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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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CNF_snd_lt {b o : ordinal.{u}} (hb : 1 < b) {x : ordinal × ordinal} :
x ∈ CNF b o → x.2 < b | begin
refine CNF_rec b _ (λ o ho IH, _) o,
{ simp },
{ rw CNF_ne_zero ho,
rintro (rfl | h),
{ simpa using div_opow_log_lt o hb },
{ exact IH h } }
end | theorem | ordinal.CNF_snd_lt | set_theory.ordinal | src/set_theory/ordinal/cantor_normal_form.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | Every coefficient in the Cantor normal form `CNF b o` is less than `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CNF_sorted (b o : ordinal) : ((CNF b o).map prod.fst).sorted (>) | begin
refine CNF_rec b _ (λ o ho IH, _) o,
{ simp },
{ cases le_or_lt b 1 with hb hb,
{ simp [CNF_of_le_one hb ho] },
{ cases lt_or_le o b with hob hbo,
{ simp [CNF_of_lt ho hob] },
{ rw [CNF_ne_zero ho, list.map_cons, list.sorted_cons],
refine ⟨λ a H, _, IH⟩,
rw list.mem_map a... | theorem | ordinal.CNF_sorted | set_theory.ordinal | src/set_theory/ordinal/cantor_normal_form.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"list.mem_map",
"list.sorted_cons",
"ordinal"
] | The exponents of the Cantor normal form are decreasing. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
opow_def (a b : ordinal) :
a ^ b = if a = 0 then 1 - b else limit_rec_on b 1 (λ _ IH, IH * a) (λ b _, bsup.{u u} b) | rfl | theorem | ordinal.opow_def | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_opow' (a : ordinal) : 0 ^ a = 1 - a | by simp only [opow_def, if_pos rfl] | theorem | ordinal.zero_opow' | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_opow {a : ordinal} (a0 : a ≠ 0) : 0 ^ a = 0 | by rwa [zero_opow', ordinal.sub_eq_zero_iff_le, one_le_iff_ne_zero] | theorem | ordinal.zero_opow | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal",
"ordinal.sub_eq_zero_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_zero (a : ordinal) : a ^ 0 = 1 | by by_cases a = 0; [simp only [opow_def, if_pos h, sub_zero],
simp only [opow_def, if_neg h, limit_rec_on_zero]] | theorem | ordinal.opow_zero | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_succ (a b : ordinal) : a ^ succ b = a ^ b * a | if h : a = 0 then by subst a; simp only [zero_opow (succ_ne_zero _), mul_zero]
else by simp only [opow_def, limit_rec_on_succ, if_neg h] | theorem | ordinal.opow_succ | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"mul_zero",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_limit {a b : ordinal} (a0 : a ≠ 0) (h : is_limit b) :
a ^ b = bsup.{u u} b (λ c _, a ^ c) | by simp only [opow_def, if_neg a0]; rw limit_rec_on_limit _ _ _ _ h; refl | theorem | ordinal.opow_limit | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_le_of_limit {a b c : ordinal} (a0 : a ≠ 0) (h : is_limit b) :
a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c | by rw [opow_limit a0 h, bsup_le_iff] | theorem | ordinal.opow_le_of_limit | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_opow_of_limit {a b c : ordinal} (b0 : b ≠ 0) (h : is_limit c) :
a < b ^ c ↔ ∃ c' < c, a < b ^ c' | by rw [← not_iff_not, not_exists]; simp only [not_lt, opow_le_of_limit b0 h, exists_prop, not_and] | theorem | ordinal.lt_opow_of_limit | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"exists_prop",
"not_and",
"not_exists",
"not_iff_not",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_one (a : ordinal) : a ^ 1 = a | by rw [← succ_zero, opow_succ]; simp only [opow_zero, one_mul] | theorem | ordinal.opow_one | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"one_mul",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_opow (a : ordinal) : 1 ^ a = 1 | begin
apply limit_rec_on a,
{ simp only [opow_zero] },
{ intros _ ih, simp only [opow_succ, ih, mul_one] },
refine λ b l IH, eq_of_forall_ge_iff (λ c, _),
rw [opow_le_of_limit ordinal.one_ne_zero l],
exact ⟨λ H, by simpa only [opow_zero] using H 0 l.pos,
λ H b' h, by rwa IH _ h⟩,
end | theorem | ordinal.one_opow | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"eq_of_forall_ge_iff",
"ih",
"mul_one",
"ordinal",
"ordinal.one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_pos {a : ordinal} (b)
(a0 : 0 < a) : 0 < a ^ b | begin
have h0 : 0 < a ^ 0, {simp only [opow_zero, zero_lt_one]},
apply limit_rec_on b,
{ exact h0 },
{ intros b IH, rw [opow_succ],
exact mul_pos IH a0 },
