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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map₂ (f : Type u → Type v → Type w) (hf : ∀ α β γ δ, α ≃ β → γ ≃ δ → f α γ ≃ f β δ) :
cardinal.{u} → cardinal.{v} → cardinal.{w} | quotient.map₂ f $ λ α β ⟨e₁⟩ γ δ ⟨e₂⟩, ⟨hf α β γ δ e₁ e₂⟩ | def | cardinal.map₂ | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"quotient.map₂"
] | Lift a binary operation `Type* → Type* → Type*` to a binary operation on `cardinal`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift (c : cardinal.{v}) : cardinal.{max v u} | map ulift (λ α β e, equiv.ulift.trans $ e.trans equiv.ulift.symm) c | def | cardinal.lift | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"lift"
] | The universe lift operation on cardinals. You can specify the universes explicitly with
`lift.{u v} : cardinal.{v} → cardinal.{max v u}` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_ulift (α) : #(ulift.{v u} α) = lift.{v} (#α) | rfl | theorem | cardinal.mk_ulift | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_umax : lift.{(max u v) u} = lift.{v u} | funext $ λ a, induction_on a $ λ α, (equiv.ulift.trans equiv.ulift.symm).cardinal_eq | theorem | cardinal.lift_umax | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | `lift.{(max u v) u}` equals `lift.{v u}`. Using `set_option pp.universes true` will make it much
easier to understand what's happening when using this lemma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_umax' : lift.{(max v u) u} = lift.{v u} | lift_umax | theorem | cardinal.lift_umax' | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | `lift.{(max v u) u}` equals `lift.{v u}`. Using `set_option pp.universes true` will make it much
easier to understand what's happening when using this lemma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_id' (a : cardinal.{max u v}) : lift.{u} a = a | induction_on a $ λ α, mk_congr equiv.ulift | theorem | cardinal.lift_id' | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.ulift"
] | A cardinal lifted to a lower or equal universe equals itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_id (a : cardinal) : lift.{u u} a = a | lift_id'.{u u} a | theorem | cardinal.lift_id | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | A cardinal lifted to the same universe equals itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_uzero (a : cardinal.{u}) : lift.{0} a = a | lift_id'.{0 u} a | theorem | cardinal.lift_uzero | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | A cardinal lifted to the zero universe equals itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_lift (a : cardinal) :
lift.{w} (lift.{v} a) = lift.{max v w} a | induction_on a $ λ α,
(equiv.ulift.trans $ equiv.ulift.trans equiv.ulift.symm).cardinal_eq | theorem | cardinal.lift_lift | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_def (α β : Type u) : #α ≤ #β ↔ nonempty (α ↪ β) | iff.rfl | theorem | cardinal.le_def | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_le_of_injective {α β : Type u} {f : α → β} (hf : injective f) : #α ≤ #β | ⟨⟨f, hf⟩⟩ | theorem | cardinal.mk_le_of_injective | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.function.embedding.cardinal_le {α β : Type u} (f : α ↪ β) : #α ≤ #β | ⟨f⟩ | theorem | function.embedding.cardinal_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_le_of_surjective {α β : Type u} {f : α → β} (hf : surjective f) : #β ≤ #α | ⟨embedding.of_surjective f hf⟩ | theorem | cardinal.mk_le_of_surjective | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mk_iff_exists_set {c : cardinal} {α : Type u} :
c ≤ #α ↔ ∃ p : set α, #p = c | ⟨induction_on c $ λ β ⟨⟨f, hf⟩⟩,
⟨set.range f, (equiv.of_injective f hf).cardinal_eq.symm⟩,
λ ⟨p, e⟩, e ▸ ⟨⟨subtype.val, λ a b, subtype.eq⟩⟩⟩ | theorem | cardinal.le_mk_iff_exists_set | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"equiv.of_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_subtype_le {α : Type u} (p : α → Prop) : #(subtype p) ≤ #α | ⟨embedding.subtype p⟩ | theorem | cardinal.mk_subtype_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_set_le (s : set α) : #s ≤ #α | mk_subtype_le s | theorem | cardinal.mk_set_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
out_embedding {c c' : cardinal} : c ≤ c' ↔ nonempty (c.out ↪ c'.out) | by { transitivity _, rw [←quotient.out_eq c, ←quotient.out_eq c'], refl } | theorem | cardinal.out_embedding | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_mk_le {α : Type u} {β : Type v} :
lift.{max v w} (#α) ≤ lift.{max u w} (#β) ↔ nonempty (α ↪ β) | ⟨λ ⟨f⟩, ⟨embedding.congr equiv.ulift equiv.ulift f⟩,
λ ⟨f⟩, ⟨embedding.congr equiv.ulift.symm equiv.ulift.symm f⟩⟩ | theorem | cardinal.lift_mk_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.ulift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_mk_le' {α : Type u} {β : Type v} :
lift.{v} (#α) ≤ lift.{u} (#β) ↔ nonempty (α ↪ β) | lift_mk_le.{u v 0} | theorem | cardinal.lift_mk_le' | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | A variant of `cardinal.lift_mk_le` with specialized universes.
