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
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
add_one_eq_succ : ∀ a : nat_ordinal, a + 1 = succ a | nadd_one | theorem | nat_ordinal.add_one_eq_succ | 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 | |
to_ordinal_cast_nat (n : ℕ) : to_ordinal n = n | begin
induction n with n hn,
{ refl },
{ change to_ordinal n ♯ 1 = n + 1,
rw hn, exact nadd_one n }
end | theorem | nat_ordinal.to_ordinal_cast_nat | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nadd_eq_add (a b : ordinal) : a ♯ b = (a.to_nat_ordinal + b.to_nat_ordinal).to_ordinal | rfl | theorem | ordinal.nadd_eq_add | 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_cast_nat (n : ℕ) : to_nat_ordinal n = n | by { rw ←nat_ordinal.to_ordinal_cast_nat n, refl } | theorem | ordinal.to_nat_ordinal_cast_nat | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_nadd_lt_nadd_left : ∀ {a b c}, a ♯ b < a ♯ c → b < c | @lt_of_add_lt_add_left nat_ordinal _ _ _ | theorem | ordinal.lt_of_nadd_lt_nadd_left | 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_of_nadd_lt_nadd_right : ∀ {a b c}, b ♯ a < c ♯ a → b < c | @_root_.lt_of_add_lt_add_right nat_ordinal _ _ _ | theorem | ordinal.lt_of_nadd_lt_nadd_right | 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 | |
le_of_nadd_le_nadd_left : ∀ {a b c}, a ♯ b ≤ a ♯ c → b ≤ c | @le_of_add_le_add_left nat_ordinal _ _ _ | theorem | ordinal.le_of_nadd_le_nadd_left | 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 | |
le_of_nadd_le_nadd_right : ∀ {a b c}, b ♯ a ≤ c ♯ a → b ≤ c | @le_of_add_le_add_right nat_ordinal _ _ _ | theorem | ordinal.le_of_nadd_le_nadd_right | 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 | |
nadd_lt_nadd_iff_left : ∀ a {b c}, a ♯ b < a ♯ c ↔ b < c | @add_lt_add_iff_left nat_ordinal _ _ _ _ | theorem | ordinal.nadd_lt_nadd_iff_left | 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 | |
nadd_lt_nadd_iff_right : ∀ a {b c}, b ♯ a < c ♯ a ↔ b < c | @add_lt_add_iff_right nat_ordinal _ _ _ _ | theorem | ordinal.nadd_lt_nadd_iff_right | 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 | |
nadd_le_nadd_iff_left : ∀ a {b c}, a ♯ b ≤ a ♯ c ↔ b ≤ c | @add_le_add_iff_left nat_ordinal _ _ _ _ | theorem | ordinal.nadd_le_nadd_iff_left | 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 | |
nadd_le_nadd_iff_right : ∀ a {b c}, b ♯ a ≤ c ♯ a ↔ b ≤ c | @_root_.add_le_add_iff_right nat_ordinal _ _ _ _ | theorem | ordinal.nadd_le_nadd_iff_right | 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 | |
nadd_le_nadd : ∀ {a b c d}, a ≤ b → c ≤ d → a ♯ c ≤ b ♯ d | @add_le_add nat_ordinal _ _ _ _ | theorem | ordinal.nadd_le_nadd | 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 | |
nadd_lt_nadd : ∀ {a b c d}, a < b → c < d → a ♯ c < b ♯ d | @add_lt_add nat_ordinal _ _ _ _ | theorem | ordinal.nadd_lt_nadd | 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 | |
nadd_lt_nadd_of_lt_of_le : ∀ {a b c d}, a < b → c ≤ d → a ♯ c < b ♯ d | @add_lt_add_of_lt_of_le nat_ordinal _ _ _ _ | theorem | ordinal.nadd_lt_nadd_of_lt_of_le | 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 | |
nadd_lt_nadd_of_le_of_lt : ∀ {a b c d}, a ≤ b → c < d → a ♯ c < b ♯ d | @add_lt_add_of_le_of_lt nat_ordinal _ _ _ _ | theorem | ordinal.nadd_lt_nadd_of_le_of_lt | 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 | |
