Documentation Verification Report

CoprodI

📁 Source: Mathlib/GroupTheory/CoprodI.lean

Statistics

MetricCount
DefinitionsCoprodI, coprodI, NeWord, head, inv, last, mulHead, prod, replaceHead, toList, toWord, Word, Pair, head, tail, cons, consRecOn, empty, equiv, equivPair, fstIdx, instDecidableEq, instDecidableEq_1, instInhabited, instInhabitedPair, instMulAction, prod, rcons, summandAction, toList, instGroup, instInv, of, instCoprodI, instInhabitedCoprodI, freeGroupEquivCoprodI
36
Theoremsinjective_lift_of_ping_pong, append_head, append_last, append_prod, inv_head, inv_last, inv_prod, mulHead_head, mulHead_prod, of_word, prod_singleton, replaceHead_head, singleton_head, singleton_last, toList_getLast?, toList_head?, toList_ne_nil, ext, ext_iff, fstIdx_ne, chain_ne, consRecOn_cons, consRecOn_empty, cons_eq_smul, cons_toList, empty_toList, equivPair_eq_of_fstIdx_ne, equivPair_head, equivPair_head_smul_equivPair_tail, equivPair_smul_same, equivPair_symm, equivPair_tail, equivPair_tail_eq_inv_smul, ext, ext_iff, fstIdx_cons, fstIdx_ne_iff, mem_equivPair_tail_iff, mem_of_mem_equivPair_tail, mem_rcons_iff, mem_smul_iff, mem_smul_iff_of_ne, ne_one, of_smul_def, prod_cons, prod_empty, prod_rcons, prod_smul, rcons_eq_smul, rcons_inj, smul_def, smul_eq_of_smul, smul_induction, empty_of_word_prod_eq_one, ext_hom, ext_hom_iff, iSup_mrange_of, induction_left, induction_on, instIsFreeGroup, inv_def, lift_comp_of, lift_comp_of', lift_injective_of_ping_pong, lift_mrange_le, lift_of, lift_of', lift_range_le, lift_symm_apply, lift_word_ping_pong, lift_word_prod_nontrivial_of_head_card, lift_word_prod_nontrivial_of_head_eq_last, lift_word_prod_nontrivial_of_not_empty, lift_word_prod_nontrivial_of_other_i, mclosure_iUnion_range_of, mrange_eq_iSup, of_apply, of_injective, of_leftInverse, range_eq_iSup, freeGroupEquivCoprodI_apply, freeGroupEquivCoprodI_symm_apply
82
Total118

FreeGroup

Theorems

NameKindAssumesProvesValidatesDepends On
injective_lift_of_ping_pong 📖mathematicalSet.Nonempty
Pairwise
Function.onFun
Set
Disjoint
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
Set.instHasSubset
Set.smulSet
SemigroupAction.toSMul
Monoid.toSemigroup
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulAction.toSemigroupAction
Compl.compl
Set.instCompl
Pi.instInv
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
FreeGroup
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
instGroup
MonoidHom.instFunLike
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
lift
ext_hom
lift_apply_of
Monoid.CoprodI.lift_of
MonoidHom.coe_comp
Monoid.CoprodI.lift_injective_of_ping_pong
Nat.instAtLeastTwoHAddOfNat
Nontrivial.to_nonempty
Equiv.cardinal_eq
Cardinal.mk_denumerable
Cardinal.natCast_le_aleph0
Set.Nonempty.inl
Disjoint.union_left
Disjoint.union_right
Disjoint.symm
Equiv.forall_congr_left
map_zpow
MonoidHom.instMonoidHomClass
zpow_zero
lt_or_gt_of_ne
Set.smul_set_mono
Disjoint.subset_compl_right
Int.le_induction
zpow_one
zpow_add
SemigroupAction.mul_smul
Set.subset_union_left
neg_add_cancel
Int.le_induction_down
zpow_neg
zpow_sub
Set.subset_union_right
MulEquiv.injective

