statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
nfp_bfamily_eq_self {a} (h : ∀ i hi, f i hi a = a) : nfp_bfamily o f a = a | nfp_family_eq_self (λ _, h _ _) | theorem | ordinal.nfp_bfamily_eq_self | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fp_bfamily_unbounded (H : ∀ i hi, is_normal (f i hi)) :
(⋂ i hi, function.fixed_points (f i hi)).unbounded (<) | λ a, ⟨_, by { rw set.mem_Inter₂, exact λ i hi, nfp_bfamily_fp (H i hi) _ },
(le_nfp_bfamily f a).not_lt⟩ | theorem | ordinal.fp_bfamily_unbounded | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"function.fixed_points",
"set.mem_Inter₂"
] | A generalization of the fixed point lemma for normal functions: any family of normal functions
has an unbounded set of common fixed points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
deriv_bfamily (o : ordinal) (f : Π b < o, ordinal → ordinal) : ordinal → ordinal | deriv_family (family_of_bfamily o f) | def | ordinal.deriv_bfamily | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | The derivative of a family of normal functions is the sequence of their common fixed points.
This is defined as `ordinal.deriv_family` of the type-indexed family associated to `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
deriv_bfamily_eq_deriv_family {o : ordinal} (f : Π b < o, ordinal → ordinal) :
deriv_bfamily o f = deriv_family (family_of_bfamily o f) | rfl | theorem | ordinal.deriv_bfamily_eq_deriv_family | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_bfamily_is_normal {o : ordinal} (f : Π b < o, ordinal → ordinal) :
is_normal (deriv_bfamily o f) | deriv_family_is_normal _ | theorem | ordinal.deriv_bfamily_is_normal | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_bfamily_fp {i hi} (H : is_normal (f i hi)) (a : ordinal) :
f i hi (deriv_bfamily o f a) = deriv_bfamily o f a | by { rw ←family_of_bfamily_enum o f, apply deriv_family_fp, rw family_of_bfamily_enum, exact H } | theorem | ordinal.deriv_bfamily_fp | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_deriv_bfamily (H : ∀ i hi, is_normal (f i hi)) {a} :
(∀ i hi, f i hi a ≤ a) ↔ ∃ b, deriv_bfamily o f b = a | begin
unfold deriv_bfamily,
rw ←le_iff_deriv_family,
{ refine ⟨λ h i, h _ _, λ h i hi, _⟩,
rw ←family_of_bfamily_enum o f,
apply h },
{ exact λ _, H _ _ }
end | theorem | ordinal.le_iff_deriv_bfamily | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fp_iff_deriv_bfamily (H : ∀ i hi, is_normal (f i hi)) {a} :
(∀ i hi, f i hi a = a) ↔ ∃ b, deriv_bfamily o f b = a | begin
rw ←le_iff_deriv_bfamily H,
refine ⟨λ h i hi, le_of_eq (h i hi), λ h i hi, _⟩,
rw ←(H i hi).le_iff_eq,
exact h i hi
end | theorem | ordinal.fp_iff_deriv_bfamily | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_bfamily_eq_enum_ord (H : ∀ i hi, is_normal (f i hi)) :
deriv_bfamily o f = enum_ord (⋂ i hi, function.fixed_points (f i hi)) | begin
rw ←eq_enum_ord _ (fp_bfamily_unbounded H),
use (deriv_bfamily_is_normal f).strict_mono,
rw set.range_eq_iff,
refine ⟨λ a, set.mem_Inter₂.2 (λ i hi, deriv_bfamily_fp (H i hi) a), λ a ha, _⟩,
rw set.mem_Inter₂ at ha,
rwa ←fp_iff_deriv_bfamily H
end | theorem | ordinal.deriv_bfamily_eq_enum_ord | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"function.fixed_points",
"set.mem_Inter₂",
"set.range_eq_iff",
"strict_mono"
] | For a family of normal functions, `ordinal.deriv_bfamily` enumerates the common fixed points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nfp (f : ordinal → ordinal) : ordinal → ordinal | nfp_family (λ _ : unit, f) | def | ordinal.nfp | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | The next fixed point function, the least fixed point of the normal function `f`, at least `a`.
