Documentation Verification Report

FinitelyGenerated

📁 Source: Mathlib/ModelTheory/FinitelyGenerated.lean

Statistics

MetricCount
DefinitionsCG, CG, instInhabited_fg
3
Theoremscg_iff, fg_iff, map_of_surjective, out, range, cg, countable_embedding, countable_hom, finite, instCountable_hom, map_of_surjective, of_finite, out, range, instCountable_embedding, cg_def, cg_iff, cg_iff_countable, cg_of_countable, cg_of_fg, fg_def, fg_iff, fg_iff_finite, map, of_map_embedding, sup, cg, finite, map, of_finite, of_map_embedding, sup, cg_bot, cg_closure, cg_closure_singleton, cg_def, cg_iff_countable, cg_iff_empty_or_exists_nat_generating_family, cg_iff_structure_cg, cg_of_countable, countable_fg_substructures_of_countable, fg_bot, fg_closure, fg_closure_singleton, fg_def, fg_iff_exists_fin_generating_family, fg_iff_finite, fg_iff_structure_fg, instCountable_fg_substructures_of_countable
49
Total52

FirstOrder.Language.Equiv

Theorems

NameKindAssumesProvesValidatesDepends On
cg_iff 📖mathematicalFirstOrder.Language.Structure.CGFirstOrder.Language.Structure.CG.map_of_surjective
Equiv.surjective
fg_iff 📖mathematicalFirstOrder.Language.Structure.FGFirstOrder.Language.Structure.FG.map_of_surjective
Equiv.surjective

FirstOrder.Language.Structure

Definitions

NameCategoryTheorems
CG 📖CompData
10 mathmath: CG.map_of_surjective, FG.cg, FirstOrder.Language.Substructure.cg_iff_structure_cg, FirstOrder.Language.exists_cg_is_age_of, cg_def, cg_iff, cg_of_fg, cg_iff_countable, FirstOrder.Language.Equiv.cg_iff, cg_of_countable

Theorems

NameKindAssumesProvesValidatesDepends On
cg_def 📖mathematicalCG
FirstOrder.Language.Substructure.CG
Top.top
FirstOrder.Language.Substructure
FirstOrder.Language.Substructure.instTop
CG.out
cg_iff 📖mathematicalCG
Set.Countable
LowerAdjoint.toFun
Set
FirstOrder.Language.Substructure
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
FirstOrder.Language.Substructure.instPartialOrder
SetLike.coe
FirstOrder.Language.Substructure.instSetLike
FirstOrder.Language.Substructure.closure
Top.top
FirstOrder.Language.Substructure.instTop
cg_def
FirstOrder.Language.Substructure.cg_def
cg_iff_countable 📖mathematicalCG
Countable
cg_def
FirstOrder.Language.Substructure.cg_iff_countable
Equiv.countable_iff
cg_of_countable 📖mathematicalCGSubtype.countable
cg_of_fg 📖mathematicalCGFG.cg
fg_def 📖mathematicalFG
FirstOrder.Language.Substructure.FG
Top.top
FirstOrder.Language.Substructure
FirstOrder.Language.Substructure.instTop
FG.out
fg_iff 📖mathematicalFG
Set.Finite
LowerAdjoint.toFun
Set
FirstOrder.Language.Substructure
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
FirstOrder.Language.Substructure.instPartialOrder
SetLike.coe
FirstOrder.Language.Substructure.instSetLike
FirstOrder.Language.Substructure.closure
Top.top
FirstOrder.Language.Substructure.instTop
fg_def
FirstOrder.Language.Substructure.fg_def
fg_iff_finite 📖mathematicalFirstOrder.Language.IsRelationalFG
Finite
FG.finite
FG.of_finite

FirstOrder.Language.Structure.CG

Theorems

NameKindAssumesProvesValidatesDepends On
map_of_surjective 📖mathematicalDFunLike.coe
FirstOrder.Language.Hom
FirstOrder.Language.Hom.instFunLike
FirstOrder.Language.Structure.CGFirstOrder.Language.Structure.cg_def
FirstOrder.Language.Hom.range_eq_top
range
out 📖mathematicalFirstOrder.Language.Substructure.CG
Top.top
FirstOrder.Language.Substructure
FirstOrder.Language.Substructure.instTop
range 📖mathematicalFirstOrder.Language.Substructure.CG
FirstOrder.Language.Hom.range
FirstOrder.Language.Hom.range_eq_map
FirstOrder.Language.Substructure.CG.map
FirstOrder.Language.Structure.cg_def

