Documentation

Mathlib.ModelTheory.Syntax

Basics on First-Order Syntax #

This file defines first-order terms, formulas, sentences, and theories in a style inspired by the Flypitch project.

Main Definitions #

Implementation Notes #

References #

For the Flypitch project:

source
inductive FirstOrder.Language.Term (L : Language) (α : Type u') :
Type (max u u')

A term on α is either a variable indexed by an element of α or a function symbol applied to simpler terms.

Instances For
source
instance FirstOrder.Language.Term.instDecidableEq {L : Language} {α : Type u'} [DecidableEq α] [(n : ℕ) → DecidableEq (L.Functions n)] :
DecidableEq (L.Term α)
Equations
source
def FirstOrder.Language.Term.varFinset {L : Language} {α : Type u'} [DecidableEq α] :
L.Term α → Finset α

The Finset of variables used in a given term.

Equations
source
def FirstOrder.Language.Term.varFinsetLeft {L : Language} {α : Type u'} {β : Type v'} [DecidableEq α] :
L.Term (α ⊕ β) → Finset α

The Finset of variables from the left side of a sum used in a given term.

Equations
source
def FirstOrder.Language.Term.relabel {L : Language} {α : Type u'} {β : Type v'} (g : α → β) :
L.Term α → L.Term β

Relabels a term's variables along a particular function.

Equations
source
theorem FirstOrder.Language.Term.relabel_id {L : Language} {α : Type u'} (t : L.Term α) :
relabel id t = t
source
@[simp]
theorem FirstOrder.Language.Term.relabel_id_eq_id {L : Language} {α : Type u'} :
relabel id = id
source
@[simp]
theorem FirstOrder.Language.Term.relabel_relabel {L : Language} {α : Type u'} {β : Type v'} {γ : Type u_1} (f : α → β) (g : β → γ) (t : L.Term α) :
relabel g (relabel f t) = relabel (g ∘ f) t
source
@[simp]
theorem FirstOrder.Language.Term.relabel_comp_relabel {L : Language} {α : Type u'} {β : Type v'} {γ : Type u_1} (f : α → β) (g : β → γ) :
relabel g ∘ relabel f = relabel (g ∘ f)
source
def FirstOrder.Language.Term.relabelEquiv {L : Language} {α : Type u'} {β : Type v'} (g : α ≃ β) :
L.Term α ≃ L.Term β

Relabels a term's variables along a bijection.

Equations
source
@[simp]
theorem FirstOrder.Language.Term.relabelEquiv_symm_apply {L : Language} {α : Type u'} {β : Type v'} (g : α ≃ β) (a✝ : L.Term β) :
(relabelEquiv g).symm a✝ = relabel (⇑g.symm) a✝
source
@[simp]
theorem FirstOrder.Language.Term.relabelEquiv_apply {L : Language} {α : Type u'} {β : Type v'} (g : α ≃ β) (a✝ : L.Term α) :
(relabelEquiv g) a✝ = relabel (⇑g) a✝
source
def FirstOrder.Language.Term.restrictVar {L : Language} {α : Type u'} {β : Type v'} [DecidableEq α] (t : L.Term α) (_f : { x : α // x ∈ t.varFinset } → β) :
L.Term β

Restricts a term to use only a set of the given variables.

Equations
source
def FirstOrder.Language.Term.restrictVarLeft {L : Language} {α : Type u'} {β : Type v'} [DecidableEq α] {γ : Type u_2} (t : L.Term (α ⊕ γ)) (_f : { x : α // x ∈ t.varFinsetLeft } → β) :
L.Term (β ⊕ γ)

Restricts a term to use only a set of the given variables on the left side of a sum.

Equations
source
def FirstOrder.Language.Constants.term {L : Language} {α : Type u'} (c : L.Constants) :
L.Term α

The representation of a constant symbol as a term.

Equations
source
def FirstOrder.Language.Functions.apply₁ {L : Language} {α : Type u'} (f : L.Functions 1) (t : L.Term α) :
L.Term α

Applies a unary function to a term.

Equations
source
def FirstOrder.Language.Functions.apply₂ {L : Language} {α : Type u'} (f : L.Functions 2) (t₁ t₂ : L.Term α) :
L.Term α

Applies a binary function to two terms.

Equations
source
def FirstOrder.Language.Term.constantsToVars {L : Language} {α : Type u'} {γ : Type u_1} :
(L.withConstants γ).Term α → L.Term (γ ⊕ α)

Sends a term with constants to a term with extra variables.

Equations
source
def FirstOrder.Language.Term.varsToConstants {L : Language} {α : Type u'} {γ : Type u_1} :
L.Term (γ ⊕ α) → (L.withConstants γ).Term α

Sends a term with extra variables to a term with constants.

Equations
source
def FirstOrder.Language.Term.constantsVarsEquiv {L : Language} {α : Type u'} {γ : Type u_1} :
(L.withConstants γ).Term α ≃ L.Term (γ ⊕ α)

A bijection between terms with constants and terms with extra variables.