Monoid

Definitions

NameCategoryTheorems
CoprodI 📖CompOp
40 mathmath: CoprodI.NeWord.append_prod, CoprodI.NeWord.mulHead_prod, CoprodI.lift_of, CoprodI.inv_def, CoprodI.mrange_eq_iSup, CoprodI.Word.prod_smul, CoprodI.mclosure_iUnion_range_of, CoprodI.lift_mrange_le, CoprodI.range_eq_iSup, CoprodI.Word.mem_smul_iff_of_ne, CoprodI.Word.of_smul_def, CoprodI.Word.cons_eq_smul, PushoutI.ofCoprodI_of, CoprodI.of_apply, freeGroupEquivCoprodI_symm_apply, CoprodI.NeWord.prod_singleton, CoprodI.lift_of', PushoutI.Reduced.exists_normalWord_prod_eq, CoprodI.iSup_mrange_of, CoprodI.Word.equivPair_head_smul_equivPair_tail, CoprodI.Word.equivPair_smul_same, CoprodI.Word.prod_cons, CoprodI.instIsFreeGroup, CoprodI.NeWord.inv_prod, CoprodI.Word.prod_empty, CoprodI.Word.prod_rcons, CoprodI.lift_symm_apply, CoprodI.Word.mem_smul_iff, CoprodI.Word.smul_eq_of_smul, CoprodI.ext_hom_iff, CoprodI.of_leftInverse, freeGroupEquivCoprodI_apply, CoprodI.Word.rcons_eq_smul, CoprodI.lift_word_ping_pong, CoprodI.lift_comp_of', CoprodI.lift_comp_of, CoprodI.lift_range_le, CoprodI.lift_injective_of_ping_pong, CoprodI.Word.equivPair_tail_eq_inv_smul, CoprodI.of_injective
instCoprodI 📖CompOp
38 mathmath: CoprodI.NeWord.append_prod, CoprodI.NeWord.mulHead_prod, CoprodI.lift_of, CoprodI.inv_def, CoprodI.mrange_eq_iSup, CoprodI.Word.prod_smul, CoprodI.mclosure_iUnion_range_of, CoprodI.lift_mrange_le, CoprodI.range_eq_iSup, CoprodI.Word.mem_smul_iff_of_ne, CoprodI.Word.of_smul_def, CoprodI.Word.cons_eq_smul, PushoutI.ofCoprodI_of, CoprodI.of_apply, freeGroupEquivCoprodI_symm_apply, CoprodI.NeWord.prod_singleton, CoprodI.lift_of', PushoutI.Reduced.exists_normalWord_prod_eq, CoprodI.iSup_mrange_of, CoprodI.Word.equivPair_head_smul_equivPair_tail, CoprodI.Word.equivPair_smul_same, CoprodI.Word.prod_cons, CoprodI.Word.prod_empty, CoprodI.Word.prod_rcons, CoprodI.lift_symm_apply, CoprodI.Word.mem_smul_iff, CoprodI.Word.smul_eq_of_smul, CoprodI.ext_hom_iff, CoprodI.of_leftInverse, freeGroupEquivCoprodI_apply, CoprodI.Word.rcons_eq_smul, CoprodI.lift_word_ping_pong, CoprodI.lift_comp_of', CoprodI.lift_comp_of, CoprodI.lift_range_le, CoprodI.lift_injective_of_ping_pong, CoprodI.Word.equivPair_tail_eq_inv_smul, CoprodI.of_injective
instInhabitedCoprodI 📖CompOp

Monoid.CoprodI

Definitions

NameCategoryTheorems
NeWord 📖CompData
Word 📖CompData
17 mathmath: Word.prod_smul, Word.mem_smul_iff_of_ne, Word.of_smul_def, Word.cons_eq_smul, Word.equivPair_symm, Word.mem_equivPair_tail_iff, Word.equivPair_tail, Word.rcons_inj, Word.equivPair_head_smul_equivPair_tail, Word.equivPair_smul_same, Word.mem_smul_iff, Word.smul_eq_of_smul, Word.equivPair_head, Word.smul_def, Word.rcons_eq_smul, Word.equivPair_eq_of_fstIdx_ne, Word.equivPair_tail_eq_inv_smul
instGroup 📖CompOp
4 mathmath: range_eq_iSup, instIsFreeGroup, freeGroupEquivCoprodI_apply, lift_range_le
instInv 📖CompOp
3 mathmath: inv_def, NeWord.inv_prod, Word.equivPair_tail_eq_inv_smul
of 📖CompOp
27 mathmath: NeWord.mulHead_prod, lift_of, inv_def, mclosure_iUnion_range_of, Word.mem_smul_iff_of_ne, Word.of_smul_def, Word.cons_eq_smul, Monoid.PushoutI.ofCoprodI_of, of_apply, NeWord.prod_singleton, lift_of', iSup_mrange_of, Word.equivPair_head_smul_equivPair_tail, Word.equivPair_smul_same, Word.prod_cons, Word.prod_rcons, lift_symm_apply, Word.mem_smul_iff, Word.smul_eq_of_smul, ext_hom_iff, of_leftInverse, freeGroupEquivCoprodI_apply, Word.rcons_eq_smul, lift_comp_of', lift_comp_of, Word.equivPair_tail_eq_inv_smul, of_injective