{ exact λ b l _, (lt_opow_of_limit (ordinal.pos_iff_ne_zero.1 a0) l).2
⟨0, l.pos, h0⟩ },
end | theorem | ordinal.opow_pos | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_ne_zero {a : ordinal} (b)
(a0 : a ≠ 0) : a ^ b ≠ 0 | ordinal.pos_iff_ne_zero.1 $ opow_pos b $ ordinal.pos_iff_ne_zero.2 a0 | theorem | ordinal.opow_ne_zero | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_is_normal {a : ordinal} (h : 1 < a) : is_normal ((^) a) | have a0 : 0 < a, from zero_lt_one.trans h,
⟨λ b, by simpa only [mul_one, opow_succ] using
(mul_lt_mul_iff_left (opow_pos b a0)).2 h,
λ b l c, opow_le_of_limit (ne_of_gt a0) l⟩ | theorem | ordinal.opow_is_normal | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"mul_lt_mul_iff_left",
"mul_one",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_lt_opow_iff_right {a b c : ordinal}
(a1 : 1 < a) : a ^ b < a ^ c ↔ b < c | (opow_is_normal a1).lt_iff | theorem | ordinal.opow_lt_opow_iff_right | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_le_opow_iff_right {a b c : ordinal}
(a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c | (opow_is_normal a1).le_iff | theorem | ordinal.opow_le_opow_iff_right | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_right_inj {a b c : ordinal}
(a1 : 1 < a) : a ^ b = a ^ c ↔ b = c | (opow_is_normal a1).inj | theorem | ordinal.opow_right_inj | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_is_limit {a b : ordinal}
(a1 : 1 < a) : is_limit b → is_limit (a ^ b) | (opow_is_normal a1).is_limit | theorem | ordinal.opow_is_limit | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_is_limit_left {a b : ordinal}
(l : is_limit a) (hb : b ≠ 0) : is_limit (a ^ b) | begin
rcases zero_or_succ_or_limit b with e|⟨b,rfl⟩|l',
{ exact absurd e hb },
{ rw opow_succ,
exact mul_is_limit (opow_pos _ l.pos) l },
{ exact opow_is_limit l.one_lt l' }
end | theorem | ordinal.opow_is_limit_left | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_le_opow_right {a b c : ordinal}
(h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c | begin
cases lt_or_eq_of_le (one_le_iff_pos.2 h₁) with h₁ h₁,
{ exact (opow_le_opow_iff_right h₁).2 h₂ },
{ subst a, simp only [one_opow] }
end | theorem | ordinal.opow_le_opow_right | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_le_opow_left {a b : ordinal} (c)
(ab : a ≤ b) : a ^ c ≤ b ^ c | begin
by_cases a0 : a = 0,
{ subst a, by_cases c0 : c = 0,
{ subst c, simp only [opow_zero] },
{ simp only [zero_opow c0, ordinal.zero_le] } },
{ apply limit_rec_on c,
{ simp only [opow_zero] },
{ intros c IH, simpa only [opow_succ] using mul_le_mul' IH ab },
{ exact λ c l IH, (opow_le_of_limi... | theorem | ordinal.opow_le_opow_left | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"mul_le_mul'",
"ordinal",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_le_opow (a : ordinal) {b : ordinal} (b1 : 0 < b) : a ≤ a ^ b | begin
nth_rewrite 0 ←opow_one a,
cases le_or_gt a 1 with a1 a1,
{ cases lt_or_eq_of_le a1 with a0 a1,
{ rw lt_one_iff_zero at a0,
rw [a0, zero_opow ordinal.one_ne_zero],
exact ordinal.zero_le _ },
rw [a1, one_opow, one_opow] },
rwa [opow_le_opow_iff_right a1, one_le_iff_pos]
end | theorem | ordinal.left_le_opow | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal",
"ordinal.one_ne_zero",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_le_opow {a : ordinal} (b) (a1 : 1 < a) : b ≤ a ^ b | (opow_is_normal a1).self_le _ | theorem | ordinal.right_le_opow | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_lt_opow_left_of_succ {a b c : ordinal}
(ab : a < b) : a ^ succ c < b ^ succ c | by { rw [opow_succ, opow_succ], exact
(mul_le_mul_right' (opow_le_opow_left c ab.le) a).trans_lt
(mul_lt_mul_of_pos_left ab (opow_pos c ((ordinal.zero_le a).trans_lt ab))) } | theorem | ordinal.opow_lt_opow_left_of_succ | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"mul_le_mul_right'",
"mul_lt_mul_of_pos_left",
"ordinal",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_add (a b c : ordinal) : a ^ (b + c) = a ^ b * a ^ c | begin
rcases eq_or_ne a 0 with rfl | a0,
{ rcases eq_or_ne c 0 with rfl | c0, { simp },
have : b + c ≠ 0 := ((ordinal.pos_iff_ne_zero.2 c0).trans_le (le_add_left _ _)).ne',
simp only [zero_opow c0, zero_opow this, mul_zero] },
rcases eq_or_lt_of_le (one_le_iff_ne_zero.2 a0) with rfl | a1,
{ simp only [o... | theorem | ordinal.opow_add | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"eq_of_forall_ge_iff",
