Documentation Verification Report

Basic

πŸ“ Source: Mathlib/ModelTheory/Basic.lean

Statistics

MetricCount
DefinitionsinducedStructure, inducedStructureEquiv, Constants, comp, funLike, instInhabited, ofInjective, refl, toEmbedding, toHom, comp, instEquivLike, instInhabited, refl, toEmbedding, toEquiv, toHom, comp, id, instFunLike, instInhabited, toFun, HomClass, toHom, trivialStructure, IsAlgebraic, IsRelational, StrongHomClass, toEmbedding, toEquiv, Structure, RelMap, funMap, card, constantMap, empty, emptyStructure, instCoeTCConstants, instDecidableEqFunctions, instDecidableEqRelations, instInhabited, instIsAlgebraicEmpty, instIsRelationalEmpty, instUniqueStructureEmpty, sum, sumStructure, Β«term_β†’[_]_Β», Β«term_β†ͺ[_]_Β», Β«term_≃[_]_Β», emptyHom
50
TheoremsinducedStructure_RelMap, inducedStructure_funMap, toEquiv_inducedStructureEquiv, toFun_inducedStructureEquiv, toFun_inducedStructureEquiv_Symm, countable_functions, coeFn_ofInjective, coe_injective, coe_toHom, comp_apply, comp_assoc, comp_inj, comp_injective, comp_refl, comp_toHom, embeddingLike, ext, ext_iff, injective, map_constants, map_fun, map_fun', map_rel, map_rel', ofInjective_toFun, ofInjective_toHom, refl_apply, refl_comp, refl_toHom, strongHomClass, toHom_comp_inj, toHom_comp_injective, toHom_inj, toHom_injective, apply_symm_apply, bijective, coe_injective, coe_toEmbedding, coe_toHom, comp_apply, comp_assoc, comp_refl, comp_right_inj, comp_right_injective, comp_symm, comp_toEmbedding, comp_toHom, ext, ext_iff, injective, injective_comp, injective_toEmbedding, instStrongHomClass, map_constants, map_fun, map_fun', map_rel, map_rel', refl_apply, refl_comp, refl_toEmbedding, refl_toHom, self_comp_symm, self_comp_symm_toEmbedding, self_comp_symm_toHom, surjective, symm_apply_apply, symm_bijective, symm_comp_self, symm_comp_self_toEmbedding, symm_comp_self_toHom, symm_symm, toEmbedding_toHom, comp_apply, comp_assoc, comp_id, ext, ext_iff, homClass, id_apply, id_comp, instStrongHomClassOfIsAlgebraic, map_constants, map_fun, map_fun', map_rel, map_rel', toFun_eq_coe, map_constants, map_fun, map_rel, strongHomClassOfIsAlgebraic, toHom_toFun, homClass, map_fun, map_rel, toEmbedding_toFun, toEquiv_invFun, toEquiv_toFun, ext, ext_iff, card_empty, card_eq_card_functions_add_card_relations, card_functions_sum, card_relations_sum, card_sum, nonempty_embedding_iff, nonempty_equiv_iff, funMap_eq_coe_constants, funMap_sumInl, funMap_sumInr, isAlgebraic_sum, isEmpty_empty, isRelational_sum, nonempty_of_nonempty_constants, relMap_sumInl, relMap_sumInr, strongHomClassEmpty, emptyHom_toFun
119
Total169

Equiv

Definitions

NameCategoryTheorems
inducedStructure πŸ“–CompOp
6 mathmath: toEquiv_inducedStructureEquiv, bundledInduced_str, inducedStructure_funMap, toFun_inducedStructureEquiv, inducedStructure_RelMap, toFun_inducedStructureEquiv_Symm
inducedStructureEquiv πŸ“–CompOp
3 mathmath: toEquiv_inducedStructureEquiv, toFun_inducedStructureEquiv, toFun_inducedStructureEquiv_Symm

Theorems

NameKindAssumesProvesValidatesDepends On
inducedStructure_RelMap πŸ“–mathematicalβ€”FirstOrder.Language.Structure.RelMap
inducedStructure
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
symm
β€”β€”
inducedStructure_funMap πŸ“–mathematicalβ€”FirstOrder.Language.Structure.funMap
inducedStructure
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
symm
β€”β€”
toEquiv_inducedStructureEquiv πŸ“–mathematicalβ€”FirstOrder.Language.Equiv.toEquiv
inducedStructure
inducedStructureEquiv
β€”β€”
toFun_inducedStructureEquiv πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Equiv
inducedStructure
EquivLike.toFunLike
FirstOrder.Language.Equiv.instEquivLike
inducedStructureEquiv
Equiv
instEquivLike
β€”β€”
toFun_inducedStructureEquiv_Symm πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Equiv
inducedStructure
EquivLike.toFunLike
FirstOrder.Language.Equiv.instEquivLike
FirstOrder.Language.Equiv.symm
inducedStructureEquiv
Equiv
instEquivLike
symm
β€”β€”

FirstOrder

Definitions

NameCategoryTheorems
Β«term_β†’[_]_Β» πŸ“–CompOpβ€”
Β«term_β†ͺ[_]_Β» πŸ“–CompOpβ€”
Β«term_≃[_]_Β» πŸ“–CompOpβ€”

