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Proof step 1 from proof swap-to-from-app (conclusion) (back to tree)

⊦ m ≤ n ∧ ¬ # σ < ⌜n⌝ → (σ to n) + σ0 from m = ((σ from m) to (n - m)) + σ0

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof swap-to-from-app (previous step) (back to tree)

(σ to n) + σ0 from m ≠ ((σ from m) to (n - m)) + σ0, m ≤ n, ¬ # σ < ⌜n⌝ ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: nat, name:elim-sub, seq:n ≤ m ⊦ n0 = m - n ↔ m = n0 + n, subst:[m, n, n0] → [n, m, n0]

Additional information

Used simplifier rules: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: n, m, n0

Proof step 3 from proof swap-to-from-app (previous step) (back to tree)

n = n0 + m, (σ to n) + σ0 from m ≠ ((σ from m) to n0) + σ0, m ≤ n, ¬ # σ < ⌜n⌝ ⊦

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⌝ + ⌜n⌝ ; (s) ⊦ m ≤ m + n; (s)

Proof step 4 from proof swap-to-from-app (previous step) (back to tree)

(σ to n0 + m) + σ0 from m ≠ ((σ from m) to n0) + σ0, ¬ # σ < ⌜n0⌝ + ⌜m⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:extensionality, seq: ⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0[n] = σ1[n]), subst:[σ0, σ1] → [(σ to n0 + m) + σ0 from m, ((σ from m) to n0) + σ0], <all paths>

Additional information

Used terms: (σ to n0 + m) + σ0 from m, ((σ from m) to n0) + σ0

Proof step 5 from proof swap-to-from-app (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0[n] = σ1[n])

Lemma extensionality

Proof step 6 from proof swap-to-from-app (previous step) (back to tree)

¬ # σ < ⌜n0⌝ + ⌜m⌝ , #((σ to n0 + m) + σ0 from m) ≠ #(((σ from m) to n0) + σ0) ∨ (∃ n. ¬(⌜n⌝ < #((σ to n0 + m) + σ0 from m) → ((σ to n0 + m) + σ0 from m)[n] = (((σ from m) to n0) + σ0)[n])) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: size-append: ⊦ #(σ + σ1) = # σ + # σ1; append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2;

Proof step 7 from proof swap-to-from-app (previous step) (back to tree)

¬ # σ < ⌜n0⌝ + ⌜m⌝ , #((σ to n0 + m) + σ0 from m) ≠ #((σ from m) to n0) + # σ0 ∨ ¬(⌜n⌝ < #((σ to n0 + m) + σ0 from m) → ((σ to n0 + m) + σ0 from m)[n] = (((σ from m) to n0) + σ0)[n]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:get-from, seq: ⊦ ⌜n⌝ + ⌜m⌝ < # σ → (σ from n)[m] = σ[n + m], subst:[σ, n, m] → [(σ to n0 + m) + σ0, m, n], <all paths>

Additional information

Used terms: (σ to n0 + m) + σ0, m, n

Proof step 8 from proof swap-to-from-app (previous step) (back to tree)

⊦ ⌜n⌝ + ⌜m⌝ < # σ → (σ from n)[m] = σ[n + m]

Lemma get-from

Proof step 9 from proof swap-to-from-app (previous step) (back to tree)

¬ ⌜m⌝ + ⌜n⌝ < #((σ to n0 + m) + σ0), ¬ # σ < ⌜n0⌝ + ⌜m⌝ , #((σ to n0 + m) + σ0 from m) ≠ #((σ from m) to n0) + # σ0 ∨ ¬(⌜n⌝ < #((σ to n0 + m) + σ0 from m) → ((σ to n0 + m) + σ0 from m)[n] = (((σ from m) to n0) + σ0)[n]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: size-append: ⊦ #(σ + σ1) = # σ + # σ1; size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; 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⌝ + n∞1 < ⌜n⌝ + n∞2 ↔ n∞1 < n∞2; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

Proof step 10 from proof swap-to-from-app (previous step) (back to tree)

¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ ⌜n⌝ < ⌜n0⌝ + # σ0, #((σ to n0 + m) + σ0 from m) ≠ #((σ from m) to n0) + # σ0 ∨ ¬(⌜n⌝ < #((σ to n0 + m) + σ0 from m) → ((σ to n0 + m) + σ0 from m)[n] = (((σ from m) to n0) + σ0)[n]) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: σ0

Additional information

Used simplifier rules: size-inf: ⊦ # ⌜nf⌝ = ∞; stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;; ⊦ ⌜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 ⊦ ⌜m⌝ < ∞; (s)