Because Lean often can not realize it should use this specialization itself,
we provide this statement separately so you don't have to solve the specialization problem either. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_mk_eq {α : Type u} {β : Type v} :
lift.{max v w} (#α) = lift.{max u w} (#β) ↔ nonempty (α ≃ β) | quotient.eq.trans
⟨λ ⟨f⟩, ⟨equiv.ulift.symm.trans $ f.trans equiv.ulift⟩,
λ ⟨f⟩, ⟨equiv.ulift.trans $ f.trans equiv.ulift.symm⟩⟩ | theorem | cardinal.lift_mk_eq | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_mk_eq' {α : Type u} {β : Type v} :
lift.{v} (#α) = lift.{u} (#β) ↔ nonempty (α ≃ β) | lift_mk_eq.{u v 0} | theorem | cardinal.lift_mk_eq' | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | A variant of `cardinal.lift_mk_eq` with specialized universes.
Because Lean often can not realize it should use this specialization itself,
we provide this statement separately so you don't have to solve the specialization problem either. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_le {a b : cardinal} : lift a ≤ lift b ↔ a ≤ b | induction_on₂ a b $ λ α β, by { rw ← lift_umax, exact lift_mk_le } | theorem | cardinal.lift_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_order_embedding : cardinal.{v} ↪o cardinal.{max v u} | order_embedding.of_map_le_iff lift (λ _ _, lift_le) | def | cardinal.lift_order_embedding | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"lift",
"order_embedding.of_map_le_iff"
] | `cardinal.lift` as an `order_embedding`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_injective : injective lift.{u v} | lift_order_embedding.injective | theorem | cardinal.lift_injective | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"lift_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_inj {a b : cardinal} : lift a = lift b ↔ a = b | lift_injective.eq_iff | theorem | cardinal.lift_inj | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_lt {a b : cardinal} : lift a < lift b ↔ a < b | lift_order_embedding.lt_iff_lt | theorem | cardinal.lift_lt | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_strict_mono : strict_mono lift | λ a b, lift_lt.2 | theorem | cardinal.lift_strict_mono | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"lift",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_monotone : monotone lift | lift_strict_mono.monotone | theorem | cardinal.lift_monotone | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"lift",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_zero (α : Type u) [is_empty α] : #α = 0 | (equiv.equiv_pempty α).cardinal_eq | lemma | cardinal.mk_eq_zero | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.equiv_pempty",
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_zero : lift 0 = 0 | mk_congr (equiv.equiv_pempty _) | theorem | cardinal.lift_zero | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.equiv_pempty",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_eq_zero {a : cardinal.{v}} : lift.{u} a = 0 ↔ a = 0 | lift_injective.eq_iff' lift_zero | theorem | cardinal.lift_eq_zero | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_zero_iff {α : Type u} : #α = 0 ↔ is_empty α | ⟨λ e, let ⟨h⟩ := quotient.exact e in h.is_empty, @mk_eq_zero α⟩ | lemma | cardinal.mk_eq_zero_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_ne_zero_iff {α : Type u} : #α ≠ 0 ↔ nonempty α | (not_iff_not.2 mk_eq_zero_iff).trans not_is_empty_iff | theorem | cardinal.mk_ne_zero_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"not_is_empty_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_ne_zero (α : Type u) [nonempty α] : #α ≠ 0 | mk_ne_zero_iff.2 ‹_› | lemma | cardinal.mk_ne_zero | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_one (α : Type u) [unique α] : #α = 1 | (equiv.equiv_punit α).cardinal_eq | lemma | cardinal.mk_eq_one | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.equiv_punit",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_one_iff_subsingleton {α : Type u} : #α ≤ 1 ↔ subsingleton α | ⟨λ ⟨f⟩, ⟨λ a b, f.injective (subsingleton.elim _ _)⟩,