FirstOrder.Language

Definitions

NameCategoryTheorems
Constants πŸ“–CompOp
2 mathmath: constantsOn_constants, ElementaryEmbedding.ofModelsElementaryDiagram_toFun
HomClass πŸ“–CompData
3 mathmath: StrongHomClass.homClass, order.instHomClassOfOrderHomClass, Hom.homClass
IsAlgebraic πŸ“–MathDef
1 mathmath: isAlgebraic_constantsOn
IsRelational πŸ“–MathDef
1 mathmath: isRelational_constantsOn
StrongHomClass πŸ“–CompData
8 mathmath: Hom.instStrongHomClassOfIsAlgebraic, HomClass.strongHomClassOfIsAlgebraic, Embedding.strongHomClass, order.instStrongHomClassOrderEmbedding, order.instStrongHomClassOfOrderIsoClass, Equiv.instStrongHomClass, strongHomClassEmpty, ElementaryEmbedding.strongHomClass
Structure πŸ“–CompData
22 mathmath: IsFraisse.is_essentially_countable, empty.isFraisse_finite, IsFraisse.is_nonempty, Equiv.bundledInduced_Ξ±, exists_elementarilyEquivalent_card_eq, dlo_age, age.nonempty, empty.isFraisseLimit_of_countable_infinite, isFraisse_finite_linear_order, Embedding.age_subset_age, isFraisseLimit_of_countable_nonempty_dlo, exists_countable_is_age_of_iff, age.fg_substructure, Equiv.bundledInduced_str, IsFraisse.is_equiv_invariant, age.countable_quotient, exists_elementaryEmbedding_card_eq, age.is_equiv_invariant, exists_elementaryEmbedding_card_eq_of_le, exists_elementaryEmbedding_card_eq_of_ge, age_directLimit, Structure.FG.mem_age_of_equiv
card πŸ“–CompOp
9 mathmath: card_eq_card_functions_add_card_relations, card_functions_sum_skolem₁_le, order.card_eq_one, card_empty, card_constantsOn, card_withConstants, card_sum, FirstOrder.Ring.card_ring, BoundedFormula.card_le
constantMap πŸ“–CompOp
20 mathmath: Set.TermDefinable₁.const, coe_con, Substructure.constants_mem, ElementaryEmbedding.map_constants, Equiv.map_constants, Term.realize_constantsToVars, HomClass.map_constants, Term.realize_constants, Term.realize_constantsVarsEquivLeft, Formula.realize_equivSentence_symm_con, Embedding.map_constants, model_distinctConstantsTheory, ElementaryEmbedding.ofModelsElementaryDiagram_toFun, withConstants_funMap_sumInr, funMap_eq_coe_constants, Set.TermDefinable.const, BoundedFormula.realize_constantsVarsEquiv, Hom.map_constants, Term.realize_varsToConstants, Formula.realize_equivSentence
empty πŸ“–CompOp
10 mathmath: isEmpty_empty, empty.isFraisse_finite, Function.emptyHom_toFun, empty.isFraisseLimit_of_countable_infinite, Cardinal.empty_theory_categorical, card_empty, empty.nonempty_equiv_iff, strongHomClassEmpty, Cardinal.empty_infinite_Theory_isComplete, empty.nonempty_embedding_iff
emptyStructure πŸ“–CompOpβ€”
instCoeTCConstants πŸ“–CompOpβ€”
instDecidableEqFunctions πŸ“–CompOpβ€”
instDecidableEqRelations πŸ“–CompOpβ€”
instInhabited πŸ“–CompOpβ€”
instIsAlgebraicEmpty πŸ“–CompOp
1 mathmath: empty.isFraisseLimit_of_countable_infinite
instIsRelationalEmpty πŸ“–CompOp
1 mathmath: empty.isFraisseLimit_of_countable_infinite
instUniqueStructureEmpty πŸ“–CompOpβ€”
sum πŸ“–CompOp
36 mathmath: LHom.sumElim_onFunction, LHom.sumElim_isExpansionOn, LHom.sumInl_injective, LHom.sumMap_comp_inl, LHom.sumMap_onFunction, LHom.funMap_sumInl, LHom.sumElim_comp_inr, LHom.sumElim_inl_inr, LHom.sumMap_isExpansionOn, LHom.funMap_sumInr, isRelational_sum, card_functions_sum_skolem₁_le, LHom.sumInr_onRelation, Substructure.skolem₁_reduct_isElementary, LHom.comp_sumElim, LHom.sumInr_onFunction, LHom.sumElim_comp_inl, card_functions_sum_skolem₁, LHom.sumElim_onRelation, LHom.sumInl_onFunction, funMap_sumInr, isAlgebraic_sum, LHom.sumInl_isExpansionOn, LHom.sumInl_onRelation, relMap_sumInl, LHom.sumMap_comp_inr, Substructure.coeSort_elementarySkolem₁Reduct, LHom.sumMap_onRelation, Substructure.elementarySkolem₁Reduct.instSmall, card_functions_sum, card_sum, LHom.sumInr_isExpansionOn, funMap_sumInl, card_relations_sum, LHom.sumInr_injective, relMap_sumInr
sumStructure πŸ“–CompOp
11 mathmath: LHom.sumElim_isExpansionOn, LHom.sumMap_isExpansionOn, Substructure.skolem₁_reduct_isElementary, funMap_sumInr, LHom.sumInl_isExpansionOn, relMap_sumInl, Substructure.coeSort_elementarySkolem₁Reduct, Substructure.elementarySkolem₁Reduct.instSmall, LHom.sumInr_isExpansionOn, funMap_sumInl, relMap_sumInr