This is defined as `ordinal.nfp_family` applied to a family consisting only of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nfp_eq_nfp_family (f : ordinal → ordinal) : nfp f = nfp_family (λ _ : unit, f) | rfl | theorem | ordinal.nfp_eq_nfp_family | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_iterate_eq_nfp (f : ordinal.{u} → ordinal.{u}) :
(λ a, sup (λ n : ℕ, f^[n] a)) = nfp f | begin
refine funext (λ a, le_antisymm _ (sup_le (λ l, _))),
{ rw sup_le_iff,
intro n,
rw [←list.length_replicate n unit.star, ←list.foldr_const f a],
apply le_sup },
{ rw list.foldr_const f a l,
exact le_sup _ _ },
end | theorem | ordinal.sup_iterate_eq_nfp | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"list.foldr_const",
"sup_le",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_le_nfp (f a n) : f^[n] a ≤ nfp f a | by { rw ←sup_iterate_eq_nfp, exact le_sup _ n } | theorem | ordinal.iterate_le_nfp | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_nfp (f a) : a ≤ nfp f a | iterate_le_nfp f a 0 | theorem | ordinal.le_nfp | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_nfp {a b} : a < nfp f b ↔ ∃ n, a < (f^[n]) b | by { rw ←sup_iterate_eq_nfp, exact lt_sup } | theorem | ordinal.lt_nfp | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_le_iff {a b} : nfp f a ≤ b ↔ ∀ n, (f^[n]) a ≤ b | by { rw ←sup_iterate_eq_nfp, exact sup_le_iff } | theorem | ordinal.nfp_le_iff | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_le {a b} : (∀ n, (f^[n]) a ≤ b) → nfp f a ≤ b | nfp_le_iff.2 | theorem | ordinal.nfp_le | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_id : nfp id = id | funext (λ a, begin
simp_rw [←sup_iterate_eq_nfp, iterate_id],
exact sup_const a
end) | theorem | ordinal.nfp_id | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_monotone (hf : monotone f) : monotone (nfp f) | nfp_family_monotone (λ i, hf) | theorem | ordinal.nfp_monotone | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.apply_lt_nfp {f} (H : is_normal f) {a b} :
f b < nfp f a ↔ b < nfp f a | begin
unfold nfp,
rw ←@apply_lt_nfp_family_iff unit (λ _, f) _ (λ _, H) a b,
exact ⟨λ h _, h, λ h, h unit.star⟩
end | theorem | ordinal.is_normal.apply_lt_nfp | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.nfp_le_apply {f} (H : is_normal f) {a b} : nfp f a ≤ f b ↔ nfp f a ≤ b | le_iff_le_iff_lt_iff_lt.2 H.apply_lt_nfp | theorem | ordinal.is_normal.nfp_le_apply | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_le_fp {f} (H : monotone f) {a b} (ab : a ≤ b) (h : f b ≤ b) : nfp f a ≤ b | nfp_family_le_fp (λ _, H) ab (λ _, h) | theorem | ordinal.nfp_le_fp | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.nfp_fp {f} (H : is_normal f) : ∀ a, f (nfp f a) = nfp f a | @nfp_family_fp unit (λ _, f) unit.star H | theorem | ordinal.is_normal.nfp_fp | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.apply_le_nfp {f} (H : is_normal f) {a b} :
f b ≤ nfp f a ↔ b ≤ nfp f a | ⟨le_trans (H.self_le _), λ h, by simpa only [H.nfp_fp] using H.le_iff.2 h⟩ | theorem | ordinal.is_normal.apply_le_nfp | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_eq_self {f : ordinal → ordinal} {a} (h : f a = a) : nfp f a = a | nfp_family_eq_self (λ _, h) | theorem | ordinal.nfp_eq_self | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fp_unbounded (H : is_normal f) : (function.fixed_points f).unbounded (<) | by { convert fp_family_unbounded (λ _ : unit, H), exact (set.Inter_const _).symm } | theorem | ordinal.fp_unbounded | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"function.fixed_points",
"set.Inter_const"
] | The fixed point lemma for normal functions: any normal function has an unbounded set of
fixed points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
deriv (f : ordinal → ordinal) : ordinal → ordinal | deriv_family (λ _ : unit, f) | def | ordinal.deriv | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"deriv",
"ordinal"
] | The derivative of a normal function `f` is the sequence of fixed points of `f`.