nadd_left_cancel : ∀ {a b c}, a ♯ b = a ♯ c → b = c | @_root_.add_left_cancel nat_ordinal _ _ | theorem | ordinal.nadd_left_cancel | 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 | |
nadd_right_cancel : ∀ {a b c}, a ♯ b = c ♯ b → a = c | @_root_.add_right_cancel nat_ordinal _ _ | theorem | ordinal.nadd_right_cancel | 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 | |
nadd_left_cancel_iff : ∀ {a b c}, a ♯ b = a ♯ c ↔ b = c | @add_left_cancel_iff nat_ordinal _ _ | theorem | ordinal.nadd_left_cancel_iff | 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 | |
nadd_right_cancel_iff : ∀ {a b c}, b ♯ a = c ♯ a ↔ b = c | @add_right_cancel_iff nat_ordinal _ _ | theorem | ordinal.nadd_right_cancel_iff | 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 | |
le_nadd_self {a b} : a ≤ b ♯ a | by simpa using nadd_le_nadd_right (ordinal.zero_le b) a | theorem | ordinal.le_nadd_self | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_nadd_left {a b c} (h : a ≤ c) : a ≤ b ♯ c | le_nadd_self.trans (nadd_le_nadd_left h b) | theorem | ordinal.le_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 | |
le_self_nadd {a b} : a ≤ a ♯ b | by simpa using nadd_le_nadd_left (ordinal.zero_le b) a | theorem | ordinal.le_self_nadd | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_nadd_right {a b c} (h : a ≤ b) : a ≤ b ♯ c | le_self_nadd.trans (nadd_le_nadd_right h c) | theorem | ordinal.le_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_left_comm : ∀ a b c, a ♯ (b ♯ c) = b ♯ (a ♯ c) | @add_left_comm nat_ordinal _ | theorem | ordinal.nadd_left_comm | 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 | |
nadd_right_comm : ∀ a b c, a ♯ b ♯ c = a ♯ c ♯ b | @add_right_comm nat_ordinal _ | theorem | ordinal.nadd_right_comm | 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 | |
nmul_def (a b : ordinal) :
a ⨳ b = Inf {c | ∀ (a' < a) (b' < b), a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} | by rw nmul | theorem | ordinal.nmul_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 | |
nmul_nonempty (a b : ordinal.{u}) :
{c : ordinal.{u} | ∀ (a' < a) (b' < b), a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'}.nonempty | ⟨_, λ a' ha b' hb, (lt_blsub₂.{u u u} _ ha hb).trans_le le_self_nadd⟩ | theorem | ordinal.nmul_nonempty | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | The set in the definition of `nmul` is nonempty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nmul_nadd_lt {a' b' : ordinal} (ha : a' < a) (hb : b' < b) :
a' ⨳ b ♯ a ⨳ b' < a ⨳ b ♯ a' ⨳ b' | by { rw nmul_def a b, exact Inf_mem (nmul_nonempty a b) a' ha b' hb } | theorem | ordinal.nmul_nadd_lt | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"Inf_mem",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_nadd_le {a' b' : ordinal} (ha : a' ≤ a) (hb : b' ≤ b) :
a' ⨳ b ♯ a ⨳ b' ≤ a ⨳ b ♯ a' ⨳ b' | begin
rcases lt_or_eq_of_le ha with ha | rfl,
{ rcases lt_or_eq_of_le hb with hb | rfl,
{ exact (nmul_nadd_lt ha hb).le },
{ rw nadd_comm } },
{ exact le_rfl }
end | theorem | ordinal.nmul_nadd_le | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"le_rfl",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_nmul_iff : c < a ⨳ b ↔ ∃ (a' < a) (b' < b), c ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' | begin
