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Proof step 1 from proof conttr-rel-thm (conclusion) (back to tree)

⊦ pcont(lift(trlp))(βf) ∧ mon(⌜λ n. ⌜slf(n)⌝ ⌝ ) ∧ mon(⌜λ n. ⌜alf(n)⌝ ⌝ ) ∧ (∀ n. trlp(slf(n), alf(n))) ∧ lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )) ∧ pfxclosed(trlp) ∧ (∀ n. # slf(n) = # alf(n) + 1) → lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) = βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ ))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof conttr-rel-thm (previous step) (back to tree)

lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), pcont(lift(trlp))(βf), pfxclosed(trlp), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: conttr-relative-def: ⊦ pcont(trp)(βf) ↔ pmon(trp)(βf) ∧ ppbd(trp)(βf); montr-relative-def: ⊦ pmon(trp)(βf) ↔ (∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ trp(σ1, α1) ∧ trp(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2));

Proof step 3 from proof conttr-rel-thm (previous step) (back to tree)

lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), pfxclosed(trlp), ppbd(lift(trlp))(βf), ∀ n. # slf(n) = # alf(n) + 1, ∀ n. trlp(slf(n), alf(n)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:conttr-relative-thm, seq:⊦ pcont(trp)(βf) ∧ mon(⌜σf⌝ ) ∧ mon(⌜αf⌝ ) ∧ (∀ n. trp(σf(n), αf(n))) ∧ trp(lim(⌜σf⌝ ), lim(⌜αf⌝ )) ∧ pfxclosed(trp) ∧ (∀ n. # σf(n) = # αf(n) + ⌜1⌝ ) → lim(⌜λ n. βf(σf(n), αf(n))⌝ ) = βf(lim(⌜σf⌝ ), lim(⌜αf⌝ )), subst:[trp, βf, σf, αf] → [lift(trlp), βf, λ n. ⌜slf(n)⌝ , λ n. ⌜alf(n)⌝ ]; [S, A, B] → [S, A, B]

Additional information

Used terms: lift(trlp), βf, λ n. ⌜slf(n)⌝ , λ n. ⌜alf(n)⌝

Proof step 4 from proof conttr-rel-thm (previous step) (back to tree)

⊦ pcont(trp)(βf) ∧ mon(⌜σf⌝ ) ∧ mon(⌜αf⌝ ) ∧ (∀ n. trp(σf(n), αf(n))) ∧ trp(lim(⌜σf⌝ ), lim(⌜αf⌝ )) ∧ pfxclosed(trp) ∧ (∀ n. # σf(n) = # αf(n) + ⌜1⌝ ) → lim(⌜λ n. βf(σf(n), αf(n))⌝ ) = βf(lim(⌜σf⌝ ), lim(⌜αf⌝ ))

Lemma conttr-relative-thm

Proof step 5 from proof conttr-rel-thm (previous step) (back to tree)

lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), pfxclosed(trlp), ppbd(lift(trlp))(βf), pcont(lift(trlp))(βf) ∧ mon(⌜λ n. ⌜slf(n)⌝ ⌝ ) ∧ mon(⌜λ n. ⌜alf(n)⌝ ⌝ ) ∧ (∀ n. lift(trlp)(⌜slf(n)⌝ , ⌜alf(n)⌝ )) ∧ lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )) ∧ pfxclosed(lift(trlp)) ∧ (∀ n. # ⌜slf(n)⌝ = # ⌜alf(n)⌝ + ⌜1⌝ ) → lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) = βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: conttr-relative-def: ⊦ pcont(trp)(βf) ↔ pmon(trp)(βf) ∧ ppbd(trp)(βf); montr-relative-def: ⊦ pmon(trp)(βf) ↔ (∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ trp(σ1, α1) ∧ trp(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2)); lift-trp-fin: ⊦ lift(trlp)(⌜sl⌝ , ⌜al⌝ ) ↔ trlp(sl, al); add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ = ⌜n0⌝ ↔ n + m = n0; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