This is defined as `ordinal.deriv_family` applied to a trivial family consisting only of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
deriv_eq_deriv_family (f : ordinal → ordinal) : deriv f = deriv_family (λ _ : unit, f) | rfl | theorem | ordinal.deriv_eq_deriv_family | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"deriv",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_zero (f) : deriv f 0 = nfp f 0 | deriv_family_zero _ | theorem | ordinal.deriv_zero | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_succ (f o) : deriv f (succ o) = nfp f (succ (deriv f o)) | deriv_family_succ _ _ | theorem | ordinal.deriv_succ | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_limit (f) {o} : is_limit o → deriv f o = bsup.{u 0} o (λ a _, deriv f a) | deriv_family_limit _ | theorem | ordinal.deriv_limit | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_is_normal (f) : is_normal (deriv f) | deriv_family_is_normal _ | theorem | ordinal.deriv_is_normal | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_id_of_nfp_id {f : ordinal → ordinal} (h : nfp f = id) : deriv f = id | ((deriv_is_normal _).eq_iff_zero_and_succ is_normal.refl).2 (by simp [h]) | theorem | ordinal.deriv_id_of_nfp_id | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"deriv",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.deriv_fp {f} (H : is_normal f) : ∀ o, f (deriv f o) = deriv f o | @deriv_family_fp unit (λ _, f) unit.star H | theorem | ordinal.is_normal.deriv_fp | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.le_iff_deriv {f} (H : is_normal f) {a} : f a ≤ a ↔ ∃ o, deriv f o = a | begin
unfold deriv,
rw ←le_iff_deriv_family (λ _ : unit, H),
exact ⟨λ h _, h, λ h, h unit.star⟩
end | theorem | ordinal.is_normal.le_iff_deriv | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.fp_iff_deriv {f} (H : is_normal f) {a} : f a = a ↔ ∃ o, deriv f o = a | by rw [←H.le_iff_eq, H.le_iff_deriv] | theorem | ordinal.is_normal.fp_iff_deriv | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_eq_enum_ord (H : is_normal f) : deriv f = enum_ord (function.fixed_points f) | by { convert deriv_family_eq_enum_ord (λ _ : unit, H), exact (set.Inter_const _).symm } | theorem | ordinal.deriv_eq_enum_ord | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"deriv",
"function.fixed_points",
"set.Inter_const"
] | `ordinal.deriv` enumerates the fixed points of a normal function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
deriv_eq_id_of_nfp_eq_id {f : ordinal → ordinal} (h : nfp f = id) : deriv f = id | (is_normal.eq_iff_zero_and_succ (deriv_is_normal _) is_normal.refl).2 $ by simp [h] | theorem | ordinal.deriv_eq_id_of_nfp_eq_id | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"deriv",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_add_zero (a) : nfp ((+) a) 0 = a * omega | begin
simp_rw [←sup_iterate_eq_nfp, ←sup_mul_nat],
congr, funext,
induction n with n hn,
{ rw [nat.cast_zero, mul_zero, iterate_zero_apply] },
{ nth_rewrite 1 nat.succ_eq_one_add,
rw [nat.cast_add, nat.cast_one, mul_one_add, iterate_succ_apply', hn] }
end | theorem | ordinal.nfp_add_zero | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"mul_one_add",
"mul_zero",
"nat.cast_add",
"nat.cast_one",
"nat.cast_zero",
"nat.succ_eq_one_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_add_eq_mul_omega {a b} (hba : b ≤ a * omega) :
nfp ((+) a) b = a * omega | begin
apply le_antisymm (nfp_le_fp (add_is_normal a).monotone hba _),
{ rw ←nfp_add_zero,
exact nfp_monotone (add_is_normal a).monotone (ordinal.zero_le b) },
{ rw [←mul_one_add, one_add_omega] }
end | theorem | ordinal.nfp_add_eq_mul_omega | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"monotone",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_right_iff_mul_omega_le {a b : ordinal} : a + b = b ↔ a * omega ≤ b | begin
