Documentation Verification Report

Multiplication

πŸ“ Source: Mathlib/SetTheory/Surreal/Multiplication.lean

Statistics

MetricCount
DefinitionsArgs, Numeric, toMultiset, ArgsRel, IH1, IH24, IH3, IH4, MulOptionsLTMul, P1, P124, P2, P24, P3, P4, instCommRing
16
Theoremsmul_congr, mul_congr_left, mul_congr_right, mul, mul_pos, P24, P3_of_lt_of_lt, numeric_P1, numeric_P24, numeric_closed, P1_of_eq, P1_of_ih, P1_of_lt, P24_neg_left, P24_neg_right, P24_of_ih, P2_neg_left, P2_neg_right, trans, P3_comm, P3_neg, P3_of_ih, P3_of_le_left, P3_of_lt, P4_neg_left, P4_neg_right, argsRel_wf, ih1, ih1_neg_left, ih1_neg_right, ih1_swap, ih24_neg, ih3_of_ih, ih4, ih4_neg, ih₁₂, ih₂₁, main, mulOption_lt, mulOption_lt_iff_P1, mulOption_lt_mul_iff_P3, mulOption_lt_mul_of_equiv, mulOption_lt_of_lt, mulOptionsLTMul_of_numeric, mul_right_le_of_equiv, numeric_mul_option, numeric_of_ih, numeric_option_mul, numeric_option_mul_option, instIsStrictOrderedRing, instNontrivial, instZeroLEOneClass
52
Total68

SetTheory.PGame

Theorems

NameKindAssumesProvesValidatesDepends On
P24 πŸ“–mathematicalNumericSurreal.Multiplication.P24β€”Surreal.Multiplication.main
Surreal.Multiplication.Args.numeric_P24
P3_of_lt_of_lt πŸ“–mathematicalNumeric
SetTheory.PGame
Preorder.toLT
instPreorder
Surreal.Multiplication.P3β€”Prod.forall'
WellFounded.prod_gameAdd
wf_isOption
Surreal.Multiplication.P3_of_lt
Numeric.moveLeft
P24
Surreal.Multiplication.P3_comm
Numeric.neg
moveLeft_neg
Surreal.Multiplication.P3_neg
neg_lt_neg_iff
Numeric.moveRight

SetTheory.PGame.Equiv

Theorems

NameKindAssumesProvesValidatesDepends On
mul_congr πŸ“–mathematicalSetTheory.PGame.Numeric
SetTheory.PGame
SetTheory.PGame.setoid
SetTheory.PGame.instMulβ€”trans
mul_congr_left
mul_congr_right
mul_congr_left πŸ“–mathematicalSetTheory.PGame.Numeric
SetTheory.PGame
SetTheory.PGame.setoid
SetTheory.PGame.instMulβ€”SetTheory.PGame.equiv_iff_game_eq
SetTheory.PGame.P24
mul_congr_right πŸ“–mathematicalSetTheory.PGame.Numeric
SetTheory.PGame
SetTheory.PGame.setoid
SetTheory.PGame.instMulβ€”trans
SetTheory.PGame.mul_comm_equiv
mul_congr_left

SetTheory.PGame.Numeric

Theorems

NameKindAssumesProvesValidatesDepends On
mul πŸ“–mathematicalSetTheory.PGame.NumericSetTheory.PGame
SetTheory.PGame.instMul
β€”Surreal.Multiplication.main
Surreal.Multiplication.Args.numeric_P1
mul_pos πŸ“–mathematicalSetTheory.PGame.Numeric
SetTheory.PGame
Preorder.toLT
SetTheory.PGame.instPreorder
SetTheory.PGame.instZero
SetTheory.PGame.instMulβ€”SetTheory.PGame.lt_iff_game_lt
SetTheory.PGame.P3_of_lt_of_lt
SetTheory.PGame.numeric_zero
SetTheory.Game.addLeftStrictMono
IsOrderedCancelAddMonoid.toAddLeftReflectLT
IsOrderedAddMonoid.toIsOrderedCancelAddMonoid
SetTheory.Game.isOrderedAddMonoid
SetTheory.PGame.quot_mul_zero
SetTheory.PGame.quot_zero_mul

Surreal

Definitions

NameCategoryTheorems
instCommRing πŸ“–CompOp
9 mathmath: nsmul_pow_two_powHalf, zsmul_pow_two_powHalf, dyadicMap_apply_pow', instIsStrictOrderedRing, dyadicMap_apply, dyadic_aux, double_powHalf_succ_eq_powHalf, dyadicMap_apply_pow, nsmul_pow_two_powHalf'

