Documentation Verification Report

Lemmas

📁 Source: Mathlib/Data/Num/Lemmas.lean

Statistics

MetricCount
DefinitionsofSnum, snumCoe, addMonoid, addMonoidWithOne, commSemiring, linearOrder, partialOrder, transfer, transfer_rw, addCommSemigroup, commMonoid, distrib, dvd, linearOrder, transfer, transfer_rw, le, lt
18
Theoremsadd_ofNat', add_of_nat, add_one, add_succ, add_toZNum, add_to_nat, add_zero, bit0_of_bit0, bit1_of_bit1, bit1_succ, bit_to_nat, castNum_and, castNum_eq_bitwise, castNum_ldiff, castNum_or, castNum_shiftLeft, castNum_shiftRight, castNum_testBit, castNum_xor, cast_add, cast_bit0, cast_bit1, cast_inj, cast_le, cast_lt, cast_mul, cast_one, cast_pos, cast_succ, cast_succ', cast_toZNum, cast_toZNumNeg, cast_to_int, cast_to_nat, cast_zero, cast_zero', cmp_eq, cmp_swap, cmp_to_nat, dvd_to_nat, isOrderedCancelAddMonoid, isStrictOrderedRing, le_iff_cmp, le_to_nat, lt_iff_cmp, lt_to_nat, mul_to_nat, natSize_to_nat, ofNat'_bit, ofNat'_eq, ofNat'_one, ofNat'_succ, ofNat'_zero, of_natCast, of_nat_inj, of_to_nat, of_to_nat', ppred_to_nat, pred_to_nat, size_eq_natSize, size_to_nat, succ'_to_nat, succ_to_nat, toNat_injective, toZNum_inj, to_nat_inj, to_nat_to_int, to_of_nat, zero_add, zneg_toZNum, zneg_toZNumNeg, add_one, add_succ, add_to_nat, bit0_of_bit0, bit1_of_bit1, bit_to_nat, cast_add, cast_bit0, cast_bit1, cast_inj, cast_le, cast_lt, cast_mul, cast_one, cast_one', cast_pos, cast_succ, cast_to_int, cast_to_nat, cast_to_num, cmp_eq, cmp_swap, cmp_to_nat, cmp_to_nat_lemma, dvd_to_nat, le_iff_cmp, le_to_nat, lt_iff_cmp, lt_to_nat, mul_to_nat, natSize_pos, natSize_to_nat, of_to_nat, of_to_nat', one_add, one_le_cast, one_sub', pred'_succ', pred'_to_nat, pred_to_nat, size_eq_natSize, size_to_nat, sub'_one, succ'_pred', succ_to_nat, to_nat_inj, to_nat_pos, to_nat_to_int
119
Total137