"eq_or_lt_of_le",
"eq_or_ne",
"mul_assoc",
"mul_one",
"mul_zero",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_one_add (a b : ordinal) : a ^ (1 + b) = a * a ^ b | by rw [opow_add, opow_one] | theorem | ordinal.opow_one_add | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_dvd_opow (a) {b c : ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c | ⟨a ^ (c - b), by rw [←opow_add, ordinal.add_sub_cancel_of_le h] ⟩ | theorem | ordinal.opow_dvd_opow | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal",
"ordinal.add_sub_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_dvd_opow_iff {a b c : ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c | ⟨λ h, le_of_not_lt $ λ hn,
not_le_of_lt ((opow_lt_opow_iff_right a1).2 hn) $
le_of_dvd (opow_ne_zero _ $ one_le_iff_ne_zero.1 $ a1.le) h,
opow_dvd_opow _⟩ | theorem | ordinal.opow_dvd_opow_iff | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"not_le_of_lt",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_mul (a b c : ordinal) : a ^ (b * c) = (a ^ b) ^ c | begin
by_cases b0 : b = 0, {simp only [b0, zero_mul, opow_zero, one_opow]},
by_cases a0 : a = 0,
{ subst a,
by_cases c0 : c = 0, {simp only [c0, mul_zero, opow_zero]},
simp only [zero_opow b0, zero_opow c0, zero_opow (mul_ne_zero b0 c0)] },
cases eq_or_lt_of_le (one_le_iff_ne_zero.2 a0) with a1 a1,
{ ... | theorem | ordinal.opow_mul | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"eq_of_forall_ge_iff",
"eq_or_lt_of_le",
"mul_ne_zero",
"mul_zero",
"ordinal",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log (b : ordinal) (x : ordinal) : ordinal | if h : 1 < b then pred (Inf {o | x < b ^ o}) else 0 | def | ordinal.log | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | The ordinal logarithm is the solution `u` to the equation `x = b ^ u * v + w` where `v < b` and
`w < b ^ u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
log_nonempty {b x : ordinal} (h : 1 < b) : {o | x < b ^ o}.nonempty | ⟨_, succ_le_iff.1 (right_le_opow _ h)⟩ | theorem | ordinal.log_nonempty | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | The set in the definition of `log` is nonempty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
log_def {b : ordinal} (h : 1 < b) (x : ordinal) : log b x = pred (Inf {o | x < b ^ o}) | by simp only [log, dif_pos h] | theorem | ordinal.log_def | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_of_not_one_lt_left {b : ordinal} (h : ¬ 1 < b) (x : ordinal) : log b x = 0 | by simp only [log, dif_neg h] | theorem | ordinal.log_of_not_one_lt_left | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_of_left_le_one {b : ordinal} (h : b ≤ 1) : ∀ x, log b x = 0 | log_of_not_one_lt_left h.not_lt | theorem | ordinal.log_of_left_le_one | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_zero_left : ∀ b, log 0 b = 0 | log_of_left_le_one zero_le_one | lemma | ordinal.log_zero_left | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_zero_right (b : ordinal) : log b 0 = 0 | if b1 : 1 < b then begin
rw [log_def b1, ← ordinal.le_zero, pred_le],
apply cInf_le',
dsimp,
rw [succ_zero, opow_one],
exact zero_lt_one.trans b1
end
else by simp only [log_of_not_one_lt_left b1] | theorem | ordinal.log_zero_right | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"cInf_le'",
"ordinal",
"ordinal.le_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_one_left : ∀ b, log 1 b = 0 | log_of_left_le_one le_rfl | theorem | ordinal.log_one_left | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_log_def {b x : ordinal} (hb : 1 < b) (hx : x ≠ 0) :
succ (log b x) = Inf {o | x < b ^ o} | begin
let t := Inf {o | x < b ^ o},
have : x < b ^ t := Inf_mem (log_nonempty hb),
rcases zero_or_succ_or_limit t with h|h|h,
{ refine ((one_le_iff_ne_zero.2 hx).not_lt _).elim,
simpa only [h, opow_zero] },
{ rw [show log b x = pred t, from log_def hb x,
succ_pred_iff_is_succ.2 h] },
{ rcases (l... | theorem | ordinal.succ_log_def | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"Inf_mem",
"le_cInf_iff''",
"le_rfl",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_opow_succ_log_self {b : ordinal} (hb : 1 < b) (x : ordinal) : x < b ^ succ (log b x) | begin
rcases eq_or_ne x 0 with rfl | hx,
{ apply opow_pos _ (zero_lt_one.trans hb) },
{ rw succ_log_def hb hx,
exact Inf_mem (log_nonempty hb) }
end | theorem | ordinal.lt_opow_succ_log_self | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"Inf_mem",