Theorems

NameKindAssumesProvesValidatesDepends On
card_empty πŸ“–mathematicalβ€”card
empty
Cardinal
Cardinal.instZero
β€”Cardinal.mk_sum
Cardinal.mk_sigma
Cardinal.mk_eq_zero
Cardinal.sum_const
Cardinal.mk_eq_aleph0
instCountableNat
instInfiniteNat
Cardinal.lift_id'
MulZeroClass.mul_zero
add_zero
card_eq_card_functions_add_card_relations πŸ“–mathematicalβ€”card
Cardinal
Cardinal.instAdd
Cardinal.sum
Cardinal.lift
Functions
Relations
β€”Cardinal.mk_sum
Cardinal.mk_sigma
Cardinal.lift_sum
card_functions_sum πŸ“–mathematicalβ€”Functions
sum
Cardinal
Cardinal.instAdd
Cardinal.lift
β€”Cardinal.mk_sum
card_relations_sum πŸ“–mathematicalβ€”Relations
sum
Cardinal
Cardinal.instAdd
Cardinal.lift
β€”Cardinal.mk_sum
card_sum πŸ“–mathematicalβ€”card
sum
Cardinal
Cardinal.instAdd
Cardinal.lift
β€”Cardinal.mk_sum
Cardinal.mk_sigma
card_functions_sum
Cardinal.sum_add_distrib'
Cardinal.lift_add
Cardinal.lift_sum
Cardinal.lift_lift
card_relations_sum
add_assoc
add_comm
funMap_eq_coe_constants πŸ“–mathematicalβ€”Structure.funMap
constantMap
β€”Fin.isEmpty'
funMap_sumInl πŸ“–mathematicalβ€”Structure.funMap
sum
sumStructure
Functions
β€”β€”
funMap_sumInr πŸ“–mathematicalβ€”Structure.funMap
sum
sumStructure
Functions
β€”β€”
isAlgebraic_sum πŸ“–mathematicalIsAlgebraicsumβ€”instIsEmptySum
isEmpty_empty πŸ“–mathematicalβ€”IsEmpty
Symbols
empty
β€”β€”
isRelational_sum πŸ“–mathematicalIsRelationalsumβ€”instIsEmptySum
nonempty_of_nonempty_constants πŸ“–β€”β€”β€”β€”Nonempty.map
relMap_sumInl πŸ“–mathematicalβ€”Structure.RelMap
sum
sumStructure
Relations
β€”β€”
relMap_sumInr πŸ“–mathematicalβ€”Structure.RelMap
sum
sumStructure
Relations
β€”β€”
strongHomClassEmpty πŸ“–mathematicalβ€”StrongHomClass
empty
β€”β€”

FirstOrder.Language.Countable

Theorems

NameKindAssumesProvesValidatesDepends On
countable_functions πŸ“–mathematicalβ€”Countable
FirstOrder.Language.Functions
β€”Function.Injective.countable
Sum.inl_injective

FirstOrder.Language.Embedding

Definitions

NameCategoryTheorems
comp πŸ“–CompOp
23 mathmath: FirstOrder.Language.Equiv.comp_toEmbedding, FirstOrder.Language.DirectLimit.lift_unique, FirstOrder.Language.DirectLimit.partialEquivLimit_comp_inclusion, subtype_equivRange, FirstOrder.Language.PartialEquiv.subtype_toEquiv_inclusion, FirstOrder.Language.isExtensionPair_iff_exists_embedding_closure_singleton_sup, comp_toHom, FirstOrder.Language.IsUltrahomogeneous.extend_embedding, refl_comp, FirstOrder.Language.DirectLimit.Equiv_isup_symm_inclusion, comp_refl, FirstOrder.Language.Substructure.subtype_comp_inclusion, FirstOrder.Language.Equiv.self_comp_symm_toEmbedding, FirstOrder.Language.PartialEquiv.le_def, comp_inj, comp_codRestrict, comp_injective, subtype_comp_codRestrict, comp_assoc, comp_apply, subtype_substructureEquivMap, FirstOrder.Language.Equiv.symm_comp_self_toEmbedding, FirstOrder.Language.PartialEquiv.toEquiv_inclusion
funLike πŸ“–CompOp
43 mathmath: FirstOrder.Language.PartialEquiv.toEmbedding_apply, map_rel, ofInjective_toFun, FirstOrder.Language.DirectLimit.of_f, equivRange_toEquiv_apply, ext_iff, coe_injective, FirstOrder.Language.DirectLimit.lift_unique, FirstOrder.Language.Substructure.subtype_apply, FirstOrder.Language.Substructure.coe_inclusion, FirstOrder.Language.Substructure.coe_subtype, FirstOrder.Language.PartialEquiv.le_iff, FirstOrder.Language.DirectLimit.liftInclusion_of, embeddingLike, substructureEquivMap_apply, strongHomClass, FirstOrder.Language.DirectLimit.equiv_iff, FirstOrder.Language.PartialEquiv.toEquiv_inclusion_apply, map_constants, map_fun, FirstOrder.Language.Substructure.subtype_injective, coe_toHom, FirstOrder.Language.DirectLimit.of_apply, coeFn_ofInjective, FirstOrder.Language.DirectLimit.instDirectedSystemSubtypeMemSubstructureDomCoeOrderHomPartialEquivEmbeddingInclusion, injective, FirstOrder.Language.DirectLimit.Equiv_isup_of_apply, FirstOrder.Language.PartialEquiv.toEmbeddingOfEqTop_apply, FirstOrder.Language.instDirectedSystemSubtypeMemSubstructureCoeOrderHomEmbeddingInclusion, comp_apply, FirstOrder.Language.DirectLimit.instDirectedSystemSubtypeMemSubstructureCodCoeOrderHomPartialEquivEmbeddingInclusion, FirstOrder.Language.StrongHomClass.toEmbedding_toFun, FirstOrder.Language.DirectLimit.Equiv_isup_symm_inclusion_apply, FirstOrder.Language.DirectLimit.exists_of, refl_apply, FirstOrder.Language.DirectedSystem.coe_natLERec, FirstOrder.Language.ElementaryEmbedding.coe_toEmbedding, FirstOrder.Language.DirectLimit.comp_unify, FirstOrder.Language.Equiv.coe_toEmbedding, FirstOrder.Language.PartialEquiv.ext_iff, FirstOrder.Language.DirectedSystem.natLERec.directedSystem, domRestrict_apply, equivRange_apply
instInhabited πŸ“–CompOpβ€”
ofInjective πŸ“–CompOp
3 mathmath: ofInjective_toFun, ofInjective_toHom, coeFn_ofInjective
refl πŸ“–CompOp
8 mathmath: FirstOrder.Language.Substructure.inclusion_self, FirstOrder.Language.Equiv.refl_toEmbedding, refl_comp, comp_refl, FirstOrder.Language.Equiv.self_comp_symm_toEmbedding, refl_apply, FirstOrder.Language.Equiv.symm_comp_self_toEmbedding, refl_toHom
toEmbedding πŸ“–CompOp
2 mathmath: map_fun', map_rel'
toHom πŸ“–CompOp
21 mathmath: FirstOrder.Language.Equiv.toEmbedding_toHom, equivRange_toEquiv_apply, ofInjective_toHom, subtype_equivRange, FirstOrder.Language.Substructure.range_subtype, substructureEquivMap_apply, FirstOrder.Language.DirectLimit.exists_fg_substructure_in_Sigma, toHom_injective, toHom_comp_inj, comp_toHom, coe_toHom, FirstOrder.Language.Hom.subtype_comp_codRestrict, toHom_comp_injective, FirstOrder.Language.ElementaryEmbedding.toEmbedding_toHom, FirstOrder.Language.DirectLimit.iSup_range_of_eq_top, subtype_substructureEquivMap, toHom_inj, FirstOrder.Language.DirectLimit.range_lift, refl_toHom, equivRange_apply, FirstOrder.Language.DirectLimit.rangeLiftInclusion

