refactor(library): rename exists_elim and exists_intro to exists.elim

and exists.intro

doc/lean/test_single.sh

@@ -22,7 +22,7 @@ while read -r line; do
         rm -f $f.$i.lean
     elif [[ $line =~ ^#\+END_SRC ]]; then
         if [[ $in_code_block -eq 1 ]]; then
-            if $LEAN $f.$i.lean > $f.$i.produced.out; then
+            if $LEAN -t 100000 $f.$i.lean > $f.$i.produced.out; then
                 echo "code fragment #$i worked"
             else
                 echo "ERROR executing $f.$i.lean, for in_code_block block starting at $lastbegin, produced output:"

doc/lean/tutorial.org

@@ -685,8 +685,8 @@ In Lean, the existential quantifier can be written as =exists x : A, B
 x= or =∃ x : A, B x=. Actually both versions are just
 notational convenience for =Exists (fun x : A, B x)=. That is, the existential quantifier
 is actually a constant defined in the file =logic.lean=.
-This file also defines the =exists_intro= and =exists_elim=.
+This file also defines the =exists.intro= and =exists.elim=.
-To build a proof for =∃ x : A, B x=, we should provide a term =w : A= and a proof term =Hw : B w= to =exists_intro=.
+To build a proof for =∃ x : A, B x=, we should provide a term =w : A= and a proof term =Hw : B w= to =exists.intro=.
 We say =w= is the witness for the existential introduction. In previous examples,
 =nat_trans3i Hxy Hzy Hzw= was a proof term for =x = w=. Then, we can create a proof term
 for =∃ a : nat, a = w= using
@@ -703,32 +703,32 @@ for =∃ a : nat, a = w= using
   axiom Hzy : z = y
   axiom Hzw : z = w
 
-  theorem ex_a_eq_w : exists a, a = w  := exists_intro x (nat_trans3i Hxy Hzy Hzw)
+  theorem ex_a_eq_w : exists a, a = w  := exists.intro x (nat_trans3i Hxy Hzy Hzw)
   check ex_a_eq_w
 #+END_SRC
 
 
-Note that =exists_intro= also has implicit arguments. For example, Lean has to infer the implicit argument
+Note that =exists.intro= also has implicit arguments. For example, Lean has to infer the implicit argument
 =P : A → Bool=, a predicate (aka function to Prop). This creates complications. For example, suppose
-we have =Hg : g 0 0 = 0= and we invoke =exists_intro 0 Hg=. There are different possible values for =P=.
+we have =Hg : g 0 0 = 0= and we invoke =exists.intro 0 Hg=. There are different possible values for =P=.
 Each possible value corresponds to a different theorem: =∃ x, g x x = x=, =∃ x, g x x = 0=,
-=∃ x, g x 0 = x=, etc. Lean uses the context where =exists_intro= occurs to infer the users intent.
+=∃ x, g x 0 = x=, etc. Lean uses the context where =exists.intro= occurs to infer the users intent.
 In the example above, we were trying to prove the theorem =∃ a, a = w=. So, we are implicitly telling
 Lean how to choose =P=. In the following example, we demonstrate this issue. We ask Lean to display
-the implicit arguments using the option =pp.implicit=. We see that each instance of =exists_intro 0 Hg=
+the implicit arguments using the option =pp.implicit=. We see that each instance of =exists.intro 0 Hg=
 has different values for the implicit argument =P=.
 
 #+BEGIN_SRC lean
   import data.nat
   open nat
 
-  check @exists_intro
+  check @exists.intro
   constant g : nat → nat → nat
   axiom Hg : g 0 0 = 0
-  theorem gex1 : ∃ x, g x x = x := exists_intro 0 Hg
+  theorem gex1 : ∃ x, g x x = x := exists.intro 0 Hg
-  theorem gex2 : ∃ x, g x 0 = x := exists_intro 0 Hg
+  theorem gex2 : ∃ x, g x 0 = x := exists.intro 0 Hg
-  theorem gex3 : ∃ x, g 0 0 = x := exists_intro 0 Hg
+  theorem gex3 : ∃ x, g 0 0 = x := exists.intro 0 Hg
-  theorem gex4 : ∃ x, g x x = 0 := exists_intro 0 Hg
+  theorem gex4 : ∃ x, g x x = 0 := exists.intro 0 Hg
   set_option pp.implicit true  -- display implicit arguments
   check gex1
   check gex2
@@ -736,14 +736,14 @@ has different values for the implicit argument =P=.
   check gex4
 #+END_SRC
 
-We can view =exists_intro= (aka existential introduction) as an information hiding procedure.
+We can view =exists.intro= (aka existential introduction) as an information hiding procedure.
 We are "hiding" what is the witness for some fact. The existential elimination performs the opposite
-operation. The =exists_elim= theorem allows us to prove some proposition =B= from =∃ x : A, B x=
+operation. The =exists.elim= theorem allows us to prove some proposition =B= from =∃ x : A, B x=
 if we can derive =B= using an "abstract" witness =w= and a proof term =Hw : B w=.
 