Theorems

NameKindAssumesProvesValidatesDepends On
empty_of_word_prod_eq_one 📖mathematicalCardinal
Cardinal.instLE
instOfNatAtLeastTwo
Cardinal.instNatCast
Nat.instAtLeastTwoHAddOfNat
Set.Nonempty
Pairwise
Function.onFun
Set
Disjoint
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
DFunLike.coe
MonoidHom
Monoid.CoprodI
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
MonoidHom.instFunLike
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
lift
Word.prod
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
Word.emptyNat.instAtLeastTwoHAddOfNat
NeWord.of_word
lift_word_prod_nontrivial_of_not_empty
ext_hom 📖MonoidHom.comp
Monoid.CoprodI
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
of
MonoidHom.cancel_right
Con.mk'_surjective
FreeMonoid.hom_eq
MonoidHom.comp_apply
of_apply
ext_hom_iff 📖mathematicalMonoidHom.comp
Monoid.CoprodI
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
of
ext_hom
iSup_mrange_of 📖mathematicaliSup
Submonoid
Monoid.CoprodI
Monoid.toMulOneClass
Monoid.instCoprodI
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
Submonoid.instCompleteLattice
MonoidHom.mrange
MonoidHom
MulOneClass.toMulOne
MonoidHom.instFunLike
MonoidHom.instMonoidHomClass
of
Top.top
Submonoid.instTop
MonoidHom.instMonoidHomClass
MonoidHom.mrange.congr_simp
lift_of'
MonoidHom.mrange_id
induction_left 📖Monoid.CoprodI
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
MulOne.toMul
DFunLike.coe
MonoidHom
MonoidHom.instFunLike
of
Submonoid.induction_of_closure_eq_top_left
mclosure_iUnion_range_of
induction_on 📖Monoid.CoprodI
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
DFunLike.coe
MonoidHom
MonoidHom.instFunLike
of
MulOne.toMul
induction_left
instIsFreeGroup 📖mathematicalIsFreeGroupMonoid.CoprodI
DivInvMonoid.toMonoid
Group.toDivInvMonoid
instGroup
FreeGroupBasis.isFreeGroup
inv_def 📖mathematicalMonoid.CoprodI
DivInvMonoid.toMonoid
Group.toDivInvMonoid
instInv
MulOpposite.unop
DFunLike.coe
MonoidHom
MulOpposite
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
MulOpposite.instMonoid
MonoidHom.instFunLike
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
lift
MonoidHom.comp
MulOpposite.instMulOne
MonoidHom.op
of
MulEquiv.toMonoidHom
MulOpposite.instMulOneClass
MulEquiv.inv'
Group.toDivisionMonoid
lift_comp_of 📖mathematicalMonoidHom.comp
Monoid.CoprodI
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
DFunLike.coe
Equiv
MonoidHom
EquivLike.toFunLike
Equiv.instEquivLike
lift
of
Equiv.symm_apply_apply
lift_comp_of' 📖mathematicalDFunLike.coe
Equiv
MonoidHom
Monoid.CoprodI
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
EquivLike.toFunLike
Equiv.instEquivLike
lift
MonoidHom.comp
of
Equiv.apply_symm_apply
lift_injective_of_ping_pong 📖mathematicalCardinal
Cardinal.instLE
instOfNatAtLeastTwo
Cardinal.instNatCast
Nat.instAtLeastTwoHAddOfNat
Set.Nonempty
Pairwise
Function.onFun
Set
Disjoint
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
Monoid.CoprodI
DivInvMonoid.toMonoid
Group.toDivInvMonoid
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
MonoidHom.instFunLike
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
lift
Nat.instAtLeastTwoHAddOfNat
injective_iff_map_eq_one
MonoidHom.instMonoidHomClass
Equiv.forall_congr_left
empty_of_word_prod_eq_one
Word.prod_empty
lift_mrange_le 📖mathematicalSubmonoid
Monoid.toMulOneClass
Preorder.toLE
PartialOrder.toPreorder
Submonoid.instPartialOrder
MonoidHom.mrange
Monoid.CoprodI
Monoid.instCoprodI
MonoidHom
MulOneClass.toMulOne
MonoidHom.instFunLike
MonoidHom.instMonoidHomClass
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
lift
MonoidHom.instMonoidHomClass
mrange_eq_iSup
lift_of 📖mathematicalDFunLike.coe
MonoidHom
Monoid.CoprodI
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
MonoidHom.instFunLike
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
lift
of
DFunLike.congr_fun
lift_comp_of
lift_of' 📖mathematicalDFunLike.coe
Equiv
MonoidHom
Monoid.CoprodI
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
EquivLike.toFunLike
Equiv.instEquivLike
lift
of
MonoidHom.id
lift_comp_of'
lift_range_le 📖mathematicalSubgroup
Preorder.toLE
PartialOrder.toPreorder
Subgroup.instPartialOrder
MonoidHom.range
Monoid.CoprodI
DivInvMonoid.toMonoid