Int

Definitions

NameCategoryTheorems
ofSnum 📖CompOp
snumCoe 📖CompOp

Num

Definitions

NameCategoryTheorems
addMonoid 📖CompOp
addMonoidWithOne 📖CompOp
10 mathmath: ZNum.of_nat_toZNumNeg, PosNum.of_to_nat, to_of_nat, of_to_nat, ofInt'_toZNum, add_of_nat, ZNum.of_nat_toZNum, of_natCast, ofNat'_eq, of_nat_inj
commSemiring 📖CompOp
4 mathmath: isStrictOrderedRing, dvd_iff_mod_eq_zero, dvd_to_nat, isOrderedCancelAddMonoid
linearOrder 📖CompOp
partialOrder 📖CompOp
2 mathmath: isStrictOrderedRing, isOrderedCancelAddMonoid
transfer 📖CompOp
transfer_rw 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
add_ofNat' 📖mathematicalofNat'
Num
instAdd
ofNat'_zero
add_zero
ofNat'_succ
add_one
add_succ
add_of_nat 📖mathematicalNum
AddMonoidWithOne.toNatCast
addMonoidWithOne
instAdd
add_ofNat'
add_one 📖mathematicalNum
instAdd
instOneNum
succ
add_succ 📖mathematicalNum
instAdd
succ
zero_add
PosNum.add_one
add_zero
succ.eq_1
succ'.eq_2
PosNum.add_succ
add_toZNum 📖mathematicaltoZNum
Num
instAdd
ZNum
ZNum.instAdd
add_to_nat 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
Num
instAdd
PosNum.add_to_nat
add_zero 📖mathematicalNum
instAdd
instZeroNum
bit0_of_bit0 📖mathematicalNum
instAdd
bit0
PosNum.bit0_of_bit0
bit1_of_bit1 📖mathematicalNum
instAdd
instOneNum
bit1
PosNum.bit1_of_bit1
bit1_succ 📖mathematicalsucc
bit1
bit0
bit_to_nat 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
bit
Nat.bit
MulZeroClass.mul_zero
two_mul
zero_add
add_assoc
castNum_and 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
Num
instAndOp
castNum_eq_bitwise
castNum_eq_bitwise 📖mathematicalNum
instZeroNum
pos
PosNum
instOnePosNum
instOneNum
PosNum.bit
bit
castNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
Nat.bitwise_zero
Nat.bitwise_zero_left
Nat.bitwise_zero_right
PosNum.bit_to_nat
Nat.bitwise_bit
bit_to_nat
castNum_ldiff 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
ldiff
Nat.ldiff
castNum_eq_bitwise
castNum_or 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
Num
instOrOp
castNum_eq_bitwise
castNum_shiftLeft 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
Num
instHShiftLeftNat
PosNum.shiftl_succ_eq_bit0_shiftl
mul_comm
mul_two
castNum_shiftRight 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
Num
instHShiftRightNat
Nat.div2_val
IsOrderedAddMonoid.toAddLeftMono
Nat.instIsOrderedAddMonoid
Nat.instZeroLEOneClass
Nat.instCharZero
Nat.instAtLeastTwoHAddOfNat
add_comm
Nat.shiftRight_eq
Nat.div2_succ
Nat.bodd_add
castNum_testBit 📖mathematicaltestBit
castNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
cast_pos
PosNum.cast_bit1
Nat.instAtLeastTwoHAddOfNat
two_mul
Nat.bit_true
Nat.testBit_bit_zero
PosNum.cast_bit0
Nat.bit_false
PosNum.testBit.eq_2
Nat.testBit_bit_succ
castNum_xor 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
Num
instXorOp
castNum_eq_bitwise
cast_add 📖mathematicalcastNum
AddMonoidWithOne.toOne
AddSemigroup.toAdd
AddMonoid.toAddSemigroup
AddMonoidWithOne.toAddMonoid
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
Num
instAdd
cast_to_nat
add_to_nat
Nat.cast_add
cast_bit0 📖mathematicalcastNum
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
bit0
Distrib.toMul
instOfNatAtLeastTwo
AddMonoidWithOne.toNatCast
Nat.instAtLeastTwoHAddOfNat
Nat.instAtLeastTwoHAddOfNat
bit0_of_bit0
two_mul
cast_add
cast_bit1 📖mathematicalcastNum
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
bit1
Distrib.toMul
instOfNatAtLeastTwo