Theorems

NameKindAssumesProvesValidatesDepends On
coeFn_ofInjective πŸ“–mathematicalFirstOrder.Language.IsAlgebraic
DFunLike.coe
FirstOrder.Language.Hom
FirstOrder.Language.Hom.instFunLike
FirstOrder.Language.Embedding
funLike
ofInjective
β€”β€”
coe_injective πŸ“–mathematicalβ€”FirstOrder.Language.Embedding
DFunLike.coe
funLike
β€”Function.Embedding.ext
coe_toHom πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Hom
FirstOrder.Language.Hom.instFunLike
toHom
FirstOrder.Language.Embedding
funLike
β€”β€”
comp_apply πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Embedding
funLike
comp
β€”β€”
comp_assoc πŸ“–mathematicalβ€”compβ€”β€”
comp_inj πŸ“–mathematicalβ€”compβ€”comp_injective
comp_injective πŸ“–mathematicalβ€”FirstOrder.Language.Embedding
comp
β€”ext
injective
DFunLike.congr_fun
comp_refl πŸ“–mathematicalβ€”comp
refl
β€”DFunLike.coe_injective
comp_toHom πŸ“–mathematicalβ€”toHom
comp
FirstOrder.Language.Hom.comp
β€”β€”
embeddingLike πŸ“–mathematicalβ€”EmbeddingLike
FirstOrder.Language.Embedding
funLike
β€”Function.Embedding.injective
ext πŸ“–β€”DFunLike.coe
FirstOrder.Language.Embedding
funLike
β€”β€”coe_injective
ext_iff πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Embedding
funLike
β€”ext
injective πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Embedding
funLike
β€”Function.Embedding.injective
map_constants πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Embedding
funLike
FirstOrder.Language.constantMap
β€”FirstOrder.Language.HomClass.map_constants
FirstOrder.Language.StrongHomClass.homClass
strongHomClass
map_fun πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Embedding
funLike
FirstOrder.Language.Structure.funMap
β€”FirstOrder.Language.HomClass.map_fun
FirstOrder.Language.StrongHomClass.homClass
strongHomClass
map_fun' πŸ“–mathematicalβ€”Function.Embedding.toFun
toEmbedding
FirstOrder.Language.Structure.funMap
β€”β€”
map_rel πŸ“–mathematicalβ€”FirstOrder.Language.Structure.RelMap
DFunLike.coe
FirstOrder.Language.Embedding
funLike
β€”FirstOrder.Language.StrongHomClass.map_rel
strongHomClass
map_rel' πŸ“–mathematicalβ€”FirstOrder.Language.Structure.RelMap
Function.Embedding.toFun
toEmbedding
β€”β€”
ofInjective_toFun πŸ“–mathematicalFirstOrder.Language.IsAlgebraic
DFunLike.coe
FirstOrder.Language.Hom
FirstOrder.Language.Hom.instFunLike
FirstOrder.Language.Embedding
funLike
ofInjective
β€”β€”
ofInjective_toHom πŸ“–mathematicalFirstOrder.Language.IsAlgebraic
DFunLike.coe
FirstOrder.Language.Hom
FirstOrder.Language.Hom.instFunLike
toHom
ofInjective
β€”FirstOrder.Language.Hom.ext
refl_apply πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Embedding
funLike
refl
β€”β€”
refl_comp πŸ“–mathematicalβ€”comp
refl
β€”DFunLike.coe_injective
refl_toHom πŸ“–mathematicalβ€”toHom
refl
FirstOrder.Language.Hom.id
β€”β€”
strongHomClass πŸ“–mathematicalβ€”FirstOrder.Language.StrongHomClass
FirstOrder.Language.Embedding
funLike
β€”map_fun'
map_rel'
toHom_comp_inj πŸ“–mathematicalβ€”FirstOrder.Language.Hom.comp
toHom
β€”toHom_comp_injective
toHom_comp_injective πŸ“–mathematicalβ€”FirstOrder.Language.Hom
FirstOrder.Language.Hom.comp
toHom
β€”FirstOrder.Language.Hom.ext
injective
DFunLike.congr_fun
toHom_inj πŸ“–mathematicalβ€”toHomβ€”toHom_injective
toHom_injective πŸ“–mathematicalβ€”FirstOrder.Language.Embedding
FirstOrder.Language.Hom
toHom
β€”ext