Group.toDivInvMonoid
instGroup
DFunLike.coe
Equiv
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
EquivLike.toFunLike
Equiv.instEquivLike
lift
induction_on
Subgroup.one_mem
lift_of
Set.mem_range_self
map_mul
MonoidHomClass.toMulHomClass
MonoidHom.instMonoidHomClass
Subgroup.mul_mem
lift_symm_apply 📖mathematicalDFunLike.coe
Equiv
MonoidHom
Monoid.CoprodI
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
lift
MonoidHom.comp
of
lift_word_ping_pong 📖mathematicalPairwiseSet
Set.instHasSubset
Set.smulSet
SemigroupAction.toSMul
Monoid.toSemigroup
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulAction.toSemigroupAction
DFunLike.coe
MonoidHom
Monoid.CoprodI
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
MonoidHom.instFunLike
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
lift
NeWord.prod
NeWord.prod_singleton
lift_of
NeWord.append_prod
map_mul
MonoidHomClass.toMulHomClass
MonoidHom.instMonoidHomClass
SemigroupAction.mul_smul
Set.smul_set_subset_smul_set_iff
lift_word_prod_nontrivial_of_head_card 📖Set.Nonempty
Pairwise
Function.onFun
Set
Disjoint
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
Nat.instAtLeastTwoHAddOfNat
Cardinal.three_le
div_inv_eq_mul
div_ne_one_of_ne
inv_ne_one
lift_word_prod_nontrivial_of_head_eq_last
NeWord.append_prod
NeWord.mulHead_prod
NeWord.prod_singleton
map_inv
MonoidHom.instMonoidHomClass
map_mul
MonoidHomClass.toMulHomClass
lift_of
mul_one
mul_inv_cancel
lift_word_prod_nontrivial_of_head_eq_last 📖Set.Nonempty
Pairwise
Function.onFun
Set
Disjoint
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
exists_ne
lift_word_prod_nontrivial_of_other_i
lift_word_prod_nontrivial_of_not_empty 📖Cardinal
Cardinal.instLE
instOfNatAtLeastTwo
Cardinal.instNatCast
Nat.instAtLeastTwoHAddOfNat
Set.Nonempty
Pairwise
Function.onFun
Set
Disjoint
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
Nat.instAtLeastTwoHAddOfNat
Cardinal.three_le
lift_word_prod_nontrivial_of_other_i
lift_word_prod_nontrivial_of_head_eq_last
lift_word_prod_nontrivial_of_head_card
NeWord.inv_prod
map_inv
MonoidHom.instMonoidHomClass
lift_word_prod_nontrivial_of_other_i 📖Set.Nonempty
Pairwise
Function.onFun
Set
Disjoint
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
one_smul
lift_word_ping_pong
Disjoint.le_bot
mclosure_iUnion_range_of 📖mathematicalSubmonoid.closure
Monoid.CoprodI
Monoid.toMulOneClass
Monoid.instCoprodI
Set.iUnion
Set.range
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
MonoidHom.instFunLike
of
Top.top
Submonoid
Submonoid.instTop
Submonoid.closure_iUnion
MonoidHom.instMonoidHomClass
MonoidHom.mclosure_range
iSup_mrange_of
mrange_eq_iSup 📖mathematicalMonoidHom.mrange
Monoid.CoprodI
Monoid.toMulOneClass
Monoid.instCoprodI
MonoidHom
MulOneClass.toMulOne
MonoidHom.instFunLike
MonoidHom.instMonoidHomClass
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
lift
iSup
Submonoid
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
Submonoid.instCompleteLattice
MonoidHom.instMonoidHomClass
lift.eq_1
Con.lift_range
FreeMonoid.mrange_lift
Set.range_sigma_eq_iUnion_range
Submonoid.closure_iUnion
MonoidHom.mclosure_range
of_apply 📖mathematicalDFunLike.coe
MonoidHom
Monoid.CoprodI
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
MonoidHom.instFunLike
of
FreeMonoid
Con.Quotient
MulOne.toMul
RightCancelMonoid.toMonoid
CancelMonoid.toRightCancelMonoid
FreeMonoid.instCancelMonoid
conGen
Rel
Con.mulOneClass
Con.mk'
FreeMonoid.of
of_injective 📖mathematicalMonoid.CoprodI
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
MonoidHom.instFunLike
of
of_leftInverse
of_leftInverse 📖mathematicalMonoid.CoprodI
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
MonoidHom.instFunLike
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
lift
Pi.mulSingle
instOneMonoidHom
MonoidHom.id
of
lift_of
Pi.mulSingle_eq_same
range_eq_iSup 📖mathematicalMonoidHom.range
Monoid.CoprodI
DivInvMonoid.toMonoid
Group.toDivInvMonoid
instGroup
DFunLike.coe
Equiv
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
EquivLike.toFunLike
Equiv.instEquivLike
lift
iSup
Subgroup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
Subgroup.instCompleteLattice
le_antisymm
lift_range_le
le_iSup
iSup_le
lift_of