Proof step 6 from proof conttr-rel-thm (previous step) (back to tree)

lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), pfxclosed(trlp), ppbd(lift(trlp))(βf), ∀ n. # slf(n) = # alf(n) + 1, ∀ n. trlp(slf(n), alf(n)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2) ⊦ trlp(slf(n), alf(n)) ∧ pfxclosed(lift(trlp)) ∧ # alf(n0) + 1 = # slf(n0)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 7 from proof conttr-rel-thm (previous step) (back to tree)

lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), pfxclosed(trlp), ppbd(lift(trlp))(βf), ¬ trlp(slf(n), alf(n)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 9 terms: n

Additional information

Used terms: n

Proof step 8 from proof conttr-rel-thm (previous step) (back to tree)

lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), pfxclosed(trlp), ppbd(lift(trlp))(βf), ¬ pfxclosed(lift(trlp)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:pdfclosedtr-def, seq:⊦ pfxclosed(trp) ↔ (∀ σ1, σ2, α1, α2. trp(σ2, α2) ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ → trp(σ1, α1)), subst:[trp] → [lift(trlp)]; [S, A] → [S, A], lhs:pfxclosed(lift(trlp)), <all paths>

Additional information

Used terms: lift(trlp)

Proof step 9 from proof conttr-rel-thm (previous step) (back to tree)

⊦ pfxclosed(trp) ↔ (∀ σ1, σ2, α1, α2. trp(σ2, α2) ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ → trp(σ1, α1))

Lemma pdfclosedtr-def

Proof step 10 from proof conttr-rel-thm (previous step) (back to tree)

lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), pfxclosed(trlp), ppbd(lift(trlp))(βf), ∀ n. # slf(n) = # alf(n) + 1, ∀ n. trlp(slf(n), alf(n)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∃ σ1, σ2, α1, α2. ¬(lift(trlp)(σ2, α2) ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ → lift(trlp)(σ1, α1)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 11 from proof conttr-rel-thm (previous step) (back to tree)

# σ1 = # α1 + ⌜1⌝ , # σ2 = # α2 + ⌜1⌝ , lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(σ2, α2), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), pfxclosed(trlp), ppbd(lift(trlp))(βf), σ1 ⊑ σ2, α1 ⊑ α2, ¬ lift(trlp)(σ1, α1), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, 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] → [trlp]; [S, A] → [S, A], lhs:pfxclosed(trlp), <all paths>

Additional information

Used terms: trlp

Proof step 12 from proof conttr-rel-thm (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))

Lemma pdfclosedtrl-def

Proof step 13 from proof conttr-rel-thm (previous step) (back to tree)

# σ2 = # α2 + ⌜1⌝ , # σ1 = # α1 + ⌜1⌝ , lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(σ2, α2), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), α1 ⊑ α2, σ1 ⊑ σ2, ¬ lift(trlp)(σ1, α1), ∀ n. # slf(n) = # alf(n) + 1, ∀ n. trlp(slf(n), alf(n)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: σ2

Additional information

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

Proof step 14 from proof conttr-rel-thm (previous step) (back to tree)

# σ1 = # α1 + ⌜1⌝ , # σ2 = # α2 + ⌜1⌝ , σ2 = ⌜sl⌝ , lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(σ2, α2), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), σ1 ⊑ σ2, α1 ⊑ α2, ¬ lift(trlp)(σ1, α1), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

Proof step 15 from proof conttr-rel-thm (previous step) (back to tree)

# σ1 = # α1 + ⌜1⌝ , ⌜# sl⌝ = # α2 + ⌜1⌝ , lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(⌜sl⌝ , α2), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), α1 ⊑ α2, σ1 ⊑ ⌜sl⌝ , ¬ lift(trlp)(σ1, α1), ∀ n. # slf(n) = # alf(n) + 1, ∀ n. trlp(slf(n), alf(n)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, 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⌝ = ∞; (s) ⊦ ⌜m⌝ + ∞ = ∞; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s)

Proof step 16 from proof conttr-rel-thm (previous step) (back to tree)