FirstOrder.Language.Structure.FG

Theorems

NameKindAssumesProvesValidatesDepends On
cg 📖mathematicalFirstOrder.Language.Structure.CGFirstOrder.Language.Structure.cg_def
FirstOrder.Language.Substructure.FG.cg
FirstOrder.Language.Structure.fg_def
countable_embedding 📖mathematicalCountable
FirstOrder.Language.Embedding
Function.Embedding.countable
instCountable_hom
FirstOrder.Language.Embedding.toHom_injective
countable_hom 📖mathematicalCountable
FirstOrder.Language.Hom
FirstOrder.Language.Structure.fg_iff
FirstOrder.Language.Hom.eq_of_eqOn_dense
Set.finite_coe_iff
Function.Embedding.countable
instCountableForallOfFinite
finite 📖mathematicalFirstOrder.Language.IsRelationalFiniteFinite.of_finite_univ
FirstOrder.Language.Substructure.FG.finite
FirstOrder.Language.Structure.fg_def
instCountable_hom 📖mathematicalCountable
FirstOrder.Language.Hom
countable_hom
map_of_surjective 📖mathematicalDFunLike.coe
FirstOrder.Language.Hom
FirstOrder.Language.Hom.instFunLike
FirstOrder.Language.Structure.FGFirstOrder.Language.Structure.fg_def
FirstOrder.Language.Hom.range_eq_top
range
of_finite 📖mathematicalFirstOrder.Language.Structure.FGSubtype.finite
out 📖mathematicalFirstOrder.Language.Substructure.FG
Top.top
FirstOrder.Language.Substructure
FirstOrder.Language.Substructure.instTop
range 📖mathematicalFirstOrder.Language.Substructure.FG
FirstOrder.Language.Hom.range
FirstOrder.Language.Hom.range_eq_map
FirstOrder.Language.Substructure.FG.map
FirstOrder.Language.Structure.fg_def

FirstOrder.Language.Structure.Fg

Theorems

NameKindAssumesProvesValidatesDepends On
instCountable_embedding 📖mathematicalCountable
FirstOrder.Language.Embedding
FirstOrder.Language.Structure.FG.countable_embedding

FirstOrder.Language.Substructure

Definitions

NameCategoryTheorems
CG 📖MathDef
12 mathmath: cg_def, cg_of_countable, cg_bot, cg_iff_structure_cg, FirstOrder.Language.Structure.CG.range, FirstOrder.Language.Structure.cg_def, FirstOrder.Language.Structure.CG.out, cg_iff_empty_or_exists_nat_generating_family, FG.cg, cg_iff_countable, cg_closure_singleton, cg_closure
instInhabited_fg 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
cg_bot 📖mathematicalCG
Bot.bot
FirstOrder.Language.Substructure
OrderBot.toBot
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
instCompleteLattice
BoundedOrder.toOrderBot
CompleteLattice.toBoundedOrder
FG.cg
fg_bot
cg_closure 📖mathematicalCG
LowerAdjoint.toFun
Set
FirstOrder.Language.Substructure
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
instPartialOrder
SetLike.coe
instSetLike
closure
cg_closure_singleton 📖mathematicalCG
LowerAdjoint.toFun
Set
FirstOrder.Language.Substructure
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
instPartialOrder
SetLike.coe
instSetLike
closure
Set.instSingletonSet
FG.cg
fg_closure_singleton
cg_def 📖mathematicalCG
Set.Countable
LowerAdjoint.toFun
Set
FirstOrder.Language.Substructure
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
instPartialOrder
SetLike.coe
instSetLike
closure
cg_iff_countable 📖mathematicalCG
Countable
FirstOrder.Language.Substructure
SetLike.instMembership
instSetLike
Set.Countable.substructure_closure
Countable.to_set
closure_eq
cg_iff_empty_or_exists_nat_generating_family 📖mathematicalCG
SetLike.coe
FirstOrder.Language.Substructure
instSetLike
Set
Set.instEmptyCollection
LowerAdjoint.toFun
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
instPartialOrder
closure
Set.range
cg_def
Set.eq_empty_or_nonempty
Set.Countable.exists_eq_range
Set.Countable.union
Set.countable_singleton
Set.Nonempty.inr
Set.singleton_nonempty
closure_union
sup_eq_left
closure_le
Set.singleton_subset_iff
Set.Nonempty.some_mem
Set.countable_empty
closure_eq_of_le
Set.empty_subset
SetLike.coe_subset_coe
instIsConcreteLE
Set.countable_range
instCountableNat
cg_iff_structure_cg 📖mathematicalCG
FirstOrder.Language.Structure.CG
FirstOrder.Language.Substructure
SetLike.instMembership
instSetLike
inducedStructure
FirstOrder.Language.Structure.cg_def
CG.of_map_embedding
FirstOrder.Language.Hom.range_eq_map
range_subtype
CG.map
cg_of_countable 📖mathematicalCGCountable.to_set
closure_eq
countable_fg_substructures_of_countable 📖mathematicalCountable
FirstOrder.Language.Substructure
FG
Subtype.prop
Function.Embedding.countable
Finset.countable
fg_bot 📖mathematicalFG
Bot.bot
FirstOrder.Language.Substructure
OrderBot.toBot
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
instCompleteLattice
BoundedOrder.toOrderBot
CompleteLattice.toBoundedOrder
Finset.coe_empty
closure_empty
fg_closure 📖mathematicalFG
LowerAdjoint.toFun
Set
FirstOrder.Language.Substructure
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
instPartialOrder
SetLike.coe
instSetLike
closure
Set.Finite.coe_toFinset
fg_closure_singleton 📖mathematicalFG
LowerAdjoint.toFun
Set
FirstOrder.Language.Substructure
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
instPartialOrder
SetLike.coe
instSetLike
closure
Set.instSingletonSet
fg_closure
Set.finite_singleton
fg_def 📖mathematicalFG
Set.Finite
LowerAdjoint.toFun
Set
FirstOrder.Language.Substructure
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
instPartialOrder
SetLike.coe
instSetLike
closure
Finset.finite_toSet
Set.Finite.exists_finset_coe
fg_iff_exists_fin_generating_family 📖mathematicalFG
LowerAdjoint.toFun
Set
FirstOrder.Language.Substructure
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
instPartialOrder
SetLike.coe
instSetLike
closure
Set.range
fg_def
Set.Finite.fin_embedding
Set.finite_range
Finite.of_fintype
fg_iff_finite 📖mathematicalFirstOrder.Language.IsRelationalFG
Finite
FirstOrder.Language.Substructure
SetLike.instMembership
instSetLike
FG.finite
FG.of_finite
fg_iff_structure_fg 📖mathematicalFG
FirstOrder.Language.Structure.FG
FirstOrder.Language.Substructure
SetLike.instMembership
instSetLike
inducedStructure
FirstOrder.Language.Structure.fg_def
FG.of_map_embedding
FirstOrder.Language.Hom.range_eq_map
range_subtype
FG.map
instCountable_fg_substructures_of_countable 📖mathematicalCountable
FirstOrder.Language.Substructure
FG
countable_fg_substructures_of_countable