refine ⟨λ h, _, _⟩,
{ rw nmul at h,
simpa using not_mem_of_lt_cInf h ⟨0, λ _ _, bot_le⟩ },
{ rintros ⟨a', ha, b', hb, h⟩,
have := h.trans_lt (nmul_nadd_lt ha hb),
rwa nadd_lt_nadd_iff_right at this }
end | theorem | ordinal.lt_nmul_iff | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"not_mem_of_lt_cInf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_le_iff : a ⨳ b ≤ c ↔ ∀ (a' < a) (b' < b), a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b' | by { rw ←not_iff_not, simp [lt_nmul_iff] } | theorem | ordinal.nmul_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 | |
nmul_comm : ∀ (a b), a ⨳ b = b ⨳ a | | a b := begin
rw [nmul, nmul],
congr, ext x, split;
intros H c hc d hd,
{ rw [nadd_comm, ←nmul_comm, ←nmul_comm a, ←nmul_comm d],
exact H _ hd _ hc },
{ rw [nadd_comm, nmul_comm, nmul_comm c, nmul_comm c],
exact H _ hd _ hc }
end
using_well_founded { dec_tac := `[solve_by_elim [psigma.lex.left, psigm... | theorem | ordinal.nmul_comm | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_zero (a) : a ⨳ 0 = 0 | by { rw [←ordinal.le_zero, nmul_le_iff], exact λ _ _ a ha, (ordinal.not_lt_zero a ha).elim } | theorem | ordinal.nmul_zero | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"ordinal.not_lt_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_nmul (a) : 0 ⨳ a = 0 | by rw [nmul_comm, nmul_zero] | theorem | ordinal.zero_nmul | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_one : ∀ a, a ⨳ 1 = a | | a := begin
rw nmul,
simp only [lt_one_iff_zero, forall_eq, nmul_zero, nadd_zero],
convert cInf_Ici,
ext b,
refine ⟨λ H, le_of_forall_lt (λ c hc, _), λ ha c hc, _⟩,
{ simpa only [nmul_one] using H c hc },
{ simpa only [nmul_one] using hc.trans_le ha }
end
using_well_founded { dec_tac := `[assumption] } | theorem | ordinal.nmul_one | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"cInf_Ici",
"forall_eq",
"le_of_forall_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_nmul (a) : 1 ⨳ a = a | by rw [nmul_comm, nmul_one] | theorem | ordinal.one_nmul | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_lt_nmul_of_pos_left (h₁ : a < b) (h₂ : 0 < c) : c ⨳ a < c ⨳ b | lt_nmul_iff.2 ⟨0, h₂, a, h₁, by simp⟩ | theorem | ordinal.nmul_lt_nmul_of_pos_left | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_lt_nmul_of_pos_right (h₁ : a < b) (h₂ : 0 < c) : a ⨳ c < b ⨳ c | lt_nmul_iff.2 ⟨a, h₁, 0, h₂, by simp⟩ | theorem | ordinal.nmul_lt_nmul_of_pos_right | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_le_nmul_of_nonneg_left (h₁ : a ≤ b) (h₂ : 0 ≤ c) : c ⨳ a ≤ c ⨳ b | begin
rcases lt_or_eq_of_le h₁ with h₁|rfl;
rcases lt_or_eq_of_le h₂ with h₂|rfl,
{ exact (nmul_lt_nmul_of_pos_left h₁ h₂).le },
all_goals { simp }
end | theorem | ordinal.nmul_le_nmul_of_nonneg_left | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_le_nmul_of_nonneg_right (h₁ : a ≤ b) (h₂ : 0 ≤ c) : a ⨳ c ≤ b ⨳ c | begin
rw [nmul_comm, nmul_comm b],
exact nmul_le_nmul_of_nonneg_left h₁ h₂
end | theorem | ordinal.nmul_le_nmul_of_nonneg_right | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_nadd : ∀ (a b c), a ⨳ (b ♯ c) = a ⨳ b ♯ a ⨳ c | | a b c := begin
apply le_antisymm (nmul_le_iff.2 $ λ a' ha d hd, _) (nadd_le_iff.2 ⟨λ d hd, _, λ d hd, _⟩),