Equations
source
@[simp]
theorem FirstOrder.Language.Term.constantsVarsEquiv_symm_apply {L : Language} {α : Type u'} {γ : Type u_1} (a✝ : L.Term (γ ⊕ α)) :
constantsVarsEquiv.symm a✝ = a✝.varsToConstants
source
@[simp]
theorem FirstOrder.Language.Term.constantsVarsEquiv_apply {L : Language} {α : Type u'} {γ : Type u_1} (a✝ : (L.withConstants γ).Term α) :
constantsVarsEquiv a✝ = a✝.constantsToVars
source
def FirstOrder.Language.Term.constantsVarsEquivLeft {L : Language} {α : Type u'} {β : Type v'} {γ : Type u_1} :
(L.withConstants γ).Term (α ⊕ β) ≃ L.Term ((γ ⊕ α) ⊕ β)

A bijection between terms with constants and terms with extra variables.

Equations
source
@[simp]
theorem FirstOrder.Language.Term.constantsVarsEquivLeft_apply {L : Language} {α : Type u'} {β : Type v'} {γ : Type u_1} (t : (L.withConstants γ).Term (α ⊕ β)) :
constantsVarsEquivLeft t = relabel (⇑(Equiv.sumAssoc γ α β).symm) t.constantsToVars
source
@[simp]
theorem FirstOrder.Language.Term.constantsVarsEquivLeft_symm_apply {L : Language} {α : Type u'} {β : Type v'} {γ : Type u_1} (t : L.Term ((γ ⊕ α) ⊕ β)) :
constantsVarsEquivLeft.symm t = (relabel (⇑(Equiv.sumAssoc γ α β)) t).varsToConstants
source
instance FirstOrder.Language.Term.inhabitedOfVar {L : Language} {α : Type u'} [Inhabited α] :
Inhabited (L.Term α)
Equations
source
instance FirstOrder.Language.Term.inhabitedOfConstant {L : Language} {α : Type u'} [Inhabited L.Constants] :
Inhabited (L.Term α)
Equations
source
def FirstOrder.Language.Term.liftAt {L : Language} {α : Type u'} {n : ℕ} (n' m : ℕ) :
L.Term (α ⊕ Fin n) → L.Term (α ⊕ Fin (n + n'))

Raises all of the Fin-indexed variables of a term greater than or equal to m by n'.

Equations
source
def FirstOrder.Language.Term.subst {L : Language} {α : Type u'} {β : Type v'} :
L.Term α → (α → L.Term β) → L.Term β

Substitutes the variables in a given term with terms.

Equations
source
def FirstOrder.«term&_» :
Lean.ParserDescr

&n is notation for the n-th free variable of a bounded formula.

Equations
source
def FirstOrder.Language.LHom.onTerm {L : Language} {L' : Language} {α : Type u'} (φ : L →ᴸ L') :
L.Term α → L'.Term α

Maps a term's symbols along a language map.

Equations
source
@[simp]
theorem FirstOrder.Language.LHom.id_onTerm {L : Language} {α : Type u'} :
(LHom.id L).onTerm = id
source
@[simp]
theorem FirstOrder.Language.LHom.comp_onTerm {L : Language} {L' : Language} {α : Type u'} {L'' : Language} (φ : L' →ᴸ L'') (ψ : L →ᴸ L') :
(φ.comp ψ).onTerm = φ.onTerm ∘ ψ.onTerm
source
def FirstOrder.Language.LEquiv.onTerm {L : Language} {L' : Language} {α : Type u'} (φ : L ≃ᴸ L') :
L.Term α ≃ L'.Term α

Maps a term's symbols along a language equivalence.

Equations
source
@[simp]
theorem FirstOrder.Language.LEquiv.onTerm_apply {L : Language} {L' : Language} {α : Type u'} (φ : L ≃ᴸ L') (a✝ : L.Term α) :
φ.onTerm a✝ = φ.toLHom.onTerm a✝
source
@[simp]
theorem FirstOrder.Language.LEquiv.onTerm_symm_apply {L : Language} {L' : Language} {α : Type u'} (φ : L ≃ᴸ L') (a✝ : L'.Term α) :
φ.onTerm.symm a✝ = φ.invLHom.onTerm a✝
source
@[deprecated FirstOrder.Language.LEquiv.onTerm (since := "2025-03-31")]
def FirstOrder.Language.Lequiv.onTerm {L : Language} {L' : Language} {α : Type u'} (φ : L ≃ᴸ L') :
L.Term α ≃ L'.Term α

Maps a term's symbols along a language equivalence. Deprecated in favor of LEquiv.onTerm.

Equations
source
inductive FirstOrder.Language.BoundedFormula (L : Language) (α : Type u') :
ℕ → Type (max u v u')

BoundedFormula α n is the type of formulas with free variables indexed by α and up to n additional free variables.