Theorems

NameKindAssumesProvesValidatesDepends On
instIsStrictOrderedRing πŸ“–mathematicalβ€”IsStrictOrderedRing
Surreal
CommSemiring.toSemiring
CommRing.toCommSemiring
instCommRing
partialOrder
β€”IsStrictOrderedRing.of_mul_pos
isOrderedAddMonoid
instZeroLEOneClass
instNontrivial
SetTheory.PGame.Numeric.mul_pos
instNontrivial πŸ“–mathematicalβ€”Nontrivial
Surreal
β€”ne_of_lt
SetTheory.PGame.zero_lt_one
instZeroLEOneClass πŸ“–mathematicalβ€”ZeroLEOneClass
Surreal
instZero
instOne
instLE
β€”LT.lt.le
SetTheory.PGame.zero_lt_one

Surreal.Multiplication

Definitions

NameCategoryTheorems
Args πŸ“–CompData
1 mathmath: argsRel_wf
ArgsRel πŸ“–MathDef
1 mathmath: argsRel_wf
IH1 πŸ“–MathDef
2 mathmath: ih1_swap, ih1
IH24 πŸ“–MathDef
2 mathmath: ih₁₂, ih₂₁
IH3 πŸ“–MathDef
1 mathmath: ih3_of_ih
IH4 πŸ“–MathDef
1 mathmath: ih4
MulOptionsLTMul πŸ“–MathDef
1 mathmath: mulOptionsLTMul_of_numeric
P1 πŸ“–MathDef
3 mathmath: P1_of_eq, mulOption_lt_iff_P1, P1_of_lt
P124 πŸ“–MathDef
1 mathmath: main
P2 πŸ“–MathDef
2 mathmath: P2_neg_left, P2_neg_right
P24 πŸ“–MathDef
4 mathmath: P24_neg_right, P24_neg_left, P24_of_ih, SetTheory.PGame.P24
P3 πŸ“–MathDef
7 mathmath: mulOption_lt_mul_iff_P3, P3_of_lt, P3_comm, SetTheory.PGame.P3_of_lt_of_lt, P3_of_ih, P3_of_le_left, P3_neg
P4 πŸ“–MathDef
2 mathmath: P4_neg_left, P4_neg_right