"eq_or_ne",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_log_le_self (b) {x : ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x | begin
rcases eq_or_ne b 0 with rfl | b0,
{ rw zero_opow',
refine (sub_le_self _ _).trans (one_le_iff_ne_zero.2 hx) },
rcases lt_or_eq_of_le (one_le_iff_ne_zero.2 b0) with hb | rfl,
{ refine le_of_not_lt (λ h, (lt_succ (log b x)).not_le _),
have := @cInf_le' _ _ {o | x < b ^ o} _ h,
rwa ←succ_log_def... | theorem | ordinal.opow_log_le_self | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"cInf_le'",
"eq_or_ne",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_le_iff_le_log {b x c : ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x | ⟨λ h, le_of_not_lt $ λ hn,
(lt_opow_succ_log_self hb x).not_le $
((opow_le_opow_iff_right hb).2 (succ_le_of_lt hn)).trans h,
λ h, ((opow_le_opow_iff_right hb).2 h).trans (opow_log_le_self b hx)⟩ | theorem | ordinal.opow_le_iff_le_log | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | `opow b` and `log b` (almost) form a Galois connection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_opow_iff_log_lt {b x c : ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c | lt_iff_lt_of_le_iff_le (opow_le_iff_le_log hb hx) | theorem | ordinal.lt_opow_iff_log_lt | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"lt_iff_lt_of_le_iff_le",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_pos {b o : ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o | by rwa [←succ_le_iff, succ_zero, ←opow_le_iff_le_log hb ho, opow_one] | theorem | ordinal.log_pos | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_eq_zero {b o : ordinal} (hbo : o < b) : log b o = 0 | begin
rcases eq_or_ne o 0 with rfl | ho,
{ exact log_zero_right b },
cases le_or_lt b 1 with hb hb,
{ rcases le_one_iff.1 hb with rfl | rfl,
{ exact log_zero_left o },
{ exact log_one_left o } },
{ rwa [←ordinal.le_zero, ←lt_succ_iff, succ_zero, ←lt_opow_iff_log_lt hb ho, opow_one] }
end | theorem | ordinal.log_eq_zero | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"eq_or_ne",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_mono_right (b) {x y : ordinal} (xy : x ≤ y) : log b x ≤ log b y | if hx : x = 0 then by simp only [hx, log_zero_right, ordinal.zero_le] else
if hb : 1 < b then
(opow_le_iff_le_log hb (lt_of_lt_of_le (ordinal.pos_iff_ne_zero.2 hx) xy).ne').1 $
(opow_log_le_self _ hx).trans xy
else by simp only [log_of_not_one_lt_left hb, ordinal.zero_le] | theorem | ordinal.log_mono_right | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_le_self (b x : ordinal) : log b x ≤ x | if hx : x = 0 then by simp only [hx, log_zero_right, ordinal.zero_le] else
if hb : 1 < b then (right_le_opow _ hb).trans (opow_log_le_self b hx)
else by simp only [log_of_not_one_lt_left hb, ordinal.zero_le] | theorem | ordinal.log_le_self | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_one_right (b : ordinal) : log b 1 = 0 | if hb : 1 < b then log_eq_zero hb else log_of_not_one_lt_left hb 1 | theorem | ordinal.log_one_right | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_opow_log_lt_self (b : ordinal) {o : ordinal} (ho : o ≠ 0) : o % b ^ log b o < o | begin
rcases eq_or_ne b 0 with rfl | hb,
{ simpa using ordinal.pos_iff_ne_zero.2 ho },
{ exact (mod_lt _ $ opow_ne_zero _ hb).trans_le (opow_log_le_self _ ho) }
end | theorem | ordinal.mod_opow_log_lt_self | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"eq_or_ne",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_mod_opow_log_lt_log_self {b o : ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) :
log b (o % b ^ log b o) < log b o | begin
cases eq_or_ne (o % b ^ log b o) 0,
{ rw [h, log_zero_right],
apply log_pos hb ho hbo },
{ rw [←succ_le_iff, succ_log_def hb h],
apply cInf_le',
apply mod_lt,
rw ←ordinal.pos_iff_ne_zero,
exact opow_pos _ (zero_lt_one.trans hb) }
end | theorem | ordinal.log_mod_opow_log_lt_log_self | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"cInf_le'",
"eq_or_ne",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_mul_add_pos {b v : ordinal} (hb : b ≠ 0) (u) (hv : v ≠ 0) (w) : 0 < b ^ u * v + w | (opow_pos u $ ordinal.pos_iff_ne_zero.2 hb).trans_le $
(le_mul_left _ $ ordinal.pos_iff_ne_zero.2 hv).trans $ le_add_right _ _ | lemma | ordinal.opow_mul_add_pos | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"le_mul_left",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_mul_add_lt_opow_mul_succ {b u w : ordinal} (v : ordinal) (hw : w < b ^ u) :
