Documentation

Foundation.FirstOrder.Arithmetic.HFS.Seq

Sequence #

def FFL.FirstOrder.Arithmetic.Seq {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (s : V) :

Equations

FFL.FirstOrder.Arithmetic.Seq s = (FFL.FirstOrder.Arithmetic.IsMapping s ∧ ∃ (l : V), FFL.FirstOrder.Arithmetic.domain s = FFL.FirstOrder.Arithmetic.under l)

Instances For

theorem FFL.FirstOrder.Arithmetic.Seq.isMapping {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} (h : Seq s) :

def FFL.FirstOrder.Arithmetic.seqDef :

𝚺₀.Semisentence 1

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One or more equations did not get rendered due to their size.

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instance FFL.FirstOrder.Arithmetic.seq_defined {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] :

𝚺₀ -Predicate Seq via seqDef

instance FFL.FirstOrder.Arithmetic.seq_definable {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] :

𝚺₀ -Predicate Seq

instance FFL.FirstOrder.Arithmetic.seq_definable' {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (ℌ : HierarchySymbol) :

ℌ-Predicate Seq

def FFL.FirstOrder.Arithmetic.«first_order_formula:Seq_» :

Lean.ParserDescr

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One or more equations did not get rendered due to their size.

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theorem FFL.FirstOrder.Arithmetic.lh_exists_uniq {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (s : V) :

∃! l : V, (Seq s → domain s = under l) ∧ (¬Seq s → l = 0)

noncomputable def FFL.FirstOrder.Arithmetic.lh {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (s : V) :

V

Equations

FFL.FirstOrder.Arithmetic.lh s = Classical.choose! ⋯

Instances For

theorem FFL.FirstOrder.Arithmetic.lh_prop {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (s : V) :

(Seq s → domain s = under (lh s)) ∧ (¬Seq s → lh s = 0)

theorem FFL.FirstOrder.Arithmetic.lh_prop_of_not_seq {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} (h : ¬Seq s) :

lh s = 0

theorem FFL.FirstOrder.Arithmetic.Seq.domain_eq {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} (h : Seq s) :

domain s = under (lh s)

@[simp]

theorem FFL.FirstOrder.Arithmetic.lh_bound {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (s : V) :

lh s ≤ 2 * s

def FFL.FirstOrder.Arithmetic.lhDef :

𝚺₀.Semisentence 2

Equations

One or more equations did not get rendered due to their size.

Instances For

instance FFL.FirstOrder.Arithmetic.lh_defined {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] :

𝚺₀ -Function₁ lh via lhDef

instance FFL.FirstOrder.Arithmetic.lh_definable {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] :

𝚺₀ -Function₁ lh

instance FFL.FirstOrder.Arithmetic.lh_definable' {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (ℌ : HierarchySymbol) :

ℌ-Function₁ lh

instance FFL.FirstOrder.Arithmetic.instBounded₁Lh {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] :

theorem FFL.FirstOrder.Arithmetic.Seq.exists {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} (h : Seq s) {x : V} (hx : x < lh s) :

∃ (y : V), ⟪x, y⟫ ∈ s

theorem FFL.FirstOrder.Arithmetic.Seq.nth_exists_uniq {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} (h : Seq s) {x : V} (hx : x < lh s) :

∃! y : V, ⟪x, y⟫ ∈ s

noncomputable def FFL.FirstOrder.Arithmetic.Seq.nth {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} (h : Seq s) {x : V} (hx : x < lh s) :

V

Equations

h.nth hx = Classical.choose! ⋯

Instances For

@[simp]

theorem FFL.FirstOrder.Arithmetic.Seq.nth_mem {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} (h : Seq s) {x : V} (hx : x < lh s) :

⟪x, h.nth hx⟫ ∈ s

theorem FFL.FirstOrder.Arithmetic.Seq.nth_uniq {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} (h : Seq s) {x y : V} (hx : x < lh s) (hy : ⟪x, y⟫ ∈ s) :

y = h.nth hx

@[simp]

theorem FFL.FirstOrder.Arithmetic.Seq.nth_lt {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} (h : Seq s) {x : V} (hx : x < lh s) :

h.nth hx < s

theorem FFL.FirstOrder.Arithmetic.Seq.lh_eq_of {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} (H : Seq s) {l : V} (h : domain s = under l) :

lh s = l

theorem FFL.FirstOrder.Arithmetic.Seq.lt_lh_iff {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} (h : Seq s) {i : V} :

i < lh s ↔ i ∈ domain s

theorem FFL.FirstOrder.Arithmetic.Seq.lt_lh_of_mem {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} (h : Seq s) {i x : V} (hix : ⟪i, x⟫ ∈ s) :

i < lh s

noncomputable def FFL.FirstOrder.Arithmetic.seqCons {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (s x : V) :