Proof step 11 from proof swap-to-from-app (previous step) (back to tree)

σ0 = ⌜x⌝ , ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ ⌜n⌝ < ⌜n0⌝ + # σ0, #((σ to n0 + m) + σ0 from m) ≠ #((σ from m) to n0) + # σ0 ∨ ¬(⌜n⌝ < #((σ to n0 + m) + σ0 from m) → ((σ to n0 + m) + σ0 from m)[n] = (((σ from m) to n0) + σ0)[n]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: size-fin: ⊦ # ⌜x⌝ = ⌜# x⌝ ; 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 ≤ m; (ss) ⊦ ⌜n⌝ < ⌜m⌝ + ⌜n0⌝ ↔ n < m + n0; (s)

Proof step 12 from proof swap-to-from-app (previous step) (back to tree)

n0 + # x ≤ n, ¬ # σ < ⌜n0⌝ + ⌜m⌝ , #((σ to n0 + m) + ⌜x⌝ from m) ≠ #((σ from m) to n0) + ⌜# x⌝ ∨ ¬(⌜n⌝ < #((σ to n0 + m) + ⌜x⌝ from m) → ((σ to n0 + m) + ⌜x⌝ from m)[n] = (((σ from m) to n0) + ⌜x⌝ )[n]) ⊦

User applied rule constructor cut

And closed the goal!

Rule argument

var: σ

Additional information

Used simplifier rules: from-fin: ⊦ ⌜x⌝ from n = ⌜x from n⌝ ; to-fin: ⊦ ⌜x⌝ to n = ⌜x to n⌝ ; from-appfin: ⊦ ⌜x⌝ + ⌜y⌝ from n = ⌜x + y from n⌝ ; append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; size-fin: ⊦ # ⌜x⌝ = ⌜# x⌝ ; length-from: ¬ # x < n ⊦ #(x from n) = # x - n; length-to: ¬ # x < n ⊦ #(x to n) = n; size-inf: ⊦ # ⌜nf⌝ = ∞; from-inf: ⊦ ⌜nf⌝ from n = ⌜nf from n⌝ ; inf-size-to: ⊦ # ⌜nf to n⌝ = ⌜n⌝ ; to-inf: ⊦ ⌜nf⌝ to n = ⌜nf to n⌝ ; from-appfin: ⊦ ⌜x⌝ + ⌜y⌝ from n = ⌜x + y from n⌝ ; append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; size-fin: ⊦ # ⌜x⌝ = ⌜# x⌝ ; length-from: ¬ # x < n ⊦ #(x from n) = # x - n; size-inf-to: ⊦ #(nf to n) = n; stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;; ⊦ ¬ ∞ < n∞; (s) ⊦ ⌜n⌝ + ⌜m⌝ = ⌜n0⌝ + ⌜n1⌝ ↔ n + m = n0 + n1; (s) ⊦ ¬ m + n < m; (s) ⊦ #(x + y) = # x + # y; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m + n - m = n; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ ⌜n⌝ < ⌜m⌝ + ⌜n0⌝ ↔ n < m + n0; (s) ⊦ m ≤ n → (n < m ↔ false); (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 ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m ≤ n ↔ ¬ n < m; (ss) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m < n → n ≠ 0; (s) n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s)

Proof step 13 from proof swap-to-from-app (previous step) (back to tree)

⌜m⌝ + ⌜n⌝ < #((σ to n0 + m) + σ0), ¬ # σ < ⌜n0⌝ + ⌜m⌝ , #((σ to n0 + m) + σ0 from m) ≠ #((σ from m) to n0) + # σ0 ∨ ¬(⌜n⌝ < #((σ to n0 + m) + σ0 from m) → ((σ to n0 + m) + σ0)[m + n] = (((σ from m) to n0) + σ0)[n]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: size-append: ⊦ #(σ + σ1) = # σ + # σ1; size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; 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⌝ + n∞1 < ⌜n⌝ + n∞2 ↔ n∞1 < n∞2; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

Proof step 14 from proof swap-to-from-app (previous step) (back to tree)

⌜n⌝ < ⌜n0⌝ + # σ0, ¬ # σ < ⌜n0⌝ + ⌜m⌝ , #((σ to n0 + m) + σ0 from m) ≠ #((σ from m) to n0) + # σ0 ∨ ¬(⌜n⌝ < #((σ to n0 + m) + σ0 from m) → ((σ to n0 + m) + σ0)[m + n] = (((σ from m) to n0) + σ0)[n]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:size-to, seq:¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ , subst:[σ, n] → [σ from m, n0], <all paths>

