def Matrix.iget {α : Type u_1} {k : ℕ} [Inhabited α] (v : Fin k → α) (x : ℕ) : α

Equations

• Matrix.iget v x = if h : x < k then v ⟨x, h⟩ else default

class LO.FirstOrder.Theory.Sub {L : Language} (T U : Theory L) : Prop

• sub : T ⊆ U

inductive LO.FirstOrder.Theory.eqAxiom {L : Language} [Semiformula.Operator.Eq L] : Theory L

• refl {L : Language} [Semiformula.Operator.Eq L] : 𝐄𝐐 (“!!(Semiterm.fvar 0) = !!(Semiterm.fvar 0)”)
• symm {L : Language} [Semiformula.Operator.Eq L] : 𝐄𝐐 (“(!!(Semiterm.fvar 0) = !!(Semiterm.fvar 1) → !!(Semiterm.fvar 1) = !!(Semiterm.fvar 0))”)
• trans {L : Language} [Semiformula.Operator.Eq L] : 𝐄𝐐 (“(!!(Semiterm.fvar 0) = !!(Semiterm.fvar 1) → (!!(Semiterm.fvar 1) = !!(Semiterm.fvar 2) → !!(Semiterm.fvar 0) = !!(Semiterm.fvar 2)))”)
• funcExt {L : Language} [Semiformula.Operator.Eq L] {k : ℕ} (f : L.Func k) : 𝐄𝐐 (“(!(Matrix.conj fun (i : Fin k) => “!!(Semiterm.fvar ↑i) = !!(Semiterm.fvar (k + ↑i))”) → !!(Semiterm.func f fun (i : Fin k) => Semiterm.fvar ↑i) = !!(Semiterm.func f fun (i : Fin k) => Semiterm.fvar (k + ↑i)))”)
• relExt {L : Language} [Semiformula.Operator.Eq L] {k : ℕ} (r : L.Rel k) : 𝐄𝐐 ((Matrix.conj fun (i : Fin k) => “!!(Semiterm.fvar ↑i) = !!(Semiterm.fvar (k + ↑i))”) ➝ (Semiformula.rel r fun (i : Fin k) => Semiterm.fvar ↑i) ➝ Semiformula.rel r fun (i : Fin k) => Semiterm.fvar (k + ↑i))

def LO.FirstOrder.Theory.«term𝐄𝐐» : Lean.ParserDescr

Equations

• LO.FirstOrder.Theory.«term𝐄𝐐» = Lean.ParserDescr.node `LO.FirstOrder.Theory.«term𝐄𝐐» 1024 (Lean.ParserDescr.symbol "𝐄𝐐")

@[simp]

theorem LO.FirstOrder.Structure.Eq.models_eqAxiom {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [Structure.Eq L M] : M ⊧ₘ* 𝐄𝐐

instance LO.FirstOrder.Structure.Eq.models_eqAxiom' (L : Language) [Semiformula.Operator.Eq L] (M : Type u) [Nonempty M] [Structure L M] [Structure.Eq L M] : M ⊧ₘ* 𝐄𝐐

def LO.FirstOrder.Structure.Eq.eqv (L : Language) [Semiformula.Operator.Eq L] {M : Type u} [Structure L M] (a b : M) : Prop

Equations

• LO.FirstOrder.Structure.Eq.eqv L a b = op(=).val ![a, b]

theorem LO.FirstOrder.Structure.Eq.eqv_refl {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] (a : M) : eqv L a a

theorem LO.FirstOrder.Structure.Eq.eqv_symm {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] {a b : M} : eqv L a b → eqv L b a

theorem LO.FirstOrder.Structure.Eq.eqv_trans {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] {a b c : M} : eqv L a b → eqv L b c → eqv L a c

theorem LO.FirstOrder.Structure.Eq.eqv_funcExt {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] {k : ℕ} (f : L.Func k) {v w : Fin k → M} (h : ∀ (i : Fin k), eqv L (v i) (w i)) : eqv L (func f v) (func f w)