 #+BEGIN_SRC lean
   import logic
-   check @exists_elim
+   check @exists.elim
 #+END_SRC
 
 In the following example, we define =even a= as =∃ b, a = 2*b=, and then we show that the sum
@@ -755,9 +755,9 @@ of two even numbers is an even number.
 
   definition even (a : nat) := ∃ b, a = 2*b
   theorem EvenPlusEven {a b : nat} (H1 : even a) (H2 : even b) : even (a + b) :=
-  exists_elim H1 (fun (w1 : nat) (Hw1 : a = 2*w1),
+  exists.elim H1 (fun (w1 : nat) (Hw1 : a = 2*w1),
-  exists_elim H2 (fun (w2 : nat) (Hw2 : b = 2*w2),
+  exists.elim H2 (fun (w2 : nat) (Hw2 : b = 2*w2),
-    exists_intro (w1 + w2)
+    exists.intro (w1 + w2)
       (calc a + b  =  2*w1 + b      : {Hw1}
               ...  =  2*w1 + 2*w2   : {Hw2}
               ...  =  2*(w1 + w2)   : eq.symm !mul.distr_left)))
@@ -767,7 +767,7 @@ of two even numbers is an even number.
 The example above also uses [[./calc.org][calculational proofs]] to show that =a + b = 2*(w1 + w2)=.
 The =calc= construct is just syntax sugar for creating proofs using transitivity and substitution.
 
-In Lean, we can use =obtain _, from _, _= as syntax sugar for =exists_elim=.
+In Lean, we can use =obtain _, from _, _= as syntax sugar for =exists.elim=.
 With this macro we can write the example above in a more natural way
 
 #+BEGIN_SRC lean
@@ -777,7 +777,7 @@ With this macro we can write the example above in a more natural way
   theorem EvenPlusEven {a b : nat} (H1 : even a) (H2 : even b) : even (a + b) :=
   obtain (w1 : nat) (Hw1 : a = 2*w1), from H1,
   obtain (w2 : nat) (Hw2 : b = 2*w2), from H2,
-    exists_intro (w1 + w2)
+    exists.intro (w1 + w2)
       (calc a + b  =  2*w1 + b      : {Hw1}
               ...  =  2*w1 + 2*w2   : {Hw2}
               ...  =  2*(w1 + w2)   : eq.symm !mul.distr_left)

library/algebra/ring.lean

@@ -66,7 +66,7 @@ theorem dvd.intro [s : has_dvd A] {a b c : A} : a * b = c → a | c := !has_dvd.
 theorem dvd.ex [s : has_dvd A] {a b : A} : a | b → ∃c, a * c = b := !has_dvd.dvd_ex
 
 theorem dvd.elim [s : has_dvd A] {P : Prop} {a b : A} (H₁ : a | b) (H₂ : ∀c, a * c = b → P) : P :=
-exists_elim (dvd.ex H₁) H₂
+exists.elim (dvd.ex H₁) H₂
 
 structure comm_semiring_dvd [class] (A : Type) extends comm_semiring A, has_dvd A
 

library/data/int/basic.lean

@@ -385,8 +385,8 @@ nat.cases_on m
 
 theorem cases (a : ℤ) : (∃n : ℕ, a = of_nat n) ∨ (∃n : ℕ, a = - n) :=
 cases_on a
-  (take n, or.inl (exists_intro n rfl))
+  (take n, or.inl (exists.intro n rfl))
-  (take n', or.inr (exists_intro (succ n') rfl))
+  (take n', or.inr (exists.intro (succ n') rfl))
 