AddMonoidWithOne.toNatCast
Nat.instAtLeastTwoHAddOfNat
Nat.instAtLeastTwoHAddOfNat
bit1_of_bit1
bit0_of_bit0
cast_add
cast_bit0
cast_inj 📖mathematicalcastNum
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
cast_to_nat
IsStrictOrderedRing.toCharZero
cast_le 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
castNum
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
Num
instLE
not_lt
cast_lt
cast_lt 📖mathematicalPreorder.toLT
PartialOrder.toPreorder
castNum
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
Num
instLT
cast_to_nat
Nat.cast_lt
IsOrderedAddMonoid.toAddLeftMono
IsOrderedRing.toIsOrderedAddMonoid
IsStrictOrderedRing.toIsOrderedRing
IsStrictOrderedRing.toZeroLEOneClass
IsStrictOrderedRing.toCharZero
lt_to_nat
cast_mul 📖mathematicalcastNum
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
Num
instMul
Distrib.toMul
MulZeroClass.zero_mul
MulZeroClass.mul_zero
PosNum.cast_mul
cast_one 📖mathematicalcastNum
Num
instOneNum
cast_pos 📖mathematicalcastNum
pos
castPosNum
cast_succ 📖mathematicalcastNum
AddMonoidWithOne.toOne
AddSemigroup.toAdd
AddMonoid.toAddSemigroup
AddMonoidWithOne.toAddMonoid
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
succ
cast_succ'
cast_succ' 📖mathematicalcastPosNum
AddMonoidWithOne.toOne
AddSemigroup.toAdd
AddMonoid.toAddSemigroup
AddMonoidWithOne.toAddMonoid
succ'
castNum
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
PosNum.cast_to_nat
succ'_to_nat
Nat.cast_add_one
cast_to_nat
cast_toZNum 📖mathematicalcastZNum
toZNum
castNum
cast_toZNumNeg 📖mathematicalcastZNum
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddSemigroup.toAdd
AddMonoid.toAddSemigroup
SubNegMonoid.toAddMonoid
SubtractionMonoid.toSubNegMonoid
NegZeroClass.toNeg
toZNumNeg
castNum
neg_zero
cast_to_int 📖mathematicalAddGroupWithOne.toIntCast
castNum
AddMonoidWithOne.toOne
AddGroupWithOne.toAddMonoidWithOne
Ring.toAddGroupWithOne
Int.instRing
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Int.instCommRing
AddSemigroup.toAdd
AddMonoid.toAddSemigroup
AddMonoidWithOne.toAddMonoid
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
AddGroupWithOne.toAddGroup
to_nat_to_int
Int.cast_natCast
cast_to_nat
cast_to_nat 📖mathematicalAddMonoidWithOne.toNatCast
castNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
AddMonoidWithOne.toOne
AddSemigroup.toAdd
AddMonoid.toAddSemigroup
AddMonoidWithOne.toAddMonoid
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
Nat.cast_zero
PosNum.cast_to_nat
cast_zero 📖mathematicalcastNum
Num
instZeroNum
cast_zero' 📖mathematicalcastNum
zero
cmp_eq 📖mathematicalcmpcmp_to_nat
lt_irrefl
cmp_swap 📖mathematicalcmpPosNum.cmp_swap
cmp_to_nat 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
Num
cmp
PosNum.to_nat_pos
PosNum.cmp_to_nat
dvd_to_nat 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
Num
semigroupDvd
SemigroupWithZero.toSemigroup
NonUnitalSemiring.toSemigroupWithZero
NonUnitalCommSemiring.toNonUnitalSemiring
CommSemiring.toNonUnitalCommSemiring
commSemiring
of_to_nat
Nat.cast_mul
mul_to_nat
isOrderedCancelAddMonoid 📖mathematicalIsOrderedCancelAddMonoid
Num
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
commSemiring
partialOrder
le_to_nat
add_to_nat
add_le_add_left
covariant_swap_add_of_covariant_add
IsOrderedAddMonoid.toAddLeftMono
Nat.instIsOrderedAddMonoid
le_of_add_le_add_left
IsLeftCancelAdd.addLeftReflectLE_of_addLeftReflectLT
AddLeftCancelSemigroup.toIsLeftCancelAdd