Additional information

Used terms: σ from m, n0

Proof step 15 from proof swap-to-from-app (previous step) (back to tree)

¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝

Lemma size-to

Proof step 16 from proof swap-to-from-app (previous step) (back to tree)

#(σ from m) < ⌜n0⌝ , ⌜n⌝ < ⌜n0⌝ + # σ0, ¬ # σ < ⌜n0⌝ + ⌜m⌝ , #((σ to n0 + m) + σ0 from m) ≠ #((σ from m) to n0) + # σ0 ∨ ¬(⌜n⌝ < #((σ to n0 + m) + σ0 from m) → ((σ to n0 + m) + σ0)[m + n] = (((σ from m) to n0) + σ0)[n]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:size-from-less, seq: ⊦ ¬ # σ < ⌜n⌝ → (#(σ from n) < ⌜n0⌝ ↔ # σ < ⌜n + n0⌝ ), subst:[σ, n, n0] → [σ, m, n0], <all paths>

Additional information

Used simplifier rules: size-from-less: ⊦ ¬ # σ < ⌜n⌝ → (#(σ from n) < ⌜n0⌝ ↔ # σ < ⌜n + n0⌝ ); less-size-from: ⌜n⌝ < # σ ⊦ ⌜n0⌝ < #(σ from n) ↔ ⌜n0 + n⌝ < # σ; size-append: ⊦ #(σ + σ1) = # σ + # σ1; size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; ⊦ ⌜n⌝ < n∞ + ⌜n⌝ ↔ n∞ ≠ ⌜0⌝ ; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ ⌜n⌝ + n∞1 < ⌜n⌝ + n∞2 ↔ n∞1 < n∞2; (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 lels-trans: ⊦ ¬ n∞1 < n∞ ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) ⊦ ¬ m + n < m; (s) Used terms: σ, m, n0

Proof step 17 from proof swap-to-from-app (previous step) (back to tree)

⊦ ¬ # σ < ⌜n⌝ → (#(σ from n) < ⌜n0⌝ ↔ # σ < ⌜n + n0⌝ )

Lemma size-from-less

Proof step 18 from proof swap-to-from-app (previous step) (back to tree)

⌜n⌝ < ⌜n0⌝ + # σ0, ¬ #(σ from m) < ⌜n0⌝ , ¬ # σ < ⌜n0⌝ + ⌜m⌝ , #((σ to n0 + m) + σ0 from m) ≠ ⌜n0⌝ + # σ0 ∨ ¬(⌜n⌝ < #((σ to n0 + m) + σ0 from m) → ((σ to n0 + m) + σ0)[m + n] = (((σ from m) to n0) + σ0)[n]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:size-from-less, seq: ⊦ ¬ # σ < ⌜n⌝ → (#(σ from n) < ⌜n0⌝ ↔ # σ < ⌜n + n0⌝ ), subst:[σ, n, n0] → [σ, m, n0], <all paths>

Additional information

Used simplifier rules: lels-trans: ⊦ ¬ n∞1 < n∞ ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) ⊦ ¬ m + n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic 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 Used terms: σ, m, n0

Proof step 19 from proof swap-to-from-app (previous step) (back to tree)

⊦ ¬ # σ < ⌜n⌝ → (#(σ from n) < ⌜n0⌝ ↔ # σ < ⌜n + n0⌝ )

Lemma size-from-less

Proof step 20 from proof swap-to-from-app (previous step) (back to tree)

⌜n⌝ < ⌜n0⌝ + # σ0, ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m + n0⌝ , ¬ # σ < ⌜n0⌝ + ⌜m⌝ , #((σ to n0 + m) + σ0 from m) ≠ ⌜n0⌝ + # σ0 ∨ ¬(⌜n⌝ < #((σ to n0 + m) + σ0 from m) → ((σ to n0 + m) + σ0)[m + n] = (((σ from m) to n0) + σ0)[n]) ⊦

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⌝ + ⌜n⌝ ; (s)

Proof step 21 from proof swap-to-from-app (previous step) (back to tree)

⌜n⌝ < ⌜n0⌝ + # σ0, ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ , #((σ to n0 + m) + σ0 from m) ≠ ⌜n0⌝ + # σ0 ∨ ¬(⌜n⌝ < #((σ to n0 + m) + σ0 from m) → ((σ to n0 + m) + σ0)[m + n] = (((σ from m) to n0) + σ0)[n]) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: σ0

Additional information

Used simplifier rules: stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;;

Proof step 22 from proof swap-to-from-app (previous step) (back to tree)