Monoid.CoprodI.FreeGroupBasis

Definitions

NameCategoryTheorems
coprodI 📖CompOp

Monoid.CoprodI.NeWord

Definitions

NameCategoryTheorems
head 📖CompOp
7 mathmath: inv_head, replaceHead_head, toList_head?, inv_last, mulHead_head, append_head, singleton_head
inv 📖CompOp
3 mathmath: inv_head, inv_last, inv_prod
last 📖CompOp
5 mathmath: inv_head, append_last, inv_last, singleton_last, toList_getLast?
mulHead 📖CompOp
2 mathmath: mulHead_prod, mulHead_head
prod 📖CompOp
5 mathmath: append_prod, mulHead_prod, prod_singleton, inv_prod, Monoid.CoprodI.lift_word_ping_pong
replaceHead 📖CompOp
1 mathmath: replaceHead_head
toList 📖CompOp
2 mathmath: toList_head?, toList_getLast?
toWord 📖CompOp
1 mathmath: of_word

Theorems

NameKindAssumesProvesValidatesDepends On
append_head 📖mathematicalhead
append
append_last 📖mathematicallast
append
append_prod 📖mathematicalprod
append
Monoid.CoprodI
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
Semigroup.to_isLawfulIdentity
Semigroup.to_isAssociative
inv_head 📖mathematicalhead
DivInvMonoid.toMonoid
Group.toDivInvMonoid
inv
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
last
inv_last 📖mathematicallast
DivInvMonoid.toMonoid
Group.toDivInvMonoid
inv
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
head
inv_prod 📖mathematicalprod
DivInvMonoid.toMonoid
Group.toDivInvMonoid
inv
Monoid.CoprodI
Monoid.CoprodI.instInv
prod_singleton
map_inv
MonoidHom.instMonoidHomClass
append_prod
mul_inv_rev
mulHead_head 📖mathematicalhead
mulHead
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
replaceHead_head
mulHead_prod 📖mathematicalprod
mulHead
Monoid.CoprodI
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
DFunLike.coe
MonoidHom
MonoidHom.instFunLike
Monoid.CoprodI.of
prod_singleton
map_mul
MonoidHomClass.toMulHomClass
MonoidHom.instMonoidHomClass
append_prod
of_word 📖mathematicaltoWordSigma.eta
toList_head?
Monoid.CoprodI.Word.ext
prod_singleton 📖mathematicalprod
singleton
DFunLike.coe
MonoidHom
Monoid.CoprodI
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
MonoidHom.instFunLike
Monoid.CoprodI.of
mul_one
replaceHead_head 📖mathematicalhead
replaceHead
singleton_head 📖mathematicalhead
singleton
singleton_last 📖mathematicallast
singleton
toList_getLast? 📖mathematicaltoList
last
List.mem_getLast?_append_of_mem_getLast?
toList_head? 📖mathematicaltoList
head
toList.induct_unfolding
toList_ne_nil 📖