FirstOrder.Language.Equiv

Definitions

NameCategoryTheorems
comp πŸ“–CompOp
12 mathmath: comp_toEmbedding, self_comp_symm, comp_assoc, symm_comp_self, comp_toHom, injective_comp, comp_symm, refl_comp, comp_right_injective, comp_apply, comp_refl, comp_right_inj
instEquivLike πŸ“–CompOp
30 mathmath: FirstOrder.Language.PartialEquiv.toEmbedding_apply, map_fun, bijective, symm_apply_apply, coe_toElementaryEmbedding, refl_apply, ext_iff, map_constants, FirstOrder.Language.PartialEquiv.le_iff, FirstOrder.Language.Embedding.substructureEquivMap_apply, apply_symm_apply, surjective, injective, FirstOrder.Language.PartialEquiv.toEquiv_inclusion_apply, map_rel, instStrongHomClass, Equiv.toFun_inducedStructureEquiv, coe_injective, FirstOrder.Language.Substructure.coe_topEquiv, coe_toHom, FirstOrder.Language.PartialEquiv.symm_apply, FirstOrder.Language.DirectLimit.Equiv_isup_of_apply, comp_apply, FirstOrder.Language.PartialEquiv.toEmbeddingOfEqTop_apply, FirstOrder.Language.DirectLimit.Equiv_isup_symm_inclusion_apply, coe_toEmbedding, FirstOrder.Language.PartialEquiv.ext_iff, Equiv.toFun_inducedStructureEquiv_Symm, FirstOrder.Language.Embedding.equivRange_apply, FirstOrder.Language.StrongHomClass.toEquiv_toFun
instInhabited πŸ“–CompOpβ€”
refl πŸ“–CompOp
7 mathmath: self_comp_symm, refl_apply, symm_comp_self, refl_toEmbedding, refl_toHom, refl_comp, comp_refl
toEmbedding πŸ“–CompOp
17 mathmath: toEmbedding_toHom, injective_toEmbedding, comp_toEmbedding, toElementaryEmbedding_toEmbedding, FirstOrder.Language.DirectLimit.partialEquivLimit_comp_inclusion, FirstOrder.Language.Embedding.subtype_equivRange, refl_toEmbedding, FirstOrder.Language.PartialEquiv.subtype_toEquiv_inclusion, FirstOrder.Language.PartialEquiv.toEquivOfEqTop_toEmbedding, FirstOrder.Language.DirectLimit.Equiv_isup_symm_inclusion, self_comp_symm_toEmbedding, FirstOrder.Language.PartialEquiv.le_def, FirstOrder.Language.equiv_between_cg, FirstOrder.Language.Embedding.subtype_substructureEquivMap, symm_comp_self_toEmbedding, coe_toEmbedding, FirstOrder.Language.PartialEquiv.toEquiv_inclusion
toEquiv πŸ“–CompOp
5 mathmath: Equiv.toEquiv_inducedStructureEquiv, FirstOrder.Language.Embedding.equivRange_toEquiv_apply, map_rel', map_fun', FirstOrder.Language.StrongHomClass.toEquiv_invFun
toHom πŸ“–CompOp
7 mathmath: toEmbedding_toHom, self_comp_symm_toHom, comp_toHom, symm_comp_self_toHom, refl_toHom, toHom_range, coe_toHom