b ^ u * v + w < b ^ u * (succ v) | by rwa [mul_succ, add_lt_add_iff_left] | lemma | ordinal.opow_mul_add_lt_opow_mul_succ | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_mul_add_lt_opow_succ {b u v w : ordinal} (hvb : v < b) (hw : w < b ^ u) :
b ^ u * v + w < b ^ (succ u) | begin
convert (opow_mul_add_lt_opow_mul_succ v hw).trans_le (mul_le_mul_left' (succ_le_of_lt hvb) _),
exact opow_succ b u
end | lemma | ordinal.opow_mul_add_lt_opow_succ | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"mul_le_mul_left'",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_opow_mul_add {b u v w : ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)
(hw : w < b ^ u) : log b (b ^ u * v + w) = u | begin
have hne' := (opow_mul_add_pos (zero_lt_one.trans hb).ne' u hv w).ne',
by_contra' hne,
cases lt_or_gt_of_ne hne with h h,
{ rw ←lt_opow_iff_log_lt hb hne' at h,
exact h.not_le ((le_mul_left _ (ordinal.pos_iff_ne_zero.2 hv)).trans (le_add_right _ _)) },
{ change _ < _ at h,
rw [←succ_le_iff, ←opo... | theorem | ordinal.log_opow_mul_add | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"le_mul_left",
"not_lt_of_le",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_opow {b : ordinal} (hb : 1 < b) (x : ordinal) : log b (b ^ x) = x | begin
convert log_opow_mul_add hb zero_ne_one.symm hb (opow_pos x (zero_lt_one.trans hb)),
rw [add_zero, mul_one]
end | theorem | ordinal.log_opow | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"mul_one",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_opow_log_pos (b : ordinal) {o : ordinal} (ho : o ≠ 0) : 0 < o / b ^ log b o | begin
rcases eq_zero_or_pos b with (rfl | hb),
{ simpa using ordinal.pos_iff_ne_zero.2 ho },
{ rw div_pos (opow_ne_zero _ hb.ne'),
exact opow_log_le_self b ho }
end | theorem | ordinal.div_opow_log_pos | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"div_pos",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_opow_log_lt {b : ordinal} (o : ordinal) (hb : 1 < b) : o / b ^ log b o < b | begin
rw [div_lt (opow_pos _ (zero_lt_one.trans hb)).ne', ←opow_succ],
exact lt_opow_succ_log_self hb o
end | theorem | ordinal.div_opow_log_lt | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_log_le_log_mul {x y : ordinal} (b : ordinal) (hx : x ≠ 0) (hy : y ≠ 0) :
log b x + log b y ≤ log b (x * y) | begin
by_cases hb : 1 < b,
{ rw [←opow_le_iff_le_log hb (mul_ne_zero hx hy), opow_add],
exact mul_le_mul' (opow_log_le_self b hx) (opow_log_le_self b hy) },
simp only [log_of_not_one_lt_left hb, zero_add]
end | theorem | ordinal.add_log_le_log_mul | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"mul_le_mul'",
"mul_ne_zero",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_opow (m : ℕ) : ∀ n : ℕ, ((pow m n : ℕ) : ordinal) = m ^ n | | 0 := by simp
| (n+1) := by rw [pow_succ', nat_cast_mul, nat_cast_opow, nat.cast_succ, add_one_eq_succ, opow_succ] | theorem | ordinal.nat_cast_opow | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"nat.cast_succ",
"ordinal",
"pow_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_opow_nat {o : ordinal} (ho : 0 < o) : sup (λ n : ℕ, o ^ n) = o ^ ω | begin
rcases lt_or_eq_of_le (one_le_iff_pos.2 ho) with ho₁ | rfl,
{ exact (opow_is_normal ho₁).apply_omega },
{ rw one_opow,
refine le_antisymm (sup_le (λ n, by rw one_opow)) _,
convert le_sup _ 0,
rw [nat.cast_zero, opow_zero] }
end | theorem | ordinal.sup_opow_nat | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [
"nat.cast_zero",
"ordinal",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
positivity_opow : expr → tactic strictness | | `(@has_pow.pow _ _ %%inst %%a %%b) := do
strictness_a ← core a,
match strictness_a with
| positive p := positive <$> mk_app ``opow_pos [b, p]
| _ := failed -- We already know that `0 ≤ x` for all `x : ordinal`
end
| _ := failed | def | tactic.positivity_opow | set_theory.ordinal | src/set_theory/ordinal/exponential.lean | [
"set_theory.ordinal.arithmetic"
] | [] | Extension for the `positivity` tactic: `ordinal.opow` takes positive values on positive inputs. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nfp_family (f : ι → ordinal → ordinal) (a) : ordinal | sup (list.foldr f a) | def | ordinal.nfp_family | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | The next common fixed point, at least `a`, for a family of normal functions.