Theorems

NameKindAssumesProvesValidatesDepends On
P1_of_eq πŸ“–mathematicalSetTheory.PGame
SetTheory.PGame.setoid
P2
P3
P1β€”P1.eq_1
sub_lt_sub_iff
SetTheory.Game.addLeftStrictMono
Mathlib.Tactic.Abel.subst_into_addg
Mathlib.Tactic.Abel.term_atomg
Mathlib.Tactic.Abel.term_add_constg
zero_add
Mathlib.Tactic.Abel.const_add_termg
add_zero
add_lt_add_left
SetTheory.Game.addRightStrictMono
P1_of_ih πŸ“–mathematicalP124
SetTheory.PGame.Numeric
SetTheory.PGame
SetTheory.PGame.instMul
β€”ih1
ih1_swap
ih1_neg_left
ih1_neg_right
SetTheory.PGame.numeric_def
SetTheory.PGame.rightMoves_mul_iff
SetTheory.PGame.leftMoves_mul_iff
mulOption_lt
SetTheory.PGame.mulOption_symm
SetTheory.PGame.mulOption_neg_neg
SetTheory.PGame.Numeric.neg
SetTheory.PGame.Numeric.sub
SetTheory.PGame.Numeric.add
numeric_option_mul
SetTheory.PGame.IsOption.mk_left
numeric_mul_option
numeric_option_mul_option
SetTheory.PGame.IsOption.mk_right
P1_of_lt πŸ“–mathematicalP3P1β€”P1.eq_1
sub_lt_sub_iff
SetTheory.Game.addLeftStrictMono
add_lt_add_iff_left
IsOrderedCancelAddMonoid.toAddLeftReflectLT
IsOrderedAddMonoid.toIsOrderedCancelAddMonoid
SetTheory.Game.isOrderedAddMonoid
Mathlib.Tactic.Abel.subst_into_addg
Mathlib.Tactic.Abel.term_atomg
Mathlib.Tactic.Abel.term_add_constg
zero_add
Mathlib.Tactic.Abel.const_add_termg
add_zero
add_lt_add
SetTheory.Game.addRightStrictMono
P24_neg_left πŸ“–mathematicalβ€”P24
SetTheory.PGame
SetTheory.PGame.instNeg
β€”P24.eq_1
P2_neg_left
P4_neg_left
P24_neg_right πŸ“–mathematicalβ€”P24
SetTheory.PGame
SetTheory.PGame.instNeg
β€”P24.eq_1
P2_neg_right
P4_neg_right
P24_of_ih πŸ“–mathematicalIH1P24
SetTheory.PGame.moveLeft
β€”β€”
P2_neg_left πŸ“–mathematicalβ€”P2
SetTheory.PGame
SetTheory.PGame.instNeg
β€”P2.eq_1
SetTheory.PGame.quot_neg_mul
SetTheory.PGame.neg_equiv_neg_iff
SetTheory.PGame.equiv_comm
P2_neg_right πŸ“–mathematicalβ€”P2
SetTheory.PGame
SetTheory.PGame.instNeg
β€”P2.eq_1
SetTheory.PGame.quot_mul_neg
P3_comm πŸ“–mathematicalβ€”P3β€”P3.eq_1
add_comm
SetTheory.PGame.quot_mul_comm
P3_neg πŸ“–mathematicalβ€”P3
SetTheory.PGame
SetTheory.PGame.instNeg
β€”SetTheory.PGame.quot_neg_mul
neg_lt_neg_iff
SetTheory.Game.addLeftStrictMono
SetTheory.Game.addRightStrictMono
Mathlib.Tactic.Abel.subst_into_negg
Mathlib.Tactic.Abel.subst_into_addg
Mathlib.Tactic.Abel.term_atomg
Mathlib.Tactic.Abel.term_add_constg
zero_add
Mathlib.Tactic.Abel.term_neg
neg_zero
Mathlib.Tactic.Abel.termg_eq
add_zero
Mathlib.Tactic.Abel.const_add_termg
P3_of_ih πŸ“–mathematicalSetTheory.PGame.Numeric
IH1
P3
SetTheory.PGame.moveLeft
SetTheory.PGame
SetTheory.PGame.instNeg
β€”P3_comm
SetTheory.PGame.isOption_neg
SetTheory.PGame.moveLeft_neg
neg_neg
SetTheory.PGame.Numeric.left_lt_right
P3_of_le_left πŸ“–mathematicalIH3
SetTheory.PGame.moveLeft
SetTheory.PGame
SetTheory.PGame.le
P3β€”SetTheory.PGame.lt_or_equiv_of_le
P3.trans
P3.eq_1
P3_of_lt πŸ“–mathematicalIH3
SetTheory.PGame.moveLeft
SetTheory.PGame
SetTheory.PGame.instNeg
Preorder.toLT
SetTheory.PGame.instPreorder
P3β€”SetTheory.PGame.lf_iff_exists_le
SetTheory.PGame.lf_of_lt
P3_of_le_left
P3_neg
SetTheory.PGame.moveLeft_neg
Equiv.symm_apply_apply
P4_neg_left πŸ“–mathematicalβ€”P4
SetTheory.PGame
SetTheory.PGame.instNeg
β€”SetTheory.PGame.moveLeft_neg
P4_neg_right πŸ“–mathematicalβ€”P4
SetTheory.PGame
SetTheory.PGame.instNeg
β€”P4.eq_1
neg_neg
argsRel_wf πŸ“–mathematicalβ€”Args
ArgsRel
β€”WellFounded.cutExpand
SetTheory.PGame.wf_isOption
ih1 πŸ“–mathematicalP124IH1β€”Relation.cutExpand_double_left
Relation.cutExpand_pair_right
ih1_neg_left πŸ“–mathematicalIH1SetTheory.PGame
SetTheory.PGame.instNeg
β€”P24_neg_left
SetTheory.PGame.isOption_neg
ih1_neg_right πŸ“–mathematicalIH1SetTheory.PGame
SetTheory.PGame.instNeg
β€”neg_eq_iff_eq_neg
SetTheory.PGame.isOption_neg
P24_neg_right
ih1_swap πŸ“–mathematicalP124IH1β€”ih1
Multiset.pair_comm
ih24_neg πŸ“–mathematicalIH24SetTheory.PGame
SetTheory.PGame.instNeg
β€”P24_neg_left
neg_neg
P24_neg_right
ih3_of_ih πŸ“–mathematicalIH24
IH4
MulOptionsLTMul
IH3
SetTheory.PGame.moveLeft
β€”mulOption_lt_mul_iff_P3
ih4 πŸ“–mathematicalP124IH4β€”Relation.cutExpand_add_right
Relation.cutExpand_pair_left
Relation.cutExpand_add_left
Relation.cutExpand_pair_right
ih4_neg πŸ“–mathematicalIH4SetTheory.PGame
SetTheory.PGame.instNeg
β€”P2_neg_left