 theorem by_cases {P : ℤ → Prop} (a : ℤ) (H1 : ∀n : ℕ, P (of_nat n)) (H2 : ∀n : ℕ, P (-n)) :
   P a :=
@@ -407,11 +407,11 @@ or.elim (cases a)
             a = -(of_nat n) : H2
           ... = -(of_nat 0) : {H3}
           ... = of_nat 0 : neg_zero,
-        or.inl (exists_intro 0 H4))
+        or.inl (exists.intro 0 H4))
       (take k : ℕ,
         assume H3 : n = succ k,
         have H4 : a = -(of_nat (succ k)), from H3 ▸ H2,
-        or.inr (exists_intro k H4)))
+        or.inr (exists.intro k H4)))
 
 theorem int_by_cases_succ {P : ℤ → Prop} (a : ℤ)
   (H1 : ∀n : ℕ, P (of_nat n)) (H2 : ∀n : ℕ, P (-(of_nat (succ n)))) : P a :=

library/data/int/order.lean

@@ -18,7 +18,7 @@ open int eq.ops
 namespace int
 
 theorem nonneg_elim {a : ℤ} : nonneg a → ∃n : ℕ, a = n :=
-cases_on a (take n H, exists_intro n rfl) (take n' H, false.elim H)
+cases_on a (take n H, exists.intro n rfl) (take n' H, false.elim H)
 
 theorem le_intro {a b : ℤ} {n : ℕ} (H : a + n = b) : a ≤ b :=
 have H1 : b - a = n, from add_imp_sub_right (!add_comm ▸ H),
@@ -27,7 +27,7 @@ show nonneg (b - a), from H1⁻¹ ▸ H2
 
 theorem le_elim {a b : ℤ} (H : a ≤ b) : ∃n : ℕ, a + n = b :=
 obtain (n : ℕ) (H1 : b - a = n), from nonneg_elim H,
-exists_intro n (!add_comm ▸ sub_imp_add H1)
+exists.intro n (!add_comm ▸ sub_imp_add H1)
 
 -- ### partial order
 
@@ -216,7 +216,7 @@ have H2 : a + succ n = b, from
   calc
     a + succ n = a + 1 + n : by simp
       ... = b : Hn,
-exists_intro n H2
+exists.intro n H2
 
 -- -- ### basic facts
 
@@ -409,13 +409,13 @@ or_resolve_right (le_or_gt a b) H
 
 theorem pos_imp_exists_nat {a : ℤ} (H : a ≥ 0) : ∃n : ℕ, a = n :=
 obtain (n : ℕ) (Hn : of_nat 0 + n = a), from le_elim H,
-exists_intro n (Hn⁻¹ ⬝ add_zero_left n)
+exists.intro n (Hn⁻¹ ⬝ add_zero_left n)
 
 theorem neg_imp_exists_nat {a : ℤ} (H : a ≤ 0) : ∃n : ℕ, a = -n :=
 have H2 : -a ≥ 0, from zero_le_neg H,
 obtain (n : ℕ) (Hn : -a = n), from pos_imp_exists_nat H2,
 have H3 : a = -n, from (neg_move Hn)⁻¹,
-exists_intro n H3
+exists.intro n H3
 
 theorem nat_abs_nonneg_eq {a : ℤ} (H : a ≥ 0) : (nat_abs a) = a :=
 obtain (n : ℕ) (Hn : a = n), from pos_imp_exists_nat H,

library/data/list/basic.lean

@@ -173,13 +173,13 @@ induction_on l
     assume H : x ∈ y::l,
     or.elim H
       (assume H1 : x = y,
-        exists_intro nil (!exists_intro (H1 ▸ rfl)))
+        exists.intro nil (!exists.intro (H1 ▸ rfl)))
       (assume H1 : x ∈ l,
         obtain s (H2 : ∃t : list T, l = s ++ (x::t)), from IH H1,
         obtain t (H3 : l = s ++ (x::t)), from H2,
         have H4 : y :: l = (y::s) ++ (x::t),
           from H3 ▸ rfl,
-        !exists_intro (!exists_intro H4)))
+        !exists.intro (!exists.intro H4)))
 
 definition mem.is_decidable [instance] (H : decidable_eq T) (x : T) (l : list T) : decidable (x ∈ l) :=
 rec_on l

library/data/nat/basic.lean

@@ -64,7 +64,7 @@ induction_on n
 
 theorem zero_or_exists_succ (n : ℕ) : n = 0 ∨ ∃k, n = succ k :=
 or_of_or_of_imp_of_imp (zero_or_succ_pred n) (assume H, H)
-    (assume H : n = succ (pred n), exists_intro (pred n) H)
+    (assume H : n = succ (pred n), exists.intro (pred n) H)
 
