feat(library/data/{finset,list}/comb.lean): add 'any' for finsets

library/data/finset/comb.lean

@@ -110,16 +110,18 @@ quot.lift_on s
   (λ l, all (elt_of l) p)
   (λ l₁ l₂ p, foldr_eq_of_perm (λ a₁ a₂ q, propext !and.left_comm) p true)
 
--- notation for bounded quantifiers
-notation `forallb` binders `∈` a `,` r:(scoped:1 P, P) := all a r
-notation `∀₀` binders `∈` a `,` r:(scoped:1 P, P) := all a r
-
 theorem all_empty (p : A → Prop) : all ∅ p = true :=
 rfl
 
 theorem of_mem_of_all {p : A → Prop} {a : A} {s : finset A} : a ∈ s → all s p → p a :=
 quot.induction_on s (λ l i h, list.of_mem_of_all i h)
 
+theorem forall_of_all {p : A → Prop} {s : finset A} (H : all s p) : ∀{a}, a ∈ s → p a :=
+λ a H', of_mem_of_all H' H
+
+definition decidable_all (p : A → Prop) [h : decidable_pred p] (s : finset A) : decidable (all s p) :=
+quot.rec_on_subsingleton s (λ l, list.decidable_all p (elt_of l))
+
 theorem all_implies {p q : A → Prop} {s : finset A} : all s p → (∀ x, p x → q x) → all s q :=
 quot.induction_on s (λ l h₁ h₂, list.all_implies h₁ h₂)
 
@@ -148,6 +150,40 @@ theorem all_inter_of_all_right {p : A → Prop} {s₁ : finset A} (s₂ : finset
 quot.induction_on₂ s₁ s₂ (λ l₁ l₂ h, list.all_inter_of_all_right _ h)
 end all
 
+/- any -/
+section any
+variables {A : Type}
+definition any (s : finset A) (p : A → Prop) : Prop :=
+quot.lift_on s
+  (λ l, any (elt_of l) p)
+  (λ l₁ l₂ p, foldr_eq_of_perm (λ a₁ a₂ q, propext !or.left_comm) p false)
+
+theorem any_empty (p : A → Prop) : any ∅ p = false := rfl
+
+theorem exists_of_any {p : A → Prop} {s : finset A} : any s p → ∃a, a ∈ s ∧ p a :=
+quot.induction_on s (λ l H, list.exists_of_any H)
+
+theorem any_of_mem {p : A → Prop} {s : finset A} {a : A} : a ∈ s → p a → any s p :=
+quot.induction_on s (λ l H1 H2, list.any_of_mem H1 H2)
+
+theorem any_of_insert [h : decidable_eq A] {p : A → Prop} (s : finset A) {a : A} (H : p a) :
+  any (insert a s) p :=
+any_of_mem (mem_insert a s) H
+
+theorem any_of_insert_right [h : decidable_eq A] {p : A → Prop} {s : finset A} (a : A) (H : any s p) :
+  any (insert a s) p :=
+obtain b (H' : b ∈ s ∧ p b), from exists_of_any H,
+any_of_mem (mem_insert_of_mem a (and.left H')) (and.right H')
+
+theorem any_of_exists {p : A → Prop} {s : finset A} (H : ∃a, a ∈ s ∧ p a) : any s p :=
+obtain a H', from H,
+any_of_mem (and.left H') (and.right H')
+
+definition decidable_any (p : A → Prop) [h : decidable_pred p] (s : finset A) : decidable (any s p) :=
+quot.rec_on_subsingleton s (λ l, list.decidable_any p (elt_of l))
+
+end any
+
 section product
 variables {A B : Type}
 definition product (s₁ : finset A) (s₂ : finset B) : finset (A × B) :=

library/data/list/comb.lean

@@ -168,15 +168,20 @@ foldr (λ a r, p a ∧ r) true l
 definition any (l : list A) (p : A → Prop) : Prop :=
 foldr (λ a r, p a ∨ r) false l
 
