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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of_state_aux : Π (n : ℕ) (s : S) (h : turn_bound s ≤ n), pgame | | 0 s h := pgame.mk {t // t ∈ L s} {t // t ∈ R s}
(λ t, begin exfalso, exact turn_bound_ne_zero_of_left_move t.2 (nonpos_iff_eq_zero.mp h) end)
(λ t, begin exfalso, exact turn_bound_ne_zero_of_right_move t.2 (nonpos_iff_eq_zero.mp h) end)
| (n+1) s h :=
pgame.mk {t // t ∈ L s} {t // t ∈ R s}
(λ t, of_... | def | pgame.of_state_aux | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [
"pgame"
] | Construct a `pgame` from a state and a (not necessarily optimal) bound on the number of
turns remaining. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_state_aux_relabelling : Π (s : S) (n m : ℕ) (hn : turn_bound s ≤ n) (hm : turn_bound s ≤ m),
relabelling (of_state_aux n s hn) (of_state_aux m s hm) | | s 0 0 hn hm :=
begin
dsimp [pgame.of_state_aux],
fsplit, refl, refl,
{ intro i, dsimp at i, exfalso,
exact turn_bound_ne_zero_of_left_move i.2 (nonpos_iff_eq_zero.mp hn) },
{ intro j, dsimp at j, exfalso,
exact turn_bound_ne_zero_of_right_move j.2 (nonpos_iff_eq_zero.mp hm) }
end
| s 0... | def | pgame.of_state_aux_relabelling | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [
"pgame.of_state_aux"
] | Two different (valid) turn bounds give equivalent games. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_state (s : S) : pgame | of_state_aux (turn_bound s) s (refl _) | def | pgame.of_state | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [
"pgame"
] | Construct a combinatorial `pgame` from a state. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_moves_of_state_aux (n : ℕ) {s : S} (h : turn_bound s ≤ n) :
left_moves (of_state_aux n s h) ≃ {t // t ∈ L s} | by induction n; refl | def | pgame.left_moves_of_state_aux | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [] | The equivalence between `left_moves` for a `pgame` constructed using `of_state_aux _ s _`, and
`L s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_moves_of_state (s : S) : left_moves (of_state s) ≃ {t // t ∈ L s} | left_moves_of_state_aux _ _ | def | pgame.left_moves_of_state | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [] | The equivalence between `left_moves` for a `pgame` constructed using `of_state s`, and `L s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_moves_of_state_aux (n : ℕ) {s : S} (h : turn_bound s ≤ n) :
right_moves (of_state_aux n s h) ≃ {t // t ∈ R s} | by induction n; refl | def | pgame.right_moves_of_state_aux | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [] | The equivalence between `right_moves` for a `pgame` constructed using `of_state_aux _ s _`, and
`R s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_moves_of_state (s : S) : right_moves (of_state s) ≃ {t // t ∈ R s} | right_moves_of_state_aux _ _ | def | pgame.right_moves_of_state | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [] | The equivalence between `right_moves` for a `pgame` constructed using `of_state s`, and
`R s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
relabelling_move_left_aux (n : ℕ) {s : S} (h : turn_bound s ≤ n)
(t : left_moves (of_state_aux n s h)) :
relabelling
(move_left (of_state_aux n s h) t)
(of_state_aux (n-1) (((left_moves_of_state_aux n h) t) : S)
((turn_bound_of_left ((left_moves_of_state_aux n h) t).2 (n-1)
(nat.le_trans h le_... | begin
induction n,
{ have t' := (left_moves_of_state_aux 0 h) t,
exfalso, exact turn_bound_ne_zero_of_left_move t'.2 (nonpos_iff_eq_zero.mp h), },
{ refl },
end | def | pgame.relabelling_move_left_aux | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [
"le_tsub_add"
] | The relabelling showing `move_left` applied to a game constructed using `of_state_aux`