{ rw nmul_nadd,
rcases lt_nadd_iff.1 hd with ⟨b', hb, hd⟩ | ⟨c', hc, hd⟩,
{ have := nadd_lt_nadd_of_lt_of_le (nmul_nadd_lt ha hb) (nmul_nadd_le ha.le hd),
rw [nmul_nadd, nmul_nadd] at this,
... | theorem | ordinal.nmul_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_nmul (a b c) : (a ♯ b) ⨳ c = a ⨳ c ♯ b ⨳ c | by rw [nmul_comm, nmul_nadd, nmul_comm, nmul_comm c] | theorem | ordinal.nadd_nmul | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_nadd_lt₃ {a' b' c' : ordinal} (ha : a' < a) (hb : b' < b) (hc : c' < c) :
a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' <
a ⨳ b ⨳ c ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' | by simpa only [nadd_nmul, ←nadd_assoc] using nmul_nadd_lt (nmul_nadd_lt ha hb) hc | theorem | ordinal.nmul_nadd_lt₃ | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_nadd_le₃ {a' b' c' : ordinal} (ha : a' ≤ a) (hb : b' ≤ b) (hc : c' ≤ c) :
a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' ≤
a ⨳ b ⨳ c ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' | by simpa only [nadd_nmul, ←nadd_assoc] using nmul_nadd_le (nmul_nadd_le ha hb) hc | theorem | ordinal.nmul_nadd_le₃ | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_nadd_lt₃' {a' b' c' : ordinal} (ha : a' < a) (hb : b' < b) (hc : c' < c) :
a' ⨳ (b ⨳ c) ♯ a ⨳ (b' ⨳ c) ♯ a ⨳ (b ⨳ c') ♯ a' ⨳ (b' ⨳ c') <
a ⨳ (b ⨳ c) ♯ a' ⨳ (b' ⨳ c) ♯ a' ⨳ (b ⨳ c') ♯ a ⨳ (b' ⨳ c') | begin
simp only [nmul_comm _ (_ ⨳ _)],
convert nmul_nadd_lt₃ hb hc ha using 1;
{ simp only [nadd_eq_add, nat_ordinal.to_ordinal_to_nat_ordinal], abel }
end | theorem | ordinal.nmul_nadd_lt₃' | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"nat_ordinal.to_ordinal_to_nat_ordinal",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_nadd_le₃' {a' b' c' : ordinal} (ha : a' ≤ a) (hb : b' ≤ b) (hc : c' ≤ c) :
a' ⨳ (b ⨳ c) ♯ a ⨳ (b' ⨳ c) ♯ a ⨳ (b ⨳ c') ♯ a' ⨳ (b' ⨳ c') ≤
a ⨳ (b ⨳ c) ♯ a' ⨳ (b' ⨳ c) ♯ a' ⨳ (b ⨳ c') ♯ a ⨳ (b' ⨳ c') | begin
simp only [nmul_comm _ (_ ⨳ _)],
convert nmul_nadd_le₃ hb hc ha using 1;
{ simp only [nadd_eq_add, nat_ordinal.to_ordinal_to_nat_ordinal], abel }
end | theorem | ordinal.nmul_nadd_le₃' | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"nat_ordinal.to_ordinal_to_nat_ordinal",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_nmul_iff₃ : d < a ⨳ b ⨳ c ↔ ∃ (a' < a) (b' < b) (c' < c),
d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ≤
a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' | begin
refine ⟨λ h, _, _⟩,
{ rcases lt_nmul_iff.1 h with ⟨e, he, c', hc, H₁⟩,
rcases lt_nmul_iff.1 he with ⟨a', ha, b', hb, H₂⟩,
refine ⟨a', ha, b', hb, c', hc, _⟩,
have := nadd_le_nadd H₁ (nmul_nadd_le H₂ hc.le),
simp only [nadd_nmul, nadd_assoc] at this,
rw [nadd_left_comm, nadd_left_comm d, na... | theorem | ordinal.lt_nmul_iff₃ | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_le_iff₃ : a ⨳ b ⨳ c ≤ d ↔ ∀ (a' < a) (b' < b) (c' < c),
a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' <
d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' | by { rw ←not_iff_not, simp [lt_nmul_iff₃] } | theorem | ordinal.nmul_le_iff₃ | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_nmul_iff₃' : d < a ⨳ (b ⨳ c) ↔ ∃ (a' < a) (b' < b) (c' < c),
d ♯ a' ⨳ (b' ⨳ c) ♯ a' ⨳ (b ⨳ c') ♯ a ⨳ (b' ⨳ c') ≤