λ ⟨h⟩, ⟨⟨λ a, punit.star, λ a b _, h _ _⟩⟩⟩ | theorem | cardinal.le_one_iff_subsingleton | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_le_one_iff_set_subsingleton {s : set α} : #s ≤ 1 ↔ s.subsingleton | le_one_iff_subsingleton.trans s.subsingleton_coe | lemma | cardinal.mk_le_one_iff_set_subsingleton | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_def (α β : Type u) : #α + #β = #(α ⊕ β) | rfl | theorem | cardinal.add_def | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_sum (α : Type u) (β : Type v) :
#(α ⊕ β) = lift.{v u} (#α) + lift.{u v} (#β) | mk_congr ((equiv.ulift).symm.sum_congr (equiv.ulift).symm) | lemma | cardinal.mk_sum | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.ulift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_option {α : Type u} : #(option α) = #α + 1 | (equiv.option_equiv_sum_punit α).cardinal_eq | theorem | cardinal.mk_option | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.option_equiv_sum_punit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_psum (α : Type u) (β : Type v) : #(psum α β) = lift.{v} (#α) + lift.{u} (#β) | (mk_congr (equiv.psum_equiv_sum α β)).trans (mk_sum α β) | lemma | cardinal.mk_psum | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.psum_equiv_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_fintype (α : Type u) [fintype α] : #α = fintype.card α | begin
refine fintype.induction_empty_option _ _ _ α,
{ introsI α β h e hα, letI := fintype.of_equiv β e.symm,
rwa [mk_congr e, fintype.card_congr e] at hα },
{ refl },
{ introsI α h hα, simp [hα], refl }
end | lemma | cardinal.mk_fintype | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"fintype",
"fintype.card",
"fintype.card_congr",
"fintype.induction_empty_option",
"fintype.of_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_def (α β : Type u) : #α * #β = #(α × β) | rfl | theorem | cardinal.mul_def | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_prod (α : Type u) (β : Type v) :
#(α × β) = lift.{v u} (#α) * lift.{u v} (#β) | mk_congr (equiv.ulift.symm.prod_congr (equiv.ulift).symm) | lemma | cardinal.mk_prod | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.ulift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_comm' (a b : cardinal.{u}) : a * b = b * a | induction_on₂ a b $ λ α β, mk_congr $ equiv.prod_comm α β | theorem | cardinal.mul_comm' | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.prod_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_def (α β) : #α ^ #β = #(β → α) | rfl | theorem | cardinal.power_def | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_arrow (α : Type u) (β : Type v) : #(α → β) = lift.{u} (#β) ^ lift.{v} (#α) | mk_congr (equiv.ulift.symm.arrow_congr equiv.ulift.symm) | theorem | cardinal.mk_arrow | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_power (a b) : lift (a ^ b) = lift a ^ lift b | induction_on₂ a b $ λ α β,
mk_congr $ equiv.ulift.trans (equiv.ulift.arrow_congr equiv.ulift).symm | theorem | cardinal.lift_power | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.ulift",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_zero {a : cardinal} : a ^ 0 = 1 | induction_on a $ λ α, mk_congr $ equiv.pempty_arrow_equiv_punit α | theorem | cardinal.power_zero | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"equiv.pempty_arrow_equiv_punit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_one {a : cardinal} : a ^ 1 = a | induction_on a $ λ α, mk_congr $ equiv.punit_arrow_equiv α | theorem | cardinal.power_one | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"equiv.punit_arrow_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_add {a b c : cardinal} : a ^ (b + c) = a ^ b * a ^ c | induction_on₃ a b c $ λ α β γ, mk_congr $ equiv.sum_arrow_equiv_prod_arrow β γ α | theorem | cardinal.power_add | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"equiv.sum_arrow_equiv_prod_arrow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_bit0 (a b : cardinal) : a ^ (bit0 b) = a ^ b * a ^ b | power_add | theorem | cardinal.power_bit0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_bit1 (a b : cardinal) : a ^ (bit1 b) = a ^ b * a ^ b * a | by rw [bit1, ←power_bit0, power_add, power_one] | theorem | cardinal.power_bit1 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_power {a : cardinal} : 1 ^ a = 1 | induction_on a $ λ α, (equiv.arrow_punit_equiv_punit α).cardinal_eq | theorem | cardinal.one_power | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"equiv.arrow_punit_equiv_punit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_bool : #bool = 2 | by simp | theorem | cardinal.mk_bool | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_Prop : #(Prop) = 2 | by simp | theorem | cardinal.mk_Prop | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_power {a : cardinal} : a ≠ 0 → 0 ^ a = 0 | induction_on a $ λ α heq, mk_eq_zero_iff.2 $ is_empty_pi.2 $
let ⟨a⟩ := mk_ne_zero_iff.1 heq in ⟨a, pempty.is_empty⟩ | theorem | cardinal.zero_power | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_ne_zero {a : cardinal} (b) : a ≠ 0 → a ^ b ≠ 0 | induction_on₂ a b $ λ α β h,
let ⟨a⟩ := mk_ne_zero_iff.1 h in mk_ne_zero_iff.2 ⟨λ _, a⟩ | theorem | cardinal.power_ne_zero | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_power {a b c : cardinal} : (a * b) ^ c = a ^ c * b ^ c | induction_on₃ a b c $ λ α β γ, mk_congr $ equiv.arrow_prod_equiv_prod_arrow α β γ | theorem | cardinal.mul_power | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"equiv.arrow_prod_equiv_prod_arrow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_mul {a b c : cardinal} : a ^ (b * c) = (a ^ b) ^ c | by { rw [mul_comm b c], exact induction_on₃ a b c (λ α β γ, mk_congr $ equiv.curry γ β α) } | theorem | cardinal.power_mul | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"equiv.curry",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_cast_right (a : cardinal.{u}) (n : ℕ) : (a ^ (↑n : cardinal.{u})) = a ^ℕ n | rfl | lemma | cardinal.pow_cast_right | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_one : lift 1 = 1 | mk_congr $ equiv.ulift.trans equiv.punit_equiv_punit | theorem | cardinal.lift_one | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.punit_equiv_punit",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_add (a b) : lift (a + b) = lift a + lift b | induction_on₂ a b $ λ α β,
mk_congr $ equiv.ulift.trans (equiv.sum_congr equiv.ulift equiv.ulift).symm | theorem | cardinal.lift_add | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.sum_congr",
"equiv.ulift",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_mul (a b) : lift (a * b) = lift a * lift b | induction_on₂ a b $ λ α β,
mk_congr $ equiv.ulift.trans (equiv.prod_congr equiv.ulift equiv.ulift).symm | theorem | cardinal.lift_mul | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.prod_congr",
"equiv.ulift",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_bit0 (a : cardinal) : lift (bit0 a) = bit0 (lift a) | lift_add a a | theorem | cardinal.lift_bit0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_bit1 (a : cardinal) : lift (bit1 a) = bit1 (lift a) | by simp [bit1] | theorem | cardinal.lift_bit1 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_two : lift.{u v} 2 = 2 | by simp | theorem | cardinal.lift_two | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_set {α : Type u} : #(set α) = 2 ^ #α | by simp [set, mk_arrow] | theorem | cardinal.mk_set | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_powerset {α : Type u} (s : set α) : #↥(𝒫 s) = 2 ^ #↥s | (mk_congr (equiv.set.powerset s)).trans mk_set | theorem | cardinal.mk_powerset | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.set.powerset"
