@[simp]
theorem
LO.FirstOrder.Arith.oringEmb_operator_zero_val
{L : Language}
[L.ORing]
:
Semiterm.lMap Language.oringEmb op(0).term = op(0).term
@[simp]
theorem
LO.FirstOrder.Arith.oringEmb_operator_one_val
{L : Language}
[L.ORing]
:
Semiterm.lMap Language.oringEmb op(1).term = op(1).term
@[simp]
theorem
LO.FirstOrder.Arith.oringEmb_operator_add_val
{L : Language}
[L.ORing]
:
Semiterm.lMap Language.oringEmb op(+).term = op(+).term
@[simp]
theorem
LO.FirstOrder.Arith.oringEmb_operator_mul_val
{L : Language}
[L.ORing]
:
Semiterm.lMap Language.oringEmb op(*).term = op(*).term
@[simp]
theorem
LO.FirstOrder.Arith.oringEmb_operator_eq_val
{L : Language}
[L.ORing]
:
(Semiformula.lMap Language.oringEmb) op(=).sentence = op(=).sentence
@[simp]
theorem
LO.FirstOrder.Arith.oringEmb_operator_lt_val
{L : Language}
[L.ORing]
:
(Semiformula.lMap Language.oringEmb) op(<).sentence = op(<).sentence
Equations
- One or more equations did not get rendered due to their size.
theorem
LO.FirstOrder.Arith.standardModel_unique'
(M : Type u_1)
[ORingStruc M]
(s : Structure ββα΅£ M)
(hZero : Structure.Zero ββα΅£ M)
(hOne : Structure.One ββα΅£ M)
(hAdd : Structure.Add ββα΅£ M)
(hMul : Structure.Mul ββα΅£ M)
(hEq : Structure.Eq ββα΅£ M)
(hLT : Structure.LT ββα΅£ M)
:
s = standardModel M
theorem
LO.FirstOrder.Arith.standardModel_unique
(M : Type u_1)
[ORingStruc M]
(s : Structure ββα΅£ M)
[hZero : Structure.Zero ββα΅£ M]
[hOne : Structure.One ββα΅£ M]
[hAdd : Structure.Add ββα΅£ M]
[hMul : Structure.Mul ββα΅£ M]
[hEq : Structure.Eq ββα΅£ M]
[hLT : Structure.LT ββα΅£ M]
:
s = standardModel M
theorem
LO.FirstOrder.Arith.standardModel_lMap_oringEmb_eq_standardModel
(M : Type u_1)
[ORingStruc M]
{L : Language}
[L.ORing]
[s : Structure L M]
[Structure.Zero L M]
[Structure.One L M]
[Structure.Add L M]
[Structure.Mul L M]
[Structure.Eq L M]
[Structure.LT L M]
:
@[simp]
theorem
LO.FirstOrder.Arith.val_lMap_oringEmb
{n : β}
{ΞΎ : Type u_2}
{M : Type u_1}
[ORingStruc M]
{L : Language}
[L.ORing]
[s : Structure L M]
[Structure.Zero L M]
[Structure.One L M]
[Structure.Add L M]
[Structure.Mul L M]
[Structure.Eq L M]
[Structure.LT L M]
{e : Fin n β M}
{Ξ΅ : ΞΎ β M}
{t : Semiterm ββα΅£ ΞΎ n}
:
Semiterm.valm M e Ξ΅ (Semiterm.lMap Language.oringEmb t) = Semiterm.valm M e Ξ΅ t
@[simp]
theorem
LO.FirstOrder.Arith.eval_lMap_oringEmb
{n : β}
{ΞΎ : Type u_2}
{M : Type u_1}
[ORingStruc M]
{L : Language}
[L.ORing]
[s : Structure L M]
[Structure.Zero L M]
[Structure.One L M]
[Structure.Add L M]
[Structure.Mul L M]
[Structure.Eq L M]
[Structure.LT L M]
{e : Fin n β M}
{Ξ΅ : ΞΎ β M}
{Ο : Semiformula ββα΅£ ΞΎ n}
:
(Semiformula.Evalm M e Ξ΅) ((Semiformula.lMap Language.oringEmb) Ο) β (Semiformula.Evalm M e Ξ΅) Ο
@[simp]
theorem
LO.FirstOrder.Arith.modelsTheory_lMap_oringEmb
{L : Language}
[L.ORing]
{M : Type u_1}
[ORingStruc M]
[s : Structure L M]
[Structure.Zero L M]
[Structure.One L M]
[Structure.Add L M]
[Structure.Mul L M]
[Structure.Eq L M]
[Structure.LT L M]
(T : Theory ββα΅£)
:
M β§β* Theory.lMap Language.oringEmb T β M β§β* T
instance
LO.FirstOrder.Arith.instModelsTheoryPAMinusOfIOpen
{M : Type u_1}
[ORingStruc M]
[M β§β* πopen]
:
instance
LO.FirstOrder.Arith.instModelsTheoryIndSchemeORingOpenNatOfIOpen
{M : Type u_1}
[ORingStruc M]
[M β§β* πopen]
:
def
LO.FirstOrder.Arith.models_PAMinus_of_models_indH
{M : Type u_1}
[ORingStruc M]
(Ξ : Polarity)
(n : β)
[M β§β* πππΞ n]
:
Equations
- β― = β―
Instances For
def
LO.FirstOrder.Arith.models_indScheme_of_models_indH
{M : Type u_1}
[ORingStruc M]
(Ξ : Polarity)
(n : β)
[M β§β* πππΞ n]
:
M β§β* Theory.indScheme ββα΅£ (Hierarchy Ξ n)
Equations
- β― = β―
Instances For
instance
LO.FirstOrder.Arith.models_PAMinus_of_models_peano
{M : Type u_1}
[ORingStruc M]
[M β§β* ππ]
:
instance
LO.FirstOrder.Arith.Standard.models_iSigma
(Ξ : Polarity)
(k : β)
:
β β§β* πππΞ k
- domain : Set M
- closedSucc (x : M) : x β self.domain β Semiterm.valb s ![x] (β(!!(Semiterm.bvar 0) + !!β1)β) β self.domain
- closedLt (x y : M) : (Semiformula.Evalb s ![x, y]) (β!!(Semiterm.bvar 0) < !!(Semiterm.bvar 1)β) β y β self.domain β x β self.domain
Instances For
structure
LO.FirstOrder.Arith.ClosedCut
(L : Language)
[L.ORing]
(M : Type w)
[s : Structure L M]
extends LO.FirstOrder.Structure.ClosedSubset L M :
Type w
- closedLt (x y : M) : (Semiformula.Evalb s ![x, y]) (β!!(Semiterm.bvar 0) < !!(Semiterm.bvar 1)β) β y β self.domain β x β self.domain
Instances For
@[reducible, inline]
Equations
Instances For
Equations
- LO.FirstOrder.Arith.Β«termππΒ» = Lean.ParserDescr.node `LO.FirstOrder.Arith.Β«termππΒ» 1024 (Lean.ParserDescr.symbol "ππ")
Instances For
theorem
LO.FirstOrder.Arith.oRing_consequence_of
(T : Theory ββα΅£)
[ππ βΌ T]
(Ο : SyntacticFormula ββα΅£)
(H : β (M : Type u_1) [inst : ORingStruc M] [inst_1 : M β§β* T], M β§β Ο)
:
T β¨ Ο