 theorem case {P : ℕ → Prop} (n : ℕ) (H1: P 0) (H2 : ∀m, P (succ m)) : P n :=
 induction_on n H1 (take m IH, H2 m)

library/data/nat/bquant.lean

@@ -29,16 +29,16 @@ namespace nat
 
   definition bex_succ {P : nat → Prop} {n : nat} (H : bex n P) : bex (succ n) P :=
   obtain (w : nat) (Hw : w < n ∧ P w), from H,
-    and.rec_on Hw (λ hlt hp, exists_intro w (and.intro (lt.step hlt) hp))
+    and.rec_on Hw (λ hlt hp, exists.intro w (and.intro (lt.step hlt) hp))
 
   definition bex_succ_of_pred {P : nat → Prop} {a : nat} (H : P a) : bex (succ a) P :=
-  exists_intro a (and.intro (lt.base a) H)
+  exists.intro a (and.intro (lt.base a) H)
 
   definition not_bex_succ {P : nat → Prop} {n : nat} (H₁ : ¬ bex n P) (H₂ : ¬ P n) : ¬ bex (succ n) P :=
   λ H, obtain (w : nat) (Hw : w < succ n ∧ P w), from H,
     and.rec_on Hw (λ hltsn hp, or.rec_on (eq_or_lt_of_le hltsn)
       (λ heq : w = n, absurd (eq.rec_on heq hp) H₂)
-      (λ hltn : w < n, absurd (exists_intro w (and.intro hltn hp)) H₁))
+      (λ hltn : w < n, absurd (exists.intro w (and.intro hltn hp)) H₁))
 
   definition ball_zero (P : nat → Prop) : ball zero P :=
   λ x Hlt, absurd Hlt !not_lt_zero

library/data/nat/div.lean

@@ -300,7 +300,7 @@ show x mod y = 0, from
 theorem dvd_iff_exists_mul (x y : ℕ) : x | y ↔ ∃z, z * x = y :=
 iff.intro
   (assume H : x | y,
-    show ∃z, z * x = y, from exists_intro _ (dvd_imp_div_mul_eq H))
+    show ∃z, z * x = y, from exists.intro _ (dvd_imp_div_mul_eq H))
   (assume H : ∃z, z * x = y,
     obtain (z : ℕ) (zx_eq : z * x = y), from H,
     show x | y, from mul_eq_imp_dvd zx_eq)
@@ -341,14 +341,14 @@ have H4 : k = k div n * (n div m) * m, from calc
   k     = k div n * n             : dvd_imp_div_mul_eq H2
     ... = k div n * (n div m * m) : H3
     ... = k div n * (n div m) * m : mul.assoc,
-mp (!dvd_iff_exists_mul⁻¹) (exists_intro (k div n * (n div m)) (H4⁻¹))
+mp (!dvd_iff_exists_mul⁻¹) (exists.intro (k div n * (n div m)) (H4⁻¹))
 
 theorem dvd_add {m n1 n2 : ℕ} (H1 : m | n1) (H2 : m | n2) : m | (n1 + n2) :=
 have H : (n1 div m + n2 div m) * m = n1 + n2, from calc
   (n1 div m + n2 div m) * m = n1 div m * m + n2 div m * m : mul.distr_right
                        ...  = n1 + n2 div m * m           : dvd_imp_div_mul_eq H1
                        ...  = n1 + n2                     : dvd_imp_div_mul_eq H2,
-mp (!dvd_iff_exists_mul⁻¹) (exists_intro _ H)
+mp (!dvd_iff_exists_mul⁻¹) (exists.intro _ H)
 
 theorem dvd_add_cancel_left {m n1 n2 : ℕ} : m | (n1 + n2) → m | n1 → m | n2 :=
 case_zero_pos m
@@ -373,7 +373,7 @@ case_zero_pos m
           ... = n1 div m * m + n2 div m * m   : add.comm
           ... = n1 + n2 div m * m             : dvd_imp_div_mul_eq H2,
     have H4 : n2 = n2 div m * m, from add.cancel_left H3,
-    mp (!dvd_iff_exists_mul⁻¹) (exists_intro _ (H4⁻¹)))
+    mp (!dvd_iff_exists_mul⁻¹) (exists.intro _ (H4⁻¹)))
 
 theorem dvd_add_cancel_right {m n1 n2 : ℕ} (H : m | (n1 + n2)) : m | n2 → m | n1 :=
 dvd_add_cancel_left (!add.comm ▸ H)