theorem LO.FirstOrder.Structure.Eq.eqv_relExt_aux {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] {k : ℕ} (r : L.Rel k) {v w : Fin k → M} (h : ∀ (i : Fin k), eqv L (v i) (w i)) : rel r v → rel r w

theorem LO.FirstOrder.Structure.Eq.eqv_relExt {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] {k : ℕ} (r : L.Rel k) {v w : Fin k → M} (h : ∀ (i : Fin k), eqv L (v i) (w i)) : rel r v = rel r w

theorem LO.FirstOrder.Structure.Eq.eqv_equivalence {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] : Equivalence (eqv L)

def LO.FirstOrder.Structure.Eq.eqvSetoid (L : Language) [Semiformula.Operator.Eq L] (M : Type u) [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] : Setoid M

Equations

• LO.FirstOrder.Structure.Eq.eqvSetoid L M = { r := LO.FirstOrder.Structure.Eq.eqv L, iseqv := ⋯ }

def LO.FirstOrder.Structure.Eq.QuotEq (L : Language) [Semiformula.Operator.Eq L] (M : Type u) [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] : Type u

Equations

• LO.FirstOrder.Structure.Eq.QuotEq L M = Quotient (LO.FirstOrder.Structure.Eq.eqvSetoid L M)

instance LO.FirstOrder.Structure.Eq.QuotEq.inhabited {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] : Nonempty (QuotEq L M)

theorem LO.FirstOrder.Structure.Eq.of_eq_of {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] {a b : M} : ⟦a⟧ = ⟦b⟧ ↔ eqv L a b

def LO.FirstOrder.Structure.Eq.QuotEq.func {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] ⦃k : ℕ⦄ (f : L.Func k) (v : Fin k → QuotEq L M) : QuotEq L M

Equations

• LO.FirstOrder.Structure.Eq.QuotEq.func f v = Quotient.liftVec (fun (x : Fin k → M) => ⟦LO.FirstOrder.Structure.func f x⟧) ⋯ v

def LO.FirstOrder.Structure.Eq.QuotEq.rel {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] ⦃k : ℕ⦄ (r : L.Rel k) (v : Fin k → QuotEq L M) : Prop

Equations

• LO.FirstOrder.Structure.Eq.QuotEq.rel r v = Quotient.liftVec (LO.FirstOrder.Structure.rel r) ⋯ v

instance LO.FirstOrder.Structure.Eq.QuotEq.struc {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] : Structure L (QuotEq L M)

Equations

• LO.FirstOrder.Structure.Eq.QuotEq.struc = { func := LO.FirstOrder.Structure.Eq.QuotEq.func, rel := LO.FirstOrder.Structure.Eq.QuotEq.rel }

theorem LO.FirstOrder.Structure.Eq.QuotEq.funk_mk {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] {k : ℕ} (f : L.Func k) (v : Fin k → M) : (Structure.func f fun (i : Fin k) => ⟦v i⟧) = ⟦Structure.func f v⟧

theorem LO.FirstOrder.Structure.Eq.QuotEq.rel_mk {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] {k : ℕ} (r : L.Rel k) (v : Fin k → M) : (Structure.rel r fun (i : Fin k) => ⟦v i⟧) ↔ Structure.rel r v

theorem LO.FirstOrder.Structure.Eq.QuotEq.val_mk {L : Language} {μ : Type u_1} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] {n : ℕ} {e : Fin n → M} {ε : μ → M} (t : Semiterm L μ n) : Semiterm.valm (QuotEq L M) (fun (i : Fin n) => ⟦e i⟧) (fun (i : μ) => ⟦ε i⟧) t = ⟦Semiterm.valm M e ε t⟧

theorem LO.FirstOrder.Structure.Eq.QuotEq.eval_mk {L : Language} {μ : Type u_1} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] {n : ℕ} {e : Fin n → M} {ε : μ → M} {φ : Semiformula L μ n} : (Semiformula.Evalm (QuotEq L M) (fun (i : Fin n) => ⟦e i⟧) fun (i : μ) => ⟦ε i⟧) φ ↔ (Semiformula.Evalm M e ε) φ