Monoid.CoprodI.Word

Definitions

NameCategoryTheorems
Pair 📖CompData
9 mathmath: equivPair_symm, mem_equivPair_tail_iff, equivPair_tail, rcons_inj, equivPair_head_smul_equivPair_tail, equivPair_smul_same, equivPair_head, equivPair_eq_of_fstIdx_ne, equivPair_tail_eq_inv_smul
cons 📖CompOp
5 mathmath: cons_toList, fstIdx_cons, cons_eq_smul, prod_cons, consRecOn_cons
consRecOn 📖CompOp
2 mathmath: consRecOn_empty, consRecOn_cons
empty 📖CompOp
5 mathmath: Monoid.PushoutI.Reduced.eq_empty_of_mem_range, Monoid.CoprodI.empty_of_word_prod_eq_one, prod_empty, consRecOn_empty, empty_toList
equiv 📖CompOp
equivPair 📖CompOp
8 mathmath: equivPair_symm, mem_equivPair_tail_iff, equivPair_tail, equivPair_head_smul_equivPair_tail, equivPair_smul_same, equivPair_head, equivPair_eq_of_fstIdx_ne, equivPair_tail_eq_inv_smul
fstIdx 📖CompOp
1 mathmath: fstIdx_cons
instDecidableEq 📖CompOp
instDecidableEq_1 📖CompOp
instInhabited 📖CompOp
instInhabitedPair 📖CompOp
instMulAction 📖CompOp
10 mathmath: prod_smul, mem_smul_iff_of_ne, of_smul_def, cons_eq_smul, equivPair_head_smul_equivPair_tail, equivPair_smul_same, mem_smul_iff, smul_eq_of_smul, rcons_eq_smul, equivPair_tail_eq_inv_smul
prod 📖CompOp
5 mathmath: prod_smul, Monoid.PushoutI.Reduced.exists_normalWord_prod_eq, prod_cons, prod_empty, prod_rcons
rcons 📖CompOp
7 mathmath: of_smul_def, equivPair_symm, rcons_inj, mem_rcons_iff, prod_rcons, smul_def, rcons_eq_smul
summandAction 📖CompOp
2 mathmath: smul_eq_of_smul, smul_def
toList 📖CompOp
13 mathmath: ext_iff, cons_toList, mem_smul_iff_of_ne, mem_equivPair_tail_iff, Monoid.PushoutI.Reduced.exists_normalWord_prod_eq, mem_rcons_iff, Monoid.PushoutI.NormalWord.ext_iff, mem_smul_iff, equivPair_head, Monoid.PushoutI.NormalWord.empty_toList, empty_toList, Monoid.PushoutI.NormalWord.cons_toList, chain_ne