contravariant_lt_of_covariant_le
isStrictOrderedRing 📖mathematicalIsStrictOrderedRing
Num
CommSemiring.toSemiring
commSemiring
partialOrder
isOrderedCancelAddMonoid
lt_to_nat
mul_to_nat
cast_zero
mul_lt_mul_of_pos_left
LinearOrderedCommMonoidWithZero.toPosMulStrictMono
mul_lt_mul_of_pos_right
IsStrictOrderedRing.toMulPosStrictMono
Nat.instIsStrictOrderedRing
le_iff_cmp 📖mathematicalNum
instLE
lt_iff_cmp
cmp_swap
le_to_nat 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
Num
instLE
not_lt
lt_to_nat
lt_iff_cmp 📖mathematicalNum
instLT
cmp
lt_to_nat 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
Num
instLT
cmp_to_nat
not_lt_of_gt
mul_to_nat 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
Num
instMul
MulZeroClass.zero_mul
PosNum.mul_to_nat
natSize_to_nat 📖mathematicalnatSize
Nat.size
castNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
size_eq_natSize
size_to_nat
ofNat'_bit 📖mathematicalofNat'
Nat.bit
bit1
bit0
Nat.binaryRec_eq
ofNat'_eq 📖mathematicalofNat'
Num
AddMonoidWithOne.toNatCast
addMonoidWithOne
ofNat'_zero
Nat.cast_zero
ofNat'_one 📖mathematicalofNat'
Num
instOneNum
ofNat'_succ 📖mathematicalofNat'
Num
instAdd
instOneNum
zero_add
ofNat'_one
ofNat'_zero
zero_add
ofNat'_bit
add_assoc
mul_add
Distrib.leftDistribClass
mul_one
add_one
bit1_succ
ofNat'_zero 📖mathematicalofNat'
Num
instZeroNum
of_natCast 📖mathematicalcastNum
AddMonoidWithOne.toOne
AddSemigroup.toAdd
AddMonoid.toAddSemigroup
AddMonoidWithOne.toAddMonoid
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
Num
AddMonoidWithOne.toNatCast
addMonoidWithOne
cast_to_nat
to_of_nat
of_nat_inj 📖mathematicalNum
AddMonoidWithOne.toNatCast
addMonoidWithOne
to_of_nat
of_to_nat 📖mathematicalNum
AddMonoidWithOne.toNatCast
addMonoidWithOne
castNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
of_to_nat'
of_to_nat' 📖mathematicalofNat'
castNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
ofNat'_zero
PosNum.of_to_nat'
ppred_to_nat 📖mathematicalNum
castNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
ppred
Nat.ppred
ppred.eq_2
Nat.ppred_eq_some
PosNum.pred'_to_nat
PosNum.to_nat_pos
pred_to_nat 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
pred
pred.eq_2
PosNum.pred'_to_nat
size_eq_natSize 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
size
natSize
PosNum.size_eq_natSize
size_to_nat 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
size
Nat.size
Nat.size_zero
PosNum.size_to_nat
succ'_to_nat 📖mathematicalcastPosNum
Nat.instOne
succ'
castNum
MulZeroClass.toZero
Nat.instMulZeroClass
PosNum.succ_to_nat
succ_to_nat 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
succ
succ'_to_nat
toNat_injective 📖mathematicalNum
castNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
of_to_nat'
toZNum_inj 📖mathematicaltoZNum
to_nat_inj 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
toNat_injective
to_nat_to_int 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
AddMonoidWithOne.toOne
AddGroupWithOne.toAddMonoidWithOne
Ring.toAddGroupWithOne
Int.instRing
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Int.instCommRing
cast_to_nat
to_of_nat 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
Num
AddMonoidWithOne.toNatCast
addMonoidWithOne
Nat.cast_zero
cast_zero
Nat.cast_succ
add_one
succ_to_nat
zero_add 📖mathematicalNum
instAdd
instZeroNum
zneg_toZNum 📖mathematicalZNum
ZNum.instNeg
toZNum
toZNumNeg
zneg_toZNumNeg 📖mathematicalZNum
ZNum.instNeg
toZNumNeg
toZNum