a' ⨳ (b ⨳ c) ♯ a ⨳ (b' ⨳ c) ♯ a ⨳ (b ⨳ c') ♯ a' ⨳ (b' ⨳ c') | begin
simp only [nmul_comm _ (_ ⨳ _), lt_nmul_iff₃, nadd_eq_add, nat_ordinal.to_ordinal_to_nat_ordinal],
split; rintro ⟨b', hb, c', hc, a', ha, h⟩,
{ use [a', ha, b', hb, c', hc], convert h using 1; abel },
{ use [c', hc, a', ha, b', hb], convert h using 1; abel }
end | theorem | ordinal.lt_nmul_iff₃' | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"nat_ordinal.to_ordinal_to_nat_ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_le_iff₃' : a ⨳ (b ⨳ c) ≤ d ↔ ∀ (a' < a) (b' < b) (c' < c),
a' ⨳ (b ⨳ c) ♯ a ⨳ (b' ⨳ c) ♯ a ⨳ (b ⨳ c') ♯ a' ⨳ (b' ⨳ c') <
d ♯ a' ⨳ (b' ⨳ c) ♯ a' ⨳ (b ⨳ c') ♯ a ⨳ (b' ⨳ c') | by { rw ←not_iff_not, simp [lt_nmul_iff₃'] } | theorem | ordinal.nmul_le_iff₃' | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_assoc : ∀ a b c, a ⨳ b ⨳ c = a ⨳ (b ⨳ c) | | a b c := begin
apply le_antisymm,
{ rw nmul_le_iff₃,
intros a' ha b' hb c' hc,
repeat { rw nmul_assoc },
exact nmul_nadd_lt₃' ha hb hc },
{ rw nmul_le_iff₃',
intros a' ha b' hb c' hc,
repeat { rw ←nmul_assoc },
exact nmul_nadd_lt₃ ha hb hc },
end
using_well_founded { dec_tac := `[solve_b... | theorem | ordinal.nmul_assoc | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_eq_mul (a b) : a ⨳ b = (a.to_nat_ordinal * b.to_nat_ordinal).to_ordinal | rfl | theorem | ordinal.nmul_eq_mul | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_nadd_one : ∀ a b, a ⨳ (b ♯ 1) = a ⨳ b ♯ a | @mul_add_one nat_ordinal _ _ _ | theorem | ordinal.nmul_nadd_one | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"mul_add_one",
"nat_ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nadd_one_nmul : ∀ a b, (a ♯ 1) ⨳ b = a ⨳ b ♯ b | @add_one_mul nat_ordinal _ _ _ | theorem | ordinal.nadd_one_nmul | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [
"add_one_mul",
"nat_ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_succ (a b) : a ⨳ succ b = a ⨳ b ♯ a | by rw [←nadd_one, nmul_nadd_one] | theorem | ordinal.nmul_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_nmul (a b) : succ a ⨳ b = a ⨳ b ♯ b | by rw [←nadd_one, nadd_one_nmul] | theorem | ordinal.succ_nmul | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmul_add_one : ∀ a b, a ⨳ (b + 1) = a ⨳ b ♯ a | nmul_succ | theorem | ordinal.nmul_add_one | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_one_nmul : ∀ a b, (a + 1) ⨳ b = a ⨳ b ♯ b | succ_nmul | theorem | ordinal.add_one_nmul | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_nmul (a b : ordinal.{u}) : a * b ≤ a ⨳ b | begin
apply b.limit_rec_on,
{ simp },
{ intros c h,
rw [mul_succ, nmul_succ],
exact (add_le_nadd _ a).trans (nadd_le_nadd_right h a) },
{ intros c hc H,
rcases eq_zero_or_pos a with rfl | ha,
{ simp },
{ rw [←is_normal.blsub_eq.{u u} (mul_is_normal ha) hc, blsub_le_iff],
exact λ i hi, ... | theorem | nat_ordinal.mul_le_nmul | set_theory.ordinal | src/set_theory/ordinal/natural_ops.lean | [
"set_theory.ordinal.arithmetic",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
onote : Type
| zero : onote
| oadd : onote → ℕ+ → onote → onote | inductive | onote | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | Recursive definition of an ordinal notation. `zero` denotes the
ordinal 0, and `oadd e n a` is intended to refer to `ω^e * n + a`.