refine ⟨λ h, _, λ h, _⟩,
{ rw [←nfp_add_zero a, ←deriv_zero],
cases (add_is_normal a).fp_iff_deriv.1 h with c hc,
rw ←hc,
exact (deriv_is_normal _).monotone (ordinal.zero_le _) },
{ have := ordinal.add_sub_cancel_of_le h,
nth_rewrite 0 ←this,
rwa [←add_assoc, ←mul_one_add, one_add_omega]... | theorem | ordinal.add_eq_right_iff_mul_omega_le | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"monotone",
"ordinal",
"ordinal.add_sub_cancel_of_le",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_right_iff_mul_omega_le {a b : ordinal} : a + b ≤ b ↔ a * omega ≤ b | by { rw ←add_eq_right_iff_mul_omega_le, exact (add_is_normal a).le_iff_eq } | theorem | ordinal.add_le_right_iff_mul_omega_le | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_add_eq_mul_omega_add (a b : ordinal.{u}) : deriv ((+) a) b = a * omega + b | begin
revert b,
rw [←funext_iff, is_normal.eq_iff_zero_and_succ (deriv_is_normal _) (add_is_normal _)],
refine ⟨_, λ a h, _⟩,
{ rw [deriv_zero, add_zero],
exact nfp_add_zero a },
{ rw [deriv_succ, h, add_succ],
exact nfp_eq_self (add_eq_right_iff_mul_omega_le.2 ((le_add_right _ _).trans (le_succ _))) ... | theorem | ordinal.deriv_add_eq_mul_omega_add | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_mul_one {a : ordinal} (ha : 0 < a) : nfp ((*) a) 1 = a ^ omega | begin
rw [←sup_iterate_eq_nfp, ←sup_opow_nat],
{ dsimp, congr, funext,
induction n with n hn,
{ rw [nat.cast_zero, opow_zero, iterate_zero_apply] },
nth_rewrite 1 nat.succ_eq_one_add,
rw [nat.cast_add, nat.cast_one, opow_add, opow_one, iterate_succ_apply', hn] },
{ exact ha }
end | theorem | ordinal.nfp_mul_one | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"nat.cast_add",
"nat.cast_one",
"nat.cast_zero",
"nat.succ_eq_one_add",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_mul_zero (a : ordinal) : nfp ((*) a) 0 = 0 | begin
rw [←ordinal.le_zero, nfp_le_iff],
intro n,
induction n with n hn, { refl },
rwa [iterate_succ_apply, mul_zero]
end | theorem | ordinal.nfp_mul_zero | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"mul_zero",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_zero_mul : nfp ((*) 0) = id | begin
rw ←sup_iterate_eq_nfp,
refine funext (λ a, (sup_le (λ n, _)).antisymm (le_sup (λ n, ((*) 0)^[n] a) 0)),
induction n with n hn, { refl },
rw function.iterate_succ',
change 0 * _ ≤ a,
rw zero_mul,
exact ordinal.zero_le a
end | theorem | ordinal.nfp_zero_mul | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"function.iterate_succ'",
"ordinal.zero_le",
"sup_le",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_mul_zero : deriv ((*) 0) = id | deriv_eq_id_of_nfp_eq_id nfp_zero_mul | theorem | ordinal.deriv_mul_zero | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_mul_eq_opow_omega {a b : ordinal} (hb : 0 < b) (hba : b ≤ a ^ omega) :
nfp ((*) a) b = a ^ omega.{u} | begin
cases eq_zero_or_pos a with ha ha,
{ rw [ha, zero_opow omega_ne_zero] at *,
rw [ordinal.le_zero.1 hba, nfp_zero_mul],
refl },
apply le_antisymm,
{ apply nfp_le_fp (mul_is_normal ha).monotone hba,
rw [←opow_one_add, one_add_omega] },
rw ←nfp_mul_one ha,
exact nfp_monotone (mul_is_normal ha)... | theorem | ordinal.nfp_mul_eq_opow_omega | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"monotone",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_or_opow_omega_le_of_mul_eq_right {a b : ordinal} (hab : a * b = b) :
b = 0 ∨ a ^ omega.{u} ≤ b | begin
cases eq_zero_or_pos a with ha ha,
{ rw [ha, zero_opow omega_ne_zero],
exact or.inr (ordinal.zero_le b) },
rw or_iff_not_imp_left,
intro hb,
change b ≠ 0 at hb,
rw ←nfp_mul_one ha,
rw ←one_le_iff_ne_zero at hb,
exact nfp_le_fp (mul_is_normal ha).monotone hb (le_of_eq hab)
