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

⊦ pfxclosed(alp) → pfxclosed(lift(alp))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

pfxclosed(alp), ¬ pfxclosed(lift(alp)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:pfxclosed-def, seq: ⊦ pfxclosed(αp) ↔ (∀ α1, α2. αp(α2) ∧ α1 ⊑ α2 → αp(α1)), subst:[αp] → [lift(alp)]; [A] → [A], lhs:pfxclosed(lift(alp)), <all paths>

Additional information

Used terms: lift(alp)

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

⊦ pfxclosed(αp) ↔ (∀ α1, α2. αp(α2) ∧ α1 ⊑ α2 → αp(α1))

Lemma pfxclosed-def

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

pfxclosed(alp), ∃ α1, α2. ¬(lift(alp)(α2) ∧ α1 ⊑ α2 → lift(alp)(α1)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

lift(alp)(α2), pfxclosed(alp), α1 ⊑ α2, ¬ lift(alp)(α1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:pfxclosedl-def, seq: ⊦ pfxclosed(alp) ↔ (∀ al1, al2. alp(al2) ∧ al1 ⊑ al2 → alp(al1)), subst:[alp] → [alp]; [A] → [A], lhs:pfxclosed(alp), <all paths>

Additional information

Used terms: alp

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

⊦ pfxclosed(alp) ↔ (∀ al1, al2. alp(al2) ∧ al1 ⊑ al2 → alp(al1))

Lemma pfxclosedl-def

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

lift(alp)(α2), α1 ⊑ α2, ¬ lift(alp)(α1), ∀ al1, al2. alp(al2) ∧ al1 ⊑ al2 → alp(al1) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: α1

Additional information

Used simplifier rules: Stream[A] freely generated by ⌝, ⌝;

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

α1 = ⌜al⌝ , lift(alp)(α2), α1 ⊑ α2, ¬ lift(alp)(α1), ∀ al1, al2. alp(al2) ∧ al1 ⊑ al2 → alp(al1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-pred-fin: ⊦ lift(alp)(⌜al⌝ ) ↔ alp(al);

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

lift(alp)(α2), ⌜al⌝ ⊑ α2, ¬ alp(al), ∀ al1, al2. alp(al2) ∧ al1 ⊑ al2 → alp(al1) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: α2

Additional information

Used simplifier rules: Stream[A] freely generated by ⌝, ⌝;

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

α2 = ⌜al0⌝ , lift(alp)(α2), ⌜al⌝ ⊑ α2, ¬ alp(al), ∀ al1, al2. alp(al2) ∧ al1 ⊑ al2 → alp(al1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-pred-fin: ⊦ lift(alp)(⌜al⌝ ) ↔ alp(al); ⊦ ⌜x⌝ ⊑ ⌜y⌝ ↔ x ⊑ y; (s)

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

alp(al0), al ⊑ al0, ¬ alp(al), ∀ al1, al2. alp(al2) ∧ al1 ⊑ al2 → alp(al1) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: al, al0

Additional information

Used terms: al, al0

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

α2 = ⌜af⌝ , lift(alp)(α2), ⌜al⌝ ⊑ α2, ¬ alp(al), ∀ al1, al2. alp(al2) ∧ al1 ⊑ al2 → alp(al1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-pred-inf: ⊦ lift(alp)(⌜af⌝ ) ↔ (∀ n. alp(af.take(n)));

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

⌜al⌝ ⊑ ⌜af⌝ , ¬ alp(al), ∀ al1, al2. alp(al2) ∧ al1 ⊑ al2 → alp(al1), ∀ n. alp(af.take(n)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name:prefix-def, seq: ⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0), subst:[σ, σ0] → [⌜al⌝ , ⌜af⌝ ]; [A] → [A], lhs:⌜al⌝ ⊑ ⌜af⌝ , <all paths>

Additional information

Used terms: ⌜al⌝ , ⌜af⌝

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

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

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

¬ alp(al), ∀ n. alp(af.take(n)), ∀ al1, al2. alp(al2) ∧ al1 ⊑ al2 → alp(al1), ∃ α. ⌜al⌝ + α = ⌜af⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ .f = nf; (s) ⊦ finp(σ) ↔ ¬(∃ nf. ⌜nf⌝ = σ); (ss) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜x⌝ + σ = ⌜nf⌝ ↔ ¬ finp(σ) ∧ x + σ.f = nf; (as)

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

¬ alp(al), ∀ al1, al2. alp(al2) ∧ al1 ⊑ al2 → alp(al1), ∀ n. alp((al + af0).take(n)) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 3 terms: # al abort: F discard: T computed: F

Additional information

Used terms: # al

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

alp((al + af0).take(# al)), ¬ alp(al), ∀ al1, al2. alp(al2) ∧ al1 ⊑ al2 → alp(al1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name:infapp-to-ge, seq: ⊦ # x ≤ n → (x + nf).take(n) = x + nf.take(n - # x), subst:[x, nf, n] → [al, af0, # al]; [A] → [A], lhs:(al + af0).take(# al), <all paths>

Additional information

Used simplifier rules: ⊦ m ≤ m; (s) ⊦ m - m = 0; (s) ⊦ nf.take(0) = []; (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ ¬ n < n; (s) Used terms: al, af0, # al

Proof step 18 from proof pfxclosed-lift (previous step) (back to tree)

⊦ # x ≤ n → (x + nf).take(n) = x + nf.take(n - # x)

A spec-lemma

Proof step 19 from proof pfxclosed-lift (previous step) (back to tree)

α1 = ⌜af⌝ , lift(alp)(α2), α1 ⊑ α2, ¬ lift(alp)(α1), ∀ al1, al2. alp(al2) ∧ al1 ⊑ al2 → alp(al1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-pred-inf: ⊦ lift(alp)(⌜af⌝ ) ↔ (∀ n. alp(af.take(n)));

Proof step 20 from proof pfxclosed-lift (previous step) (back to tree)

lift(alp)(α2), ⌜af⌝ ⊑ α2, ¬ alp(af.take(n)), ∀ al1, al2. alp(al2) ∧ al1 ⊑ al2 → alp(al1) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: α2

Additional information

Used simplifier rules: Stream[A] freely generated by ⌝, ⌝; ⊦ ¬ ⌜nf⌝ ⊑ ⌜x⌝ ; (s)

Proof step 21 from proof pfxclosed-lift (previous step) (back to tree)

α2 = ⌜af0⌝ , lift(alp)(α2), ⌜af⌝ ⊑ α2, ¬ alp(af.take(n)), ∀ al1, al2. alp(al2) ∧ al1 ⊑ al2 → alp(al1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-pred-inf: ⊦ lift(alp)(⌜af⌝ ) ↔ (∀ n. alp(af.take(n))); ⊦ ⌜nf⌝ ⊑ ⌜nf0⌝ ↔ nf = nf0; (s)

Proof step 22 from proof pfxclosed-lift (previous step) (back to tree)

¬ alp(af0.take(n)), ∀ n. alp(af0.take(n)), ∀ al1, al2. alp(al2) ∧ al1 ⊑ al2 → alp(al1) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 2 terms: n

Additional information

Used terms: n

Proof Tree for pfxclosed-lift (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