For this to be valid Cantor normal form, we must have the exponents
decrease to the right, but we can't state this condition until we've
defined `repr`, so it is a separate definiti... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_def : zero = 0 | rfl | theorem | onote.zero_def | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
omega : onote | oadd 1 1 0 | def | onote.omega | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [
"onote"
] | Notation for ω | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
repr : onote → ordinal.{0} | | 0 := 0
| (oadd e n a) := ω ^ repr e * n + repr a | def | onote.repr | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [
"onote"
] | The ordinal denoted by a notation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_string_aux1 (e : onote) (n : ℕ) (s : string) : string | if e = 0 then _root_.to_string n else
(if e = 1 then "ω" else "ω^(" ++ s ++ ")") ++
if n = 1 then "" else "*" ++ _root_.to_string n | def | onote.to_string_aux1 | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [
"onote"
] | Auxiliary definition to print an ordinal notation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_string : onote → string | | zero := "0"
| (oadd e n 0) := to_string_aux1 e n (to_string e)
| (oadd e n a) := to_string_aux1 e n (to_string e) ++ " + " ++ to_string a | def | onote.to_string | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [
"onote"
] | Print an ordinal notation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
repr' : onote → string | | zero := "0"
| (oadd e n a) := "(oadd " ++ repr' e ++ " " ++ _root_.to_string (n:ℕ) ++ " " ++ repr' a ++ ")" | def | onote.repr' | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [
"onote"
] | Print an ordinal notation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_def {x y : onote} : x < y ↔ repr x < repr y | iff.rfl | theorem | onote.lt_def | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [
"onote"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_def {x y : onote} : x ≤ y ↔ repr x ≤ repr y | iff.rfl | theorem | onote.le_def | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [
"onote"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_nat : ℕ → onote | | 0 := 0
| (nat.succ n) := oadd 0 n.succ_pnat 0 | def | onote.of_nat | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [
"onote"
] | Convert a `nat` into an ordinal | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_nat_one : of_nat 1 = 1 | rfl | theorem | onote.of_nat_one | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
repr_of_nat (n : ℕ) : repr (of_nat n) = n | by cases n; simp | theorem | onote.repr_of_nat | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
repr_one : repr 1 = 1 | by simpa using repr_of_nat 1 | theorem | onote.repr_one | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
omega_le_oadd (e n a) : ω ^ repr e ≤ repr (oadd e n a) | begin
unfold repr,
refine le_trans _ (le_add_right _ _),
simpa using (mul_le_mul_iff_left $ opow_pos (repr e) omega_pos).2 (nat_cast_le.2 n.2)
end | theorem | onote.omega_le_oadd | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [
"mul_le_mul_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
oadd_pos (e n a) : 0 < oadd e n a | @lt_of_lt_of_le _ _ _ _ _ (opow_pos _ omega_pos)