end | theorem | ordinal.eq_zero_or_opow_omega_le_of_mul_eq_right | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"monotone",
"or_iff_not_imp_left",
"ordinal",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_right_iff_opow_omega_dvd {a b : ordinal} : a * b = b ↔ a ^ omega ∣ b | begin
cases eq_zero_or_pos a with ha ha,
{ rw [ha, zero_mul, zero_opow omega_ne_zero, zero_dvd_iff],
exact eq_comm },
refine ⟨λ hab, _, λ h, _⟩,
{ rw dvd_iff_mod_eq_zero,
rw [←div_add_mod b (a ^ omega), mul_add, ←mul_assoc, ←opow_one_add, one_add_omega,
add_left_cancel] at hab,
cases eq_zero_o... | theorem | ordinal.mul_eq_right_iff_opow_omega_dvd | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"not_lt_of_le",
"ordinal",
"zero_dvd_iff",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_right_iff_opow_omega_dvd {a b : ordinal} (ha : 0 < a) : a * b ≤ b ↔ a ^ omega ∣ b | by { rw ←mul_eq_right_iff_opow_omega_dvd, exact (mul_is_normal ha).le_iff_eq } | theorem | ordinal.mul_le_right_iff_opow_omega_dvd | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_mul_opow_omega_add {a c : ordinal} (b) (ha : 0 < a) (hc : 0 < c) (hca : c ≤ a ^ omega) :
nfp ((*) a) (a ^ omega * b + c) = a ^ omega.{u} * (succ b) | begin
apply le_antisymm,
{ apply nfp_le_fp (mul_is_normal ha).monotone,
{ rw mul_succ,
apply add_le_add_left hca },
{ rw [←mul_assoc, ←opow_one_add, one_add_omega] } },
{ cases mul_eq_right_iff_opow_omega_dvd.1 ((mul_is_normal ha).nfp_fp (a ^ omega * b + c))
with d hd,
rw hd,
apply mul... | theorem | ordinal.nfp_mul_opow_omega_add | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"monotone",
"mul_le_mul_left'",
"mul_lt_mul_iff_left",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_mul_eq_opow_omega_mul {a : ordinal.{u}} (ha : 0 < a) (b) :
deriv ((*) a) b = a ^ omega * b | begin
revert b,
rw [←funext_iff,
is_normal.eq_iff_zero_and_succ (deriv_is_normal _) (mul_is_normal (opow_pos omega ha))],
refine ⟨_, λ c h, _⟩,
{ rw [deriv_zero, nfp_mul_zero, mul_zero] },
{ rw [deriv_succ, h],
exact nfp_mul_opow_omega_add c ha zero_lt_one (one_le_iff_pos.2 (opow_pos _ ha)) },
end | theorem | ordinal.deriv_mul_eq_opow_omega_mul | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"deriv",
"mul_zero",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_ordinal : Type* | ordinal | def | nat_ordinal | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"ordinal"
] | A type synonym for ordinals with natural addition and multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordinal.to_nat_ordinal : ordinal ≃o nat_ordinal | order_iso.refl _ | def | ordinal.to_nat_ordinal | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"nat_ordinal",
"order_iso.refl",
"ordinal"
] | The identity function between `ordinal` and `nat_ordinal`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_ordinal.to_ordinal : nat_ordinal ≃o ordinal | order_iso.refl _ | def | nat_ordinal.to_ordinal | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"nat_ordinal",
"order_iso.refl",
"ordinal"
] | The identity function between `nat_ordinal` and `ordinal`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ordinal_symm_eq : nat_ordinal.to_ordinal.symm = ordinal.to_nat_ordinal | rfl | theorem | nat_ordinal.to_ordinal_symm_eq | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"ordinal.to_nat_ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ordinal_to_nat_ordinal (a : nat_ordinal) : a.to_ordinal.to_nat_ordinal = a | rfl | theorem | nat_ordinal.to_ordinal_to_nat_ordinal | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"nat_ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_wf : @well_founded nat_ordinal (<) | ordinal.lt_wf | theorem | nat_ordinal.lt_wf | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"nat_ordinal",