has itself been constructed using `of_state_aux`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
relabelling_move_left (s : S) (t : left_moves (of_state s)) :
relabelling
(move_left (of_state s) t)
(of_state (((left_moves_of_state s).to_fun t) : S)) | begin
transitivity,
apply relabelling_move_left_aux,
apply of_state_aux_relabelling,
end | def | pgame.relabelling_move_left | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [] | The relabelling showing `move_left` applied to a game constructed using `of`
has itself been constructed using `of`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
relabelling_move_right_aux (n : ℕ) {s : S} (h : turn_bound s ≤ n)
(t : right_moves (of_state_aux n s h)) :
relabelling
(move_right (of_state_aux n s h) t)
(of_state_aux (n-1) (((right_moves_of_state_aux n h) t) : S)
((turn_bound_of_right ((right_moves_of_state_aux n h) t).2 (n-1)
(nat.le_trans... | begin
induction n,
{ have t' := (right_moves_of_state_aux 0 h) t,
exfalso, exact turn_bound_ne_zero_of_right_move t'.2 (nonpos_iff_eq_zero.mp h), },
{ refl },
end | def | pgame.relabelling_move_right_aux | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [
"le_tsub_add"
] | The relabelling showing `move_right` applied to a game constructed using `of_state_aux`
has itself been constructed using `of_state_aux`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
relabelling_move_right (s : S) (t : right_moves (of_state s)) :
relabelling
(move_right (of_state s) t)
(of_state (((right_moves_of_state s).to_fun t) : S)) | begin
transitivity,
apply relabelling_move_right_aux,
apply of_state_aux_relabelling,
end | def | pgame.relabelling_move_right | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [] | The relabelling showing `move_right` applied to a game constructed using `of`
has itself been constructed using `of`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fintype_left_moves_of_state_aux (n : ℕ) (s : S) (h : turn_bound s ≤ n) :
fintype (left_moves (of_state_aux n s h)) | begin
apply fintype.of_equiv _ (left_moves_of_state_aux _ _).symm,
apply_instance,
end | instance | pgame.fintype_left_moves_of_state_aux | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [
"fintype",
"fintype.of_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fintype_right_moves_of_state_aux (n : ℕ) (s : S) (h : turn_bound s ≤ n) :
fintype (right_moves (of_state_aux n s h)) | begin
apply fintype.of_equiv _ (right_moves_of_state_aux _ _).symm,
apply_instance,
end | instance | pgame.fintype_right_moves_of_state_aux | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [
"fintype",
"fintype.of_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
short_of_state_aux : Π (n : ℕ) {s : S} (h : turn_bound s ≤ n), short (of_state_aux n s h) | | 0 s h :=
short.mk'
(λ i, begin
have i := (left_moves_of_state_aux _ _).to_fun i,
exfalso,
exact turn_bound_ne_zero_of_left_move i.2 (nonpos_iff_eq_zero.mp h),
end)
(λ j, begin
have j := (right_moves_of_state_aux _ _).to_fun j,
exfalso,
exact turn_bound_ne_zero_of_right_move j.2 (nonpos... | instance | pgame.short_of_state_aux | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
short_of_state (s : S) : short (of_state s) | begin
dsimp [pgame.of_state],
apply_instance
end | instance | pgame.short_of_state | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [
"pgame.of_state"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_state {S : Type u} [pgame.state S] (s : S) : game | ⟦pgame.of_state s⟧ | def | game.of_state | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [
"game",
"pgame.state"
] | Construct a combinatorial `game` from a state. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_add (a b) : lift (a + b) = lift a + lift b | quotient.induction_on₂ a b $ λ ⟨α, r, _⟩ ⟨β, s, _⟩,
quotient.sound ⟨(rel_iso.preimage equiv.ulift _).trans