Theorems

NameKindAssumesProvesValidatesDepends On
apply_symm_apply πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Equiv
EquivLike.toFunLike
instEquivLike
symm
β€”Equiv.apply_symm_apply
bijective πŸ“–mathematicalβ€”Function.Bijective
DFunLike.coe
FirstOrder.Language.Equiv
EquivLike.toFunLike
instEquivLike
β€”EquivLike.bijective
coe_injective πŸ“–mathematicalβ€”FirstOrder.Language.Equiv
DFunLike.coe
EquivLike.toFunLike
instEquivLike
β€”DFunLike.coe_injective
coe_toEmbedding πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Embedding
FirstOrder.Language.Embedding.funLike
toEmbedding
FirstOrder.Language.Equiv
EquivLike.toFunLike
instEquivLike
β€”β€”
coe_toHom πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Hom
FirstOrder.Language.Hom.instFunLike
toHom
FirstOrder.Language.Equiv
EquivLike.toFunLike
instEquivLike
β€”β€”
comp_apply πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Equiv
EquivLike.toFunLike
instEquivLike
comp
β€”β€”
comp_assoc πŸ“–mathematicalβ€”compβ€”β€”
comp_refl πŸ“–mathematicalβ€”comp
refl
β€”β€”
comp_right_inj πŸ“–mathematicalβ€”compβ€”comp_right_injective
comp_right_injective πŸ“–mathematicalβ€”FirstOrder.Language.Equiv
comp
β€”comp_assoc
self_comp_symm
comp_refl
comp_symm πŸ“–mathematicalβ€”symm
comp
β€”β€”
comp_toEmbedding πŸ“–mathematicalβ€”toEmbedding
comp
FirstOrder.Language.Embedding.comp
β€”β€”
comp_toHom πŸ“–mathematicalβ€”toHom
comp
FirstOrder.Language.Hom.comp
β€”β€”
ext πŸ“–β€”DFunLike.coe
FirstOrder.Language.Equiv
EquivLike.toFunLike
instEquivLike
β€”β€”coe_injective
ext_iff πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Equiv
EquivLike.toFunLike
instEquivLike
β€”ext
injective πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Equiv
EquivLike.toFunLike
instEquivLike
β€”EquivLike.injective
injective_comp πŸ“–mathematicalβ€”FirstOrder.Language.Equiv
comp
β€”ext
injective
injective_toEmbedding πŸ“–mathematicalβ€”FirstOrder.Language.Equiv
FirstOrder.Language.Embedding
toEmbedding
β€”DFunLike.coe_injective
instStrongHomClass πŸ“–mathematicalβ€”FirstOrder.Language.StrongHomClass
FirstOrder.Language.Equiv
EquivLike.toFunLike
instEquivLike
β€”map_fun'
map_rel'
map_constants πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Equiv
EquivLike.toFunLike
instEquivLike
FirstOrder.Language.constantMap
β€”FirstOrder.Language.HomClass.map_constants
FirstOrder.Language.StrongHomClass.homClass
instStrongHomClass
map_fun πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Equiv
EquivLike.toFunLike
instEquivLike
FirstOrder.Language.Structure.funMap
β€”FirstOrder.Language.HomClass.map_fun
FirstOrder.Language.StrongHomClass.homClass
instStrongHomClass
map_fun' πŸ“–mathematicalβ€”Equiv.toFun
toEquiv
FirstOrder.Language.Structure.funMap
β€”β€”
map_rel πŸ“–mathematicalβ€”FirstOrder.Language.Structure.RelMap
DFunLike.coe
FirstOrder.Language.Equiv
EquivLike.toFunLike
instEquivLike
β€”FirstOrder.Language.StrongHomClass.map_rel
instStrongHomClass
map_rel' πŸ“–mathematicalβ€”FirstOrder.Language.Structure.RelMap
Equiv.toFun
toEquiv
β€”β€”
refl_apply πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Equiv
EquivLike.toFunLike
instEquivLike
refl
β€”β€”
refl_comp πŸ“–mathematicalβ€”comp
refl
β€”β€”
refl_toEmbedding πŸ“–mathematicalβ€”toEmbedding
refl
FirstOrder.Language.Embedding.refl
β€”β€”
refl_toHom πŸ“–mathematicalβ€”toHom
refl
FirstOrder.Language.Hom.id
β€”β€”
self_comp_symm πŸ“–mathematicalβ€”comp
symm
refl
β€”ext
comp_apply
apply_symm_apply
refl_apply
self_comp_symm_toEmbedding πŸ“–mathematicalβ€”FirstOrder.Language.Embedding.comp
toEmbedding
symm
FirstOrder.Language.Embedding.refl
β€”comp_toEmbedding
self_comp_symm
refl_toEmbedding
self_comp_symm_toHom πŸ“–mathematicalβ€”FirstOrder.Language.Hom.comp
toHom
symm
FirstOrder.Language.Hom.id
β€”comp_toHom
self_comp_symm
refl_toHom
surjective πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Equiv
EquivLike.toFunLike
instEquivLike
β€”EquivLike.surjective
symm_apply_apply πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Equiv
EquivLike.toFunLike
instEquivLike
symm
β€”Equiv.symm_apply_apply
symm_bijective πŸ“–mathematicalβ€”Function.Bijective
FirstOrder.Language.Equiv
symm
β€”Function.bijective_iff_has_inverse
symm_symm
symm_comp_self πŸ“–mathematicalβ€”comp
symm
refl
β€”ext
comp_apply
symm_apply_apply
refl_apply
symm_comp_self_toEmbedding πŸ“–mathematicalβ€”FirstOrder.Language.Embedding.comp
toEmbedding
symm
FirstOrder.Language.Embedding.refl
β€”comp_toEmbedding
symm_comp_self
refl_toEmbedding
symm_comp_self_toHom πŸ“–mathematicalβ€”FirstOrder.Language.Hom.comp
toHom
symm
FirstOrder.Language.Hom.id
β€”comp_toHom
symm_comp_self
refl_toHom
symm_symm πŸ“–mathematicalβ€”symmβ€”β€”
toEmbedding_toHom πŸ“–mathematicalβ€”FirstOrder.Language.Embedding.toHom
toEmbedding
toHom
β€”β€”