library/data/nat/order.lean

@@ -32,12 +32,12 @@ h ▸ le.add_right n k
 
 theorem le_elim {n m : ℕ} (h : n ≤ m) : ∃k, n + k = m :=
 le.rec_on h
-  (exists_intro 0 rfl)
+  (exists.intro 0 rfl)
   (λ m (h : n < m), lt.rec_on h
-    (exists_intro 1 rfl)
+    (exists.intro 1 rfl)
     (λ b hlt (ih : ∃ (k : ℕ), n + k = b),
       obtain (k : ℕ) (h : n + k = b), from ih,
-      exists_intro (succ k) (calc
+      exists.intro (succ k) (calc
         n + succ k = succ (n + k) : add.succ_right
                ... = succ b       : h)))
 
@@ -261,7 +261,7 @@ theorem succ_pos (n : ℕ) : 0 < succ n :=
 theorem lt_imp_eq_succ {n m : ℕ} (H : n < m) : exists k, m = succ k :=
 discriminate
   (take (Hm : m = 0), absurd (Hm ▸ H) !not_lt_zero)
-  (take (l : ℕ) (Hm : m = succ l), exists_intro l Hm)
+  (take (l : ℕ) (Hm : m = succ l), exists.intro l Hm)
 
 -- ### interaction with le
 

library/data/nat/sub.lean

@@ -224,7 +224,7 @@ or.elim !le_total
 
 theorem le_elim_sub {n m : ℕ} (H : n ≤ m) : ∃k, m - k = n :=
 obtain (k : ℕ) (Hk : n + k = m), from le_elim H,
-exists_intro k
+exists.intro k
   (calc
     m - k = n + k - k : {Hk⁻¹}
       ... = n         : !sub_add_left)

library/data/quotient/basic.lean

@@ -200,16 +200,16 @@ comp_quotient_map_binary Q H (Hrefl a) (Hrefl b)
 definition image {A B : Type} (f : A → B) := subtype (fun b, ∃a, f a = b)
 
 theorem image_inhabited {A B : Type} (f : A → B) (H : inhabited A) : inhabited (image f) :=
-inhabited.mk (tag (f (default A)) (exists_intro (default A) rfl))
+inhabited.mk (tag (f (default A)) (exists.intro (default A) rfl))
 
 theorem image_inhabited2 {A B : Type} (f : A → B) (a : A) : inhabited (image f) :=
 image_inhabited f (inhabited.mk a)
 
 definition fun_image {A B : Type} (f : A → B) (a : A) : image f :=
-tag (f a) (exists_intro a rfl)
+tag (f a) (exists.intro a rfl)
 
 theorem fun_image_def {A B : Type} (f : A → B) (a : A) :
-  fun_image f a = tag (f a) (exists_intro a rfl) := rfl
+  fun_image f a = tag (f a) (exists.intro a rfl) := rfl
 
 theorem elt_of_fun_image {A B : Type} (f : A → B) (a : A) : elt_of (fun_image f a) = f a :=
 elt_of.tag _ _
@@ -221,11 +221,11 @@ theorem fun_image_surj {A B : Type} {f : A → B} (u : image f) : ∃a, fun_imag
 subtype.destruct u
   (take (b : B) (H : ∃a, f a = b),
     obtain a (H': f a = b), from H,
-    (exists_intro a (tag_eq H')))
+    (exists.intro a (tag_eq H')))
 
 theorem image_tag {A B : Type} {f : A → B} (u : image f) : ∃a H, tag (f a) H = u :=
 obtain a (H : fun_image f a = u), from fun_image_surj u,
-exists_intro a (exists_intro (exists_intro a rfl) H)
+exists.intro a (exists.intro (exists.intro a rfl) H)
 
 open eq.ops
 

library/data/quotient/classical.lean

@@ -21,7 +21,7 @@ definition prelim_map {A : Type} (R : A → A → Prop) (a : A) :=
 theorem prelim_map_rel {A : Type} {R : A → A → Prop} (H : is_equivalence R) (a : A)
   : R a (prelim_map R a) :=
 have reflR : reflexive R, from is_equivalence.refl R,
-epsilon_spec (exists_intro a (reflR a))
+epsilon_spec (exists.intro a (reflR a))
 
 -- TODO: only needed: R PER
 theorem prelim_map_congr {A : Type} {R : A → A → Prop} (H1 : is_equivalence R) {a b : A}

library/init/logic.lean

@@ -238,12 +238,12 @@ calc_trans iff.trans
 inductive Exists {A : Type} (P : A → Prop) : Prop :=
 intro : ∀ (a : A), P a → Exists P
 