# σ1 = # α1 + ⌜1⌝ , α2 = ⌜al⌝ , ⌜# sl⌝ = # α2 + ⌜1⌝ , lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(⌜sl⌝ , α2), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), σ1 ⊑ ⌜sl⌝ , α1 ⊑ α2, ¬ lift(trlp)(σ1, α1), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-trp-fin: ⊦ lift(trlp)(⌜sl⌝ , ⌜al⌝ ) ↔ trlp(sl, al); add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ = ⌜n0⌝ ↔ n + m = n0; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

Proof step 17 from proof conttr-rel-thm (previous step) (back to tree)

# σ1 = # α1 + ⌜1⌝ , # al + 1 = # sl, lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), trlp(sl, al), α1 ⊑ ⌜al⌝ , σ1 ⊑ ⌜sl⌝ , ¬ lift(trlp)(σ1, α1), ∀ n. # slf(n) = # alf(n) + 1, ∀ n. trlp(slf(n), alf(n)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: σ1

Additional information

Used simplifier rules: Stream[A] freely generated by ⌝, ⌝; ⊦ # ⌜nf⌝ = ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ ¬ ⌜nf⌝ ⊑ ⌜x⌝ ; (s)

Proof step 18 from proof conttr-rel-thm (previous step) (back to tree)

# σ1 = # α1 + ⌜1⌝ , # al + 1 = # sl, σ1 = ⌜sl0⌝ , lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), trlp(sl, al), σ1 ⊑ ⌜sl⌝ , α1 ⊑ ⌜al⌝ , ¬ lift(trlp)(σ1, α1), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ ⊑ ⌜y⌝ ↔ x ⊑ y; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

Proof step 19 from proof conttr-rel-thm (previous step) (back to tree)

# al + 1 = # sl, ⌜# sl0⌝ = # α1 + ⌜1⌝ , lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), trlp(sl, al), α1 ⊑ ⌜al⌝ , sl0 ⊑ sl, ¬ lift(trlp)(⌜sl0⌝ , α1), ∀ n. # slf(n) = # alf(n) + 1, ∀ n. trlp(slf(n), alf(n)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, 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 ⌝, ⌝; ⊦ # ⌜nf⌝ = ∞; (s) ⊦ ⌜m⌝ + ∞ = ∞; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s)

Proof step 20 from proof conttr-rel-thm (previous step) (back to tree)

# al + 1 = # sl, α1 = ⌜al0⌝ , ⌜# sl0⌝ = # α1 + ⌜1⌝ , lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), trlp(sl, al), sl0 ⊑ sl, α1 ⊑ ⌜al⌝ , ¬ lift(trlp)(⌜sl0⌝ , α1), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-trp-fin: ⊦ lift(trlp)(⌜sl⌝ , ⌜al⌝ ) ↔ trlp(sl, al); ⊦ ⌜x⌝ ⊑ ⌜y⌝ ↔ x ⊑ y; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ = ⌜n0⌝ ↔ n + m = n0; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

Proof step 21 from proof conttr-rel-thm (previous step) (back to tree)

# al + 1 = # sl, # al0 + 1 = # sl0, lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), trlp(sl, al), sl0 ⊑ sl, al0 ⊑ al, ¬ trlp(sl0, al0), ∀ n. # slf(n) = # alf(n) + 1, ∀ n. trlp(slf(n), alf(n)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 15 terms: sl0, sl, al0, al abort: F discard: T computed: T

Additional information

Used terms: sl0, sl, al0, al

Proof step 22 from proof conttr-rel-thm (previous step) (back to tree)

# σ1 = # α1 + ⌜1⌝ , # σ2 = # α2 + ⌜1⌝ , σ2 = ⌜sf⌝ , lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(σ2, α2), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), σ1 ⊑ σ2, α1 ⊑ α2, ¬ lift(trlp)(σ1, α1), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ # ⌜nf⌝ = ∞; (s)

Proof step 23 from proof conttr-rel-thm (previous step) (back to tree)

# σ1 = # α1 + ⌜1⌝ , # α2 = ∞, lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(⌜sf⌝ , α2), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), α1 ⊑ α2, σ1 ⊑ ⌜sf⌝ , ¬ lift(trlp)(σ1, α1), ∀ n. # slf(n) = # alf(n) + 1, ∀ n. trlp(slf(n), alf(n)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, 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 ⌝, ⌝; ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜n⌝ ≠ ∞; (s)