"ordinal.lt_wf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ordinal_zero : to_ordinal 0 = 0 | rfl | theorem | nat_ordinal.to_ordinal_zero | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ordinal_one : to_ordinal 1 = 1 | rfl | theorem | nat_ordinal.to_ordinal_one | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ordinal_eq_zero (a) : to_ordinal a = 0 ↔ a = 0 | iff.rfl | theorem | nat_ordinal.to_ordinal_eq_zero | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ordinal_eq_one (a) : to_ordinal a = 1 ↔ a = 1 | iff.rfl | theorem | nat_ordinal.to_ordinal_eq_one | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ordinal_max : (max a b).to_ordinal = max a.to_ordinal b.to_ordinal | rfl | theorem | nat_ordinal.to_ordinal_max | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ordinal_min : (min a b).to_ordinal = min a.to_ordinal b.to_ordinal | rfl | theorem | nat_ordinal.to_ordinal_min | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_def (a : nat_ordinal) : succ a = (a.to_ordinal + 1).to_nat_ordinal | rfl | theorem | nat_ordinal.succ_def | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"nat_ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rec {β : nat_ordinal → Sort*} (h : Π a, β (to_nat_ordinal a)) : Π a, β a | λ a, h a.to_ordinal | def | nat_ordinal.rec | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"nat_ordinal"
] | A recursor for `nat_ordinal`. Use as `induction x using nat_ordinal.rec`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induction {p : nat_ordinal → Prop} : ∀ i (h : ∀ j, (∀ k, k < j → p k) → p j), p i | ordinal.induction | theorem | nat_ordinal.induction | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"nat_ordinal",
"ordinal.induction"
] | `ordinal.induction` but for `nat_ordinal`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_nat_ordinal_symm_eq : to_nat_ordinal.symm = nat_ordinal.to_ordinal | rfl | theorem | ordinal.to_nat_ordinal_symm_eq | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"nat_ordinal.to_ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_ordinal_to_ordinal (a : ordinal) : a.to_nat_ordinal.to_ordinal = a | rfl | theorem | ordinal.to_nat_ordinal_to_ordinal | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_ordinal_zero : to_nat_ordinal 0 = 0 | rfl | theorem | ordinal.to_nat_ordinal_zero | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_ordinal_one : to_nat_ordinal 1 = 1 | rfl | theorem | ordinal.to_nat_ordinal_one | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_ordinal_eq_zero (a) : to_nat_ordinal a = 0 ↔ a = 0 | iff.rfl | theorem | ordinal.to_nat_ordinal_eq_zero | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_ordinal_eq_one (a) : to_nat_ordinal a = 1 ↔ a = 1 | iff.rfl | theorem | ordinal.to_nat_ordinal_eq_one | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_ordinal_max (a b : ordinal) :
to_nat_ordinal (max a b) = max a.to_nat_ordinal b.to_nat_ordinal | rfl | theorem | ordinal.to_nat_ordinal_max | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_ordinal_min (a b : ordinal) :
(linear_order.min a b).to_nat_ordinal = linear_order.min a.to_nat_ordinal b.to_nat_ordinal | rfl | theorem | ordinal.to_nat_ordinal_min | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nadd : ordinal → ordinal → ordinal | | a b := max
(blsub.{u u} a $ λ a' h, nadd a' b)
(blsub.{u u} b $ λ b' h, nadd a b')
using_well_founded { dec_tac := `[solve_by_elim [psigma.lex.left, psigma.lex.right]] } | def | ordinal.nadd | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"ordinal"
] | Natural addition on ordinals `a ♯ b`, also known as the Hessenberg sum, is recursively defined
as the least ordinal greater than `a' ♯ b` and `a ♯ b'` for all `a' < a` and `b' < b`. In contrast
to normal ordinal addition, it is commutative.