-definition exists_intro := @Exists.intro
+definition exists.intro := @Exists.intro
 
 notation `exists` binders `,` r:(scoped P, Exists P) := r
 notation `∃` binders `,` r:(scoped P, Exists P) := r
 
-theorem exists_elim {A : Type} {p : A → Prop} {B : Prop} (H1 : ∃x, p x) (H2 : ∀ (a : A) (H : p a), B) : B :=
+theorem exists.elim {A : Type} {p : A → Prop} {B : Prop} (H1 : ∃x, p x) (H2 : ∀ (a : A) (H : p a), B) : B :=
 Exists.rec H2 H1
 
 inductive decidable [class] (p : Prop) : Type :=

library/logic/axioms/examples/diaconescu.lean

@@ -17,10 +17,10 @@ private definition u := epsilon (λx, x = true ∨ p)
 private definition v := epsilon (λx, x = false ∨ p)
 
 private lemma u_def : u = true ∨ p :=
-epsilon_spec (exists_intro true (or.inl rfl))
+epsilon_spec (exists.intro true (or.inl rfl))
 
 private lemma v_def : v = false ∨ p :=
-epsilon_spec (exists_intro false (or.inl rfl))
+epsilon_spec (exists.intro false (or.inl rfl))
 
 private lemma uv_implies_p : ¬(u = v) ∨ p :=
 or.elim u_def

library/logic/axioms/hilbert.lean

@@ -23,7 +23,7 @@ axiom strong_indefinite_description {A : Type} (P : A → Prop) (H : nonempty A)
 -- axiom indefinite_description {A : Type} {P : A->Prop} (H : ∃x, P x) : {x : A, P x}
 
 theorem nonempty_imp_exists_true {A : Type} (H : nonempty A) : ∃x : A, true :=
-nonempty.elim H (take x, exists_intro x trivial)
+nonempty.elim H (take x, exists.intro x trivial)
 
 theorem nonempty_imp_inhabited {A : Type} (H : nonempty A) : inhabited A :=
 let u : {x | (∃y : A, true) → true} := strong_indefinite_description (λa, true) H in
@@ -52,7 +52,7 @@ theorem epsilon_spec {A : Type} {P : A → Prop} (Hex : ∃y, P y) :
 epsilon_spec_aux (nonempty_of_exists Hex) P Hex
 
 theorem epsilon_singleton {A : Type} (a : A) : @epsilon A (nonempty.intro a) (λx, x = a) = a :=
-epsilon_spec (exists_intro a (eq.refl a))
+epsilon_spec (exists.intro a (eq.refl a))
 
 
 -- the axiom of choice
@@ -62,7 +62,7 @@ theorem axiom_of_choice {A : Type} {B : A → Type} {R : Πx, B x → Prop} (H :
   ∃f, ∀x, R x (f x) :=
 let f := λx, @epsilon _ (nonempty_of_exists (H x)) (λy, R x y),
     H := take x, epsilon_spec (H x)
-in exists_intro f H
+in exists.intro f H
 
 theorem skolem {A : Type} {B : A → Type} {P : Πx, B x → Prop} :
   (∀x, ∃y, P x y) ↔ ∃f, (∀x, P x (f x)) :=
@@ -70,4 +70,4 @@ iff.intro
   (assume H : (∀x, ∃y, P x y), axiom_of_choice H)
   (assume H : (∃f, (∀x, P x (f x))),
     take x, obtain (fw : ∀x, B x) (Hw : ∀x, P x (fw x)), from H,
-      exists_intro (fw x) (Hw x))
+      exists.intro (fw x) (Hw x))

library/logic/connectives.lean

@@ -139,7 +139,7 @@ notation `∃!` binders `,` r:(scoped P, exists_unique P) := r
 
 theorem exists_unique.intro {A : Type} {p : A → Prop} (w : A) (H1 : p w) (H2 : ∀y, p y → y = w) :
   ∃!x, p x :=
-exists_intro w (and.intro H1 H2)
+exists.intro w (and.intro H1 H2)
 
 theorem exists_unique.elim {A : Type} {p : A → Prop} {b : Prop}
     (H2 : ∃!x, p x) (H1 : ∀x, p x → (∀y, p y → y = x) → b) : b :=