This is defined for any family of functions, as the supremum of all values reachable by applying
finitely many functions in the family to `a`.
`ordinal.nfp_family_fp` shows this is a fixed point, `ordinal.le_nfp_family` shows it's at
least `... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nfp_family_eq_sup (f : ι → ordinal → ordinal) (a) :
nfp_family f a = sup (list.foldr f a) | rfl | theorem | ordinal.nfp_family_eq_sup | 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 | |
foldr_le_nfp_family (f : ι → ordinal → ordinal) (a l) :
list.foldr f a l ≤ nfp_family f a | le_sup _ _ | theorem | ordinal.foldr_le_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 | |
le_nfp_family (f : ι → ordinal → ordinal) (a) : a ≤ nfp_family f a | le_sup _ [] | theorem | ordinal.le_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 | |
lt_nfp_family {a b} : a < nfp_family f b ↔ ∃ l, a < list.foldr f b l | lt_sup | theorem | ordinal.lt_nfp_family | 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_family_le_iff {a b} : nfp_family f a ≤ b ↔ ∀ l, list.foldr f a l ≤ b | sup_le_iff | theorem | ordinal.nfp_family_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_family_le {a b} : (∀ l, list.foldr f a l ≤ b) → nfp_family f a ≤ b | sup_le | theorem | ordinal.nfp_family_le | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_family_monotone (hf : ∀ i, monotone (f i)) : monotone (nfp_family f) | λ a b h, sup_le $ λ l, (list.foldr_monotone hf l h).trans (le_sup _ l) | theorem | ordinal.nfp_family_monotone | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"list.foldr_monotone",
"monotone",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_lt_nfp_family (H : ∀ i, is_normal (f i)) {a b} (hb : b < nfp_family f a) (i) :
f i b < nfp_family f a | let ⟨l, hl⟩ := lt_nfp_family.1 hb in lt_sup.2 ⟨i :: l, (H i).strict_mono hl⟩ | theorem | ordinal.apply_lt_nfp_family | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_lt_nfp_family_iff [nonempty ι] (H : ∀ i, is_normal (f i)) {a b} :
(∀ i, f i b < nfp_family f a) ↔ b < nfp_family f a | ⟨λ h, lt_nfp_family.2 $ let ⟨l, hl⟩ := lt_sup.1 $ h $ classical.arbitrary ι in
⟨l, ((H _).self_le b).trans_lt hl⟩, apply_lt_nfp_family H⟩ | theorem | ordinal.apply_lt_nfp_family_iff | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"classical.arbitrary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_family_le_apply [nonempty ι] (H : ∀ i, is_normal (f i)) {a b} :
(∃ i, nfp_family f a ≤ f i b) ↔ nfp_family f a ≤ b | by { rw ←not_iff_not, push_neg, exact apply_lt_nfp_family_iff H } | theorem | ordinal.nfp_family_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_family_le_fp (H : ∀ i, monotone (f i)) {a b} (ab : a ≤ b) (h : ∀ i, f i b ≤ b) :
nfp_family f a ≤ b | sup_le $ λ l, begin
by_cases hι : is_empty ι,
{ resetI, rwa unique.eq_default l },
{ haveI := not_is_empty_iff.1 hι,
induction l with i l IH generalizing a, {exact ab},
exact (H i (IH ab)).trans (h i) }
end | theorem | ordinal.nfp_family_le_fp | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"is_empty",
"monotone",
"sup_le",
"unique.eq_default"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_family_fp {i} (H : is_normal (f i)) (a) : f i (nfp_family f a) = nfp_family f a | begin
unfold nfp_family,
rw @is_normal.sup _ H _ _ ⟨[]⟩,
apply le_antisymm;
refine ordinal.sup_le (λ l, _),
{ exact le_sup _ (i :: l) },
{ exact (H.self_le _).trans (le_sup _ _) }
end | theorem | ordinal.nfp_family_fp | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal.sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_le_nfp_family [hι : nonempty ι] {f : ι → ordinal → ordinal} (H : ∀ i, is_normal (f i))
{a b} : (∀ i, f i b ≤ nfp_family f a) ↔ b ≤ nfp_family f a | begin
refine ⟨λ h, _, λ h i, _⟩,
{ unfreezingI { cases hι with i },
exact ((H i).self_le b).trans (h i) },
rw ←nfp_family_fp (H i),
exact (H i).monotone h
end | theorem | ordinal.apply_le_nfp_family | 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 | |
nfp_family_eq_self {f : ι → ordinal → ordinal} {a} (h : ∀ i, f i a = a) :
nfp_family f a = a | le_antisymm (sup_le $ λ l, by rw list.foldr_fixed' h l) $ le_nfp_family f a | theorem | ordinal.nfp_family_eq_self | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"list.foldr_fixed'",