Instances For
source
@[reducible, inline]
abbrev FirstOrder.Language.Formula (L : Language) (α : Type u') :
Type (max u v u')

Formula α is the type of formulas with all free variables indexed by α.

Equations
source
@[reducible, inline]
abbrev FirstOrder.Language.Sentence (L : Language) :
Type (max u v)

A sentence is a formula with no free variables.

Equations
source
@[reducible, inline]
abbrev FirstOrder.Language.Theory (L : Language) :
Type (max u v)

A theory is a set of sentences.

Equations
source
def FirstOrder.Language.Relations.boundedFormula {L : Language} {α : Type u'} {n l : ℕ} (R : L.Relations n) (ts : Fin n → L.Term (α ⊕ Fin l)) :
L.BoundedFormula α l

Applies a relation to terms as a bounded formula.

Equations
source
def FirstOrder.Language.Relations.boundedFormula₁ {L : Language} {α : Type u'} {n : ℕ} (r : L.Relations 1) (t : L.Term (α ⊕ Fin n)) :
L.BoundedFormula α n

Applies a unary relation to a term as a bounded formula.

Equations
source
def FirstOrder.Language.Relations.boundedFormula₂ {L : Language} {α : Type u'} {n : ℕ} (r : L.Relations 2) (t₁ t₂ : L.Term (α ⊕ Fin n)) :
L.BoundedFormula α n

Applies a binary relation to two terms as a bounded formula.

Equations
source
def FirstOrder.Language.Term.bdEqual {L : Language} {α : Type u'} {n : ℕ} (t₁ t₂ : L.Term (α ⊕ Fin n)) :
L.BoundedFormula α n

The equality of two terms as a bounded formula.

Equations
source
def FirstOrder.Language.Relations.formula {L : Language} {α : Type u'} {n : ℕ} (R : L.Relations n) (ts : Fin n → L.Term α) :
L.Formula α

Applies a relation to terms as a bounded formula.

Equations
source
def FirstOrder.Language.Relations.formula₁ {L : Language} {α : Type u'} (r : L.Relations 1) (t : L.Term α) :
L.Formula α

Applies a unary relation to a term as a formula.

Equations
source
def FirstOrder.Language.Relations.formula₂ {L : Language} {α : Type u'} (r : L.Relations 2) (t₁ t₂ : L.Term α) :
L.Formula α

Applies a binary relation to two terms as a formula.

Equations
source
def FirstOrder.Language.Term.equal {L : Language} {α : Type u'} (t₁ t₂ : L.Term α) :
L.Formula α

The equality of two terms as a first-order formula.

Equations
source
instance FirstOrder.Language.BoundedFormula.instInhabited {L : Language} {α : Type u'} {n : ℕ} :
Inhabited (L.BoundedFormula α n)
Equations
source
instance FirstOrder.Language.BoundedFormula.instBot {L : Language} {α : Type u'} {n : ℕ} :
Bot (L.BoundedFormula α n)
Equations
source
@[match_pattern]
def FirstOrder.Language.BoundedFormula.not {L : Language} {α : Type u'} {n : ℕ} (φ : L.BoundedFormula α n) :
L.BoundedFormula α n

The negation of a bounded formula is also a bounded formula.

Equations
source
@[match_pattern]
def FirstOrder.Language.BoundedFormula.ex {L : Language} {α : Type u'} {n : ℕ} (φ : L.BoundedFormula α (n + 1)) :
L.BoundedFormula α n

Puts an ∃ quantifier on a bounded formula.

Equations
source
instance FirstOrder.Language.BoundedFormula.instTop {L : Language} {α : Type u'} {n : ℕ} :
Top (L.BoundedFormula α n)
Equations
source
instance FirstOrder.Language.BoundedFormula.instMin {L : Language} {α : Type u'} {n : ℕ} :
Min (L.BoundedFormula α n)
Equations
source
instance FirstOrder.Language.BoundedFormula.instMax {L : Language} {α : Type u'} {n : ℕ} :
Max (L.BoundedFormula α n)
Equations
source
def FirstOrder.Language.BoundedFormula.iff {L : Language} {α : Type u'} {n : ℕ} (φ ψ : L.BoundedFormula α n) :
L.BoundedFormula α n

The biimplication between two bounded formulas.

Equations
source
def FirstOrder.Language.BoundedFormula.freeVarFinset {L : Language} {α : Type u'} [DecidableEq α] {n : ℕ} :
L.BoundedFormula α n → Finset α

The Finset of variables used in a given formula.

Equations
source
def FirstOrder.Language.BoundedFormula.castLE {L : Language} {α : Type u'} {m n : ℕ} (_h : m ≤ n) :
L.BoundedFormula α m → L.BoundedFormula α n

Casts L.BoundedFormula α m as L.BoundedFormula α n, where m ≤ n.