(rel_iso.sum_lex_congr (rel_iso.preimage equiv.ulift _)
(rel_iso.preimage equiv.ulift _)).symm⟩ | theorem | ordinal.lift_add | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"equiv.ulift",
"lift",
"rel_iso.preimage",
"rel_iso.sum_lex_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_succ (a) : lift (succ a) = succ (lift a) | by { rw [←add_one_eq_succ, lift_add, lift_one], refl } | theorem | ordinal.lift_succ | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_contravariant_class_le : contravariant_class ordinal.{u} ordinal.{u} (+) (≤) | ⟨λ a b c, induction_on a $ λ α r hr, induction_on b $ λ β₁ s₁ hs₁, induction_on c $ λ β₂ s₂ hs₂ ⟨f⟩,
⟨have fl : ∀ a, f (sum.inl a) = sum.inl a := λ a,
by simpa only [initial_seg.trans_apply, initial_seg.le_add_apply]
using @initial_seg.eq _ _ _ _ (@sum.lex.is_well_order _ _ _ _ hr hs₂)
((initial_seg... | instance | ordinal.add_contravariant_class_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"contravariant_class",
"initial_seg.coe_fn_to_rel_embedding",
"initial_seg.eq",
"initial_seg.le_add",
"initial_seg.le_add_apply",
"initial_seg.trans_apply",
"rel_embedding.coe_fn_to_embedding",
"rel_embedding.map_rel_iff",
"sum.lex_inr_inr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left_cancel (a) {b c : ordinal} : a + b = a + c ↔ b = c | by simp only [le_antisymm_iff, add_le_add_iff_left] | theorem | ordinal.add_left_cancel | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lt_add_iff_left' (a) {b c : ordinal} : a + b < a + c ↔ b < c | by rw [← not_le, ← not_le, add_le_add_iff_left] | theorem | ordinal.add_lt_add_iff_left' | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_covariant_class_lt : covariant_class ordinal.{u} ordinal.{u} (+) (<) | ⟨λ a b c, (add_lt_add_iff_left' a).2⟩ | instance | ordinal.add_covariant_class_lt | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_contravariant_class_lt : contravariant_class ordinal.{u} ordinal.{u} (+) (<) | ⟨λ a b c, (add_lt_add_iff_left' a).1⟩ | instance | ordinal.add_contravariant_class_lt | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"contravariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_swap_contravariant_class_lt :
contravariant_class ordinal.{u} ordinal.{u} (swap (+)) (<) | ⟨λ a b c, lt_imp_lt_of_le_imp_le (λ h, add_le_add_right h _)⟩ | instance | ordinal.add_swap_contravariant_class_lt | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"contravariant_class",
"lt_imp_lt_of_le_imp_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_add_iff_right {a b : ordinal} : ∀ n : ℕ, a + n ≤ b + n ↔ a ≤ b | | 0 := by simp
| (n+1) := by rw [nat_cast_succ, add_succ, add_succ, succ_le_succ_iff, add_le_add_iff_right] | theorem | ordinal.add_le_add_iff_right | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_right_cancel {a b : ordinal} (n : ℕ) : a + n = b + n ↔ a = b | by simp only [le_antisymm_iff, add_le_add_iff_right] | theorem | ordinal.add_right_cancel | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_zero_iff {a b : ordinal} : a + b = 0 ↔ (a = 0 ∧ b = 0) | induction_on a $ λ α r _, induction_on b $ λ β s _, begin
simp_rw [←type_sum_lex, type_eq_zero_iff_is_empty],
exact is_empty_sum
end | theorem | ordinal.add_eq_zero_iff | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_empty_sum",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_eq_zero_of_add_eq_zero {a b : ordinal} (h : a + b = 0) : a = 0 | (add_eq_zero_iff.1 h).1 | theorem | ordinal.left_eq_zero_of_add_eq_zero | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_eq_zero_of_add_eq_zero {a b : ordinal} (h : a + b = 0) : b = 0 | (add_eq_zero_iff.1 h).2 | theorem | ordinal.right_eq_zero_of_add_eq_zero | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pred (o : ordinal) : ordinal | if h : ∃ a, o = succ a then classical.some h else o | def | ordinal.pred | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | The ordinal predecessor of `o` is `o'` if `o = succ o'`,