FirstOrder.Language.Hom

Definitions

NameCategoryTheorems
comp πŸ“–CompOp
16 mathmath: FirstOrder.Language.Equiv.self_comp_symm_toHom, FirstOrder.Language.Substructure.map_map, id_comp, FirstOrder.Language.Substructure.comap_comap, FirstOrder.Language.Equiv.comp_toHom, FirstOrder.Language.Equiv.symm_comp_self_toHom, FirstOrder.Language.Embedding.toHom_comp_inj, FirstOrder.Language.Embedding.comp_toHom, comp_assoc, comp_apply, subtype_comp_codRestrict, FirstOrder.Language.Embedding.toHom_comp_injective, range_comp_le_range, range_comp, comp_id, comp_codRestrict
id πŸ“–CompOp
10 mathmath: FirstOrder.Language.Equiv.self_comp_symm_toHom, id_apply, FirstOrder.Language.Substructure.comap_id, id_comp, FirstOrder.Language.Equiv.symm_comp_self_toHom, FirstOrder.Language.Equiv.refl_toHom, range_id, FirstOrder.Language.Substructure.map_id, comp_id, FirstOrder.Language.Embedding.refl_toHom
instFunLike πŸ“–CompOp
28 mathmath: domRestrict_toFun, FirstOrder.Language.Substructure.coe_comap, ext_iff, Function.emptyHom_toFun, instStrongHomClassOfIsAlgebraic, mem_range, mem_range_self, id_apply, map_fun, mem_eqLocus, FirstOrder.Language.Substructure.coe_map, comp_apply, FirstOrder.Language.Embedding.coe_toHom, toFun_eq_coe, map_rel, range_coe, FirstOrder.Language.Equiv.coe_toHom, range_eq_top, FirstOrder.Language.HomClass.toHom_toFun, FirstOrder.Language.Substructure.map_closure, map_constants, FirstOrder.Language.ElementaryEmbedding.coe_toHom, homClass, FirstOrder.Language.Substructure.mem_map_of_mem, FirstOrder.Language.Substructure.mem_map, FirstOrder.Language.Substructure.mem_comap, FirstOrder.Language.Substructure.closure_image, FirstOrder.Language.Substructure.apply_coe_mem_map
instInhabited πŸ“–CompOpβ€”
toFun πŸ“–CompOp
3 mathmath: map_fun', toFun_eq_coe, map_rel'

Theorems

NameKindAssumesProvesValidatesDepends On
comp_apply πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Hom
instFunLike
comp
β€”β€”
comp_assoc πŸ“–mathematicalβ€”compβ€”β€”
comp_id πŸ“–mathematicalβ€”comp
id
β€”β€”
ext πŸ“–β€”DFunLike.coe
FirstOrder.Language.Hom
instFunLike
β€”β€”DFunLike.ext
ext_iff πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Hom
instFunLike
β€”ext
homClass πŸ“–mathematicalβ€”FirstOrder.Language.HomClass
FirstOrder.Language.Hom
instFunLike
β€”map_fun'
map_rel'
id_apply πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Hom
instFunLike
id
β€”β€”
id_comp πŸ“–mathematicalβ€”comp
id
β€”β€”
instStrongHomClassOfIsAlgebraic πŸ“–mathematicalFirstOrder.Language.IsAlgebraicFirstOrder.Language.StrongHomClass
FirstOrder.Language.Hom
instFunLike
β€”FirstOrder.Language.HomClass.strongHomClassOfIsAlgebraic
homClass
map_constants πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Hom
instFunLike
FirstOrder.Language.constantMap
β€”FirstOrder.Language.HomClass.map_constants
homClass
map_fun πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Hom
instFunLike
FirstOrder.Language.Structure.funMap
β€”FirstOrder.Language.HomClass.map_fun
homClass
map_fun' πŸ“–mathematicalβ€”toFun
FirstOrder.Language.Structure.funMap
β€”β€”
map_rel πŸ“–mathematicalFirstOrder.Language.Structure.RelMapDFunLike.coe
FirstOrder.Language.Hom
instFunLike
β€”FirstOrder.Language.HomClass.map_rel
homClass
map_rel' πŸ“–mathematicalFirstOrder.Language.Structure.RelMaptoFunβ€”β€”
toFun_eq_coe πŸ“–mathematicalβ€”toFun
DFunLike.coe
FirstOrder.Language.Hom
instFunLike
β€”β€”

FirstOrder.Language.HomClass

Definitions

NameCategoryTheorems
toHom πŸ“–CompOp
1 mathmath: toHom_toFun

Theorems

NameKindAssumesProvesValidatesDepends On
map_constants πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.constantMap
β€”Fin.isEmpty'
map_fun
map_fun πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Structure.funMap
β€”β€”
map_rel πŸ“–mathematicalFirstOrder.Language.Structure.RelMapDFunLike.coeβ€”β€”
strongHomClassOfIsAlgebraic πŸ“–mathematicalFirstOrder.Language.IsAlgebraicFirstOrder.Language.StrongHomClassβ€”map_fun
toHom_toFun πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Hom
FirstOrder.Language.Hom.instFunLike
toHom
β€”β€”

FirstOrder.Language.Inhabited

Definitions

NameCategoryTheorems
trivialStructure πŸ“–CompOpβ€”

FirstOrder.Language.StrongHomClass

Definitions

NameCategoryTheorems
toEmbedding πŸ“–CompOp
1 mathmath: toEmbedding_toFun
toEquiv πŸ“–CompOp
2 mathmath: toEquiv_invFun, toEquiv_toFun

Theorems

NameKindAssumesProvesValidatesDepends On
homClass πŸ“–mathematicalβ€”FirstOrder.Language.HomClassβ€”map_fun
map_rel
map_fun πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Structure.funMap
β€”β€”
map_rel πŸ“–mathematicalβ€”FirstOrder.Language.Structure.RelMap
DFunLike.coe
β€”β€”
toEmbedding_toFun πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Embedding
FirstOrder.Language.Embedding.funLike
toEmbedding
β€”β€”
toEquiv_invFun πŸ“–mathematicalβ€”Equiv.invFun
FirstOrder.Language.Equiv.toEquiv
toEquiv
EquivLike.inv
β€”β€”
toEquiv_toFun πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Equiv
EquivLike.toFunLike
FirstOrder.Language.Equiv.instEquivLike
toEquiv
β€”β€”