theorem LO.FirstOrder.Structure.Eq.QuotEq.eval_mk₀ {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] {ξ : Type u_2} {ε : ξ → M} {φ : Formula L ξ} : (Semiformula.Evalfm (QuotEq L M) fun (i : ξ) => ⟦ε i⟧) φ ↔ (Semiformula.Evalfm M ε) φ

theorem LO.FirstOrder.Structure.Eq.QuotEq.models_iff {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] {φ : SyntacticFormula L} : QuotEq L M ⊧ₘ φ ↔ M ⊧ₘ φ

theorem LO.FirstOrder.Structure.Eq.QuotEq.elementaryEquiv (L : Language) [Semiformula.Operator.Eq L] (M : Type u) [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] : QuotEq L M ≡ₑ[L] M

theorem LO.FirstOrder.Structure.Eq.QuotEq.rel_eq {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] (a b : QuotEq L M) : op(=).val ![a, b] ↔ a = b

instance LO.FirstOrder.Structure.Eq.QuotEq.structureEq {L : Language} [Semiformula.Operator.Eq L] {M : Type u} [Nonempty M] [Structure L M] [H : M ⊧ₘ* 𝐄𝐐] : Structure.Eq L (QuotEq L M)

theorem LO.FirstOrder.consequence_iff_eq {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] {φ : SyntacticFormula L} : T ⊨[Struc L] φ ↔ ∀ (M : Type v) [inst : Nonempty M] [inst_1 : Structure L M] [inst_2 : Structure.Eq L M], M ⊧ₘ* T → M ⊧ₘ φ

theorem LO.FirstOrder.consequence_iff_eq' {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] {φ : SyntacticFormula L} : T ⊨[Struc L] φ ↔ ∀ (M : Type v) [inst : Nonempty M] [inst_1 : Structure L M] [inst_2 : Structure.Eq L M] [inst_3 : M ⊧ₘ* T], M ⊧ₘ φ

theorem LO.FirstOrder.satisfiable_iff_eq {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] : Semantics.Satisfiable (Struc L) T ↔ ∃ (M : Type v) (x : Nonempty M) (x_1 : Structure L M) (_ : Structure.Eq L M), M ⊧ₘ* T

instance LO.FirstOrder.instModelsTheoryModelOfSatEqAxiomOfSubtheorySyntacticFormulaTheory {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) : ModelOfSat sat ⊧ₘ* 𝐄𝐐

def LO.FirstOrder.ModelOfSatEq {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) : Type v

Equations

• LO.FirstOrder.ModelOfSatEq sat = LO.FirstOrder.Structure.Eq.QuotEq L (LO.FirstOrder.ModelOfSat sat)

instance LO.FirstOrder.ModelOfSatEq.instNonempty {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) : Nonempty (ModelOfSatEq sat)

noncomputable instance LO.FirstOrder.ModelOfSatEq.struc {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) : Structure L (ModelOfSatEq sat)

Equations

• LO.FirstOrder.ModelOfSatEq.struc sat = LO.FirstOrder.Structure.Eq.QuotEq.struc

instance LO.FirstOrder.ModelOfSatEq.instEq {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) : Structure.Eq L (ModelOfSatEq sat)

theorem LO.FirstOrder.ModelOfSatEq.models {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) : ModelOfSatEq sat ⊧ₘ* T

instance LO.FirstOrder.ModelOfSatEq.mod {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) : ModelOfSatEq sat ⊧ₘ* T

noncomputable instance LO.FirstOrder.ModelOfSatEq.instZeroOfZero {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) [Semiterm.Operator.Zero L] : Zero (ModelOfSatEq sat)

Equations

• LO.FirstOrder.ModelOfSatEq.instZeroOfZero sat = { zero := LO.FirstOrder.Semiterm.Operator.val op(0) ![] }

instance LO.FirstOrder.ModelOfSatEq.strucZero {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) [Semiterm.Operator.Zero L] : Structure.Zero L (ModelOfSatEq sat)

noncomputable instance LO.FirstOrder.ModelOfSatEq.instOneOfOne {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) [Semiterm.Operator.One L] : One (ModelOfSatEq sat)