and `o` otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pred_succ (o) : pred (succ o) = o | by have h : ∃ a, succ o = succ a := ⟨_, rfl⟩;
simpa only [pred, dif_pos h] using (succ_injective $ classical.some_spec h).symm | theorem | ordinal.pred_succ | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pred_le_self (o) : pred o ≤ o | if h : ∃ a, o = succ a then let ⟨a, e⟩ := h in
by rw [e, pred_succ]; exact le_succ a
else by rw [pred, dif_neg h] | theorem | ordinal.pred_le_self | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pred_eq_iff_not_succ {o} : pred o = o ↔ ¬ ∃ a, o = succ a | ⟨λ e ⟨a, e'⟩, by rw [e', pred_succ] at e; exact (lt_succ a).ne e,
λ h, dif_neg h⟩ | theorem | ordinal.pred_eq_iff_not_succ | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pred_eq_iff_not_succ' {o} : pred o = o ↔ ∀ a, o ≠ succ a | by simpa using pred_eq_iff_not_succ | theorem | ordinal.pred_eq_iff_not_succ' | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pred_lt_iff_is_succ {o} : pred o < o ↔ ∃ a, o = succ a | iff.trans (by simp only [le_antisymm_iff, pred_le_self, true_and, not_le])
(iff_not_comm.1 pred_eq_iff_not_succ).symm | theorem | ordinal.pred_lt_iff_is_succ | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pred_zero : pred 0 = 0 | pred_eq_iff_not_succ'.2 $ λ a, (succ_ne_zero a).symm | theorem | ordinal.pred_zero | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_pred_iff_is_succ {o} : succ (pred o) = o ↔ ∃ a, o = succ a | ⟨λ e, ⟨_, e.symm⟩, λ ⟨a, e⟩, by simp only [e, pred_succ]⟩ | theorem | ordinal.succ_pred_iff_is_succ | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_lt_of_not_succ {o b : ordinal} (h : ¬ ∃ a, o = succ a) : succ b < o ↔ b < o | ⟨(lt_succ b).trans, λ l, lt_of_le_of_ne (succ_le_of_lt l) (λ e, h ⟨_, e.symm⟩)⟩ | theorem | ordinal.succ_lt_of_not_succ | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_pred {a b} : a < pred b ↔ succ a < b | if h : ∃ a, b = succ a then let ⟨c, e⟩ := h in
by rw [e, pred_succ, succ_lt_succ_iff]
else by simp only [pred, dif_neg h, succ_lt_of_not_succ h] | theorem | ordinal.lt_pred | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pred_le {a b} : pred a ≤ b ↔ a ≤ succ b | le_iff_le_iff_lt_iff_lt.2 lt_pred | theorem | ordinal.pred_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_is_succ {o} : (∃ a, lift o = succ a) ↔ (∃ a, o = succ a) | ⟨λ ⟨a, h⟩,
let ⟨b, e⟩ := lift_down $ show a ≤ lift o, from le_of_lt $
h.symm ▸ lt_succ a in
⟨b, lift_inj.1 $ by rw [h, ← e, lift_succ]⟩,
λ ⟨a, h⟩, ⟨lift a, by simp only [h, lift_succ]⟩⟩ | theorem | ordinal.lift_is_succ | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_pred (o) : lift (pred o) = pred (lift o) | if h : ∃ a, o = succ a then
by cases h with a e; simp only [e, pred_succ, lift_succ]
else by rw [pred_eq_iff_not_succ.2 h,
pred_eq_iff_not_succ.2 (mt lift_is_succ.1 h)] | theorem | ordinal.lift_pred | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit (o : ordinal) : Prop | o ≠ 0 ∧ ∀ a < o, succ a < o | def | ordinal.is_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | A limit ordinal is an ordinal which is not zero and not a successor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit.succ_lt {o a : ordinal} (h : is_limit o) : a < o → succ a < o | h.2 a | theorem | ordinal.is_limit.succ_lt | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_zero_is_limit : ¬ is_limit 0 | | ⟨h, _⟩ := h rfl | theorem | ordinal.not_zero_is_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_succ_is_limit (o) : ¬ is_limit (succ o) | | ⟨_, h⟩ := lt_irrefl _ (h _ (lt_succ o)) | theorem | ordinal.not_succ_is_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_succ_of_is_limit {o} (h : is_limit o) : ¬ ∃ a, o = succ a | | ⟨a, e⟩ := not_succ_is_limit a (e ▸ h) | theorem | ordinal.not_succ_of_is_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_lt_of_is_limit {o a : ordinal} (h : is_limit o) : succ a < o ↔ a < o | ⟨(lt_succ a).trans, h.2 _⟩ | theorem | ordinal.succ_lt_of_is_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_succ_of_is_limit {o} (h : is_limit o) {a} : o ≤ succ a ↔ o ≤ a | le_iff_le_iff_lt_iff_lt.2 $ succ_lt_of_is_limit h | theorem | ordinal.le_succ_of_is_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_le {o} (h : is_limit o) {a} : o ≤ a ↔ ∀ x < o, x ≤ a | ⟨λ h x l, l.le.trans h,
λ H, (le_succ_of_is_limit h).1 $ le_of_not_lt $ λ hn,
not_lt_of_le (H _ hn) (lt_succ a)⟩ | theorem | ordinal.limit_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"not_lt_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_limit {o} (h : is_limit o) {a} : a < o ↔ ∃ x < o, a < x | by simpa only [not_ball, not_le] using not_congr (@limit_le _ h a) | theorem | ordinal.lt_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"not_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_is_limit (o) : is_limit (lift o) ↔ is_limit o | and_congr (not_congr $ by simpa only [lift_zero] using @lift_inj o 0)
⟨λ H a h, lift_lt.1 $ by simpa only [lift_succ] using H _ (lift_lt.2 h),
λ H a h, begin
obtain ⟨a', rfl⟩ := lift_down h.le,
rw [←lift_succ, lift_lt],
exact H a' (lift_lt.1 h)
end⟩ | theorem | ordinal.lift_is_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit.pos {o : ordinal} (h : is_limit o) : 0 < o | lt_of_le_of_ne (ordinal.zero_le _) h.1.symm | theorem | ordinal.is_limit.pos | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit.one_lt {o : ordinal} (h : is_limit o) : 1 < o | by simpa only [succ_zero] using h.2 _ h.pos | theorem | ordinal.is_limit.one_lt | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit.nat_lt {o : ordinal} (h : is_limit o) : ∀ n : ℕ, (n : ordinal) < o | | 0 := h.pos
| (n+1) := h.2 _ (is_limit.nat_lt n) | theorem | ordinal.is_limit.nat_lt | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_or_succ_or_limit (o : ordinal) :
o = 0 ∨ (∃ a, o = succ a) ∨ is_limit o | if o0 : o = 0 then or.inl o0 else
if h : ∃ a, o = succ a then or.inr (or.inl h) else
or.inr $ or.inr ⟨o0, λ a, (succ_lt_of_not_succ h).2⟩ | theorem | ordinal.zero_or_succ_or_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_rec_on {C : ordinal → Sort*}
(o : ordinal) (H₁ : C 0) (H₂ : ∀ o, C o → C (succ o))
(H₃ : ∀ o, is_limit o → (∀ o' < o, C o') → C o) : C o | lt_wf.fix (λ o IH,
if o0 : o = 0 then by rw o0; exact H₁ else
if h : ∃ a, o = succ a then
by rw ← succ_pred_iff_is_succ.2 h; exact
H₂ _ (IH _ $ pred_lt_iff_is_succ.2 h)
else H₃ _ ⟨o0, λ a, (succ_lt_of_not_succ h).2⟩ IH) o | def | ordinal.limit_rec_on | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | Main induction principle of ordinals: if one can prove a property by
induction at successor ordinals and at limit ordinals, then it holds for all ordinals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_rec_on_zero {C} (H₁ H₂ H₃) : @limit_rec_on C 0 H₁ H₂ H₃ = H₁ | by rw [limit_rec_on, lt_wf.fix_eq, dif_pos rfl]; refl | theorem | ordinal.limit_rec_on_zero | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_rec_on_succ {C} (o H₁ H₂ H₃) :
@limit_rec_on C (succ o) H₁ H₂ H₃ = H₂ o (@limit_rec_on C o H₁ H₂ H₃) | begin
have h : ∃ a, succ o = succ a := ⟨_, rfl⟩,
rw [limit_rec_on, lt_wf.fix_eq, dif_neg (succ_ne_zero o), dif_pos h],
generalize : limit_rec_on._proof_2 (succ o) h = h₂,
generalize : limit_rec_on._proof_3 (succ o) h = h₃,
revert h₂ h₃, generalize e : pred (succ o) = o', intros,
rw pred_succ at e, subst o',... | theorem | ordinal.limit_rec_on_succ | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_rec_on_limit {C} (o H₁ H₂ H₃ h) :