-theorem all_nil (p : A → Prop) : all [] p = true
+theorem all_nil_eq (p : A → Prop) : all [] p = true
 
-theorem all_cons (p : A → Prop) (a : A) (l : list A) : all (a::l) p = (p a ∧ all l p)
+theorem all_nil (p : A → Prop) : all [] p := trivial
+
+theorem all_cons_eq (p : A → Prop) (a : A) (l : list A) : all (a::l) p = (p a ∧ all l p)
+
+theorem all_cons {p : A → Prop} {a : A} {l : list A} (H1 : p a) (H2 : all l p) : all (a::l) p :=
+and.intro H1 H2
 
 theorem all_of_all_cons {p : A → Prop} {a : A} {l : list A} : all (a::l) p → all l p :=
-assume h, by rewrite [all_cons at h]; exact (and.elim_right h)
+assume h, by rewrite [all_cons_eq at h]; exact (and.elim_right h)
 
 theorem of_all_cons {p : A → Prop} {a : A} {l : list A} : all (a::l) p → p a :=
-assume h, by rewrite [all_cons at h]; exact (and.elim_left h)
+assume h, by rewrite [all_cons_eq at h]; exact (and.elim_left h)
 
 theorem all_cons_of_all {p : A → Prop} {a : A} {l : list A} : p a → all l p → all (a::l) p :=
 assume pa alllp, and.intro pa alllp
@@ -193,16 +198,21 @@ theorem of_mem_of_all {p : A → Prop} {a : A} : ∀ {l}, a ∈ l → all l p
 | (b::l) h₁ h₂ :=
   or.elim (eq_or_mem_of_mem_cons h₁)
     (λ aeqb : a = b,
-      by rewrite [all_cons at h₂, -aeqb at h₂]; exact (and.elim_left h₂))
+      by rewrite [all_cons_eq at h₂, -aeqb at h₂]; exact (and.elim_left h₂))
     (λ ainl : a ∈ l,
-      have allp : all l p, by rewrite [all_cons at h₂]; exact (and.elim_right h₂),
+      have allp : all l p, by rewrite [all_cons_eq at h₂]; exact (and.elim_right h₂),
       of_mem_of_all ainl allp)
 
+theorem all_of_forall {p : A → Prop} : ∀ {l}, (∀a, a ∈ l → p a) → all l p
+| []     H := !all_nil
+| (a::l) H := all_cons (H a !mem_cons)
+                       (all_of_forall (λ a' H', H a' (mem_cons_of_mem _ H')))
+
 theorem any_nil (p : A → Prop) : any [] p = false
 
 theorem any_cons (p : A → Prop) (a : A) (l : list A) : any (a::l) p = (p a ∨ any l p)
 
-theorem any_of_mem (p : A → Prop) {a : A} : ∀ {l}, a ∈ l → p a → any l p
+theorem any_of_mem {p : A → Prop} {a : A} : ∀ {l}, a ∈ l → p a → any l p
 | []     i h := absurd i !not_mem_nil
 | (b::l) i h :=
   or.elim (eq_or_mem_of_mem_cons i)
@@ -211,6 +221,14 @@ theorem any_of_mem (p : A → Prop) {a : A} : ∀ {l}, a ∈ l → p a → any l
       have anyl : any l p, from any_of_mem ainl h,
       or.inr anyl)
 
+theorem exists_of_any {p : A → Prop} : ∀{l : list A}, any l p → ∃a, a ∈ l ∧ p a
+| []     H := false.elim H
+| (b::l) H := or.elim H
+                (assume H1 : p b, exists.intro b (and.intro !mem_cons H1))
+                (assume H1 : any l p,
+                  obtain a (H2 : a ∈ l ∧ p a), from exists_of_any H1,
+                  exists.intro a (and.intro (mem_cons_of_mem b (and.left H2)) (and.right H2)))
+
 definition decidable_all (p : A → Prop) [H : decidable_pred p] : ∀ l, decidable (all l p)
 | []       := decidable_true
 | (a :: l) :=