Proof step 24 from proof conttr-rel-thm (previous step) (back to tree)

# α2 = ∞, # σ1 = # α1 + ⌜1⌝ , α2 = ⌜af⌝ , lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(⌜sf⌝ , α2), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), σ1 ⊑ ⌜sf⌝ , α1 ⊑ α2, ¬ lift(trlp)(σ1, α1), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-trp-inf: ⊦ lift(trlp)(⌜sf⌝ , ⌜af⌝ ) ↔ (∀ n. trlp(sf.take(n + 1), af.take(n))); ⊦ # ⌜nf⌝ = ∞; (s)

Proof step 25 from proof conttr-rel-thm (previous step) (back to tree)

# σ1 = # α1 + ⌜1⌝ , lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), α1 ⊑ ⌜af⌝ , σ1 ⊑ ⌜sf⌝ , ¬ lift(trlp)(σ1, α1), ∀ n. # slf(n) = # alf(n) + 1, ∀ n. trlp(slf(n), alf(n)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ n. trlp(sf.take(n + 1), af.take(n)) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: σ1

Additional information

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

Proof step 26 from proof conttr-rel-thm (previous step) (back to tree)

# σ1 = # α1 + ⌜1⌝ , σ1 = ⌜sl⌝ , lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), σ1 ⊑ ⌜sf⌝ , α1 ⊑ ⌜af⌝ , ¬ lift(trlp)(σ1, α1), ∀ n. trlp(sf.take(n + 1), af.take(n)), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

Proof step 27 from proof conttr-rel-thm (previous step) (back to tree)

⌜# sl⌝ = # α1 + ⌜1⌝ , lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), α1 ⊑ ⌜af⌝ , ⌜sl⌝ ⊑ ⌜sf⌝ , ¬ lift(trlp)(⌜sl⌝ , α1), ∀ n. # slf(n) = # alf(n) + 1, ∀ n. trlp(slf(n), alf(n)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ n. trlp(sf.take(n + 1), af.take(n)) ⊦

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 ⌝, ⌝; ⊦ # ⌜nf⌝ = ∞; (s) ⊦ ⌜m⌝ + ∞ = ∞; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s)

Proof step 28 from proof conttr-rel-thm (previous step) (back to tree)

α1 = ⌜al⌝ , ⌜# sl⌝ = # α1 + ⌜1⌝ , lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), ⌜sl⌝ ⊑ ⌜sf⌝ , α1 ⊑ ⌜af⌝ , ¬ lift(trlp)(⌜sl⌝ , α1), ∀ n. trlp(sf.take(n + 1), af.take(n)), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-trp-fin: ⊦ lift(trlp)(⌜sl⌝ , ⌜al⌝ ) ↔ trlp(sl, al); add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ = ⌜n0⌝ ↔ n + m = n0; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

Proof step 29 from proof conttr-rel-thm (previous step) (back to tree)

# al + 1 = # sl, lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), ⌜al⌝ ⊑ ⌜af⌝ , ⌜sl⌝ ⊑ ⌜sf⌝ , ¬ trlp(sl, al), ∀ n. # slf(n) = # alf(n) + 1, ∀ n. trlp(slf(n), alf(n)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ n. trlp(sf.take(n + 1), af.take(n)) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 14 terms: # al abort: F discard: T computed: F

Additional information

Used terms: # al

Proof step 30 from proof conttr-rel-thm (previous step) (back to tree)

# al + 1 = # sl, lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), trlp(sf.take(# al + 1), af.take(# al)), ⌜sl⌝ ⊑ ⌜sf⌝ , ⌜al⌝ ⊑ ⌜af⌝ , ¬ trlp(sl, al), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 31 from proof conttr-rel-thm (previous step) (back to tree)

# al + 1 = # sl, lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), trlp(sf.take(# sl), af.take(# al)), ⌜al⌝ ⊑ ⌜af⌝ , ⌜sl⌝ ⊑ ⌜sf⌝ , ¬ trlp(sl, al), ∀ n. # slf(n) = # alf(n) + 1, ∀ n. trlp(slf(n), alf(n)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1) ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: stream, name:elim-fromto-inf, seq: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n, subst:[nf, n, x0, nf0] → [sf, # sl, sl0, nf0]; [A] → [S]