@limit_rec_on C o H₁ H₂ H₃ = H₃ o h (λ x h, @limit_rec_on C x H₁ H₂ H₃) | by rw [limit_rec_on, lt_wf.fix_eq, dif_neg h.1, dif_neg (not_succ_of_is_limit h)]; refl | theorem | ordinal.limit_rec_on_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_top_out_succ (o : ordinal) : order_top (succ o).out.α | ⟨_, le_enum_succ⟩ | instance | ordinal.order_top_out_succ | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"order_top",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
enum_succ_eq_top {o : ordinal} :
enum (<) o (by { rw type_lt, exact lt_succ o }) = (⊤ : (succ o).out.α) | rfl | theorem | ordinal.enum_succ_eq_top | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_succ_of_type_succ_lt {α} {r : α → α → Prop} [wo : is_well_order α r]
(h : ∀ a < type r, succ a < type r) (x : α) : ∃ y, r x y | begin
use enum r (succ (typein r x)) (h _ (typein_lt_type r x)),
convert (enum_lt_enum (typein_lt_type r x) _).mpr (lt_succ _), rw [enum_typein]
end | lemma | ordinal.has_succ_of_type_succ_lt | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
out_no_max_of_succ_lt {o : ordinal} (ho : ∀ a < o, succ a < o) : no_max_order o.out.α | ⟨has_succ_of_type_succ_lt (by rwa type_lt)⟩ | theorem | ordinal.out_no_max_of_succ_lt | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"no_max_order",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_singleton {r : α → α → Prop} [is_well_order α r] (hr : (type r).is_limit) (x) :
bounded r {x} | begin
refine ⟨enum r (succ (typein r x)) (hr.2 _ (typein_lt_type r x)), _⟩,
intros b hb,
rw mem_singleton_iff.1 hb,
nth_rewrite 0 ←enum_typein r x,
rw @enum_lt_enum _ r,
apply lt_succ
end | lemma | ordinal.bounded_singleton | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type_subrel_lt (o : ordinal.{u}) :
type (subrel (<) {o' : ordinal | o' < o}) = ordinal.lift.{u+1} o | begin
refine quotient.induction_on o _,
rintro ⟨α, r, wo⟩, resetI, apply quotient.sound,
constructor, symmetry, refine (rel_iso.preimage equiv.ulift r).trans (enum_iso r).symm
end | lemma | ordinal.type_subrel_lt | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"equiv.ulift",
"ordinal",
"rel_iso.preimage",
"subrel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_initial_seg (o : ordinal.{u}) :
#{o' : ordinal | o' < o} = cardinal.lift.{u+1} o.card | by rw [lift_card, ←type_subrel_lt, card_type] | lemma | ordinal.mk_initial_seg | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal (f : ordinal → ordinal) : Prop | (∀ o, f o < f (succ o)) ∧ ∀ o, is_limit o → ∀ a, f o ≤ a ↔ ∀ b < o, f b ≤ a | def | ordinal.is_normal | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | A normal ordinal function is a strictly increasing function which is
order-continuous, i.e., the image `f o` of a limit ordinal `o` is the sup of `f a` for
`a < o`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_normal.limit_le {f} (H : is_normal f) : ∀ {o}, is_limit o →
∀ {a}, f o ≤ a ↔ ∀ b < o, f b ≤ a | H.2 | theorem | ordinal.is_normal.limit_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.limit_lt {f} (H : is_normal f) {o} (h : is_limit o) {a} :
a < f o ↔ ∃ b < o, a < f b | not_iff_not.1 $ by simpa only [exists_prop, not_exists, not_and, not_lt] using H.2 _ h a | theorem | ordinal.is_normal.limit_lt | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"exists_prop",
"not_and",
"not_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.strict_mono {f} (H : is_normal f) : strict_mono f | λ a b, limit_rec_on b (not.elim (not_lt_of_le $ ordinal.zero_le _))
(λ b IH h, (lt_or_eq_of_le (le_of_lt_succ h)).elim
(λ h, (IH h).trans (H.1 _))
(λ e, e ▸ H.1 _))
(λ b l IH h, lt_of_lt_of_le (H.1 a)
((H.2 _ l _).1 le_rfl _ (l.2 _ h))) | theorem | ordinal.is_normal.strict_mono | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"le_rfl",
"not.elim",
"not_lt_of_le",