V

Equations

s ⁀' x = insert ⟪FFL.FirstOrder.Arithmetic.lh s, x⟫ s

Instances For

theorem FFL.FirstOrder.Arithmetic.znth_existsUnique {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (s i : V) :

∃! x : V, (Seq s ∧ i < lh s → ⟪i, x⟫ ∈ s) ∧ (¬(Seq s ∧ i < lh s) → x = 0)

noncomputable def FFL.FirstOrder.Arithmetic.znth {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (s i : V) :

V

Equations

FFL.FirstOrder.Arithmetic.znth s i = Classical.choose! ⋯

Instances For

theorem FFL.FirstOrder.Arithmetic.Seq.znth {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s i : V} (h : Seq s) (hi : i < lh s) :

⟪i, znth s i⟫ ∈ s

theorem FFL.FirstOrder.Arithmetic.Seq.znth_eq_of_mem {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {x s i : V} (h : Seq s) (hi : ⟪i, x⟫ ∈ s) :

znth s i = x

theorem FFL.FirstOrder.Arithmetic.znth_prop_not {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s i : V} (h : ¬Seq s ∨ lh s ≤ i) :

znth s i = 0

def FFL.FirstOrder.Arithmetic.znthDef :

𝚺₀.Semisentence 3

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One or more equations did not get rendered due to their size.

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instance FFL.FirstOrder.Arithmetic.znth_defined {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] :

𝚺₀ -Function₂ znth via znthDef

instance FFL.FirstOrder.Arithmetic.znth_definable {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] :

𝚺₀ -Function₂ znth

instance FFL.FirstOrder.Arithmetic.znth_definable' {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (ℌ : HierarchySymbol) :

ℌ-Function₂ znth

def FFL.FirstOrder.Arithmetic.«term_⁀'_» :

Lean.TrailingParserDescr

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One or more equations did not get rendered due to their size.

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@[simp]

theorem FFL.FirstOrder.Arithmetic.seq_empty {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] :

@[simp]

theorem FFL.FirstOrder.Arithmetic.lh_empty {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] :

theorem FFL.FirstOrder.Arithmetic.Seq.isempty_of_lh_eq_zero {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} (Hs : Seq s) (h : lh s = 0) :

@[simp]

theorem FFL.FirstOrder.Arithmetic.Seq.subset_seqCons {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (s x : V) :

s ⊆ s ⁀' x

theorem FFL.FirstOrder.Arithmetic.Seq.lt_seqCons {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} (hs : Seq s) (x : V) :

s < s ⁀' x

@[simp]

theorem FFL.FirstOrder.Arithmetic.Seq.mem_seqCons {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (s x : V) :

⟪lh s, x⟫ ∈ s ⁀' x

theorem FFL.FirstOrder.Arithmetic.Seq.seqCons {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} (h : Seq s) (x : V) :

Seq (s ⁀' x)

@[simp]

theorem FFL.FirstOrder.Arithmetic.Seq.lh_seqCons {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (x : V) {s : V} (h : Seq s) :

lh (s ⁀' x) = lh s + 1

theorem FFL.FirstOrder.Arithmetic.mem_seqCons_iff {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {i x z s : V} :

⟪i, x⟫ ∈ s ⁀' z ↔ i = lh s ∧ x = z ∨ ⟪i, x⟫ ∈ s

@[simp]

theorem FFL.FirstOrder.Arithmetic.lh_mem_seqCons {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (s z : V) :

⟪lh s, z⟫ ∈ s ⁀' z

@[simp]

theorem FFL.FirstOrder.Arithmetic.lh_mem_seqCons_iff {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s x z : V} (H : Seq s) :

⟪lh s, x⟫ ∈ s ⁀' z ↔ x = z

theorem FFL.FirstOrder.Arithmetic.Seq.mem_seqCons_iff_of_lt {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {i s x z : V} (hi : i < lh s) :

⟪i, x⟫ ∈ s ⁀' z ↔ ⟪i, x⟫ ∈ s

@[simp]

theorem FFL.FirstOrder.Arithmetic.lh_not_mem {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} (Ss : Seq s) (x : V) :

⟪lh s, x⟫ ∉ s

theorem FFL.FirstOrder.Arithmetic.seqCons_graph {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (t x s : V) :

t = s ⁀' x ↔ ∃ l ≤ 2 * s, l = lh s ∧ ∃ p ≤ (2 * s + x + 1) ^ 2, p = ⟪l, x⟫ ∧ t = insert p s

def FFL.FirstOrder.Arithmetic.seqConsDef :

𝚺₀.Semisentence 3

Equations

One or more equations did not get rendered due to their size.