Additional information

Used simplifier rules: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n; (elim) Used terms: sf, # sl, sl0, nf0

Proof step 32 from proof conttr-rel-thm (previous step) (back to tree)

# sl0 = # sl, # al + 1 = # sl, sf = (⌜sl0⌝ + ⌜nf0⌝ ).f, lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), trlp(sl0, af.take(# al)), ⌜sl⌝ ⊑ ⌜sf⌝ , ⌜al⌝ ⊑ ⌜af⌝ , ¬ trlp(sl, al), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x + nf⌝ = ⌜x⌝ + ⌜nf⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ (⌜x⌝ + ⌜nf⌝ ).f = x + nf; (s)

Proof step 33 from proof conttr-rel-thm (previous step) (back to tree)

# sl0 = # sl, # al + 1 = # sl, lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), trlp(sl0, af.take(# al)), ⌜al⌝ ⊑ ⌜af⌝ , ⌜sl⌝ ⊑ ⌜sl0⌝ + ⌜nf0⌝ , ¬ trlp(sl, al), ∀ n. # slf(n) = # alf(n) + 1, ∀ n. trlp(slf(n), alf(n)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1) ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: stream, name:elim-fromto-inf, seq: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n, subst:[nf, n, x0, nf0] → [af, # al, al0, af0]; [A] → [A]

Additional information

Used simplifier rules: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n; (elim) Used terms: af, # al, al0, af0

Proof step 34 from proof conttr-rel-thm (previous step) (back to tree)

# sl0 = # sl, # al0 = # al, # al + 1 = # sl, af = (⌜al0⌝ + ⌜af0⌝ ).f, lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), trlp(sl0, al0), ⌜sl⌝ ⊑ ⌜sl0⌝ + ⌜nf0⌝ , ⌜al⌝ ⊑ ⌜af⌝ , ¬ trlp(sl, al), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x + nf⌝ = ⌜x⌝ + ⌜nf⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ (⌜x⌝ + ⌜nf⌝ ).f = x + nf; (s)

Proof step 35 from proof conttr-rel-thm (previous step) (back to tree)

# al0 = # al, # sl0 = # sl, # al + 1 = # sl, lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), trlp(sl0, al0), ⌜sl⌝ ⊑ ⌜sl0⌝ + ⌜nf0⌝ , ⌜al⌝ ⊑ ⌜al0⌝ + ⌜af0⌝ , ¬ trlp(sl, al), ∀ n. # slf(n) = # alf(n) + 1, ∀ n. trlp(slf(n), alf(n)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1) ⊦

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⌝ , ⌜al0⌝ + ⌜af0⌝ ]; [A] → [A], lhs:⌜al⌝ ⊑ ⌜al0⌝ + ⌜af0⌝ , <all paths>

Additional information

Used terms: ⌜al⌝ , ⌜al0⌝ + ⌜af0⌝

Proof step 36 from proof conttr-rel-thm (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 37 from proof conttr-rel-thm (previous step) (back to tree)

# sl0 = # sl, # al0 = # al, # al + 1 = # sl, lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), trlp(sl0, al0), ⌜sl⌝ ⊑ ⌜sl0⌝ + ⌜nf0⌝ , ¬ trlp(sl, al), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1, ∃ α. ⌜al⌝ + α = ⌜al0⌝ + ⌜af0⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # x = # y → (⌜x⌝ + σ = ⌜y⌝ + σ0 ↔ x = y ∧ σ = σ0); (s)

Proof step 38 from proof conttr-rel-thm (previous step) (back to tree)

# sl0 = # sl, # al0 + 1 = # sl, lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), trlp(sl0, al0), ⌜sl⌝ ⊑ ⌜sl0⌝ + ⌜nf0⌝ , ¬ trlp(sl, al0), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