σ0 = ⌜x⌝ , ⌜n⌝ < ⌜n0⌝ + # σ0, ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ , #((σ to n0 + m) + σ0 from m) ≠ ⌜n0⌝ + # σ0 ∨ ¬(⌜n⌝ < #((σ to n0 + m) + σ0 from m) → ((σ to n0 + m) + σ0)[m + n] = (((σ from m) to n0) + σ0)[n]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: less-size-from: ⌜n⌝ < # σ ⊦ ⌜n0⌝ < #(σ from n) ↔ ⌜n0 + n⌝ < # σ; size-append: ⊦ #(σ + σ1) = # σ + # σ1; size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; size-fin: ⊦ # ⌜x⌝ = ⌜# x⌝ ; append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic 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) ⊦ ⌜n⌝ + n∞1 < ⌜n⌝ + n∞2 ↔ n∞1 < n∞2; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ m + m0 ≤ m ↔ m0 = 0; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ ⌜n1⌝ < ⌜n0⌝ + ⌜n⌝ + ⌜m⌝ ↔ n1 < n0 + n + m; (s)

Proof step 23 from proof swap-to-from-app (previous step) (back to tree)

n < n0 + # x, ¬ # σ < ⌜m⌝ + ⌜n0⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜n0⌝ + ⌜m⌝ , #((σ to n0 + m) + ⌜x⌝ from m) ≠ ⌜n0⌝ + ⌜# x⌝ ∨ ((σ to n0 + m) + ⌜x⌝ )[m + n] ≠ (((σ from m) to n0) + ⌜x⌝ )[n] ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:fin-size-from2, seq: ⊦ ¬ # σ < ⌜n⌝ ∧ finp(σ) → #(σ from n) = ⌜(# σ).n - n⌝ , subst:[σ, n] → [(σ to n0 + m) + ⌜x⌝ , m], <all paths>

Additional information

Used simplifier rules: size-append: ⊦ #(σ + σ1) = # σ + # σ1; size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; size-fin: ⊦ # ⌜x⌝ = ⌜# x⌝ ; neq-fin-inf-03: ⊦ σ0 + ⌜x⌝ = ⌜nf0⌝ ↔ σ0 = ⌜nf0⌝ ; inf-not-to: ⊦ ⌜nf⌝ ≠ σ to n; append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ ⌜n1⌝ + ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n1 + n + m < n0; (s) ⊦ ¬ m + n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic 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 ⊦ finp(σ) ↔ ¬(∃ nf. ⌜nf⌝ = σ); (ss) Used terms: (σ to n0 + m) + ⌜x⌝ , m

Proof step 24 from proof swap-to-from-app (previous step) (back to tree)

⊦ ¬ # σ < ⌜n⌝ ∧ finp(σ) → #(σ from n) = ⌜(# σ).n - n⌝

Lemma fin-size-from2

Proof step 25 from proof swap-to-from-app (previous step) (back to tree)

n < n0 + # x, finp((σ to n0 + m) + ⌜x⌝ ), ¬ #((σ to n0 + m) + ⌜x⌝ ) < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ⌜(#((σ to n0 + m) + ⌜x⌝ )).n - m⌝ ≠ ⌜n0⌝ + ⌜# x⌝ ∨ ((σ to n0 + m) + ⌜x⌝ )[m + n] ≠ (((σ from m) to n0) + ⌜x⌝ )[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: size-append: ⊦ #(σ + σ1) = # σ + # σ1; size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; size-fin: ⊦ # ⌜x⌝ = ⌜# x⌝ ; append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; sel-fin: ⊦ ⌜x⌝ [n] = x[n]; 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⌝ + ⌜n1⌝ ↔ n + m = n0 + n1; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ m + n - m = n; (s) ⊦ ⌜n⌝ .n = n; (s) n∞ ≠ ∞ ⊦ (⌜m⌝ + n∞).n = m + n∞.n; (s) ⊦ ⌜n⌝ ≠ ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ ¬ m + n < m; (s) ⊦ ⌜n1⌝ + ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n1 + n + m < n0; (s) ⊦ finp(σ) ↔ (∃ x. ⌜x⌝ = σ); (as)

Proof step 26 from proof swap-to-from-app (previous step) (back to tree)

⌜x0⌝ = (σ to n0 + m) + ⌜x⌝ , x0[m + n] ≠ (((σ from m) to n0) + ⌜x⌝ )[n], n < n0 + # x, ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:extensionality, seq: ⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0[n] = σ1[n]), subst:[σ0, σ1] → [⌜x0⌝ , (σ to n0 + m) + ⌜x⌝ ], <all paths>