Instances For

instance FFL.FirstOrder.Arithmetic.seqCons_defined {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] :

𝚺₀ -Function₂ seqCons via seqConsDef

instance FFL.FirstOrder.Arithmetic.seqCons_definable {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] :

𝚺₀ -Function₂ seqCons

instance FFL.FirstOrder.Arithmetic.seqCons_definable' {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (ℌ : HierarchySymbol) :

ℌ-Function₂ seqCons

@[simp]

theorem FFL.FirstOrder.Arithmetic.natCast_empty {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] :

theorem FFL.FirstOrder.Arithmetic.seqCons_absolute {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (s a : ℕ) :

↑(s ⁀' a) = ↑s ⁀' ↑a

theorem FFL.FirstOrder.Arithmetic.Seq.restr {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} (H : Seq s) {i : V} (hi : i ≤ lh s) :

Seq (Arithmetic.restr s (under i))

theorem FFL.FirstOrder.Arithmetic.Seq.restr_lh {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} (H : Seq s) {i : V} (hi : i ≤ lh s) :

lh (Arithmetic.restr s (under i)) = i

theorem FFL.FirstOrder.Arithmetic.domain_bitRemove_of_isMapping_of_mem {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {x y s : V} (hs : IsMapping s) (hxy : ⟪x, y⟫ ∈ s) :

domain (bitRemove ⟪x, y⟫ s) = bitRemove x (domain s)

theorem FFL.FirstOrder.Arithmetic.Seq.eq_of_eq_of_subset {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s₁ s₂ : V} (H₁ : Seq s₁) (H₂ : Seq s₂) (hl : lh s₁ = lh s₂) (h : s₁ ⊆ s₂) :

s₁ = s₂

theorem FFL.FirstOrder.Arithmetic.subset_pair {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s t : V} (h : ∀ (i x : V), ⟪i, x⟫ ∈ s → ⟪i, x⟫ ∈ t) :

s ⊆ t

theorem FFL.FirstOrder.Arithmetic.Seq.lh_ext {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s₁ s₂ : V} (H₁ : Seq s₁) (H₂ : Seq s₂) (h : lh s₁ = lh s₂) (H : ∀ (i x₁ x₂ : V), ⟪i, x₁⟫ ∈ s₁ → ⟪i, x₂⟫ ∈ s₂ → x₁ = x₂) :

s₁ = s₂

@[simp]

theorem FFL.FirstOrder.Arithmetic.Seq.seqCons_ext {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {a₁ a₂ s₁ s₂ : V} (H₁ : Seq s₁) (H₂ : Seq s₂) :

s₁ ⁀' a₁ = s₂ ⁀' a₂ ↔ a₁ = a₂ ∧ s₁ = s₂

theorem FFL.FirstOrder.Arithmetic.ne_zero_iff_one_le {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {a : V} :

a ≠ 0 ↔ 1 ≤ a

TODO: move to Lemmata.lean

theorem FFL.FirstOrder.Arithmetic.Seq.cases_iff {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} :

Seq s ↔ s = ∅ ∨ ∃ (x : V) (s' : V), Seq s' ∧ s = s' ⁀' x

theorem FFL.FirstOrder.Arithmetic.Seq.cases {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {s : V} :

Seq s → s = ∅ ∨ ∃ (x : V) (s' : V), Seq s' ∧ s = s' ⁀' x

Alias of the forward direction of FFL.FirstOrder.Arithmetic.Seq.cases_iff.

theorem FFL.FirstOrder.Arithmetic.seq_induction {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (Γ : SigmaPiDelta) {P : V → Prop} (hP : { Γ := Γ, rank := 1 }-Predicate P) (hnil : P ∅) (hcons : ∀ (s x : V), Seq s → P s → P (s ⁀' x)) {s : V} :

Seq s → P s

def FFL.FirstOrder.Arithmetic.«term!⟦_⟧» :

Lean.ParserDescr

!⟦x, y, z, ...⟧ notation for Seq

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One or more equations did not get rendered due to their size.