Theorems

NameKindAssumesProvesValidatesDepends On
chain_ne 📖mathematicaltoList
consRecOn_cons 📖mathematicalconsRecOn
cons
consRecOn_empty 📖mathematicalconsRecOn
empty
cons_eq_smul 📖mathematicalcons
Monoid.CoprodI
Monoid.CoprodI.Word
SemigroupAction.toSMul
Monoid.toSemigroup
Monoid.instCoprodI
MulAction.toSemigroupAction
instMulAction
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
MonoidHom.instFunLike
Monoid.CoprodI.of
Pair.fstIdx_ne
of_smul_def
equivPair_eq_of_fstIdx_ne
mul_one
cons_toList 📖mathematicaltoList
cons
empty_toList 📖mathematicaltoList
empty
equivPair_eq_of_fstIdx_ne 📖mathematicalDFunLike.coe
Equiv
Monoid.CoprodI.Word
Pair
EquivLike.toFunLike
Equiv.instEquivLike
equivPair
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
Equiv.apply_eq_iff_eq_symm_apply
Pair.fstIdx_ne
equivPair_head 📖mathematicalPair.head
DFunLike.coe
Equiv
Monoid.CoprodI.Word
Pair
EquivLike.toFunLike
Equiv.instEquivLike
equivPair
toList
Exists.fst
Exists.snd
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
Exists.fst
Exists.snd
instIsEmptyFalse
equivPair_head_smul_equivPair_tail 📖mathematicalMonoid.CoprodI
Monoid.CoprodI.Word
SemigroupAction.toSMul
Monoid.toSemigroup
Monoid.instCoprodI
MulAction.toSemigroupAction
instMulAction
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
MonoidHom.instFunLike
Monoid.CoprodI.of
Pair.head
Equiv
Pair
EquivLike.toFunLike
Equiv.instEquivLike
equivPair
Pair.tail
rcons_eq_smul
equivPair_symm
Equiv.symm_apply_apply
equivPair_smul_same 📖mathematicalDFunLike.coe
Equiv
Monoid.CoprodI.Word
Pair
EquivLike.toFunLike
Equiv.instEquivLike
equivPair
Monoid.CoprodI
SemigroupAction.toSMul
Monoid.toSemigroup
Monoid.instCoprodI
MulAction.toSemigroupAction
instMulAction
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
MonoidHom.instFunLike
Monoid.CoprodI.of
MulOne.toMul
Pair.head
Pair.tail
Pair.fstIdx_ne
Pair.fstIdx_ne
of_smul_def
equivPair_symm
Equiv.apply_symm_apply
equivPair_symm 📖mathematicalDFunLike.coe
Equiv
Pair
Monoid.CoprodI.Word
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
equivPair
rcons
equivPair_tail 📖mathematicalDFunLike.coe
Equiv
Monoid.CoprodI.Word
Pair
EquivLike.toFunLike
Equiv.instEquivLike
equivPair
Pair.tail
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
Pair.fstIdx_ne
equivPair_eq_of_fstIdx_ne
Pair.fstIdx_ne
equivPair_tail_eq_inv_smul 📖mathematicalPair.tail
DivInvMonoid.toMonoid
Group.toDivInvMonoid
DFunLike.coe
Equiv
Monoid.CoprodI.Word
Pair
EquivLike.toFunLike
Equiv.instEquivLike
equivPair
Monoid.CoprodI
SemigroupAction.toSMul
Monoid.toSemigroup
Monoid.instCoprodI
MulAction.toSemigroupAction
instMulAction
Monoid.CoprodI.instInv
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
MonoidHom.instFunLike
Monoid.CoprodI.of
Pair.head
inv_smul_eq_iff
equivPair_head_smul_equivPair_tail
ext 📖toList
ext_iff 📖mathematicaltoListext
fstIdx_cons 📖mathematicalfstIdx
cons
fstIdx_ne_iff 📖not_iff_not
mem_equivPair_tail_iff 📖mathematicaltoList
Pair.tail
DFunLike.coe
Equiv
Monoid.CoprodI.Word
Pair
EquivLike.toFunLike
Equiv.instEquivLike
equivPair
instIsEmptyFalse
mem_of_mem_equivPair_tail 📖toList
Pair.tail
DFunLike.coe
Equiv
Monoid.CoprodI.Word
Pair
EquivLike.toFunLike
Equiv.instEquivLike
equivPair
mem_equivPair_tail_iff
instIsEmptyFalse
mem_rcons_iff 📖mathematicaltoList
rcons
Pair.tail
Pair.head
Pair.fstIdx_ne
mem_smul_iff 📖mathematicaltoList
Monoid.CoprodI
Monoid.CoprodI.Word
SemigroupAction.toSMul
Monoid.toSemigroup
Monoid.instCoprodI
MulAction.toSemigroupAction
instMulAction
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
MonoidHom.instFunLike
Monoid.CoprodI.of
MulOne.toMul
Pair.fstIdx_ne
of_smul_def
mem_rcons_iff
mem_equivPair_tail_iff
Exists.fst
Exists.snd
equivPair_head
ne_one
mul_dite
mul_one
fstIdx.eq_1
instIsEmptyFalse
One.instNonempty
mem_smul_iff_of_ne 📖mathematicaltoList
Monoid.CoprodI
Monoid.CoprodI.Word
SemigroupAction.toSMul
Monoid.toSemigroup
Monoid.instCoprodI
MulAction.toSemigroupAction
instMulAction
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
MonoidHom.instFunLike
Monoid.CoprodI.of
instIsEmptyFalse
ne_one 📖toList
of_smul_def 📖mathematicalMonoid.CoprodI
Monoid.CoprodI.Word
SemigroupAction.toSMul
Monoid.toSemigroup
Monoid.instCoprodI
MulAction.toSemigroupAction
instMulAction
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
MonoidHom.instFunLike
Monoid.CoprodI.of
rcons
prod_cons 📖mathematicalprod
cons
Monoid.CoprodI
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
DFunLike.coe
MonoidHom
MonoidHom.instFunLike
Monoid.CoprodI.of
prod_empty 📖mathematicalprod
empty
Monoid.CoprodI
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
prod_rcons 📖mathematicalprod
rcons
Monoid.CoprodI
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
DFunLike.coe
MonoidHom
MonoidHom.instFunLike
Monoid.CoprodI.of
Pair.head
Pair.tail
Pair.fstIdx_ne
rcons.eq_1
map_one
MonoidHomClass.toOneHomClass
MonoidHom.instMonoidHomClass
one_mul
cons.eq_1
prod.eq_1
prod_smul 📖mathematicalprod
Monoid.CoprodI
Monoid.CoprodI.Word
SemigroupAction.toSMul
Monoid.toSemigroup
Monoid.instCoprodI
MulAction.toSemigroupAction
instMulAction
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.CoprodI.induction_on
one_smul
one_mul
Pair.fstIdx_ne
of_smul_def
prod_rcons
MonoidHom.map_mul
mul_assoc
equivPair_symm
Equiv.symm_apply_apply
SemigroupAction.mul_smul
rcons_eq_smul 📖mathematicalrcons
Monoid.CoprodI
Monoid.CoprodI.Word
SemigroupAction.toSMul
Monoid.toSemigroup
Monoid.instCoprodI
MulAction.toSemigroupAction
instMulAction
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
MonoidHom.instFunLike
Monoid.CoprodI.of
Pair.head
Pair.tail
Pair.fstIdx_ne
equivPair_tail
mul_one
rcons_inj 📖mathematicalPair
Monoid.CoprodI.Word
rcons
Pair.fstIdx_ne
ext
smul_def 📖mathematicalMonoid.CoprodI.Word
SemigroupAction.toSMul
Monoid.toSemigroup
MulAction.toSemigroupAction
summandAction
rcons
smul_eq_of_smul 📖mathematicalMonoid.CoprodI.Word
SemigroupAction.toSMul
Monoid.toSemigroup
MulAction.toSemigroupAction
summandAction
Monoid.CoprodI
Monoid.instCoprodI
instMulAction
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
MonoidHom.instFunLike
Monoid.CoprodI.of
smul_induction 📖empty
Monoid.CoprodI
Monoid.CoprodI.Word
SemigroupAction.toSMul
Monoid.toSemigroup
Monoid.instCoprodI
MulAction.toSemigroupAction
instMulAction
DFunLike.coe
MonoidHom
MulOneClass.toMulOne
Monoid.toMulOneClass
MonoidHom.instFunLike
Monoid.CoprodI.of
cons_eq_smul