Additional information

Used terms: ⌜x0⌝ , (σ to n0 + m) + ⌜x⌝

Proof step 27 from proof swap-to-from-app (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0[n] = σ1[n])

Lemma extensionality

Proof step 28 from proof swap-to-from-app (previous step) (back to tree)

# ⌜x0⌝ = #((σ to n0 + m) + ⌜x⌝ ), x0[m + n] ≠ (((σ from m) to n0) + ⌜x⌝ )[n], n < n0 + # x, ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ , ∀ n. ⌜n⌝ < # ⌜x0⌝ → ⌜x0⌝ [n] = ((σ to n0 + m) + ⌜x⌝ )[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: size-append: ⊦ #(σ + σ1) = # σ + # σ1; size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; size-fin: ⊦ # ⌜x⌝ = ⌜# x⌝ ; sel-fin: ⊦ ⌜x⌝ [n] = x[n]; ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (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 ⊦ ⌜n1⌝ + ⌜n⌝ + ⌜m⌝ = ⌜n0⌝ ↔ n1 + n + m = n0; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

Proof step 29 from proof swap-to-from-app (previous step) (back to tree)

n0 + m + # x = # x0, x0[m + n] ≠ (((σ from m) to n0) + ⌜x⌝ )[n], n < n0 + # x, ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ , ∀ n1. n1 < # x0 → x0[n1] = ((σ to n0 + m) + ⌜x⌝ )[n1] ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 7 terms: m + n abort: F discard: T computed: T

Additional information

Used terms: m + n

Proof step 30 from proof swap-to-from-app (previous step) (back to tree)

n0 + m + # x = # x0, x0[m + n] ≠ (((σ from m) to n0) + ⌜x⌝ )[n], n < n0 + # x, ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ , m + n < # x0 → x0[m + n] = ((σ to n0 + m) + ⌜x⌝ )[m + n] ⊦

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 = m0 + n0 ∧ n < n0 → m0 + n < m; (s)

Proof step 31 from proof swap-to-from-app (previous step) (back to tree)

n0 + m + # x = # x0, x0[m + n] = ((σ to n0 + m) + ⌜x⌝ )[m + n], x0[m + n] ≠ (((σ from m) to n0) + ⌜x⌝ )[n], n < n0 + # x, ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:get-append-ls, seq: ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m], subst:[σ, σ1, m] → [(σ from m) to n0, ⌜x⌝ , n], <all paths>

Additional information

Used terms: (σ from m) to n0, ⌜x⌝ , n

Proof step 32 from proof swap-to-from-app (previous step) (back to tree)

⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m]

Lemma get-append-ls

Proof step 33 from proof swap-to-from-app (previous step) (back to tree)

n0 + m + # x = # x0, x0[m + n] = ((σ to n0 + m) + ⌜x⌝ )[m + n], x0[m + n] ≠ (((σ from m) to n0) + ⌜x⌝ )[n], n < n0 + # x, ¬ ⌜n⌝ < #((σ from m) to n0), ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: get-append-notls: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]; sel-fin: ⊦ ⌜x⌝ [n] = x[n]; append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; size-from-less: ⊦ ¬ # σ < ⌜n⌝ → (#(σ from n) < ⌜n0⌝ ↔ # σ < ⌜n + n0⌝ ); size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ # x = 0 ↔ x = []; (s) ⊦ m < m + n ↔ n ≠ 0; (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ ⌜n⌝ .n = n; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf

Proof step 34 from proof swap-to-from-app (previous step) (back to tree)

n0 + m + # x = # x0, x0[m + n] = ((σ to n0 + m) + ⌜x⌝ )[m + n], x0[m + n] ≠ x[n - n0], x ≠ [], n < n0 + # x, n0 ≤ n, ¬ # σ < ⌜m⌝ + ⌜n0⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜n0⌝ + ⌜m⌝ ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: nat, name:elim-sub, seq:n ≤ m ⊦ n0 = m - n ↔ m = n0 + n, subst:[m, n, n0] → [n, n0, n1]

Additional information

Used simplifier rules: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: n, n0, n1

Proof step 35 from proof swap-to-from-app (previous step) (back to tree)

n0 + m + # x = # x0, x0[m + n] = ((σ to n0 + m) + ⌜x⌝ )[m + n], n = n1 + n0, x0[m + n] ≠ x[n1], x ≠ [], n < n0 + # x, n0 ≤ n, ¬ # σ < ⌜m⌝ + ⌜n0⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜n0⌝ + ⌜m⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic 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) ⊦ m ≤ m + n; (s)