Equations
source
@[simp]
theorem FirstOrder.Language.BoundedFormula.castLE_rfl {L : Language} {α : Type u'} {n : ℕ} (h : n ≤ n) (φ : L.BoundedFormula α n) :
castLE h φ = φ
source
@[simp]
theorem FirstOrder.Language.BoundedFormula.castLE_castLE {L : Language} {α : Type u'} {k m n : ℕ} (km : k ≤ m) (mn : m ≤ n) (φ : L.BoundedFormula α k) :
castLE mn (castLE km φ) = castLE ⋯ φ
source
@[simp]
theorem FirstOrder.Language.BoundedFormula.castLE_comp_castLE {L : Language} {α : Type u'} {k m n : ℕ} (km : k ≤ m) (mn : m ≤ n) :
castLE mn ∘ castLE km = castLE ⋯
source
def FirstOrder.Language.BoundedFormula.restrictFreeVar {L : Language} {α : Type u'} {β : Type v'} [DecidableEq α] {n : ℕ} (φ : L.BoundedFormula α n) (_f : { x : α // x ∈ φ.freeVarFinset } → β) :
L.BoundedFormula β n

Restricts a bounded formula to only use a particular set of free variables.

Equations
source
def FirstOrder.Language.BoundedFormula.alls {L : Language} {α : Type u'} {n : ℕ} :
L.BoundedFormula α n → L.Formula α

Places universal quantifiers on all extra variables of a bounded formula.

Equations
source
def FirstOrder.Language.BoundedFormula.exs {L : Language} {α : Type u'} {n : ℕ} :
L.BoundedFormula α n → L.Formula α

Places existential quantifiers on all extra variables of a bounded formula.

Equations
source
def FirstOrder.Language.BoundedFormula.mapTermRel {L : Language} {L' : Language} {α : Type u'} {β : Type v'} {g : ℕ → ℕ} (ft : (n : ℕ) → L.Term (α ⊕ Fin n) → L'.Term (β ⊕ Fin (g n))) (fr : (n : ℕ) → L.Relations n → L'.Relations n) (h : (n : ℕ) → L'.BoundedFormula β (g (n + 1)) → L'.BoundedFormula β (g n + 1)) {n : ℕ} :
L.BoundedFormula α n → L'.BoundedFormula β (g n)

Maps bounded formulas along a map of terms and a map of relations.

Equations
source
def FirstOrder.Language.BoundedFormula.liftAt {L : Language} {α : Type u'} {n : ℕ} (n' _m : ℕ) :
L.BoundedFormula α n → L.BoundedFormula α (n + n')

Raises all of the Fin-indexed variables of a formula greater than or equal to m by n'.

Equations
  • One or more equations did not get rendered due to their size.
source
@[simp]
theorem FirstOrder.Language.BoundedFormula.mapTermRel_mapTermRel {L : Language} {L' : Language} {α : Type u'} {β : Type v'} {γ : Type u_1} {L'' : Language} (ft : (n : ℕ) → L.Term (α ⊕ Fin n) → L'.Term (β ⊕ Fin n)) (fr : (n : ℕ) → L.Relations n → L'.Relations n) (ft' : (n : ℕ) → L'.Term (β ⊕ Fin n) → L''.Term (γ ⊕ Fin n)) (fr' : (n : ℕ) → L'.Relations n → L''.Relations n) {n : ℕ} (φ : L.BoundedFormula α n) :
mapTermRel ft' fr' (fun (x : ℕ) => id) (mapTermRel ft fr (fun (x : ℕ) => id) φ) = mapTermRel (fun (x : ℕ) => ft' x ∘ ft x) (fun (x : ℕ) => fr' x ∘ fr x) (fun (x : ℕ) => id) φ
source
@[simp]
theorem FirstOrder.Language.BoundedFormula.mapTermRel_id_id_id {L : Language} {α : Type u'} {n : ℕ} (φ : L.BoundedFormula α n) :
mapTermRel (fun (x : ℕ) => id) (fun (x : ℕ) => id) (fun (x : ℕ) => id) φ = φ
source
def FirstOrder.Language.BoundedFormula.mapTermRelEquiv {L : Language} {L' : Language} {α : Type u'} {β : Type v'} (ft : (n : ℕ) → L.Term (α ⊕ Fin n) ≃ L'.Term (β ⊕ Fin n)) (fr : (n : ℕ) → L.Relations n ≃ L'.Relations n) {n : ℕ} :
L.BoundedFormula α n ≃ L'.BoundedFormula β n

An equivalence of bounded formulas given by an equivalence of terms and an equivalence of relations.