Monoid.CoprodI.Word.Pair

Definitions

NameCategoryTheorems
head 📖CompOp
8 mathmath: ext_iff, Monoid.CoprodI.Word.mem_rcons_iff, Monoid.CoprodI.Word.equivPair_head_smul_equivPair_tail, Monoid.CoprodI.Word.equivPair_smul_same, Monoid.CoprodI.Word.prod_rcons, Monoid.CoprodI.Word.equivPair_head, Monoid.CoprodI.Word.rcons_eq_smul, Monoid.CoprodI.Word.equivPair_tail_eq_inv_smul
tail 📖CompOp
9 mathmath: ext_iff, Monoid.CoprodI.Word.mem_equivPair_tail_iff, Monoid.CoprodI.Word.equivPair_tail, Monoid.CoprodI.Word.mem_rcons_iff, Monoid.CoprodI.Word.equivPair_head_smul_equivPair_tail, Monoid.CoprodI.Word.equivPair_smul_same, Monoid.CoprodI.Word.prod_rcons, Monoid.CoprodI.Word.rcons_eq_smul, Monoid.CoprodI.Word.equivPair_tail_eq_inv_smul

Theorems

NameKindAssumesProvesValidatesDepends On
ext 📖head
tail
ext_iff 📖mathematicalhead
tail
ext
fstIdx_ne 📖

(root)

Definitions

NameCategoryTheorems
freeGroupEquivCoprodI 📖CompOp
2 mathmath: freeGroupEquivCoprodI_symm_apply, freeGroupEquivCoprodI_apply

Theorems

NameKindAssumesProvesValidatesDepends On
freeGroupEquivCoprodI_apply 📖mathematicalDFunLike.coe
MulEquiv
FreeGroup
Monoid.CoprodI
DivInvMonoid.toMonoid
Group.toDivInvMonoid
FreeGroup.instGroup
FreeGroup.instMul
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
EquivLike.toFunLike
MulEquiv.instEquivLike
freeGroupEquivCoprodI
MonoidHom
MonoidHom.instFunLike
Equiv
Monoid.CoprodI.instGroup
Equiv.instEquivLike
FreeGroup.lift
Monoid.CoprodI.of
FreeGroup.of
freeGroupEquivCoprodI_symm_apply 📖mathematicalDFunLike.coe
MulEquiv
Monoid.CoprodI
FreeGroup
DivInvMonoid.toMonoid
Group.toDivInvMonoid
FreeGroup.instGroup
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.instCoprodI
FreeGroup.instMul
EquivLike.toFunLike
MulEquiv.instEquivLike
MulEquiv.symm
freeGroupEquivCoprodI
MonoidHom
MonoidHom.instFunLike
Equiv
Equiv.instEquivLike
Monoid.CoprodI.lift
FreeGroup.lift
FreeGroup.of

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