library/logic/examples/nuprl_examples.lean

@@ -112,7 +112,7 @@ assume (H : ∀x, C → P x) (Hc : C),
 theorem thm18a : ((∃x, P x) → C) → (∀x, P x → C) :=
 assume H : (∃x, P x) → C,
   take x, assume Hp : P x,
-  have Hex : ∃x, P x, from exists_intro x Hp,
+  have Hex : ∃x, P x, from exists.intro x Hp,
   H Hex
 
 theorem thm18b : (∀x, P x → C) → (∃x, P x) → C :=
@@ -126,28 +126,28 @@ assume (Hem : C ∨ ¬C) (Hin : ∃x : T, true) (H1 : C → ∃x, P x),
     (assume Hc : C,
       obtain (w : T) (Hw : P w), from H1 Hc,
       have Hr : C → P w, from assume Hc, Hw,
-      exists_intro w Hr)
+      exists.intro w Hr)
     (assume Hnc : ¬C,
       obtain (w : T) (Hw : true), from Hin,
       have Hr : C → P w, from assume Hc, absurd Hc Hnc,
-      exists_intro w Hr)
+      exists.intro w Hr)
 
 theorem thm19b : (∃x, C → P x) → C → (∃x, P x) :=
 assume (H : ∃x, C → P x) (Hc : C),
   obtain (w : T) (Hw : C → P w), from H,
-  exists_intro w (Hw Hc)
+  exists.intro w (Hw Hc)
 
 theorem thm20a : (C ∨ ¬C) → (∃x : T, true) → ((¬∀x, P x) → ∃x, ¬P x) → ((∀x, P x) → C) → (∃x, P x → C) :=
 assume Hem Hin Hnf H,
   or.elim Hem
     (assume Hc : C,
       obtain (w : T) (Hw : true), from Hin,
-      exists_intro w (assume H : P w, Hc))
+      exists.intro w (assume H : P w, Hc))
     (assume Hnc : ¬C,
       have H1 : ¬(∀x, P x), from mt H Hnc,
       have H2 : ∃x, ¬P x, from Hnf H1,
       obtain (w : T) (Hw : ¬P w), from H2,
-      exists_intro w (assume H : P w, absurd H Hw))
+      exists.intro w (assume H : P w, absurd H Hw))
 
 theorem thm20b : (∃x, P x → C) → (∀ x, P x) → C :=
 assume Hex Hall,
@@ -159,16 +159,16 @@ assume Hin H,
   or.elim H
     (assume Hex : ∃x, P x,
       obtain (w : T) (Hw : P w), from Hex,
-      exists_intro w (or.inl Hw))
+      exists.intro w (or.inl Hw))
     (assume Hc  : C,
       obtain (w : T) (Hw : true), from Hin,
-      exists_intro w (or.inr Hc))
+      exists.intro w (or.inr Hc))
 
 theorem thm21b : (∃x, P x ∨ C) → ((∃x, P x) ∨ C) :=
 assume H,
   obtain (w : T) (Hw : P w ∨ C), from H,
   or.elim Hw
-    (assume H : P w, or.inl (exists_intro w H))
+    (assume H : P w, or.inl (exists.intro w H))
     (assume Hc : C, or.inr Hc)
 
 theorem thm22a : (∀x, P x) ∨ C → ∀x, P x ∨ C :=
@@ -193,12 +193,12 @@ assume H,
   have Hex : ∃x, P x, from and.elim_left H,
   have Hc : C, from and.elim_right H,
   obtain (w : T) (Hw : P w), from Hex,
-  exists_intro w (and.intro Hw Hc)
+  exists.intro w (and.intro Hw Hc)
 
 theorem thm23b : (∃x, P x ∧ C) → (∃x, P x) ∧ C :=
 assume H,
   obtain (w : T) (Hw : P w ∧ C), from H,
-  have Hex : ∃x, P x, from exists_intro w (and.elim_left Hw),
+  have Hex : ∃x, P x, from exists.intro w (and.elim_left Hw),
   and.intro Hex (and.elim_right Hw)
 
 theorem thm24a : (∀x, P x) ∧ C → (∀x, P x ∧ C) :=

library/logic/identities.lean

@@ -102,7 +102,7 @@ assume H, by_contradiction (assume Hna : ¬a,
 theorem forall_not_of_not_exists {A : Type} {P : A → Prop} [D : ∀x, decidable (P x)]
     (H : ¬∃x, P x) : ∀x, ¬P x :=
 take x, or.elim (em (P x))
-  (assume Hp : P x,   absurd (exists_intro x Hp) H)
+  (assume Hp : P x,   absurd (exists.intro x Hp) H)
   (assume Hn : ¬P x, Hn)
 