"ordinal",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fp_family_unbounded (H : ∀ i, is_normal (f i)) :
(⋂ i, function.fixed_points (f i)).unbounded (<) | λ a, ⟨_, λ s ⟨i, hi⟩, begin
rw ←hi,
exact nfp_family_fp (H i) a
end, (le_nfp_family f a).not_lt⟩ | theorem | ordinal.fp_family_unbounded | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"function.fixed_points"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_family (f : ι → ordinal → ordinal) (o : ordinal) : ordinal | limit_rec_on o (nfp_family f 0)
(λ a IH, nfp_family f (succ IH))
(λ a l, bsup.{(max u v) u} a) | def | ordinal.deriv_family | 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 for all functions such that `ordinal.deriv_family_zero`,
`ordinal.deriv_family_succ`, and `ordinal.deriv_family_limit` are satisfied. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
deriv_family_zero (f : ι → ordinal → ordinal) :
deriv_family f 0 = nfp_family f 0 | limit_rec_on_zero _ _ _ | theorem | ordinal.deriv_family_zero | 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_family_succ (f : ι → ordinal → ordinal) (o) :
deriv_family f (succ o) = nfp_family f (succ (deriv_family f o)) | limit_rec_on_succ _ _ _ _ | theorem | ordinal.deriv_family_succ | 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_family_limit (f : ι → ordinal → ordinal) {o} : is_limit o →
deriv_family f o = bsup.{(max u v) u} o (λ a _, deriv_family f a) | limit_rec_on_limit _ _ _ _ | theorem | ordinal.deriv_family_limit | 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_family_is_normal (f : ι → ordinal → ordinal) : is_normal (deriv_family f) | ⟨λ o, by rw [deriv_family_succ, ← succ_le_iff]; apply le_nfp_family,
λ o l a, by rw [deriv_family_limit _ l, bsup_le_iff]⟩ | theorem | ordinal.deriv_family_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_family_fp {i} (H : is_normal (f i)) (o : ordinal.{max u v}) :
f i (deriv_family f o) = deriv_family f o | begin
refine limit_rec_on o _ (λ o IH, _) _,
{ rw [deriv_family_zero], exact nfp_family_fp H 0 },
{ rw [deriv_family_succ], exact nfp_family_fp H _ },
{ intros o l IH,
rw [deriv_family_limit _ l,
is_normal.bsup.{(max u v) u (max u v)} H (λ a _, deriv_family f a) l.1],
refine eq_of_forall_ge_iff (λ... | theorem | ordinal.deriv_family_fp | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"eq_of_forall_ge_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_deriv_family (H : ∀ i, is_normal (f i)) {a} :
(∀ i, f i a ≤ a) ↔ ∃ o, deriv_family f o = a | ⟨λ ha, begin
suffices : ∀ o (_ : a ≤ deriv_family f o), ∃ o, deriv_family f o = a,
from this a ((deriv_family_is_normal _).self_le _),
refine λ o, limit_rec_on o (λ h₁, ⟨0, le_antisymm _ h₁⟩) (λ o IH h₁, _) (λ o l IH h₁, _),
{ rw deriv_family_zero,
exact nfp_family_le_fp (λ i, (H i).monotone) (ordinal.zero_... | theorem | ordinal.le_iff_deriv_family | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"eq_or_lt_of_le",
"monotone",
"not_ball",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fp_iff_deriv_family (H : ∀ i, is_normal (f i)) {a} :
(∀ i, f i a = a) ↔ ∃ o, deriv_family f o = a | iff.trans ⟨λ h i, le_of_eq (h i), λ h i, (H i).le_iff_eq.1 (h i)⟩ (le_iff_deriv_family H) | theorem | ordinal.fp_iff_deriv_family | 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_family_eq_enum_ord (H : ∀ i, is_normal (f i)) :
deriv_family f = enum_ord (⋂ i, function.fixed_points (f i)) | begin
rw ←eq_enum_ord _ (fp_family_unbounded H),
use (deriv_family_is_normal f).strict_mono,
rw set.range_eq_iff,
refine ⟨_, λ a ha, _⟩,
{ rintros a S ⟨i, hi⟩,
rw ←hi,
exact deriv_family_fp (H i) a },
rw set.mem_Inter at ha,
rwa ←fp_iff_deriv_family H
end | theorem | ordinal.deriv_family_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_Inter",
"set.range_eq_iff",
"strict_mono"
] | For a family of normal functions, `ordinal.deriv_family` enumerates the common fixed points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nfp_bfamily (o : ordinal) (f : Π b < o, ordinal → ordinal) : ordinal → ordinal | nfp_family (family_of_bfamily o f) | def | ordinal.nfp_bfamily | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | The next common fixed point, at least `a`, for a family of normal functions indexed by ordinals.
