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Proof step 1 from proof pfxclosed-execstar (conclusion) (back to tree)

⊦ pfxclosed(exec*(ioa))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof pfxclosed-execstar (previous step) (back to tree)

¬ pfxclosed(exec*(ioa)) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:pfxclosed-fragstar, seq: ⊦ pfxclosed(frag*(ioa)), subst:[ioa] → [ioa]; ['s, 'a, 'e] → ['s, 'a, 'e]

Additional information

Used terms: ioa

Proof step 3 from proof pfxclosed-execstar (previous step) (back to tree)

⊦ pfxclosed(frag*(ioa))

Lemma pfxclosed-fragstar

Proof step 4 from proof pfxclosed-execstar (previous step) (back to tree)

pfxclosed(frag*(ioa)), ¬ pfxclosed(exec*(ioa)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: contfun-lim, name:pdfclosedtrl-def, seq:⊦ pfxclosed(trlp) ↔ (∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1)), subst:[trlp] → [frag*(ioa)]; ['s, 'a] → ['s, 'a], <all paths>

Additional information

Used terms: frag*(ioa)

Proof step 5 from proof pfxclosed-execstar (previous step) (back to tree)

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

A spec-lemma

Proof step 6 from proof pfxclosed-execstar (previous step) (back to tree)

¬ pfxclosed(exec*(ioa)), ∀ sl1, sl2, al1, al2. frag*(ioa)(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → frag*(ioa)(sl1, al1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: contfun-lim, name:pdfclosedtrl-def, seq:⊦ pfxclosed(trlp) ↔ (∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1)), subst:[trlp] → [exec*(ioa)]; ['s, 'a] → ['s, 'a], <all paths>

Additional information

Used terms: exec*(ioa)

Proof step 7 from proof pfxclosed-execstar (previous step) (back to tree)

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

A spec-lemma

Proof step 8 from proof pfxclosed-execstar (previous step) (back to tree)

∀ sl1, sl2, al1, al2. frag*(ioa)(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → frag*(ioa)(sl1, al1), ∃ sl1, sl2, al1, al2. ¬(exec*(ioa)(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → exec*(ioa)(sl1, al1)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ exec*(ioa)(sl, al) ↔ frag*(ioa)(sl, al) ∧ ioa.start(sl[0]); (s)

Proof step 9 from proof pfxclosed-execstar (previous step) (back to tree)

# sl1 = # al1 + 1, # sl2 = # al2 + 1, ioa.start(sl2[0]), frag*(ioa)(sl2, al2), sl1 ⊑ sl2, al1 ⊑ al2, ∀ sl1, sl2, al1, al2. frag*(ioa)(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → frag*(ioa)(sl1, al1) ⊦ frag*(ioa)(sl1, al1) ∧ ioa.start(sl1[0])

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: IO-Aut-data, name:elim-IOAut, seq:⊦ (start = ioaut.start ∧ step = ioaut.step) ∧ ext = ioaut.ext ↔ ioaut = IOAut(start, step, ext), subst:[ioaut, start, step, ext] → [ioa, start, step, ext]; ['s, 'a, 'e] → ['s, 'a, 'e]

Additional information

Used simplifier rules: ⊦ (start = ioaut.start ∧ step = ioaut.step) ∧ ext = ioaut.ext ↔ ioaut = IOAut(start, step, ext); (elim) Used terms: ioa, start, step, ext

Proof step 10 from proof pfxclosed-execstar (previous step) (back to tree)

# sl2 = # al2 + 1, # sl1 = # al1 + 1, ioa = IOAut(start, step, ext), frag*(ioa)(sl2, al2), start(sl2[0]), al1 ⊑ al2, sl1 ⊑ sl2, ∀ sl1, sl2, al1, al2. frag*(ioa)(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → frag*(ioa)(sl1, al1) ⊦ frag*(ioa)(sl1, al1) ∧ start(sl1[0])

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 ⊦ frag*(IOAut(start, step, ext))(sl, al) → (# sl = # al + 1 ↔ true); (ss)

Proof step 11 from proof pfxclosed-execstar (previous step) (back to tree)

# sl1 = # al1 + 1, # sl2 = # al2 + 1, frag*(IOAut(start, step, ext))(sl2, al2), start(sl2[0]), sl1 ⊑ sl2, al1 ⊑ al2, ∀ sl, sl0, al, al0. frag*(IOAut(start, step, ext))(sl0, al0) ∧ sl ⊑ sl0 ∧ al ⊑ al0 ∧ # sl = # al + 1 → frag*(IOAut(start, step, ext))(sl, al) ⊦ frag*(IOAut(start, step, ext))(sl1, al1) ∧ start(sl1[0])

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 7 terms: sl1, sl2, al1, al2 abort: F discard: T computed: T

Additional information

Used terms: sl1, sl2, al1, al2

Proof step 12 from proof pfxclosed-execstar (previous step) (back to tree)

# sl2 = # al2 + 1, # sl1 = # al1 + 1, frag*(IOAut(start, step, ext))(sl2, al2), start(sl2[0]), al1 ⊑ al2, sl1 ⊑ sl2, frag*(IOAut(start, step, ext))(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 → frag*(IOAut(start, step, ext))(sl1, al1) ⊦ frag*(IOAut(start, step, ext))(sl1, al1) ∧ start(sl1[0])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 13 from proof pfxclosed-execstar (previous step) (back to tree)

# sl1 = # al1 + 1, # sl2 = # al2 + 1, frag*(IOAut(start, step, ext))(sl2, al2), frag*(IOAut(start, step, ext))(sl1, al1), start(sl2[0]), sl1 ⊑ sl2, al1 ⊑ al2, ¬ start(sl1[0]) ⊦

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] → [sl1, sl2]; ['a] → ['s], <all paths>

Additional information

Used terms: sl1, sl2

Proof step 14 from proof pfxclosed-execstar (previous step) (back to tree)

⊦ x ⊑ y ↔ (∃ z. x + z = y)

A spec-lemma

Proof step 15 from proof pfxclosed-execstar (previous step) (back to tree)

# sl2 = # al2 + 1, # sl1 = # al1 + 1, frag*(IOAut(start, step, ext))(sl1, al1), frag*(IOAut(start, step, ext))(sl2, al2), start(sl2[0]), al1 ⊑ al2, ¬ start(sl1[0]), ∃ z. sl1 + z = sl2 ⊦

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 16 from proof pfxclosed-execstar (previous step) (back to tree)

# sl1 = # al1 + 1, # sl1 + # sl = # al2 + 1, frag*(IOAut(start, step, ext))(sl1 + sl, al2), frag*(IOAut(start, step, ext))(sl1, al1), start((sl1 + sl)[0]), al1 ⊑ al2, ¬ start(sl1[0]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name:get-append-less, seq: ⊦ n < # x → (x + y)[n] = x[n], subst:[n, x, y] → [0, sl1, sl]; ['a] → ['s], <all paths>

Additional information

Used simplifier rules: ⊦ ¬ n < # x → (x + y)[n] = y[n - # x]; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ 0 < m ↔ m ≠ 0; (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 Used terms: 0, sl1, sl

Proof step 17 from proof pfxclosed-execstar (previous step) (back to tree)

⊦ n < # x → (x + y)[n] = x[n]

A spec-lemma

Proof Tree for pfxclosed-execstar (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