michael/lean2
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

fix(library/data): powerset notation

Browse source
This commit is contained in:
Leonardo de Moura 2015-08-13 09:04:00 -07:00
parent 51c48277c8
commit f4a81fdd73
3 changed files with 19 additions and 20 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

23
library/data/finset/comb.lean
View file

@ -393,22 +393,21 @@ quot.lift_on s
(λ l, list_powerset (elt_of l))
(λ l₁ l₂ p, list_powerset_eq_list_powerset_of_perm p)
notation [priority finset.prio] `𝒫` s := powerset s
prefix [priority finset.prio] `𝒫`:100 := powerset
theorem powerset_empty : powerset (∅ : finset A) = '{∅} := rfl
theorem powerset_empty : 𝒫 (∅ : finset A) = '{∅} := rfl
theorem powerset_insert {a : A} {s : finset A} : a ∉ s →
powerset (insert a s) = powerset s ∪ image (insert a) (powerset s) :=
theorem powerset_insert {a : A} {s : finset A} : a ∉ s → 𝒫 (insert a s) = 𝒫 s ∪ image (insert a) (𝒫 s) :=
quot.induction_on s
(λ l,
assume H : a ∉ quot.mk l,
calc
powerset (insert a (quot.mk l))
𝒫 (insert a (quot.mk l))
= list_powerset (list.insert a (elt_of l)) : rfl
... = list_powerset (#list a :: elt_of l) : by rewrite [list.insert_eq_of_not_mem H]
... = powerset (quot.mk l) ∪ image (insert a) (powerset (quot.mk l)) : rfl)
... = 𝒫 (quot.mk l) ∪ image (insert a) (𝒫 (quot.mk l)) : rfl)
theorem mem_powerset_iff_subset (s : finset A) : ∀ x, x ∈ powerset s ↔ x ⊆ s :=
theorem mem_powerset_iff_subset (s : finset A) : ∀ x, x ∈ 𝒫 s ↔ x ⊆ s :=
begin
induction s with a s nains ih,
intro x,
@ -421,7 +420,7 @@ begin
(assume H,
or.elim H
(suppose x ⊆ s, subset.trans this !subset_insert)
(suppose ∃ y, y ∈ powerset s ∧ insert a y = x,
(suppose ∃ y, y ∈ 𝒫 s ∧ insert a y = x,
obtain y [yps iay], from this,
show x ⊆ insert a s,
begin
@ -436,19 +435,19 @@ begin
(suppose a ∈ x,
or.inr (exists.intro (erase a x)
(and.intro
(show erase a x ∈ powerset s, by rewrite ih; apply H')
(show erase a x ∈ 𝒫 s, by rewrite ih; apply H')
(show insert a (erase a x) = x, from insert_erase this))))
(suppose a ∉ x, or.inl
(show x ⊆ s, by rewrite [(erase_eq_of_not_mem this) at H']; apply H'))))
end
theorem subset_of_mem_powerset {s t : finset A} (H : s ∈ powerset t) : s ⊆ t :=
theorem subset_of_mem_powerset {s t : finset A} (H : s ∈ 𝒫 t) : s ⊆ t :=
iff.mp (mem_powerset_iff_subset t s) H
theorem mem_powerset_of_subset {s t : finset A} (H : s ⊆ t) : s ∈ powerset t :=
theorem mem_powerset_of_subset {s t : finset A} (H : s ⊆ t) : s ∈ 𝒫 t :=
iff.mpr (mem_powerset_iff_subset t s) H
theorem empty_mem_powerset (s : finset A) : ∅ ∈ powerset s :=
theorem empty_mem_powerset (s : finset A) : ∅ ∈ 𝒫 s :=
mem_powerset_of_subset (empty_subset s)
end powerset

14
library/data/hf.lean
View file

@ -368,19 +368,19 @@ begin unfold [image, union], rewrite [*to_finset_of_finset, finset.image_union]
definition powerset (s : hf) : hf :=
of_finset (finset.image of_finset (finset.powerset (to_finset s)))
notation [priority hf.prio] `𝒫` s := powerset s
prefix [priority hf.prio] `𝒫`:100 := powerset
theorem powerset_empty : powerset ∅ = insert ∅ ∅ :=
theorem powerset_empty : 𝒫 ∅ = insert ∅ ∅ :=
rfl
theorem powerset_insert {a : hf} {s : hf} : a ∉ s → powerset (insert a s) = powerset s ∪ image (insert a) (powerset s) :=
theorem powerset_insert {a : hf} {s : hf} : a ∉ s → 𝒫 (insert a s) = 𝒫 s ∪ image (insert a) (𝒫 s) :=
begin unfold [not_mem, mem, powerset, insert, union, image], rewrite [*to_finset_of_finset], intro h,
have (λ (x : finset hf), of_finset (finset.insert a x)) = (λ (x : finset hf), of_finset (finset.insert a (to_finset (of_finset x)))), from
funext (λ x, by rewrite to_finset_of_finset),
rewrite [finset.powerset_insert h, finset.image_union, -*finset.image_compose,↑compose,this]
end
theorem mem_powerset_iff_subset (s : hf) : ∀ x : hf, x ∈ powerset s ↔ x ⊆ s :=
theorem mem_powerset_iff_subset (s : hf) : ∀ x : hf, x ∈ 𝒫 s ↔ x ⊆ s :=
begin
intro x, unfold [mem, powerset, subset], rewrite [to_finset_of_finset, finset.mem_image_eq], apply iff.intro,
suppose (∃ (w : finset hf), finset.mem w (finset.powerset (to_finset s)) ∧ of_finset w = x),
@ -391,12 +391,12 @@ begin
exists.intro (to_finset x) (and.intro this (of_finset_to_finset x))
end
theorem subset_of_mem_powerset {s t : hf} (H : s ∈ powerset t) : s ⊆ t :=
theorem subset_of_mem_powerset {s t : hf} (H : s ∈ 𝒫 t) : s ⊆ t :=
iff.mp (mem_powerset_iff_subset t s) H
theorem mem_powerset_of_subset {s t : hf} (H : s ⊆ t) : s ∈ powerset t :=
theorem mem_powerset_of_subset {s t : hf} (H : s ⊆ t) : s ∈ 𝒫 t :=
iff.mpr (mem_powerset_iff_subset t s) H
theorem empty_mem_powerset (s : hf) : ∅ ∈ powerset s :=
theorem empty_mem_powerset (s : hf) : ∅ ∈ 𝒫 s :=
mem_powerset_of_subset (empty_subset s)
end hf

2
library/data/set/basic.lean
View file

@ -297,7 +297,7 @@ ext (take x, iff.intro
/- powerset -/
definition powerset (s : set X) : set (set X) := {x : set X | x ⊆ s}
notation `𝒫` s := powerset s
prefix `𝒫`:100 := powerset
/- large unions -/