Equations
  • One or more equations did not get rendered due to their size.
source
@[simp]
theorem FirstOrder.Language.BoundedFormula.mapTermRelEquiv_apply {L : Language} {L' : Language} {α : Type u'} {β : Type v'} (ft : (n : ℕ) → L.Term (α ⊕ Fin n) ≃ L'.Term (β ⊕ Fin n)) (fr : (n : ℕ) → L.Relations n ≃ L'.Relations n) {n : ℕ} (a✝ : L.BoundedFormula α n) :
(mapTermRelEquiv ft fr) a✝ = mapTermRel (fun (n : ℕ) => ⇑(ft n)) (fun (n : ℕ) => ⇑(fr n)) (fun (x : ℕ) => id) a✝
source
@[simp]
theorem FirstOrder.Language.BoundedFormula.mapTermRelEquiv_symm_apply {L : Language} {L' : Language} {α : Type u'} {β : Type v'} (ft : (n : ℕ) → L.Term (α ⊕ Fin n) ≃ L'.Term (β ⊕ Fin n)) (fr : (n : ℕ) → L.Relations n ≃ L'.Relations n) {n : ℕ} (a✝ : L'.BoundedFormula β n) :
(mapTermRelEquiv ft fr).symm a✝ = mapTermRel (fun (n : ℕ) => ⇑(ft n).symm) (fun (n : ℕ) => ⇑(fr n).symm) (fun (x : ℕ) => id) a✝
source
def FirstOrder.Language.BoundedFormula.relabelAux {α : Type u'} {β : Type v'} {n : ℕ} (g : α → β ⊕ Fin n) (k : ℕ) :
α ⊕ Fin k → β ⊕ Fin (n + k)

A function to help relabel the variables in bounded formulas.

Equations
source
@[simp]
theorem FirstOrder.Language.BoundedFormula.sumElim_comp_relabelAux {M : Type w} {α : Type u'} {β : Type v'} {n m : ℕ} {g : α → β ⊕ Fin n} {v : β → M} {xs : Fin (n + m) → M} :
Sum.elim v xs ∘ relabelAux g m = Sum.elim (Sum.elim v (xs ∘ Fin.castAdd m) ∘ g) (xs ∘ Fin.natAdd n)
source
@[deprecated FirstOrder.Language.BoundedFormula.sumElim_comp_relabelAux (since := "2025-02-21")]
theorem FirstOrder.Language.BoundedFormula.sum_elim_comp_relabelAux {M : Type w} {α : Type u'} {β : Type v'} {n m : ℕ} {g : α → β ⊕ Fin n} {v : β → M} {xs : Fin (n + m) → M} :
Sum.elim v xs ∘ relabelAux g m = Sum.elim (Sum.elim v (xs ∘ Fin.castAdd m) ∘ g) (xs ∘ Fin.natAdd n)

Alias of FirstOrder.Language.BoundedFormula.sumElim_comp_relabelAux.

source
@[simp]
theorem FirstOrder.Language.BoundedFormula.relabelAux_sumInl {α : Type u'} {n : ℕ} (k : ℕ) :
relabelAux Sum.inl k = Sum.map id (Fin.natAdd n)
source
@[deprecated FirstOrder.Language.BoundedFormula.relabelAux_sumInl (since := "2025-02-21")]
theorem FirstOrder.Language.BoundedFormula.relabelAux_sum_inl {α : Type u'} {n : ℕ} (k : ℕ) :
relabelAux Sum.inl k = Sum.map id (Fin.natAdd n)

Alias of FirstOrder.Language.BoundedFormula.relabelAux_sumInl.

source
def FirstOrder.Language.BoundedFormula.relabel {L : Language} {α : Type u'} {β : Type v'} {n : ℕ} (g : α → β ⊕ Fin n) {k : ℕ} (φ : L.BoundedFormula α k) :
L.BoundedFormula β (n + k)

Relabels a bounded formula's variables along a particular function.

Equations
  • One or more equations did not get rendered due to their size.
source
def FirstOrder.Language.BoundedFormula.relabelEquiv {L : Language} {α : Type u'} {β : Type v'} (g : α ≃ β) {k : ℕ} :
L.BoundedFormula α k ≃ L.BoundedFormula β k

Relabels a bounded formula's free variables along a bijection.