Proof step 36 from proof swap-to-from-app (previous step) (back to tree)

n0 + m + # x = # x0, x0[m + n1 + n0] = ((σ to n0 + m) + ⌜x⌝ )[m + n1 + n0], x0[m + n1 + n0] ≠ x[n1], x ≠ [], n1 < # x, ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:get-append-notls, seq: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n], subst:[σ, σ1, m] → [σ to n0 + m, ⌜x⌝ , m + n1 + n0], <all paths>

Additional information

Used simplifier rules: get-append-ls: ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m]; append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; sel-fin: ⊦ ⌜x⌝ [n] = x[n]; ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ ⌜n⌝ ≠ ∞; (s) n∞ ≠ ∞ ⊦ (⌜m⌝ + n∞).n = m + n∞.n; (s) ⊦ ⌜n⌝ .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 ⊦ m + n - n + n1 = m - n1; (s) ⊦ m + n - m = n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ ⌜n⌝ + n∞1 < ⌜n⌝ + n∞2 ↔ n∞1 < n∞2; (s) ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) ⊦ ¬ m + n < m; (s) Used terms: σ to n0 + m, ⌜x⌝ , m + n1 + n0

Proof step 37 from proof swap-to-from-app (previous step) (back to tree)

⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]

Lemma get-append-notls

Proof step 38 from proof swap-to-from-app (previous step) (back to tree)

n0 + m + # x = # x0, x0[m + n] = ((σ to n0 + m) + ⌜x⌝ )[m + n], x0[m + n] ≠ ((σ from m) to n0)[n], n < n0 + # x, ⌜n⌝ < #((σ from m) to n0), ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; size-from-less: ⊦ ¬ # σ < ⌜n⌝ → (#(σ from n) < ⌜n0⌝ ↔ # σ < ⌜n + n0⌝ ); get-to: m < n, ¬ # σ < ⌜n⌝ ⊦ (σ to n)[m] = σ[m]; ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf

Proof step 39 from proof swap-to-from-app (previous step) (back to tree)

n0 + m + # x = # x0, x0[m + n] = ((σ to n0 + m) + ⌜x⌝ )[m + n], x0[m + n] ≠ (σ from m)[n], n < n0, n < n0 + # x, ¬ # σ < ⌜m⌝ + ⌜n0⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜n0⌝ + ⌜m⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:get-from, seq: ⊦ ⌜n⌝ + ⌜m⌝ < # σ → (σ from n)[m] = σ[n + m], subst:[σ, n, m] → [σ, m, n], <all paths>

Additional information

Used simplifier rules: lele-trans: ⊦ ¬ n∞1 < n∞ ∧ ¬ n∞2 < n∞1 → ¬ n∞2 < n∞; (forward) from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) ⊦ ¬ m + n < m; (s) ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ + ⌜n1⌝ ↔ n + m < n0 + n1; (s) ⊦ m + n < m + n0 ↔ n < n0; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m < n → (n ≤ m ↔ false); (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic 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 Used terms: σ, m, n

Proof step 40 from proof swap-to-from-app (previous step) (back to tree)

⊦ ⌜n⌝ + ⌜m⌝ < # σ → (σ from n)[m] = σ[n + m]

Lemma get-from

Proof step 41 from proof swap-to-from-app (previous step) (back to tree)

n0 + m + # x = # x0, x0[m + n] = ((σ to n0 + m) + ⌜x⌝ )[m + n], x0[m + n] ≠ σ[m + n], n < n0 + # x, n < n0, ⌜m⌝ + ⌜n⌝ < # σ, ¬ # σ < ⌜m⌝ + ⌜n0⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜n0⌝ + ⌜m⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:get-append-notls, seq: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n], subst:[σ, σ1, m] → [σ to n0 + m, ⌜x⌝ , m + n], <all paths>

Additional information

Used simplifier rules: get-append-ls: ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m]; get-to: m < n, ¬ # σ < ⌜n⌝ ⊦ (σ to n)[m] = σ[m]; append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; sel-fin: ⊦ ⌜x⌝ [n] = x[n]; size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; ⊦ ⌜n⌝ ≠ ∞; (s) n∞ ≠ ∞ ⊦ (⌜m⌝ + n∞).n = m + n∞.n; (s) ⊦ ⌜n⌝ .n = n; (s) ⊦ m + n - n + n1 = m - n1; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ + ⌜n1⌝ ↔ n + m < n0 + n1; (s) ⊦ m + n < m + n0 ↔ n < n0; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m < n → (n ≤ m ↔ false); (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic 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 Used terms: σ to n0 + m, ⌜x⌝ , m + n