PosNum

Definitions

NameCategoryTheorems
addCommSemigroup 📖CompOp
commMonoid 📖CompOp
distrib 📖CompOp
dvd 📖CompOp
1 mathmath: dvd_to_nat
linearOrder 📖CompOp
transfer 📖CompOp
transfer_rw 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
add_one 📖mathematicalPosNum
instAdd
instOnePosNum
succ
add_succ 📖mathematicalPosNum
instAdd
succ
one_add
add_one
add_to_nat 📖mathematicalcastPosNum
Nat.instOne
PosNum
instAdd
one_add
succ_to_nat
add_comm
cast_one
add_one
add_add_add_comm
add_left_comm
add_assoc
bit0_of_bit0 📖mathematicalPosNum
instAdd
bit0
succ.eq_3
bit1_of_bit1 📖mathematicalPosNum
instAdd
instOnePosNum
bit1
add_one
bit0_of_bit0
succ.eq_3
bit_to_nat 📖mathematicalcastPosNum
Nat.instOne
bit
Nat.bit
two_mul
add_assoc
cast_add 📖mathematicalcastPosNum
AddMonoidWithOne.toOne
AddSemigroup.toAdd
AddMonoid.toAddSemigroup
AddMonoidWithOne.toAddMonoid
PosNum
instAdd
cast_to_nat
add_to_nat
Nat.cast_add
cast_bit0 📖mathematicalcastPosNum
bit0
cast_bit1 📖mathematicalcastPosNum
bit1
cast_inj 📖mathematicalcastPosNum
AddMonoidWithOne.toOne
AddSemigroup.toAdd
AddMonoid.toAddSemigroup
AddMonoidWithOne.toAddMonoid
cast_to_nat
cast_le 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
castPosNum
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
PosNum
instLE
not_lt
cast_lt
cast_lt 📖mathematicalPreorder.toLT
PartialOrder.toPreorder
castPosNum
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
PosNum
instLT
cast_to_nat
Nat.cast_lt
IsOrderedAddMonoid.toAddLeftMono
IsOrderedRing.toIsOrderedAddMonoid
IsStrictOrderedRing.toIsOrderedRing
IsStrictOrderedRing.toZeroLEOneClass
IsStrictOrderedRing.toCharZero
lt_to_nat
cast_mul 📖mathematicalcastPosNum
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
PosNum
instMul
Distrib.toMul
cast_to_nat
mul_to_nat
Nat.cast_mul
cast_one 📖mathematicalcastPosNum
PosNum
instOnePosNum
cast_one' 📖mathematicalcastPosNum
one
cast_pos 📖mathematicalPreorder.toLT
PartialOrder.toPreorder
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
castPosNum
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
lt_of_lt_of_le
zero_lt_one
IsStrictOrderedRing.toZeroLEOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
one_le_cast
cast_succ 📖mathematicalcastPosNum
AddMonoidWithOne.toOne
AddSemigroup.toAdd
AddMonoid.toAddSemigroup
AddMonoidWithOne.toAddMonoid
succ
add_one
cast_add
cast_one
cast_to_int 📖mathematicalAddGroupWithOne.toIntCast
castPosNum
AddMonoidWithOne.toOne
AddGroupWithOne.toAddMonoidWithOne
Ring.toAddGroupWithOne
Int.instRing
AddSemigroup.toAdd
AddMonoid.toAddSemigroup
AddMonoidWithOne.toAddMonoid
to_nat_to_int
Int.cast_natCast
cast_to_nat
cast_to_nat 📖mathematicalAddMonoidWithOne.toNatCast
castPosNum
Nat.instOne
AddMonoidWithOne.toOne
AddSemigroup.toAdd
AddMonoid.toAddSemigroup
AddMonoidWithOne.toAddMonoid
Nat.cast_one
Nat.cast_add
cast_to_num 📖mathematicalcastPosNum
Num
instOneNum
Num.instAdd
Num.pos
cast_to_nat
of_to_nat
cmp_eq 📖mathematicalcmpcmp_to_nat