Equations
  • One or more equations did not get rendered due to their size.
source
@[simp]
theorem FirstOrder.Language.BoundedFormula.relabel_falsum {L : Language} {α : Type u'} {β : Type v'} {n : ℕ} (g : α → β ⊕ Fin n) {k : ℕ} :
relabel g falsum = falsum
source
@[simp]
theorem FirstOrder.Language.BoundedFormula.relabel_bot {L : Language} {α : Type u'} {β : Type v'} {n : ℕ} (g : α → β ⊕ Fin n) {k : ℕ} :
relabel g ⊥ = ⊥
source
@[simp]
theorem FirstOrder.Language.BoundedFormula.relabel_imp {L : Language} {α : Type u'} {β : Type v'} {n : ℕ} (g : α → β ⊕ Fin n) {k : ℕ} (φ ψ : L.BoundedFormula α k) :
relabel g (φ.imp ψ) = (relabel g φ).imp (relabel g ψ)
source
@[simp]
theorem FirstOrder.Language.BoundedFormula.relabel_not {L : Language} {α : Type u'} {β : Type v'} {n : ℕ} (g : α → β ⊕ Fin n) {k : ℕ} (φ : L.BoundedFormula α k) :
relabel g φ.not = (relabel g φ).not
source
@[simp]
theorem FirstOrder.Language.BoundedFormula.relabel_all {L : Language} {α : Type u'} {β : Type v'} {n : ℕ} (g : α → β ⊕ Fin n) {k : ℕ} (φ : L.BoundedFormula α (k + 1)) :
relabel g φ.all = (relabel g φ).all
source
@[simp]
theorem FirstOrder.Language.BoundedFormula.relabel_ex {L : Language} {α : Type u'} {β : Type v'} {n : ℕ} (g : α → β ⊕ Fin n) {k : ℕ} (φ : L.BoundedFormula α (k + 1)) :
relabel g φ.ex = (relabel g φ).ex
source
@[simp]
theorem FirstOrder.Language.BoundedFormula.relabel_sumInl {L : Language} {α : Type u'} {n : ℕ} (φ : L.BoundedFormula α n) :
relabel Sum.inl φ = castLE ⋯ φ
source
@[deprecated FirstOrder.Language.BoundedFormula.relabel_sumInl (since := "2025-02-21")]
theorem FirstOrder.Language.BoundedFormula.relabel_sum_inl {L : Language} {α : Type u'} {n : ℕ} (φ : L.BoundedFormula α n) :
relabel Sum.inl φ = castLE ⋯ φ

Alias of FirstOrder.Language.BoundedFormula.relabel_sumInl.

source
def FirstOrder.Language.BoundedFormula.subst {L : Language} {α : Type u'} {β : Type v'} {n : ℕ} (φ : L.BoundedFormula α n) (f : α → L.Term β) :
L.BoundedFormula β n

Substitutes the variables in a given formula with terms.

Equations
  • One or more equations did not get rendered due to their size.
source
def FirstOrder.Language.BoundedFormula.constantsVarsEquiv {L : Language} {α : Type u'} {γ : Type u_1} {n : ℕ} :
(L.withConstants γ).BoundedFormula α n ≃ L.BoundedFormula (γ ⊕ α) n

A bijection sending formulas with constants to formulas with extra variables.

Equations
  • One or more equations did not get rendered due to their size.
source
def FirstOrder.Language.BoundedFormula.toFormula {L : Language} {α : Type u'} {n : ℕ} :
L.BoundedFormula α n → L.Formula (α ⊕ Fin n)

Turns the extra variables of a bounded formula into free variables.

Equations
source
noncomputable def FirstOrder.Language.BoundedFormula.iSup {L : Language} {α : Type u'} {β : Type v'} {n : ℕ} [Finite β] (f : β → L.BoundedFormula α n) :
L.BoundedFormula α n

Take the disjunction of a finite set of formulas

Equations
source
noncomputable def FirstOrder.Language.BoundedFormula.iInf {L : Language} {α : Type u'} {β : Type v'} {n : ℕ} [Finite β] (f : β → L.BoundedFormula α n) :
L.BoundedFormula α n

Take the conjunction of a finite set of formulas

Equations
source
def FirstOrder.Language.LHom.onBoundedFormula {L : Language} {L' : Language} {α : Type u'} (g : L →ᴸ L') {k : ℕ} :
L.BoundedFormula α k → L'.BoundedFormula α k

Maps a bounded formula's symbols along a language map.

Equations
source
@[simp]
theorem FirstOrder.Language.LHom.id_onBoundedFormula {L : Language} {α : Type u'} {n : ℕ} :
(LHom.id L).onBoundedFormula = id
source
@[simp]
theorem FirstOrder.Language.LHom.comp_onBoundedFormula {L : Language} {L' : Language} {α : Type u'} {n : ℕ} {L'' : Language} (φ : L' →ᴸ L'') (ψ : L →ᴸ L') :
(φ.comp ψ).onBoundedFormula = φ.onBoundedFormula ∘ ψ.onBoundedFormula
source
def FirstOrder.Language.LHom.onFormula {L : Language} {L' : Language} {α : Type u'} (g : L →ᴸ L') :
L.Formula α → L'.Formula α

Maps a formula's symbols along a language map.

Equations
source
def FirstOrder.Language.LHom.onSentence {L : Language} {L' : Language} (g : L →ᴸ L') :
L.Sentence → L'.Sentence

Maps a sentence's symbols along a language map.

Equations
source
def FirstOrder.Language.LHom.onTheory {L : Language} {L' : Language} (g : L →ᴸ L') (T : L.Theory) :
L'.Theory

Maps a theory's symbols along a language map.