Proof step 42 from proof swap-to-from-app (previous step) (back to tree)

⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]

Lemma get-append-notls

Proof step 43 from proof swap-to-from-app (previous step) (back to tree)

σ0 = ⌜nf⌝ , ⌜n⌝ < ⌜n0⌝ + # σ0, ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ , #((σ to n0 + m) + σ0 from m) ≠ ⌜n0⌝ + # σ0 ∨ ¬(⌜n⌝ < #((σ to n0 + m) + σ0 from m) → ((σ to n0 + m) + σ0)[m + n] = (((σ from m) to n0) + σ0)[n]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: less-size-from: ⌜n⌝ < # σ ⊦ ⌜n0⌝ < #(σ from n) ↔ ⌜n0 + n⌝ < # σ; size-append: ⊦ #(σ + σ1) = # σ + # σ1; size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; size-inf: ⊦ # ⌜nf⌝ = ∞; append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; 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⌝ ≠ ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ n∞ < ∞ ↔ n∞ ≠ ∞; (s) ⊦ ⌜m⌝ + ∞ = ∞; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ ⌜m⌝ < ∞; (s)

Proof step 44 from proof swap-to-from-app (previous step) (back to tree)

¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ , #((σ to n0 + m) + ⌜nf⌝ from m) ≠ ∞ ∨ ((σ to n0 + m) + ⌜nf⌝ )[m + n] ≠ (((σ from m) to n0) + ⌜nf⌝ )[n] ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:inf-size-from, seq: ⊦ # σ = ∞ → #(σ from n) = ∞, subst:[σ, n] → [(σ to n0 + m) + ⌜nf⌝ , m], <all paths>

Additional information

Used simplifier rules: size-append: ⊦ #(σ + σ1) = # σ + # σ1; size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; size-inf: ⊦ # ⌜nf⌝ = ∞; append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ ⌜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 Used terms: (σ to n0 + m) + ⌜nf⌝ , m

Proof step 45 from proof swap-to-from-app (previous step) (back to tree)

⊦ # σ = ∞ → #(σ from n) = ∞

Lemma inf-size-from

Proof step 46 from proof swap-to-from-app (previous step) (back to tree)

#((σ to n0 + m) + ⌜nf⌝ ) = ∞, ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ , ∞ ≠ ∞ ∨ ((σ to n0 + m) + ⌜nf⌝ )[m + n] ≠ (((σ from m) to n0) + ⌜nf⌝ )[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: size-append: ⊦ #(σ + σ1) = # σ + # σ1; size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; size-inf: ⊦ # ⌜nf⌝ = ∞; append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; 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⌝ + ∞ = ∞; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

Proof step 47 from proof swap-to-from-app (previous step) (back to tree)

((σ to n0 + m) + ⌜nf⌝ )[m + n] ≠ (((σ from m) to n0) + ⌜nf⌝ )[n], ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:get-append-ls, seq: ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m], subst:[σ, σ1, m] → [(σ from m) to n0, ⌜nf⌝ , n], <all paths>

Additional information

Used terms: (σ from m) to n0, ⌜nf⌝ , n

Proof step 48 from proof swap-to-from-app (previous step) (back to tree)

⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m]

Lemma get-append-ls

Proof step 49 from proof swap-to-from-app (previous step) (back to tree)

((σ to n0 + m) + ⌜nf⌝ )[m + n] ≠ (((σ from m) to n0) + ⌜nf⌝ )[n], ¬ ⌜n⌝ < #((σ from m) to n0), ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: get-append-notls: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]; sel-inf: ⊦ ⌜nf⌝ [n] = nf(n); append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; size-from-less: ⊦ ¬ # σ < ⌜n⌝ → (#(σ from n) < ⌜n0⌝ ↔ # σ < ⌜n + n0⌝ ); size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ ⌜n⌝ .n = n; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf

Proof step 50 from proof swap-to-from-app (previous step) (back to tree)

((σ to n0 + m) + ⌜nf⌝ )[m + n] ≠ nf(n - n0), n0 ≤ n, ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: nat, name:elim-sub, seq:n ≤ m ⊦ n0 = m - n ↔ m = n0 + n, subst:[m, n, n0] → [n, n0, n1]

Additional information

Used simplifier rules: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: n, n0, n1

Proof step 51 from proof swap-to-from-app (previous step) (back to tree)

n = n1 + n0, ((σ to n0 + m) + ⌜nf⌝ )[m + n] ≠ nf(n1), n0 ≤ n, ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ ⊦