FirstOrder.Language.Substructure.CG

Theorems

NameKindAssumesProvesValidatesDepends On
map 📖mathematicalFirstOrder.Language.Substructure.CGFirstOrder.Language.Substructure.mapFirstOrder.Language.Substructure.cg_def
Set.Countable.image
FirstOrder.Language.Substructure.closure_image
of_map_embedding 📖FirstOrder.Language.Substructure.CG
FirstOrder.Language.Substructure.map
FirstOrder.Language.Embedding.toHom
FirstOrder.Language.Substructure.cg_def
Set.Countable.preimage
FirstOrder.Language.Embedding.injective
FirstOrder.Language.Substructure.map_injective_of_injective
FirstOrder.Language.Substructure.map_closure
FirstOrder.Language.Embedding.coe_toHom
Set.image_preimage_eq_of_subset
FirstOrder.Language.Substructure.subset_closure
FirstOrder.Language.Hom.map_le_range
sup 📖mathematicalFirstOrder.Language.Substructure.CGFirstOrder.Language.Substructure
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
FirstOrder.Language.Substructure.instCompleteLattice
FirstOrder.Language.Substructure.cg_def
Set.Countable.union
FirstOrder.Language.Substructure.closure_union

FirstOrder.Language.Substructure.FG

Theorems

NameKindAssumesProvesValidatesDepends On
cg 📖mathematicalFirstOrder.Language.Substructure.FGFirstOrder.Language.Substructure.CGFirstOrder.Language.Substructure.fg_def
Set.Finite.countable
finite 📖mathematicalFirstOrder.Language.IsRelational
FirstOrder.Language.Substructure.FG
Finite
FirstOrder.Language.Substructure
SetLike.instMembership
FirstOrder.Language.Substructure.instSetLike
Finset.finite_toSet
FirstOrder.Language.Substructure.closure_eq_of_isRelational
map 📖mathematicalFirstOrder.Language.Substructure.FGFirstOrder.Language.Substructure.mapFirstOrder.Language.Substructure.fg_def
Set.Finite.image
FirstOrder.Language.Substructure.closure_image
of_finite 📖mathematicalFirstOrder.Language.Substructure.FGSet.Finite.coe_toFinset
FirstOrder.Language.Substructure.closure_eq
of_map_embedding 📖FirstOrder.Language.Substructure.FG
FirstOrder.Language.Substructure.map
FirstOrder.Language.Embedding.toHom
FirstOrder.Language.Substructure.fg_def
Set.Finite.preimage
Function.Injective.injOn
FirstOrder.Language.Embedding.injective
Finset.finite_toSet
FirstOrder.Language.Substructure.map_injective_of_injective
FirstOrder.Language.Substructure.map_closure
FirstOrder.Language.Embedding.coe_toHom
Set.image_preimage_eq_of_subset
FirstOrder.Language.Substructure.subset_closure
FirstOrder.Language.Hom.map_le_range
sup 📖mathematicalFirstOrder.Language.Substructure.FGFirstOrder.Language.Substructure
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
FirstOrder.Language.Substructure.instCompleteLattice
FirstOrder.Language.Substructure.fg_def
Set.Finite.union
FirstOrder.Language.Substructure.closure_union

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