(omega_le_oadd _ _ _) | theorem | onote.oadd_pos | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cmp : onote → onote → ordering | | 0 0 := ordering.eq
| _ 0 := ordering.gt
| 0 _ := ordering.lt
| o₁@(oadd e₁ n₁ a₁) o₂@(oadd e₂ n₂ a₂) :=
(cmp e₁ e₂).or_else $ (_root_.cmp (n₁:ℕ) n₂).or_else (cmp a₁ a₂) | def | onote.cmp | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [
"onote"
] | Compare ordinal notations | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_cmp_eq : ∀ {o₁ o₂}, cmp o₁ o₂ = ordering.eq → o₁ = o₂ | | 0 0 h := rfl
| (oadd e n a) 0 h := by injection h
| 0 (oadd e n a) h := by injection h
| o₁@(oadd e₁ n₁ a₁) o₂@(oadd e₂ n₂ a₂) h := begin
revert h, simp only [cmp],
cases h₁ : cmp e₁ e₂; intro h; try {cases h},
obtain rfl := eq_of_cmp_eq h₁,
revert h, cases h₂ : _root_.cmp (n₁:ℕ) n₂; intro h; try {... | theorem | onote.eq_of_cmp_eq | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_lt_one : (0 : onote) < 1 | by rw [lt_def, repr, repr_one]; exact zero_lt_one | theorem | onote.zero_lt_one | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [
"onote",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF_below : onote → ordinal.{0} → Prop
| zero {b} : NF_below 0 b
| oadd' {e n a eb b} : NF_below e eb →
NF_below a (repr e) → repr e < b → NF_below (oadd e n a) b | inductive | onote.NF_below | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [
"onote"
] | `NF_below o b` says that `o` is a normal form ordinal notation
satisfying `repr o < ω ^ b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF (o : onote) : Prop | (out : Exists (NF_below o)) | class | onote.NF | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [
"onote"
] | A normal form ordinal notation has the form
ω ^ a₁ * n₁ + ω ^ a₂ * n₂ + ... ω ^ aₖ * nₖ
where `a₁ > a₂ > ... > aₖ` and all the `aᵢ` are
also in normal form.
We will essentially only be interested in normal form
ordinal notations, but to avoid complicating the algorithms
we define everything over genera... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
NF.zero : NF 0 | ⟨⟨0, NF_below.zero⟩⟩ | instance | onote.NF.zero | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF_below.oadd {e n a b} : NF e →
NF_below a (repr e) → repr e < b → NF_below (oadd e n a) b | | ⟨⟨eb, h⟩⟩ := NF_below.oadd' h | theorem | onote.NF_below.oadd | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF_below.fst {e n a b} (h : NF_below (oadd e n a) b) : NF e | by cases h with _ _ _ _ eb _ h₁ h₂ h₃; exact ⟨⟨_, h₁⟩⟩ | theorem | onote.NF_below.fst | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF.fst {e n a} : NF (oadd e n a) → NF e | | ⟨⟨b, h⟩⟩ := h.fst | theorem | onote.NF.fst | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF_below.snd {e n a b} (h : NF_below (oadd e n a) b) : NF_below a (repr e) | by cases h with _ _ _ _ eb _ h₁ h₂ h₃; exact h₂ | theorem | onote.NF_below.snd | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF.snd' {e n a} : NF (oadd e n a) → NF_below a (repr e) | | ⟨⟨b, h⟩⟩ := h.snd | theorem | onote.NF.snd' | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF.snd {e n a} (h : NF (oadd e n a)) : NF a | ⟨⟨_, h.snd'⟩⟩ | theorem | onote.NF.snd | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF.oadd {e a} (h₁ : NF e) (n)