Natural addition can equivalently be characterized as the ordinal resulting fr... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nmul : ordinal.{u} → ordinal.{u} → ordinal.{u} | | a b := Inf {c | ∀ (a' < a) (b' < b), nmul a' b ♯ nmul a b' < c ♯ nmul a' b'}
using_well_founded { dec_tac := `[solve_by_elim [psigma.lex.left, psigma.lex.right]] } | def | ordinal.nmul | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | Natural multiplication on ordinals `a ⨳ b`, also known as the Hessenberg product, is recursively
defined as the least ordinal such that `a ⨳ b + a' ⨳ b'` is greater than `a' ⨳ b + a ⨳ b'` for all
`a' < a` and `b < b'`. In contrast to normal ordinal multiplication, it is commutative and
distributive (over natural additi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nadd_def (a b : ordinal) : a ♯ b = max
(blsub.{u u} a $ λ a' h, a' ♯ b)
(blsub.{u u} b $ λ b' h, a ♯ b') | by rw nadd | theorem | ordinal.nadd_def | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_nadd_iff : a < b ♯ c ↔ (∃ b' < b, a ≤ b' ♯ c) ∨ ∃ c' < c, a ≤ b ♯ c' | by { rw nadd_def, simp [lt_blsub_iff] } | theorem | ordinal.lt_nadd_iff | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nadd_le_iff : b ♯ c ≤ a ↔ (∀ b' < b, b' ♯ c < a) ∧ ∀ c' < c, b ♯ c' < a | by { rw nadd_def, simp [blsub_le_iff] } | theorem | ordinal.nadd_le_iff | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nadd_lt_nadd_left (h : b < c) (a) : a ♯ b < a ♯ c | lt_nadd_iff.2 (or.inr ⟨b, h, le_rfl⟩) | theorem | ordinal.nadd_lt_nadd_left | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nadd_lt_nadd_right (h : b < c) (a) : b ♯ a < c ♯ a | lt_nadd_iff.2 (or.inl ⟨b, h, le_rfl⟩) | theorem | ordinal.nadd_lt_nadd_right | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nadd_le_nadd_left (h : b ≤ c) (a) : a ♯ b ≤ a ♯ c | begin
rcases lt_or_eq_of_le h with h | rfl,
{ exact (nadd_lt_nadd_left h a).le },
{ exact le_rfl }
end | theorem | ordinal.nadd_le_nadd_left | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nadd_le_nadd_right (h : b ≤ c) (a) : b ♯ a ≤ c ♯ a | begin
rcases lt_or_eq_of_le h with h | rfl,
{ exact (nadd_lt_nadd_right h a).le },
{ exact le_rfl }
end | theorem | ordinal.nadd_le_nadd_right | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nadd_comm : ∀ a b, a ♯ b = b ♯ a | | a b := begin
rw [nadd_def, nadd_def, max_comm],
congr; ext c hc;
apply nadd_comm
end
using_well_founded { dec_tac := `[solve_by_elim [psigma.lex.left, psigma.lex.right]] } | theorem | ordinal.nadd_comm | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub_nadd_of_mono {f : Π c < a ♯ b, ordinal.{max u v}}
(hf : ∀ {i j} hi hj, i ≤ j → f i hi ≤ f j hj) : blsub _ f = max
(blsub.{u v} a (λ a' ha', f (a' ♯ b) $ nadd_lt_nadd_right ha' b))
(blsub.{u v} b (λ b' hb', f (a ♯ b') $ nadd_lt_nadd_left hb' a)) | begin
apply (blsub_le_iff.2 (λ i h, _)).antisymm (max_le _ _),
{ rcases lt_nadd_iff.1 h with ⟨a', ha', hi⟩ | ⟨b', hb', hi⟩,
{ exact lt_max_of_lt_left ((hf h (nadd_lt_nadd_right ha' b) hi).trans_lt (lt_blsub _ _ _)) },
{ exact lt_max_of_lt_right ((hf h (nadd_lt_nadd_left hb' a) hi).trans_lt (lt_blsub _ _ _))... | theorem | ordinal.blsub_nadd_of_mono | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"lt_max_of_lt_left",
"lt_max_of_lt_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nadd_assoc : ∀ a b c, a ♯ b ♯ c = a ♯ (b ♯ c) | | a b c := begin