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 ≤ m + n; (s)

Proof step 52 from proof swap-to-from-app (previous step) (back to tree)

((σ to n0 + m) + ⌜nf⌝ )[m + n1 + n0] ≠ nf(n1), ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:get-append-notls, seq: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n], subst:[σ, σ1, m] → [σ to n0 + m, ⌜nf⌝ , m + n1 + n0], <all paths>

Additional information

Used simplifier rules: get-append-ls: ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m]; append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; sel-inf: ⊦ ⌜nf⌝ [n] = nf(n); ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ ⌜n⌝ ≠ ∞; (s) n∞ ≠ ∞ ⊦ (⌜m⌝ + n∞).n = m + n∞.n; (s) ⊦ ⌜n⌝ .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 ⊦ m + n - n + n1 = m - n1; (s) ⊦ m + n - m = n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ ⌜n⌝ + n∞1 < ⌜n⌝ + n∞2 ↔ n∞1 < n∞2; (s) ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) ⊦ ¬ m + n < m; (s) Used terms: σ to n0 + m, ⌜nf⌝ , m + n1 + n0

Proof step 53 from proof swap-to-from-app (previous step) (back to tree)

⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]

Lemma get-append-notls

Proof step 54 from proof swap-to-from-app (previous step) (back to tree)

((σ to n0 + m) + ⌜nf⌝ )[m + n] ≠ ((σ from m) to n0)[n], ⌜n⌝ < #((σ from m) to n0), ¬ # σ < ⌜n0⌝ + ⌜m⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜m⌝ + ⌜n0⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; size-from-less: ⊦ ¬ # σ < ⌜n⌝ → (#(σ from n) < ⌜n0⌝ ↔ # σ < ⌜n + n0⌝ ); get-to: m < n, ¬ # σ < ⌜n⌝ ⊦ (σ to n)[m] = σ[m]; ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf

Proof step 55 from proof swap-to-from-app (previous step) (back to tree)

((σ to n0 + m) + ⌜nf⌝ )[m + n] ≠ (σ from m)[n], n < n0, ¬ # σ < ⌜m⌝ + ⌜n0⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜n0⌝ + ⌜m⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:get-from, seq: ⊦ ⌜n⌝ + ⌜m⌝ < # σ → (σ from n)[m] = σ[n + m], subst:[σ, n, m] → [σ, m, n], <all paths>

Additional information

Used simplifier rules: lele-trans: ⊦ ¬ n∞1 < n∞ ∧ ¬ n∞2 < n∞1 → ¬ n∞2 < n∞; (forward) from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) ⊦ ¬ m + n < m; (s) ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ + ⌜n1⌝ ↔ n + m < n0 + n1; (s) ⊦ m + n < m + n0 ↔ n < n0; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m < n → (n ≤ m ↔ false); (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic 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 Used terms: σ, m, n

Proof step 56 from proof swap-to-from-app (previous step) (back to tree)

⊦ ⌜n⌝ + ⌜m⌝ < # σ → (σ from n)[m] = σ[n + m]

Lemma get-from

Proof step 57 from proof swap-to-from-app (previous step) (back to tree)

((σ to n0 + m) + ⌜nf⌝ )[m + n] ≠ σ[m + n], n < n0, ⌜m⌝ + ⌜n⌝ < # σ, ¬ # σ < ⌜m⌝ + ⌜n0⌝ , ¬ # σ < ⌜m⌝ , ¬ # σ < ⌜n0⌝ + ⌜m⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:get-append-ls, seq: ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m], subst:[σ, σ1, m] → [σ to n0 + m, ⌜nf⌝ , m + n], <all paths>

Additional information

Used simplifier rules: get-append-notls: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]; sel-inf: ⊦ ⌜nf⌝ [n] = nf(n); append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; get-to: m < n, ¬ # σ < ⌜n⌝ ⊦ (σ to n)[m] = σ[m]; ⊦ m + n < m + n0 ↔ n < n0; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s) n∞ ≠ ∞ ⊦ (⌜m⌝ + n∞).n = m + n∞.n; (s) ⊦ ⌜n⌝ .n = n; (s) ⊦ m + n - n + n1 = m - n1; (s) ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ + ⌜n1⌝ ↔ n + m < n0 + n1; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m < n → (n ≤ m ↔ false); (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf Used terms: σ to n0 + m, ⌜nf⌝ , m + n

Proof step 58 from proof swap-to-from-app (previous step) (back to tree)

⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m]

Lemma get-append-ls

Proof Tree for swap-to-from-app (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