This is defined as `ordinal.nfp_family` of the type-indexed family associated to `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nfp_bfamily_eq_nfp_family {o : ordinal} (f : Π b < o, ordinal → ordinal) :
nfp_bfamily o f = nfp_family (family_of_bfamily o f) | rfl | theorem | ordinal.nfp_bfamily_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 | |
foldr_le_nfp_bfamily {o : ordinal} (f : Π b < o, ordinal → ordinal) (a l) :
list.foldr (family_of_bfamily o f) a l ≤ nfp_bfamily o f a | le_sup _ _ | theorem | ordinal.foldr_le_nfp_bfamily | 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_nfp_bfamily {o : ordinal} (f : Π b < o, ordinal → ordinal) (a) :
a ≤ nfp_bfamily o f a | le_sup _ [] | theorem | ordinal.le_nfp_bfamily | 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 | |
lt_nfp_bfamily {a b} :
a < nfp_bfamily o f b ↔ ∃ l, a < list.foldr (family_of_bfamily o f) b l | lt_sup | theorem | ordinal.lt_nfp_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 | |
nfp_bfamily_le_iff {o : ordinal} {f : Π b < o, ordinal → ordinal} {a b} :
nfp_bfamily o f a ≤ b ↔ ∀ l, list.foldr (family_of_bfamily o f) a l ≤ b | sup_le_iff | theorem | ordinal.nfp_bfamily_le_iff | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_bfamily_le {o : ordinal} {f : Π b < o, ordinal → ordinal} {a b} :
(∀ l, list.foldr (family_of_bfamily o f) a l ≤ b) → nfp_bfamily o f a ≤ b | sup_le | theorem | ordinal.nfp_bfamily_le | set_theory.ordinal | src/set_theory/ordinal/fixed_point.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_bfamily_monotone (hf : ∀ i hi, monotone (f i hi)) : monotone (nfp_bfamily o f) | nfp_family_monotone (λ i, hf _ _) | theorem | ordinal.nfp_bfamily_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 | |
apply_lt_nfp_bfamily (H : ∀ i hi, is_normal (f i hi)) {a b} (hb : b < nfp_bfamily o f a)
(i hi) : f i hi b < nfp_bfamily o f a | begin
rw ←family_of_bfamily_enum o f,
apply apply_lt_nfp_family _ hb,
exact λ _, H _ _
end | theorem | ordinal.apply_lt_nfp_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 | |
apply_lt_nfp_bfamily_iff (ho : o ≠ 0) (H : ∀ i hi, is_normal (f i hi)) {a b} :
(∀ i hi, f i hi b < nfp_bfamily o f a) ↔ b < nfp_bfamily o f a | ⟨λ h, begin
haveI := out_nonempty_iff_ne_zero.2 ho,
refine (apply_lt_nfp_family_iff _).1 (λ _, h _ _),
exact λ _, H _ _,
end, apply_lt_nfp_bfamily H⟩ | theorem | ordinal.apply_lt_nfp_bfamily_iff | 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_bfamily_le_apply (ho : o ≠ 0) (H : ∀ i hi, is_normal (f i hi)) {a b} :
(∃ i hi, nfp_bfamily o f a ≤ f i hi b) ↔ nfp_bfamily o f a ≤ b | by { rw ←not_iff_not, push_neg, convert apply_lt_nfp_bfamily_iff ho H, simp only [not_le] } | theorem | ordinal.nfp_bfamily_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_bfamily_le_fp (H : ∀ i hi, monotone (f i hi)) {a b} (ab : a ≤ b)
(h : ∀ i hi, f i hi b ≤ b) : nfp_bfamily o f a ≤ b | nfp_family_le_fp (λ _, H _ _) ab (λ i, h _ _) | theorem | ordinal.nfp_bfamily_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 | |
nfp_bfamily_fp {i hi} (H : is_normal (f i hi)) (a) :
f i hi (nfp_bfamily o f a) = nfp_bfamily o f a | by { rw ←family_of_bfamily_enum o f, apply nfp_family_fp, rw family_of_bfamily_enum, exact H } | theorem | ordinal.nfp_bfamily_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 | |
apply_le_nfp_bfamily (ho : o ≠ 0) (H : ∀ i hi, is_normal (f i hi)) {a b} :
(∀ i hi, f i hi b ≤ nfp_bfamily o f a) ↔ b ≤ nfp_bfamily o f a | begin
refine ⟨λ h, _, λ h i hi, _⟩,
{ have ho' : 0 < o := ordinal.pos_iff_ne_zero.2 ho,
exact ((H 0 ho').self_le b).trans (h 0 ho') },
{ rw ←nfp_bfamily_fp (H i hi),
exact (H i hi).monotone h }
end | theorem | ordinal.apply_le_nfp_bfamily | 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 |
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