Equations

• LO.FirstOrder.ModelOfSatEq.instOneOfOne sat = { one := LO.FirstOrder.Semiterm.Operator.val op(1) ![] }

instance LO.FirstOrder.ModelOfSatEq.instOne {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) [Semiterm.Operator.One L] : Structure.One L (ModelOfSatEq sat)

noncomputable instance LO.FirstOrder.ModelOfSatEq.instAddOfAdd {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) [Semiterm.Operator.Add L] : Add (ModelOfSatEq sat)

Equations

• LO.FirstOrder.ModelOfSatEq.instAddOfAdd sat = { add := fun (x y : LO.FirstOrder.ModelOfSatEq sat) => op(+).val ![x, y] }

instance LO.FirstOrder.ModelOfSatEq.instAdd {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) [Semiterm.Operator.Add L] : Structure.Add L (ModelOfSatEq sat)

noncomputable instance LO.FirstOrder.ModelOfSatEq.instMulOfMul {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) [Semiterm.Operator.Mul L] : Mul (ModelOfSatEq sat)

Equations

• LO.FirstOrder.ModelOfSatEq.instMulOfMul sat = { mul := fun (x y : LO.FirstOrder.ModelOfSatEq sat) => op(*).val ![x, y] }

instance LO.FirstOrder.ModelOfSatEq.instMul {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) [Semiterm.Operator.Mul L] : Structure.Mul L (ModelOfSatEq sat)

instance LO.FirstOrder.ModelOfSatEq.instLTOfLT {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) [Semiformula.Operator.LT L] : LT (ModelOfSatEq sat)

Equations

• LO.FirstOrder.ModelOfSatEq.instLTOfLT sat = { lt := fun (x y : LO.FirstOrder.ModelOfSatEq sat) => op(<).val ![x, y] }

instance LO.FirstOrder.ModelOfSatEq.instLT {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) [Semiformula.Operator.LT L] : Structure.LT L (ModelOfSatEq sat)

instance LO.FirstOrder.ModelOfSatEq.instMembershipOfMem {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) [Semiformula.Operator.Mem L] : Membership (ModelOfSatEq sat) (ModelOfSatEq sat)

Equations

• LO.FirstOrder.ModelOfSatEq.instMembershipOfMem sat = { mem := fun (x y : LO.FirstOrder.ModelOfSatEq sat) => op(∈).val ![y, x] }

instance LO.FirstOrder.ModelOfSatEq.instMem {L : Language} [Semiformula.Operator.Eq L] {T : Theory L} [𝐄𝐐 ≼ T] (sat : Semantics.Satisfiable (Struc L) T) [Semiformula.Operator.Mem L] : Structure.Mem L (ModelOfSatEq sat)

def LO.FirstOrder.Semiformula.existsUnique {L : Language} [Operator.Eq L] {n : ℕ} {ξ : Type u_3} (φ : Semiformula L ξ (n + 1)) : Semiformula L ξ n

Equations

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

def LO.FirstOrder.Semiformula.«term∃'!_» : Lean.ParserDescr

Equations

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

@[simp]

theorem LO.FirstOrder.Semiformula.eval_existsUnique {L : Language} [Operator.Eq L] {M : Type u_2} [s : Structure L M] [Structure.Eq L M] {ξ : Type u_3} {n : ℕ} {e : Fin n → M} {ε : ξ → M} {φ : Semiformula L ξ (n + 1)} : (Eval s e ε) (∃'! φ) ↔ ∃! x : M, (Eval s (x :> e) ε) φ

def LO.FirstOrder.BinderNotation.«first_order_formula∃!_» : Lean.ParserDescr

Equations

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

def LO.FirstOrder.BinderNotation.«first_order_formula∃!_,_» : Lean.ParserDescr

Equations

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