls, 'ra, 'la, 'pa, 'rs] → ['ls, 'ra, 'la, 'pa, 'rs], <all paths>

Additional information

Used simplifier rules: ⊦ #(x + y) = # x + # y; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: psl2 + psl1, rs, rstep, ra, pal3 + pal1, lren, rren, rs0, lstep, ls0

Proof step 42 from proof pmon-pia (previous step) (back to tree)

⊦ # psl = # pal ∧ rstep(rs, ra, rs0) ∧ ¬ cod(lren)(rren(ra)) → πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls, rs0) ' + psl, rren(ra) ' + pal) = ra + πra(lstep, rstep, lren, rren, (ls, rs0) + psl, pal)

Lemma projaction-rec-addR

Proof step 43 from proof pmon-pia (previous step) (back to tree)

# psl1 = # pal1, # psl2 = # pal3, #(psl2 + psl1) = #(pal3 + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls0, rs0) ' + psl2) + psl1, pal3 + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs0) ' + psl2, pal3), rstep(rs, ra, rs0), ¬ cod(lren)(rren(ra)), ¬ ra ' + πra(lstep, rstep, lren, rren, (ls0, rs0) ' + psl2, pal3) ⊑ ra + πra(lstep, rstep, lren, rren, (ls0, rs0) + (psl2 + psl1), pal3 + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren0, lext, rext. # pal0 + # pal2 < #(rren(ra) + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren0, lext, rext), Π(lext, rext, lren, rren0)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren0, lext, rext), Π(lext, rext, lren, rren0)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren0, psl, pal0) ⊑ πra(lstep, rstep, lren, rren0, psl + psl1, pal0 + pal2) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ' + x ⊑ b ' + y ↔ a = b ∧ x ⊑ y; (ws) ⊦ a + x = a ' + x; (s) ⊦ #(x + y) = # x + # y; (s)

Proof step 44 from proof pmon-pia (previous step) (back to tree)

# psl1 = # pal1, # psl2 = # pal3, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls0, rs0) ' + psl2) + psl1, pal3 + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs0) ' + psl2, pal3), rstep(rs, ra, rs0), ¬ cod(lren)(rren(ra)), ¬ πra(lstep, rstep, lren, rren, (ls0, rs0) ' + psl2, pal3) ⊑ πra(lstep, rstep, lren, rren, (ls0, rs0) ' + psl2 + psl1, pal3 + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren0, lext, rext. # pal0 + # pal2 < #(rren(ra) + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren0, lext, rext), Π(lext, rext, lren, rren0)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren0, lext, rext), Π(lext, rext, lren, rren0)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren0, psl, pal0) ⊑ πra(lstep, rstep, lren, rren0, psl + psl1, pal0 + pal2) → false ⊦

User applied rule apply induction

And closed the goal!

Rule argument

subst: [pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren0, lext, rext] → [pal3, pal1, (ls0, rs0) ' + psl2, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext]

Additional information

Used simplifier rules: ⊦ #(a ' + x) = (# x)+1; (s) ⊦ m +1 = m + 1; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ 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 ⊦ a + x = a ' + x; (s) ⊦ m + n < m + n0 ↔ n < n0; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ ¬ n < n; (s) Used terms: pal3, pal1, (ls0, rs0) ' + psl2, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext

Proof step 45 from proof pmon-pia (previous step) (back to tree)

# psl1 = # pal1, # psl2 = # pal3, pa = rren(ra0), rren(ra0) = lren(la0), lext(la0) ≠ None, rext(ra0) ≠ None, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls0, rs0) ' + psl2) + psl1, pal3 + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs0) ' + psl2, pal3), lstep(ls, la0, ls0), rstep(rs, ra0, rs0), ¬ πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs0) ' + psl2, pa ' + pal3) ⊑ πra(lstep, rstep, lren, rren, (ls, rs) ' + ((ls0, rs0) ' + psl2) + psl1, (pa ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext. # pal0 + # pal2 < #(pa + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren, psl, pal0) ⊑ πra(lstep, rstep, lren, rren, psl + psl1, pal0 + pal2) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: projaction-rec-addboth: pa = lren(la) ⊦ # psl = # pal ∧ rstep(rs, ra, rs0) ∧ rren(ra) = lren(la) ∧ lstep(ls, la, ls0) → πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs0) ' + psl, pa ' + pal) = ra + πra(lstep, rstep, lren, rren, (ls0, rs0) ' + psl, pal); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a + x = a ' + x; (s)

Proof step 46 from proof pmon-pia (previous step) (back to tree)