neg_neg
P2_neg_right
ih₁₂ πŸ“–mathematicalP124IH24β€”IH24.eq_1
Relation.cutExpand_add_right
Relation.cutExpand_pair_left
Relation.cutExpand_add_left
Relation.cutExpand_pair_right
ih₂₁ πŸ“–mathematicalP124IH24β€”ih₁₂
Multiset.pair_comm
Mathlib.Tactic.Abel.subst_into_add
Mathlib.Tactic.Abel.term_atom
Mathlib.Tactic.Abel.term_add_const
zero_add
Mathlib.Tactic.Abel.const_add_term
add_zero
main πŸ“–mathematicalArgs.NumericP124β€”argsRel_wf
ArgsRel.numeric_closed
P1_of_ih
Args.numeric_P1
ih₁₂
ih₂₁
ih4
ih24_neg
ih4_neg
mul_right_le_of_equiv
Args.numeric_P24
symm
IsEquiv.toSymm
Quotient.instIsEquivEquiv
numeric_of_ih
mulOptionsLTMul_of_numeric
P3_of_lt
ih3_of_ih
mulOption_lt πŸ“–mathematicalSetTheory.PGame.Numeric
IH1
SetTheory.PGame
SetTheory.PGame.setoid
Preorder.toLT
PartialOrder.toPreorder
SetTheory.Game.instPartialOrderGame
SetTheory.PGame.mulOption
SetTheory.Game.instNeg
SetTheory.PGame.instNeg
β€”SetTheory.PGame.lt_or_equiv_or_gt
SetTheory.PGame.Numeric.moveLeft
mulOption_lt_of_lt
mulOption_lt_iff_P1
P1_of_eq
P24_of_ih
SetTheory.PGame.isOption_neg
P3_of_ih
SetTheory.PGame.mulOption_neg_neg
lt_neg
SetTheory.Game.addLeftStrictMono
SetTheory.Game.addRightStrictMono
SetTheory.PGame.Numeric.neg
ih1_neg_right
ih1_neg_left
mulOption_lt_iff_P1 πŸ“–mathematicalβ€”SetTheory.PGame
SetTheory.PGame.setoid
Preorder.toLT
PartialOrder.toPreorder
SetTheory.Game.instPartialOrderGame
SetTheory.PGame.mulOption
SetTheory.Game.instNeg
SetTheory.PGame.instNeg
P1
SetTheory.PGame.moveLeft
β€”neg_sub'
neg_add
SetTheory.PGame.quot_mul_neg
neg_neg
mulOption_lt_mul_iff_P3 πŸ“–mathematicalβ€”SetTheory.PGame
SetTheory.PGame.setoid
Preorder.toLT
PartialOrder.toPreorder
SetTheory.Game.instPartialOrderGame
SetTheory.PGame.mulOption
SetTheory.PGame.instMul
P3
SetTheory.PGame.moveLeft
β€”sub_lt_iff_lt_add'
SetTheory.Game.addLeftStrictMono
mulOption_lt_mul_of_equiv πŸ“–mathematicalSetTheory.PGame.Numeric
IH24
SetTheory.PGame
SetTheory.PGame.setoid
Preorder.toLT
PartialOrder.toPreorder
SetTheory.Game.instPartialOrderGame
SetTheory.PGame.mulOption
SetTheory.PGame.instMul
β€”sub_lt_iff_lt_add'
SetTheory.Game.addLeftStrictMono
SetTheory.PGame.lt_congr_right
SetTheory.PGame.Numeric.moveLeft_lt
mulOption_lt_of_lt πŸ“–mathematicalSetTheory.PGame.Numeric
IH1
SetTheory.PGame
Preorder.toLT
SetTheory.PGame.instPreorder
SetTheory.PGame.moveLeft
SetTheory.PGame.setoid
PartialOrder.toPreorder
SetTheory.Game.instPartialOrderGame
SetTheory.PGame.mulOption
SetTheory.Game.instNeg
SetTheory.PGame.instNeg
β€”mulOption_lt_iff_P1
P1_of_lt
P3_of_ih
P24_of_ih
mulOptionsLTMul_of_numeric πŸ“–mathematicalSetTheory.PGame.Numeric
SetTheory.PGame
SetTheory.PGame.instMul
MulOptionsLTMul
SetTheory.PGame.instNeg
β€”SetTheory.PGame.Numeric.moveLeft_lt
SetTheory.PGame.quot_neg_mul_neg
SetTheory.PGame.leftMoves_mul_iff
SetTheory.PGame.Numeric.lt_moveRight
forallβ‚‚_imp
lt_neg
SetTheory.Game.addLeftStrictMono
SetTheory.Game.addRightStrictMono
SetTheory.PGame.quot_mul_neg
SetTheory.PGame.quot_neg_mul
mul_right_le_of_equiv πŸ“–mathematicalSetTheory.PGame.Numeric
IH24
SetTheory.PGame
SetTheory.PGame.setoid
SetTheory.PGame.le
SetTheory.PGame.instMul
β€”SetTheory.PGame.neg_equiv_neg_iff
SetTheory.PGame.le_of_forall_lt
SetTheory.PGame.leftMoves_mul_iff
mulOption_lt_mul_of_equiv
SetTheory.PGame.quot_neg_mul_neg
SetTheory.PGame.Numeric.neg
ih24_neg
SetTheory.PGame.rightMoves_mul_iff
lt_neg
SetTheory.Game.addLeftStrictMono
SetTheory.Game.addRightStrictMono
SetTheory.PGame.quot_mul_neg
symm
IsEquiv.toSymm
Quotient.instIsEquivEquiv
SetTheory.PGame.quot_neg_mul
numeric_mul_option πŸ“–mathematicalP124
SetTheory.PGame.IsOption
SetTheory.PGame.Numeric
SetTheory.PGame
SetTheory.PGame.instMul
β€”Relation.cutExpand_pair_right
numeric_of_ih πŸ“–mathematicalP124SetTheory.PGame.Numeric
SetTheory.PGame
SetTheory.PGame.instMul
β€”Relation.cutExpand_add_right
Relation.cutExpand_add_left
Relation.cutExpand_zero
numeric_option_mul πŸ“–mathematicalP124
SetTheory.PGame.IsOption
SetTheory.PGame.Numeric
SetTheory.PGame
SetTheory.PGame.instMul
β€”Relation.cutExpand_pair_left
numeric_option_mul_option πŸ“–mathematicalP124
SetTheory.PGame.IsOption
SetTheory.PGame.Numeric
SetTheory.PGame
SetTheory.PGame.instMul
β€”Relation.cutExpand_pair_right
Relation.cutExpand_pair_left