rw [nadd_def a (b ♯ c), nadd_def, blsub_nadd_of_mono, blsub_nadd_of_mono, max_assoc],
{ congr; ext d hd;
apply nadd_assoc },
{ exact λ i j _ _ h, nadd_le_nadd_left h a },
{ exact λ i j _ _ h, nadd_le_nadd_right h c }
end
using_well_founded { dec_tac := `[solve_by_elim [psigma.lex.left, psig... | theorem | ordinal.nadd_assoc | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nadd_zero : a ♯ 0 = a | begin
induction a using ordinal.induction with a IH,
rw [nadd_def, blsub_zero, max_zero_right],
convert blsub_id a,
ext b hb,
exact IH _ hb
end | theorem | ordinal.nadd_zero | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"ordinal.induction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_nadd : 0 ♯ a = a | by rw [nadd_comm, nadd_zero] | theorem | ordinal.zero_nadd | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nadd_one : a ♯ 1 = succ a | begin
induction a using ordinal.induction with a IH,
rw [nadd_def, blsub_one, nadd_zero, max_eq_right_iff, blsub_le_iff],
intros i hi,
rwa [IH i hi, succ_lt_succ_iff]
end | theorem | ordinal.nadd_one | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"max_eq_right_iff",
"ordinal.induction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_nadd : 1 ♯ a = succ a | by rw [nadd_comm, nadd_one] | theorem | ordinal.one_nadd | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nadd_succ : a ♯ succ b = succ (a ♯ b) | by rw [←nadd_one (a ♯ b), nadd_assoc, nadd_one] | theorem | ordinal.nadd_succ | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_nadd : succ a ♯ b = succ (a ♯ b) | by rw [←one_nadd (a ♯ b), ←nadd_assoc, one_nadd] | theorem | ordinal.succ_nadd | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nadd_nat (n : ℕ) : a ♯ n = a + n | begin
induction n with n hn,
{ simp },
{ rw [nat.cast_succ, add_one_eq_succ, nadd_succ, add_succ, hn] }
end | theorem | ordinal.nadd_nat | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"nat.cast_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_nadd (n : ℕ) : ↑n ♯ a = a + n | by rw [nadd_comm, nadd_nat] | theorem | ordinal.nat_nadd | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_nadd : a + b ≤ a ♯ b | begin
apply b.limit_rec_on,
{ simp },
{ intros c h,
rwa [add_succ, nadd_succ, succ_le_succ_iff] },
{ intros c hc H,
rw [←is_normal.blsub_eq.{u u} (add_is_normal a) hc, blsub_le_iff],
exact λ i hi, (H i hi).trans_lt (nadd_lt_nadd_left hi a) }
end | theorem | ordinal.add_le_nadd | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_covariant_class_lt :
covariant_class nat_ordinal.{u} nat_ordinal.{u} (+) (<) | ⟨λ a b c h, nadd_lt_nadd_left h a⟩ | instance | nat_ordinal.add_covariant_class_lt | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_covariant_class_le :
covariant_class nat_ordinal.{u} nat_ordinal.{u} (+) (≤) | ⟨λ a b c h, nadd_le_nadd_left h a⟩ | instance | nat_ordinal.add_covariant_class_le | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_contravariant_class_le :
contravariant_class nat_ordinal.{u} nat_ordinal.{u} (+) (≤) | ⟨λ a b c h, by { by_contra' h', exact h.not_lt (add_lt_add_left h' a) }⟩ | instance | nat_ordinal.add_contravariant_class_le | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"contravariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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