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] → [⌜sl⌝ , ⌜sl0⌝ + ⌜nf0⌝ ]; [A] → [S], lhs:⌜sl⌝ ⊑ ⌜sl0⌝ + ⌜nf0⌝ , <all paths>

Additional information

Used simplifier rules: ⊦ # x = # y → (⌜x⌝ + σ = ⌜y⌝ + σ0 ↔ x = y ∧ σ = σ0); (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: ⌜sl⌝ , ⌜sl0⌝ + ⌜nf0⌝

Proof step 39 from proof conttr-rel-thm (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 40 from proof conttr-rel-thm (previous step) (back to tree)

# σ1 = # α1 + ⌜1⌝ , σ1 = ⌜sf0⌝ , lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), σ1 ⊑ ⌜sf⌝ , α1 ⊑ ⌜af⌝ , ¬ lift(trlp)(σ1, α1), ∀ n. trlp(sf.take(n + 1), af.take(n)), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ ⊑ ⌜nf0⌝ ↔ nf = nf0; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ # ⌜nf⌝ = ∞; (s)

Proof step 41 from proof conttr-rel-thm (previous step) (back to tree)

# α1 = ∞, lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), α1 ⊑ ⌜af⌝ , ¬ lift(trlp)(⌜sf⌝ , α1), ∀ n. trlp(sf.take(n + 1), af.take(n)), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

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 ⌝, ⌝; ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜n⌝ ≠ ∞; (s)

Proof step 42 from proof conttr-rel-thm (previous step) (back to tree)

# α1 = ∞, α1 = ⌜af0⌝ , lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), α1 ⊑ ⌜af⌝ , ¬ lift(trlp)(⌜sf⌝ , α1), ∀ n. # slf(n) = # alf(n) + 1, ∀ n. trlp(slf(n), alf(n)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ n. trlp(sf.take(n + 1), af.take(n)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-trp-inf: ⊦ lift(trlp)(⌜sf⌝ , ⌜af⌝ ) ↔ (∀ n. trlp(sf.take(n + 1), af.take(n))); ⊦ ⌜nf⌝ ⊑ ⌜nf0⌝ ↔ nf = nf0; (s) ⊦ # ⌜nf⌝ = ∞; (s)

Proof step 43 from proof conttr-rel-thm (previous step) (back to tree)

lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), ppbd(lift(trlp))(βf), ¬ trlp(sf.take(n + 1), af.take(n)), ∀ n. # slf(n) = # alf(n) + 1, ∀ n. trlp(slf(n), alf(n)), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ sl1, sl2, al1, al2. trlp(sl2, al2) ∧ sl1 ⊑ sl2 ∧ al1 ⊑ al2 ∧ # sl1 = # al1 + 1 ∧ # sl2 = # al2 + 1 → trlp(sl1, al1), ∀ n. trlp(sf.take(n + 1), af.take(n)) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 11 terms: n

Additional information

Used terms: n

Proof step 44 from proof conttr-rel-thm (previous step) (back to tree)

# alf(n0) + 1 ≠ # slf(n0), lim(⌜λ n. βf(⌜slf(n)⌝ , ⌜alf(n)⌝ )⌝ ) ≠ βf(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), lift(trlp)(lim(⌜λ n. ⌜slf(n)⌝ ⌝ ), lim(⌜λ n. ⌜alf(n)⌝ ⌝ )), mon(⌜λ n. ⌜slf(n)⌝ ⌝ ), mon(⌜λ n. ⌜alf(n)⌝ ⌝ ), pfxclosed(trlp), ppbd(lift(trlp))(βf), ∀ σ1, α1, σ2, α2. # σ1 = # α1 + ⌜1⌝ ∧ # σ2 = # α2 + ⌜1⌝ ∧ σ1 ⊑ σ2 ∧ α1 ⊑ α2 ∧ lift(trlp)(σ1, α1) ∧ lift(trlp)(σ2, α2) → βf(σ1, α1) ⊑ βf(σ2, α2), ∀ n. trlp(slf(n), alf(n)), ∀ n. # slf(n) = # alf(n) + 1 ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 10 terms: n0

Additional information

Used terms: n0

Proof Tree for conttr-rel-thm (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