Surreal.Multiplication.Args

Definitions

NameCategoryTheorems
Numeric πŸ“–MathDef
2 mathmath: numeric_P24, numeric_P1
toMultiset πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
numeric_P1 πŸ“–mathematicalβ€”Numeric
P1
SetTheory.PGame.Numeric
β€”β€”
numeric_P24 πŸ“–mathematicalβ€”Numeric
P24
SetTheory.PGame.Numeric
β€”β€”

Surreal.Multiplication.ArgsRel

Theorems

NameKindAssumesProvesValidatesDepends On
numeric_closed πŸ“–β€”Surreal.Multiplication.ArgsRel
Surreal.Multiplication.Args.Numeric
β€”β€”Relation.TransGen.closed'
Relation.cutExpand_closed
WellFounded.irrefl
SetTheory.PGame.wf_isOption
SetTheory.PGame.Numeric.isOption

Surreal.Multiplication.P3

Theorems

NameKindAssumesProvesValidatesDepends On
trans πŸ“–β€”Surreal.Multiplication.P3β€”β€”eq_1
add_lt_add_iff_left
SetTheory.Game.addLeftStrictMono
IsOrderedCancelAddMonoid.toAddLeftReflectLT
IsOrderedAddMonoid.toIsOrderedCancelAddMonoid
SetTheory.Game.isOrderedAddMonoid
Mathlib.Tactic.Abel.subst_into_addg
Mathlib.Tactic.Abel.term_atomg
Mathlib.Tactic.Abel.term_add_constg
zero_add
Mathlib.Tactic.Abel.const_add_termg
add_zero
add_lt_add
SetTheory.Game.addRightStrictMono

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