Equations
source
@[simp]
theorem FirstOrder.Language.LHom.mem_onTheory {L : Language} {L' : Language} {g : L →ᴸ L'} {T : L.Theory} {φ : L'.Sentence} :
φ ∈ g.onTheory T ↔ ∃ φ₀ ∈ T, g.onSentence φ₀ = φ
source
def FirstOrder.Language.LEquiv.onBoundedFormula {L : Language} {L' : Language} {α : Type u'} {n : ℕ} (φ : L ≃ᴸ L') :
L.BoundedFormula α n ≃ L'.BoundedFormula α n

Maps a bounded formula's symbols along a language equivalence.

Equations
source
@[simp]
theorem FirstOrder.Language.LEquiv.onBoundedFormula_apply {L : Language} {L' : Language} {α : Type u'} {n : ℕ} (φ : L ≃ᴸ L') (a✝ : L.BoundedFormula α n) :
φ.onBoundedFormula a✝ = φ.toLHom.onBoundedFormula a✝
source
@[simp]
theorem FirstOrder.Language.LEquiv.onBoundedFormula_symm_apply {L : Language} {L' : Language} {α : Type u'} {n : ℕ} (φ : L ≃ᴸ L') (a✝ : L'.BoundedFormula α n) :
φ.onBoundedFormula.symm a✝ = φ.invLHom.onBoundedFormula a✝
source
theorem FirstOrder.Language.LEquiv.onBoundedFormula_symm {L : Language} {L' : Language} {α : Type u'} {n : ℕ} (φ : L ≃ᴸ L') :
φ.onBoundedFormula.symm = φ.symm.onBoundedFormula
source
def FirstOrder.Language.LEquiv.onFormula {L : Language} {L' : Language} {α : Type u'} (φ : L ≃ᴸ L') :
L.Formula α ≃ L'.Formula α

Maps a formula's symbols along a language equivalence.

Equations
source
@[simp]
theorem FirstOrder.Language.LEquiv.onFormula_apply {L : Language} {L' : Language} {α : Type u'} (φ : L ≃ᴸ L') :
⇑φ.onFormula = φ.toLHom.onFormula
source
@[simp]
theorem FirstOrder.Language.LEquiv.onFormula_symm {L : Language} {L' : Language} {α : Type u'} (φ : L ≃ᴸ L') :
φ.onFormula.symm = φ.symm.onFormula
source
def FirstOrder.Language.LEquiv.onSentence {L : Language} {L' : Language} (φ : L ≃ᴸ L') :
L.Sentence ≃ L'.Sentence

Maps a sentence's symbols along a language equivalence.

Equations
source
@[simp]
theorem FirstOrder.Language.LEquiv.onSentence_symm_apply {L : Language} {L' : Language} (φ : L ≃ᴸ L') (a✝ : L'.BoundedFormula Empty 0) :
φ.onSentence.symm a✝ = φ.invLHom.onBoundedFormula a✝
source
@[simp]
theorem FirstOrder.Language.LEquiv.onSentence_apply {L : Language} {L' : Language} (φ : L ≃ᴸ L') (a✝ : L.BoundedFormula Empty 0) :
φ.onSentence a✝ = φ.toLHom.onBoundedFormula a✝
source
def FirstOrder.«term_='_» :
Lean.TrailingParserDescr

The equality of two terms as a bounded formula.

Equations
source
def FirstOrder.«term_⟹_» :
Lean.TrailingParserDescr

The implication between two bounded formulas

Equations
source
def FirstOrder.«term∀'_» :
Lean.ParserDescr

The universal quantifier over bounded formulas

Equations
source
def FirstOrder.«term∼_» :
Lean.ParserDescr

The negation of a bounded formula is also a bounded formula.

Equations
source
def FirstOrder.«term_⇔_» :
Lean.TrailingParserDescr

The biimplication between two bounded formulas.

Equations
source
def FirstOrder.«term∃'_» :
Lean.ParserDescr

Puts an ∃ quantifier on a bounded formula.

Equations
source
def FirstOrder.Language.Formula.relabel {L : Language} {α : Type u'} {β : Type v'} (g : α → β) :
L.Formula α → L.Formula β

Relabels a formula's variables along a particular function.

Equations
source
def FirstOrder.Language.Formula.graph {L : Language} {n : ℕ} (f : L.Functions n) :
L.Formula (Fin (n + 1))

The graph of a function as a first-order formula.

Equations
source
@[reducible, inline]
abbrev FirstOrder.Language.Formula.not {L : Language} {α : Type u'} (φ : L.Formula α) :
L.Formula α

The negation of a formula.

Equations
source
@[reducible, inline]
abbrev FirstOrder.Language.Formula.imp {L : Language} {α : Type u'} :
L.Formula α → L.Formula α → L.Formula α

The implication between formulas, as a formula.