(h₂ : NF_below a (repr e)) : NF (oadd e n a) | ⟨⟨_, NF_below.oadd h₁ h₂ (lt_succ _)⟩⟩ | theorem | onote.NF.oadd | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF.oadd_zero (e n) [h : NF e] : NF (oadd e n 0) | h.oadd _ NF_below.zero | instance | onote.NF.oadd_zero | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF_below.lt {e n a b} (h : NF_below (oadd e n a) b) : repr e < b | by cases h with _ _ _ _ eb _ h₁ h₂ h₃; exact h₃ | theorem | onote.NF_below.lt | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF_below_zero : ∀ {o}, NF_below o 0 ↔ o = 0 | | 0 := ⟨λ _, rfl, λ _, NF_below.zero⟩
| (oadd e n a) := ⟨λ h, (not_le_of_lt h.lt).elim (ordinal.zero_le _),
λ e, e.symm ▸ NF_below.zero⟩ | theorem | onote.NF_below_zero | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [
"not_le_of_lt",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF.zero_of_zero {e n a} (h : NF (oadd e n a)) (e0 : e = 0) : a = 0 | by simpa [e0, NF_below_zero] using h.snd' | theorem | onote.NF.zero_of_zero | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF_below.repr_lt {o b} (h : NF_below o b) : repr o < ω ^ b | begin
induction h with _ e n a eb b h₁ h₂ h₃ _ IH,
{ exact opow_pos _ omega_pos },
{ rw repr,
apply ((add_lt_add_iff_left _).2 IH).trans_le,
rw ← mul_succ,
apply (mul_le_mul_left' (succ_le_of_lt (nat_lt_omega _)) _).trans,
rw ← opow_succ,
exact opow_le_opow_right omega_pos (succ_le_of_lt h₃) }... | theorem | onote.NF_below.repr_lt | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [
"mul_le_mul_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF_below.mono {o b₁ b₂} (bb : b₁ ≤ b₂) (h : NF_below o b₁) : NF_below o b₂ | begin
induction h with _ e n a eb b h₁ h₂ h₃ _ IH; constructor,
exacts [h₁, h₂, lt_of_lt_of_le h₃ bb]
end | theorem | onote.NF_below.mono | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF.below_of_lt {e n a b} (H : repr e < b) : NF (oadd e n a) → NF_below (oadd e n a) b | | ⟨⟨b', h⟩⟩ := by cases h with _ _ _ _ eb _ h₁ h₂ h₃;
exact NF_below.oadd' h₁ h₂ H | theorem | onote.NF.below_of_lt | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF.below_of_lt' : ∀ {o b}, repr o < ω ^ b → NF o → NF_below o b | | 0 b H _ := NF_below.zero
| (oadd e n a) b H h := h.below_of_lt $ (opow_lt_opow_iff_right one_lt_omega).1 $
(lt_of_le_of_lt (omega_le_oadd _ _ _) H) | theorem | onote.NF.below_of_lt' | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF_below_of_nat : ∀ n, NF_below (of_nat n) 1 | | 0 := NF_below.zero
| (nat.succ n) := NF_below.oadd NF.zero NF_below.zero zero_lt_one | theorem | onote.NF_below_of_nat | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF_of_nat (n) : NF (of_nat n) | ⟨⟨_, NF_below_of_nat n⟩⟩ | instance | onote.NF_of_nat | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
NF_one : NF 1 | by rw ← of_nat_one; apply_instance | instance | onote.NF_one | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
oadd_lt_oadd_1 {e₁ n₁ o₁ e₂ n₂ o₂} (h₁ : NF (oadd e₁ n₁ o₁)) (h : e₁ < e₂) :
oadd e₁ n₁ o₁ < oadd e₂ n₂ o₂ | @lt_of_lt_of_le _ _ _ _ _ ((h₁.below_of_lt h).repr_lt) (omega_le_oadd _ _ _) | theorem | onote.oadd_lt_oadd_1 | set_theory.ordinal | src/set_theory/ordinal/notation.lean | [
"set_theory.ordinal.principal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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