] | A variant of `cardinal.mk_set` expressed in terms of a `set` instead of a `Type`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_two_power (a) : lift (2 ^ a) = 2 ^ lift a | by simp | theorem | cardinal.lift_two_power | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_le : ∀ a : cardinal, 0 ≤ a | by rintro ⟨α⟩; exact ⟨embedding.of_is_empty⟩ | theorem | cardinal.zero_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_add' : ∀ {a b c d : cardinal}, a ≤ b → c ≤ d → a + c ≤ b + d | by rintros ⟨α⟩ ⟨β⟩ ⟨γ⟩ ⟨δ⟩ ⟨e₁⟩ ⟨e₂⟩; exact ⟨e₁.sum_map e₂⟩ | theorem | cardinal.add_le_add' | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_covariant_class : covariant_class cardinal cardinal (+) (≤) | ⟨λ a b c, add_le_add' le_rfl⟩ | instance | cardinal.add_covariant_class | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_swap_covariant_class : covariant_class cardinal cardinal (swap (+)) (≤) | ⟨λ a b c h, add_le_add' h le_rfl⟩ | instance | cardinal.add_swap_covariant_class | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_power_le (c : cardinal.{u}) : (0 : cardinal.{u}) ^ c ≤ 1 | by { by_cases h : c = 0, rw [h, power_zero], rw [zero_power h], apply zero_le } | lemma | cardinal.zero_power_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_le_power_left : ∀ {a b c : cardinal}, a ≠ 0 → b ≤ c → a ^ b ≤ a ^ c | by rintros ⟨α⟩ ⟨β⟩ ⟨γ⟩ hα ⟨e⟩; exact
let ⟨a⟩ := mk_ne_zero_iff.1 hα in
⟨@embedding.arrow_congr_left _ _ _ ⟨a⟩ e⟩ | theorem | cardinal.power_le_power_left | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_le_power (a : cardinal) {b : cardinal} (hb : 1 ≤ b) : a ≤ a ^ b | begin
rcases eq_or_ne a 0 with rfl|ha,
{ exact zero_le _ },
{ convert power_le_power_left ha hb, exact power_one.symm }
end | theorem | cardinal.self_le_power | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"eq_or_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cantor (a : cardinal.{u}) : a < 2 ^ a | begin
induction a using cardinal.induction_on with α,
rw [← mk_set],
refine ⟨⟨⟨singleton, λ a b, singleton_eq_singleton_iff.1⟩⟩, _⟩,
rintro ⟨⟨f, hf⟩⟩,
exact cantor_injective f hf
end | theorem | cardinal.cantor | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal.induction_on"
] | **Cantor's theorem** | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_lt_iff_nontrivial {α : Type u} : 1 < #α ↔ nontrivial α | by rw [← not_le, le_one_iff_subsingleton, ← not_nontrivial_iff_subsingleton, not_not] | theorem | cardinal.one_lt_iff_nontrivial | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"nontrivial",
"not_nontrivial_iff_subsingleton",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_le_max_power_one {a b c : cardinal} (h : b ≤ c) : a ^ b ≤ max (a ^ c) 1 | begin
by_cases ha : a = 0,
simp [ha, zero_power_le],
exact (power_le_power_left ha h).trans (le_max_left _ _)
end | theorem | cardinal.power_le_max_power_one | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_le_power_right {a b c : cardinal} : a ≤ b → a ^ c ≤ b ^ c | induction_on₃ a b c $ λ α β γ ⟨e⟩, ⟨embedding.arrow_congr_right e⟩ | theorem | cardinal.power_le_power_right | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_pos {a : cardinal} (b) (ha : 0 < a) : 0 < a ^ b | (power_ne_zero _ ha.ne').bot_lt | theorem | cardinal.power_pos | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_wf : @well_founded cardinal.{u} (<) | ⟨λ a, classical.by_contradiction $ λ h, begin
let ι := {c : cardinal // ¬ acc (<) c},
let f : ι → cardinal := subtype.val,
haveI hι : nonempty ι := ⟨⟨_, h⟩⟩,
obtain ⟨⟨c : cardinal, hc : ¬acc (<) c⟩, ⟨h_1 : Π j, (f ⟨c, hc⟩).out ↪ (f j).out⟩⟩ :=
embedding.min_injective (λ i, (f i).out),