Equations
source
noncomputable def FirstOrder.Language.Formula.iAlls {L : Language} {α : Type u'} (β : Type v') [Finite β] (φ : L.Formula (α ⊕ β)) :
L.Formula α

iAlls f φ transforms a L.Formula (α ⊕ β) into a L.Formula β by universally quantifying over all variables Sum.inr _.

Equations
source
noncomputable def FirstOrder.Language.Formula.iExs {L : Language} {α : Type u'} (β : Type v') [Finite β] (φ : L.Formula (α ⊕ β)) :
L.Formula α

iExs f φ transforms a L.Formula (α ⊕ β) into a L.Formula β by existentially quantifying over all variables Sum.inr _.

Equations
source
noncomputable def FirstOrder.Language.Formula.iExsUnique {L : Language} {α : Type u'} (β : Type v') [Finite β] (φ : L.Formula (α ⊕ β)) :
L.Formula α

iExsUnique f φ transforms a L.Formula (α ⊕ β) into a L.Formula β by existentially quantifying over all variables Sum.inr _ and asserting that the solution should be unique

Equations
  • One or more equations did not get rendered due to their size.
source
@[reducible, inline]
abbrev FirstOrder.Language.Formula.iff {L : Language} {α : Type u'} (φ ψ : L.Formula α) :
L.Formula α

The biimplication between formulas, as a formula.

Equations
source
def FirstOrder.Language.Formula.equivSentence {L : Language} {α : Type u'} :
L.Formula α ≃ (L.withConstants α).Sentence

A bijection sending formulas to sentences with constants.

Equations
source
theorem FirstOrder.Language.Formula.equivSentence_not {L : Language} {α : Type u'} (φ : L.Formula α) :
equivSentence φ.not = Formula.not (equivSentence φ)
source
theorem FirstOrder.Language.Formula.equivSentence_inf {L : Language} {α : Type u'} (φ ψ : L.Formula α) :
equivSentence (φ ⊓ ψ) = equivSentence φ ⊓ equivSentence ψ
source
def FirstOrder.Language.Relations.reflexive {L : Language} (r : L.Relations 2) :
L.Sentence

The sentence indicating that a basic relation symbol is reflexive.

Equations
source
def FirstOrder.Language.Relations.irreflexive {L : Language} (r : L.Relations 2) :
L.Sentence

The sentence indicating that a basic relation symbol is irreflexive.

Equations
source
def FirstOrder.Language.Relations.symmetric {L : Language} (r : L.Relations 2) :
L.Sentence

The sentence indicating that a basic relation symbol is symmetric.

Equations
  • One or more equations did not get rendered due to their size.
source
def FirstOrder.Language.Relations.antisymmetric {L : Language} (r : L.Relations 2) :
L.Sentence

The sentence indicating that a basic relation symbol is antisymmetric.

Equations
  • One or more equations did not get rendered due to their size.
source
def FirstOrder.Language.Relations.transitive {L : Language} (r : L.Relations 2) :
L.Sentence

The sentence indicating that a basic relation symbol is transitive.

Equations
  • One or more equations did not get rendered due to their size.
source
def FirstOrder.Language.Relations.total {L : Language} (r : L.Relations 2) :
L.Sentence

The sentence indicating that a basic relation symbol is total.

Equations
  • One or more equations did not get rendered due to their size.
source
def FirstOrder.Language.Sentence.cardGe (L : Language) (n : ℕ) :
L.Sentence

A sentence indicating that a structure has n distinct elements.

Equations
  • One or more equations did not get rendered due to their size.
source
def FirstOrder.Language.infiniteTheory (L : Language) :
L.Theory

A theory indicating that a structure is infinite.

Equations
Instances For
source
def FirstOrder.Language.nonemptyTheory (L : Language) :
L.Theory

A theory that indicates a structure is nonempty.

Equations
Instances For
source
def FirstOrder.Language.distinctConstantsTheory (L : Language) {α : Type u'} (s : Set α) :
(L.withConstants α).Theory

A theory indicating that each of a set of constants is distinct.

Equations
source
theorem FirstOrder.Language.distinctConstantsTheory_mono {L : Language} {α : Type u'} {s t : Set α} (h : s ⊆ t) :
L.distinctConstantsTheory s ⊆ L.distinctConstantsTheory t
source
theorem FirstOrder.Language.monotone_distinctConstantsTheory {L : Language} {α : Type u'} :
Monotone L.distinctConstantsTheory
source
theorem FirstOrder.Language.directed_distinctConstantsTheory {L : Language} {α : Type u'} :
Directed (fun (x1 x2 : (L.withConstants α).Theory) => x1 ⊆ x2) L.distinctConstantsTheory
source
theorem FirstOrder.Language.distinctConstantsTheory_eq_iUnion {L : Language} {α : Type u'} (s : Set α) :
L.distinctConstantsTheory s = ⋃ (t : Finset ↑s), L.distinctConstantsTheory ↑(Finset.map (Function.Embedding.subtype fun (x : α) => x ∈ s) t)