Instances For

def FFL.FirstOrder.Arithmetic.vecConsUnexpander :

Lean.PrettyPrinter.Unexpander

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One or more equations did not get rendered due to their size.

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@[simp]

theorem FFL.FirstOrder.Arithmetic.singleton_seq {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (x : V) :

@[simp]

theorem FFL.FirstOrder.Arithmetic.doubleton_seq {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (x y : V) :

Seq !⟦x, y⟧

@[simp]

theorem FFL.FirstOrder.Arithmetic.mem_singleton_seq_iff {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (x y : V) :

⟪0, x⟫ ∈ !⟦y⟧ ↔ x = y

def FFL.FirstOrder.Arithmetic.mkSeq₁Def :

𝚺₀.Semisentence 2

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One or more equations did not get rendered due to their size.

Instances For

instance FFL.FirstOrder.Arithmetic.mkSeq₁_defined {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] :

𝚺₀ -Function₁ fun (x : V) => !⟦x⟧ via mkSeq₁Def

instance FFL.FirstOrder.Arithmetic.mkSeq₁_definable {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] :

𝚺₀ -Function₁ fun (x : V) => !⟦x⟧

instance FFL.FirstOrder.Arithmetic.mkSeq₁_definable' {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (Γ : HierarchySymbol) :

Γ-Function₁ fun (x : V) => !⟦x⟧

def FFL.FirstOrder.Arithmetic.mkSeq₂Def :

𝚺₁.Semisentence 3

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One or more equations did not get rendered due to their size.

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instance FFL.FirstOrder.Arithmetic.mkSeq₂_defined {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] :

𝚺₁ -Function₂ fun (x y : V) => !⟦x, y⟧ via mkSeq₂Def

instance FFL.FirstOrder.Arithmetic.mkSeq₂_definable {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] :

𝚺₁ -Function₂ fun (x y : V) => !⟦x, y⟧

instance FFL.FirstOrder.Arithmetic.mkSeq₂_definable' {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] (Γ : SigmaPiDelta) (m : ℕ) :

{ Γ := Γ, rank := m + 1 }-Function₂ fun (x y : V) => !⟦x, y⟧

theorem FFL.FirstOrder.Arithmetic.sigmaOne_skolem_seq {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {R : V → V → Prop} (hP : 𝚺₁ -Relation R) {l : V} (H : ∀ x < l, ∃ (y : V), R x y) :

∃ (s : V), Seq s ∧ lh s = l ∧ ∀ (i x : V), ⟪i, x⟫ ∈ s → R i x

theorem FFL.FirstOrder.Arithmetic.sigmaOne_skolem_seq! {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {R : V → V → Prop} (hP : 𝚺₁ -Relation R) {l : V} (H : ∀ x < l, ∃! y : V, R x y) :

∃! s : V, Seq s ∧ lh s = l ∧ ∀ (i x : V), ⟪i, x⟫ ∈ s → R i x

noncomputable def FFL.FirstOrder.Arithmetic.vecToSeq {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {n : ℕ} :

(Fin n → V) → V

Equations

FFL.FirstOrder.Arithmetic.vecToSeq x_2 = ∅
FFL.FirstOrder.Arithmetic.vecToSeq v = (FFL.FirstOrder.Arithmetic.vecToSeq fun (x : Fin n) => v x.castSucc) ⁀' v (Fin.last n)

Instances For

@[simp]

theorem FFL.FirstOrder.Arithmetic.vecToSeq_nil {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] :

vecToSeq ![] = ∅

@[simp]

theorem FFL.FirstOrder.Arithmetic.vecToSeq_vecCons {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {n : ℕ} (v : Fin n → V) (a : V) :

vecToSeq (v <: a) = vecToSeq v ⁀' a

@[simp]

theorem FFL.FirstOrder.Arithmetic.vecToSeq_seq {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {n : ℕ} (v : Fin n → V) :

Seq (vecToSeq v)

@[simp]

theorem FFL.FirstOrder.Arithmetic.lh_vecToSeq {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {n : ℕ} (v : Fin n → V) :

lh (vecToSeq v) = ↑n

theorem FFL.FirstOrder.Arithmetic.mem_vectoSeq {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝗜𝚺₁] {n : ℕ} (v : Fin n → V) (i : Fin n) :

⟪↑↑i, v i⟫ ∈ vecToSeq v