# psl2 = # pal3, # psl1 = # pal1, rren(ra0) = lren(la0), lext(la0) ≠ None, rext(ra0) ≠ None, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs0) ' + psl2, pal3), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls0, rs0) ' + psl2) + psl1, pal3 + pal1), lstep(ls, la0, ls0), rstep(rs, ra0, rs0), ¬ ra0 ' + πra(lstep, rstep, lren, rren, (ls0, rs0) ' + psl2, pal3) ⊑ πra(lstep, rstep, lren, rren, (ls, rs) ' + ((ls0, rs0) ' + psl2) + psl1, (lren(la0) ' + pal3) + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren0, lext, rext. # pal0 + # pal2 < #(rren(ra0) + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren0, lext, rext), Π(lext, rext, lren, rren0)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren0, lext, rext), Π(lext, rext, lren, rren0)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren0, psl, pal0) ⊑ πra(lstep, rstep, lren, rren0, psl + psl1, pal0 + pal2) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:projaction-rec-addboth, seq:pa = lren(la) ⊦ # psl = # pal ∧ rstep(rs, ra, rs0) ∧ rren(ra) = lren(la) ∧ lstep(ls, la, ls0) → πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs0) ' + psl, pa ' + pal) = ra + πra(lstep, rstep, lren, rren, (ls0, rs0) ' + psl, pal), subst:[pa, lstep, ra, ls, rs, ls0, pal, rs0, la, lren, rstep, rren, psl] → [lren(la0), lstep, ra0, ls, rs, ls0, pal3 + pal1, rs0, la0, lren, rstep, rren, psl2 + psl1]; ['ls, 'ra, 'la, 'pa, 'rs] → ['ls, 'ra, 'la, 'pa, 'rs], <all paths>

Additional information

Used simplifier rules: ⊦ #(x + y) = # x + # y; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: psl2 + psl1, lren(la0), rs, rstep, ra0, pal3 + pal1, lren, rren, la0, rs0, lstep, ls, ls0

Proof step 47 from proof pmon-pia (previous step) (back to tree)

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

Lemma projaction-rec-addboth

Proof step 48 from proof pmon-pia (previous step) (back to tree)

# psl1 = # pal1, # psl2 = # pal3, #(psl2 + psl1) = #(pal3 + pal1), lren(la0) = lren(la0), rren(ra0) = lren(la0), lext(la0) ≠ None, rext(ra0) ≠ None, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls0, rs0) ' + psl2) + psl1, pal3 + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs0) ' + psl2, pal3), lstep(ls, la0, ls0), rstep(rs, ra0, rs0), ¬ ra0 ' + πra(lstep, rstep, lren, rren, (ls0, rs0) ' + psl2, pal3) ⊑ ra0 + πra(lstep, rstep, lren, rren, (ls0, rs0) ' + psl2 + psl1, pal3 + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren0, lext, rext. # pal0 + # pal2 < #(rren(ra0) + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren0, lext, rext), Π(lext, rext, lren, rren0)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren0, lext, rext), Π(lext, rext, lren, rren0)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren0, psl, pal0) ⊑ πra(lstep, rstep, lren, rren0, psl + psl1, pal0 + pal2) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ' + x ⊑ b ' + y ↔ a = b ∧ x ⊑ y; (ws) ⊦ a + x = a ' + x; (s) ⊦ #(x + y) = # x + # y; (s)

Proof step 49 from proof pmon-pia (previous step) (back to tree)

# psl1 = # pal1, # psl2 = # pal3, rren(ra0) = lren(la0), lext(la0) ≠ None, rext(ra0) ≠ None, frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(((ls0, rs0) ' + psl2) + psl1, pal3 + pal1), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs0) ' + psl2, pal3), lstep(ls, la0, ls0), rstep(rs, ra0, rs0), ¬ πra(lstep, rstep, lren, rren, (ls0, rs0) ' + psl2, pal3) ⊑ πra(lstep, rstep, lren, rren, (ls0, rs0) ' + psl2 + psl1, pal3 + pal1), ∀ pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren0, lext, rext. # pal0 + # pal2 < #(rren(ra0) + pal3) + # pal1 → # psl = # pal0 + 1 ∧ # psl + # psl1 = (# pal0 + # pal2) + 1 ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren0, lext, rext), Π(lext, rext, lren, rren0)))(psl, pal0) ∧ frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren0, lext, rext), Π(lext, rext, lren, rren0)))(psl + psl1, pal0 + pal2) ∧ ¬ πra(lstep, rstep, lren, rren0, psl, pal0) ⊑ πra(lstep, rstep, lren, rren0, psl + psl1, pal0 + pal2) → false ⊦

User applied rule apply induction

And closed the goal!

Rule argument

subst: [pal0, pal2, psl, psl1, lstart, rstart, lstep, rstep, lren, rren0, lext, rext] → [pal3, pal1, (ls0, rs0) ' + psl2, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext]

Additional information

Used simplifier rules: ⊦ #(a ' + x) = (# x)+1; (s) ⊦ m +1 = m + 1; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ 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 ⊦ a + x = a ' + x; (s) ⊦ m + n < m + n0 ↔ n < n0; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ ¬ n < n; (s) Used terms: pal3, pal1, (ls0, rs0) ' + psl2, psl1, lstart, rstart, lstep, rstep, lren, rren, lext, rext

Proof Tree for pmon-pia (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