apply hc (acc.intro ... | theorem | cardinal.lt_wf | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wo : @is_well_order cardinal.{u} (<) | { } | instance | cardinal.wo | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_empty : Inf (∅ : set cardinal.{u}) = 0 | dif_neg not_nonempty_empty | theorem | cardinal.Inf_empty | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"Inf_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_def (c : cardinal) : succ c = Inf {c' | c < c'} | rfl | theorem | cardinal.succ_def | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_pos : ∀ c : cardinal, 0 < succ c | bot_lt_succ | lemma | cardinal.succ_pos | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_ne_zero (c : cardinal) : succ c ≠ 0 | (succ_pos _).ne' | lemma | cardinal.succ_ne_zero | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_one_le_succ (c : cardinal.{u}) : c + 1 ≤ succ c | begin
refine (le_cInf_iff'' (exists_gt c)).2 (λ b hlt, _),
rcases ⟨b, c⟩ with ⟨⟨β⟩, ⟨γ⟩⟩,
cases le_of_lt hlt with f,
have : ¬ surjective f := λ hn, (not_le_of_lt hlt) (mk_le_of_surjective hn),
simp only [surjective, not_forall] at this,
rcases this with ⟨b, hb⟩,
calc #γ + 1 = #(option γ) : mk_option.symm
... | theorem | cardinal.add_one_le_succ | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"le_cInf_iff''",
"not_forall",
"not_le_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit (c : cardinal) : Prop | c ≠ 0 ∧ is_succ_limit c | def | cardinal.is_limit | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | A cardinal is a limit if it is not zero or a successor cardinal. Note that `ℵ₀` is a limit
cardinal by this definition, but `0` isn't.
Use `is_succ_limit` if you want to include the `c = 0` case. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit.ne_zero {c} (h : is_limit c) : c ≠ 0 | h.1 | theorem | cardinal.is_limit.ne_zero | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit.is_succ_limit {c} (h : is_limit c) : is_succ_limit c | h.2 | theorem | cardinal.is_limit.is_succ_limit | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit.succ_lt {x c} (h : is_limit c) : x < c → succ x < c | h.is_succ_limit.succ_lt | theorem | cardinal.is_limit.succ_lt | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_succ_limit_zero : is_succ_limit (0 : cardinal) | is_succ_limit_bot | theorem | cardinal.is_succ_limit_zero | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum {ι} (f : ι → cardinal) : cardinal | mk Σ i, (f i).out | def | cardinal.sum | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | The indexed sum of cardinals is the cardinality of the
indexed disjoint union, i.e. sigma type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_sum {ι} (f : ι → cardinal) (i) : f i ≤ sum f | by rw ← quotient.out_eq (f i); exact
⟨⟨λ a, ⟨i, a⟩, λ a b h, eq_of_heq $ by injection h⟩⟩ | theorem | cardinal.le_sum | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"quotient.out_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_sigma {ι} (f : ι → Type*) : #(Σ i, f i) = sum (λ i, #(f i)) | mk_congr $ equiv.sigma_congr_right $ λ i, out_mk_equiv.symm | theorem | cardinal.mk_sigma | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.sigma_congr_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_const (ι : Type u) (a : cardinal.{v}) :
sum (λ i : ι, a) = lift.{v} (#ι) * lift.{u} a | induction_on a $ λ α, mk_congr $
calc (Σ i : ι, quotient.out (#α)) ≃ ι × quotient.out (#α) : equiv.sigma_equiv_prod _ _
... ≃ ulift ι × ulift α : equiv.ulift.symm.prod_congr (out_mk_equiv.trans equiv.ulift.symm) | theorem | cardinal.sum_const | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.sigma_equiv_prod",
"quotient.out"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_const' (ι : Type u) (a : cardinal.{u}) : sum (λ _:ι, a) = #ι * a | by simp | theorem | cardinal.sum_const' | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_add_distrib {ι} (f g : ι → cardinal) :
sum (f + g) = sum f + sum g | by simpa only [mk_sigma, mk_sum, mk_out, lift_id] using
mk_congr (equiv.sigma_sum_distrib (quotient.out ∘ f) (quotient.out ∘ g)) | theorem | cardinal.sum_add_distrib | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"equiv.sigma_sum_distrib",
"quotient.out"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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