"ordinal.zero_le",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.monotone {f} (H : is_normal f) : monotone f | H.strict_mono.monotone | theorem | ordinal.is_normal.monotone | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal_iff_strict_mono_limit (f : ordinal → ordinal) :
is_normal f ↔ (strict_mono f ∧ ∀ o, is_limit o → ∀ a, (∀ b < o, f b ≤ a) → f o ≤ a) | ⟨λ hf, ⟨hf.strict_mono, λ a ha c, (hf.2 a ha c).2⟩, λ ⟨hs, hl⟩, ⟨λ a, hs (lt_succ a),
λ a ha c, ⟨λ hac b hba, ((hs hba).trans_le hac).le, hl a ha c⟩⟩⟩ | theorem | ordinal.is_normal_iff_strict_mono_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.lt_iff {f} (H : is_normal f) {a b} : f a < f b ↔ a < b | strict_mono.lt_iff_lt $ H.strict_mono | theorem | ordinal.is_normal.lt_iff | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"strict_mono.lt_iff_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.le_iff {f} (H : is_normal f) {a b} : f a ≤ f b ↔ a ≤ b | le_iff_le_iff_lt_iff_lt.2 H.lt_iff | theorem | ordinal.is_normal.le_iff | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.inj {f} (H : is_normal f) {a b} : f a = f b ↔ a = b | by simp only [le_antisymm_iff, H.le_iff] | theorem | ordinal.is_normal.inj | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.self_le {f} (H : is_normal f) (a) : a ≤ f a | lt_wf.self_le_of_strict_mono H.strict_mono a | theorem | ordinal.is_normal.self_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.le_set {f o} (H : is_normal f) (p : set ordinal) (p0 : p.nonempty) (b)
(H₂ : ∀ o, b ≤ o ↔ ∀ a ∈ p, a ≤ o) : f b ≤ o ↔ ∀ a ∈ p, f a ≤ o | ⟨λ h a pa, (H.le_iff.2 ((H₂ _).1 le_rfl _ pa)).trans h,
λ h, begin
revert H₂, refine limit_rec_on b (λ H₂, _) (λ S _ H₂, _) (λ S L _ H₂, (H.2 _ L _).2 (λ a h', _)),
{ cases p0 with x px,
have := ordinal.le_zero.1 ((H₂ _).1 (ordinal.zero_le _) _ px),
rw this at px, exact h _ px },
{ rcases not_ball.1 (mt (... | theorem | ordinal.is_normal.le_set | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"le_rfl",
"ordinal",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.le_set' {f o} (H : is_normal f) (p : set α) (p0 : p.nonempty) (g : α → ordinal)
(b) (H₂ : ∀ o, b ≤ o ↔ ∀ a ∈ p, g a ≤ o) : f b ≤ o ↔ ∀ a ∈ p, f (g a) ≤ o | by simpa [H₂] using H.le_set (g '' p) (p0.image g) b | theorem | ordinal.is_normal.le_set' | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.refl : is_normal id | ⟨lt_succ, λ o l a, limit_le l⟩ | theorem | ordinal.is_normal.refl | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.trans {f g} (H₁ : is_normal f) (H₂ : is_normal g) : is_normal (f ∘ g) | ⟨λ x, H₁.lt_iff.2 (H₂.1 _), λ o l a, H₁.le_set' (< o) ⟨_, l.pos⟩ g _ (λ c, H₂.2 _ l _)⟩ | theorem | ordinal.is_normal.trans | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.is_limit {f} (H : is_normal f) {o} (l : is_limit o) : is_limit (f o) | ⟨ne_of_gt $ (ordinal.zero_le _).trans_lt $ H.lt_iff.2 l.pos,
λ a h, let ⟨b, h₁, h₂⟩ := (H.limit_lt l).1 h in
(succ_le_of_lt h₂).trans_lt (H.lt_iff.2 h₁)⟩ | theorem | ordinal.is_normal.is_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.le_iff_eq {f} (H : is_normal f) {a} : f a ≤ a ↔ f a = a | (H.self_le a).le_iff_eq | theorem | ordinal.is_normal.le_iff_eq | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_of_limit {a b c : ordinal} (h : is_limit b) : a + b ≤ c ↔ ∀ b' < b, a + b' ≤ c | ⟨λ h b' l, (add_le_add_left l.le _).trans h,
λ H, le_of_not_lt $
induction_on a (λ α r _, induction_on b $ λ β s _ h H l, begin
resetI,
suffices : ∀ x : β, sum.lex r s (sum.inr x) (enum _ _ l),
{ cases enum _ _ l with x x,
{ cases this (enum s 0 h.pos) },
{ exact irrefl _ (this _) } },
intros x,
rw [←... | theorem | ordinal.add_le_of_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"rel_embedding.of_monotone",
"sum.lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_is_normal (a : ordinal) : is_normal ((+) a) | ⟨λ b, (add_lt_add_iff_left a).2 (lt_succ b),