lt_irrefl
cmp_swap 📖mathematicalcmpcmp.eq_def
cmp_to_nat 📖mathematicalcastPosNum
Nat.instOne
PosNum
cmp
to_nat_pos
cmp_to_nat_lemma
cmp_to_nat_lemma 📖mathematicalcastPosNum
Nat.instOne
bit1
bit0
add_assoc
dvd_to_nat 📖mathematicalcastPosNum
Nat.instOne
PosNum
dvd
Num.dvd_to_nat
le_iff_cmp 📖mathematicalPosNum
instLE
lt_iff_cmp
cmp_swap
le_to_nat 📖mathematicalcastPosNum
Nat.instOne
PosNum
instLE
not_lt
lt_to_nat
lt_iff_cmp 📖mathematicalPosNum
instLT
cmp
lt_to_nat 📖mathematicalcastPosNum
Nat.instOne
PosNum
instLT
cmp_to_nat
not_lt_of_gt
mul_to_nat 📖mathematicalcastPosNum
Nat.instOne
PosNum
instMul
mul_one
left_distrib
Distrib.leftDistribClass
add_to_nat
natSize_pos 📖mathematicalnatSize
natSize_to_nat 📖mathematicalnatSize
Nat.size
castPosNum
Nat.instOne
size_eq_natSize
size_to_nat
of_to_nat 📖mathematicalNum
AddMonoidWithOne.toNatCast
Num.addMonoidWithOne
castPosNum
Nat.instOne
Num.pos
of_to_nat'
of_to_nat' 📖mathematicalNum.ofNat'
castPosNum
Nat.instOne
Num.pos
Num.ofNat'_one
two_mul
Num.ofNat'_bit
one_add 📖mathematicalPosNum
instAdd
instOnePosNum
succ
one_le_cast 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
castPosNum
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
cast_to_nat
Nat.cast_one
Nat.cast_le
IsOrderedAddMonoid.toAddLeftMono
IsOrderedRing.toIsOrderedAddMonoid
IsStrictOrderedRing.toIsOrderedRing
IsStrictOrderedRing.toZeroLEOneClass
IsStrictOrderedRing.toCharZero
to_nat_pos
one_sub' 📖mathematicalsub'
PosNum
instOnePosNum
Num.toZNumNeg
pred'
pred'_succ' 📖mathematicalpred'
Num.succ'
pred'_to_nat
Num.succ'_to_nat
pred'_to_nat 📖mathematicalcastNum
Nat.instOne
MulZeroClass.toZero
Nat.instMulZeroClass
pred'
castPosNum
to_nat_pos
pred_to_nat 📖mathematicalPosNum
instLT
instOnePosNum
castPosNum
Nat.instOne
pred
cast_lt
Nat.instIsStrictOrderedRing
pred'_to_nat
size_eq_natSize 📖mathematicalcastPosNum
Nat.instOne
size
natSize
size.eq_2
succ_to_nat
natSize.eq_2
size.eq_3
natSize.eq_3
size_to_nat 📖mathematicalcastPosNum
Nat.instOne
size
Nat.size
Nat.size_one
size.eq_2
succ_to_nat
cast_bit0
Nat.instAtLeastTwoHAddOfNat
two_mul
Nat.bit_false_apply
Nat.size_bit
to_nat_pos
size.eq_3
cast_bit1
Nat.bit_true_apply
sub'_one 📖mathematicalsub'
PosNum
instOnePosNum
Num.toZNum
pred'
succ'_pred' 📖mathematicalNum.succ'
pred'
Num.succ'_to_nat
pred'_to_nat
to_nat_pos
succ_to_nat 📖mathematicalcastPosNum
Nat.instOne
succ
add_assoc
add_left_comm
to_nat_inj 📖mathematicalcastPosNum
Nat.instOne
of_to_nat
to_nat_pos 📖mathematicalcastPosNum
Nat.instOne
add_pos
IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono
AddLeftCancelSemigroup.toIsLeftCancelAdd
IsOrderedAddMonoid.toAddLeftMono
Nat.instIsOrderedAddMonoid
to_nat_to_int 📖mathematicalcastPosNum
Nat.instOne
AddMonoidWithOne.toOne
AddGroupWithOne.toAddMonoidWithOne
Ring.toAddGroupWithOne
Int.instRing
cast_to_nat

SNum

Definitions

NameCategoryTheorems
le 📖CompOp
lt 📖CompOp

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