FirstOrder.Language.Structure

Definitions

NameCategoryTheorems
RelMap πŸ“–MathDef
33 mathmath: FirstOrder.Language.Embedding.map_rel, FirstOrder.Language.ElementaryEmbedding.map_rel, FirstOrder.Language.relMap_quotient_mk', FirstOrder.Language.Formula.realize_rel, FirstOrder.Language.Formula.realize_rel₁, FirstOrder.Language.Equiv.map_rel', FirstOrder.Language.Relations.realize_irreflexive, FirstOrder.Language.Formula.realize_relβ‚‚, FirstOrder.Language.Prestructure.rel_equiv, FirstOrder.Language.Relations.realize_transitive, FirstOrder.Language.OrderedStructure.relMap_leSymb, ext_iff, FirstOrder.Language.DirectLimit.relMap_unify_equiv, FirstOrder.Language.BoundedFormula.realize_rel₁, FirstOrder.Language.StrongHomClass.map_rel, FirstOrder.Language.DirectLimit.relMap_quotient_mk'_sigma_mk', FirstOrder.Language.Relations.realize_total, FirstOrder.Language.LHom.IsExpansionOn.map_onRelation, FirstOrder.Language.Equiv.map_rel, FirstOrder.Language.withConstants_relMap_sumInl, FirstOrder.Language.LHom.map_onRelation, FirstOrder.Language.Relations.realize_symmetric, FirstOrder.Language.relMap_sumInl, Equiv.inducedStructure_RelMap, FirstOrder.Language.BoundedFormula.realize_rel, FirstOrder.Language.simpleGraphOfStructure_adj, FirstOrder.Language.DirectLimit.relMap_equiv_unify, FirstOrder.Language.Theory.simpleGraph_model_iff, FirstOrder.Language.Relations.realize_reflexive, FirstOrder.Language.Relations.realize_antisymmetric, FirstOrder.Language.relMap_sumInr, FirstOrder.Language.BoundedFormula.realize_relβ‚‚, FirstOrder.Language.Embedding.map_rel'
funMap πŸ“–CompOp
39 mathmath: FirstOrder.Language.Embedding.map_fun', FirstOrder.Language.Equiv.map_fun, FirstOrder.Language.presburger.funMap_one, FirstOrder.Language.LHom.funMap_sumInl, FirstOrder.Language.DirectLimit.funMap_quotient_mk'_sigma_mk', FirstOrder.Language.Term.realize_function_term, FirstOrder.Language.Equiv.map_fun', FirstOrder.Language.funMap_quotient_mk', FirstOrder.Language.LHom.funMap_sumInr, FirstOrder.Language.ElementaryEmbedding.map_fun, FirstOrder.Language.LHom.map_onFunction, ext_iff, FirstOrder.Language.LHom.IsExpansionOn.map_onFunction, FirstOrder.Ring.CompatibleRing.funMap_mul, FirstOrder.Language.DirectLimit.funMap_equiv_unify, FirstOrder.Language.Hom.map_fun', Equiv.inducedStructure_funMap, FirstOrder.Ring.CompatibleRing.funMap_one, FirstOrder.Language.presburger.funMap_zero, FirstOrder.Language.withConstants_funMap_sumInl, FirstOrder.Language.Hom.map_fun, FirstOrder.Language.funMap_sumInr, FirstOrder.Language.presburger.funMap_add, FirstOrder.Language.Embedding.map_fun, FirstOrder.Ring.CompatibleRing.funMap_add, FirstOrder.Language.Formula.realize_graph, FirstOrder.Language.Prestructure.fun_equiv, FirstOrder.Language.withConstants_funMap_sumInr, FirstOrder.Ring.CompatibleRing.funMap_neg, FirstOrder.Language.funMap_eq_coe_constants, FirstOrder.Ring.CompatibleRing.funMap_zero, FirstOrder.Language.Term.realize_func, FirstOrder.Language.StrongHomClass.map_fun, FirstOrder.Language.Ultraproduct.funMap_cast, FirstOrder.Language.funMap_sumInl, FirstOrder.Language.HomClass.map_fun, FirstOrder.Language.Term.realize_functions_applyβ‚‚, FirstOrder.Language.Term.realize_functions_apply₁, FirstOrder.Language.DirectLimit.funMap_unify_equiv

Theorems

NameKindAssumesProvesValidatesDepends On
ext πŸ“–β€”funMap
RelMap
β€”β€”β€”
ext_iff πŸ“–mathematicalβ€”funMap
RelMap
β€”ext

FirstOrder.Language.empty

Theorems

NameKindAssumesProvesValidatesDepends On
nonempty_embedding_iff πŸ“–mathematicalβ€”FirstOrder.Language.Embedding
FirstOrder.Language.empty
Cardinal
Cardinal.instLE
Cardinal.lift
β€”trans
IsPreorder.toIsTrans
IsEquiv.toIsPreorder
iff_isEquiv
Nonempty.map
Function.instEmbeddingLikeEmbedding
FirstOrder.Language.strongHomClassEmpty
Cardinal.lift_mk_le'
nonempty_equiv_iff πŸ“–mathematicalβ€”FirstOrder.Language.Equiv
FirstOrder.Language.empty
Cardinal.lift
β€”trans
IsPreorder.toIsTrans
IsEquiv.toIsPreorder
iff_isEquiv
Nonempty.map
Cardinal.lift_mk_eq'

Function

Definitions

NameCategoryTheorems
emptyHom πŸ“–CompOp
1 mathmath: emptyHom_toFun

Theorems

NameKindAssumesProvesValidatesDepends On
emptyHom_toFun πŸ“–mathematicalβ€”DFunLike.coe
FirstOrder.Language.Hom
FirstOrder.Language.empty
FirstOrder.Language.Hom.instFunLike
emptyHom
β€”β€”

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