λ b l c, add_le_of_limit l⟩ | theorem | ordinal.add_is_normal | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_is_limit (a) {b} : is_limit b → is_limit (a + b) | (add_is_normal a).is_limit | theorem | ordinal.add_is_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_nonempty {a b : ordinal} : {o | a ≤ b + o}.nonempty | ⟨a, le_add_left _ _⟩ | theorem | ordinal.sub_nonempty | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | The set in the definition of subtraction is nonempty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_add_sub (a b : ordinal) : a ≤ b + (a - b) | Inf_mem sub_nonempty | theorem | ordinal.le_add_sub | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"Inf_mem",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_le {a b c : ordinal} : a - b ≤ c ↔ a ≤ b + c | ⟨λ h, (le_add_sub a b).trans (add_le_add_left h _), λ h, cInf_le' h⟩ | theorem | ordinal.sub_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"cInf_le'",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_sub {a b c : ordinal} : a < b - c ↔ c + a < b | lt_iff_lt_of_le_iff_le sub_le | theorem | ordinal.lt_sub | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"lt_iff_lt_of_le_iff_le",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_sub_cancel (a b : ordinal) : a + b - a = b | le_antisymm (sub_le.2 $ le_rfl)
((add_le_add_iff_left a).1 $ le_add_sub _ _) | theorem | ordinal.add_sub_cancel | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"le_rfl",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_eq_of_add_eq {a b c : ordinal} (h : a + b = c) : c - a = b | h ▸ add_sub_cancel _ _ | theorem | ordinal.sub_eq_of_add_eq | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_le_self (a b : ordinal) : a - b ≤ a | sub_le.2 $ le_add_left _ _ | theorem | ordinal.sub_le_self | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_sub_cancel_of_le {a b : ordinal} (h : b ≤ a) : b + (a - b) = a | (le_add_sub a b).antisymm' begin
rcases zero_or_succ_or_limit (a-b) with e|⟨c,e⟩|l,
{ simp only [e, add_zero, h] },
{ rw [e, add_succ, succ_le_iff, ← lt_sub, e], exact lt_succ c },
{ exact (add_le_of_limit l).2 (λ c l, (lt_sub.1 l).le) }
end | theorem | ordinal.add_sub_cancel_of_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"antisymm'",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_sub_of_le {a b c : ordinal} (h : b ≤ a) : c ≤ a - b ↔ b + c ≤ a | by rw [←add_le_add_iff_left b, ordinal.add_sub_cancel_of_le h] | theorem | ordinal.le_sub_of_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"ordinal.add_sub_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_lt_of_le {a b c : ordinal} (h : b ≤ a) : a - b < c ↔ a < b + c | lt_iff_lt_of_le_iff_le (le_sub_of_le h) | theorem | ordinal.sub_lt_of_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"lt_iff_lt_of_le_iff_le",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_zero (a : ordinal) : a - 0 = a | by simpa only [zero_add] using add_sub_cancel 0 a | theorem | ordinal.sub_zero | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_sub (a : ordinal) : 0 - a = 0 | by rw ← ordinal.le_zero; apply sub_le_self | theorem | ordinal.zero_sub | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"ordinal.le_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_self (a : ordinal) : a - a = 0 | by simpa only [add_zero] using add_sub_cancel a 0 | theorem | ordinal.sub_self | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_eq_zero_iff_le {a b : ordinal} : a - b = 0 ↔ a ≤ b | ⟨λ h, by simpa only [h, add_zero] using le_add_sub a b,
λ h, by rwa [← ordinal.le_zero, sub_le, add_zero]⟩ | theorem | ordinal.sub_eq_zero_iff_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"ordinal.le_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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