 theorem exists_not_of_not_forall {A : Type} {P : A → Prop} [D : ∀x, decidable (P x)]

library/logic/quantifiers.lean

@@ -28,9 +28,9 @@ iff.intro
 theorem exists_congr {A : Type} {φ ψ : A → Prop} (H : ∀x, φ x ↔ ψ x) : (∃x, φ x) ↔ (∃x, ψ x) :=
 iff.intro
   (assume Hex, obtain w Pw, from Hex,
-    exists_intro w (iff.elim_left (H w) Pw))
+    exists.intro w (iff.elim_left (H w) Pw))
   (assume Hex, obtain w Qw, from Hex,
-    exists_intro w (iff.elim_right (H w) Qw))
+    exists.intro w (iff.elim_right (H w) Qw))
 
 theorem forall_true_iff_true (A : Type) : (∀x : A, true) ↔ true :=
 iff.intro (assume H, trivial) (assume H, take x, trivial)
@@ -41,7 +41,7 @@ iff.intro (assume Hl, inhabited.destruct H (take x, Hl x)) (assume Hr, take x, H
 theorem exists_p_iff_p (A : Type) [H : inhabited A] (p : Prop) : (∃x : A, p) ↔ p :=
 iff.intro
   (assume Hl, obtain a Hp, from Hl, Hp)
-  (assume Hr, inhabited.destruct H (take a, exists_intro a Hr))
+  (assume Hr, inhabited.destruct H (take a, exists.intro a Hr))
 
 theorem forall_and_distribute {A : Type} (φ ψ : A → Prop) :
   (∀x, φ x ∧ ψ x) ↔ (∀x, φ x) ∧ (∀x, ψ x) :=
@@ -54,13 +54,13 @@ theorem exists_or_distribute {A : Type} (φ ψ : A → Prop) :
 iff.intro
   (assume H, obtain (w : A) (Hw : φ w ∨ ψ w), from H,
     or.elim Hw
-      (assume Hw1 : φ w, or.inl (exists_intro w Hw1))
+      (assume Hw1 : φ w, or.inl (exists.intro w Hw1))
-      (assume Hw2 : ψ w, or.inr (exists_intro w Hw2)))
+      (assume Hw2 : ψ w, or.inr (exists.intro w Hw2)))
   (assume H, or.elim H
     (assume H1, obtain (w : A) (Hw : φ w), from H1,
-      exists_intro w (or.inl Hw))
+      exists.intro w (or.inl Hw))
     (assume H2, obtain (w : A) (Hw : ψ w), from H2,
-      exists_intro w (or.inr Hw)))
+      exists.intro w (or.inr Hw)))
 
 theorem nonempty_of_exists {A : Type} {P : A → Prop} (H : ∃x, P x) : nonempty A :=
 obtain w Hw, from H, nonempty.intro w
@@ -78,7 +78,7 @@ section
 
   definition decidable_exists_eq [instance] : decidable (∃ x, x = a ∧ P x) :=
   decidable.rec_on H
-     (λ pa, inl (exists_intro a (and.intro rfl pa)))
+     (λ pa, inl (exists.intro a (and.intro rfl pa)))
      (λ npa, inr (λ h,
        obtain (w : A) (Hw : w = a ∧ P w), from h,
        absurd (and.rec_on Hw (λ h₁ h₂, h₁ ▸ h₂)) npa))

src/frontends/lean/builtin_exprs.cpp

@@ -331,7 +331,7 @@ static expr parse_show(parser & p, unsigned, expr const *, pos_info const & pos)
 static name * g_exists_elim = nullptr;
 static expr parse_obtain(parser & p, unsigned, expr const *, pos_info const & pos) {
     if (!p.env().find(*g_exists_elim))
-        throw parser_error("invalid use of 'obtain' expression, environment does not contain 'exists_elim' theorem", pos);
+        throw parser_error("invalid use of 'obtain' expression, environment does not contain 'exists.elim' theorem", pos);
     // exists_elim {A : Type} {P : A → Prop} {B : Prop} (H1 : ∃ x : A, P x) (H2 : ∀ (a : A) (H : P a), B)
     buffer<expr> ps;
     auto b_pos = p.pos();
@@ -534,7 +534,7 @@ parse_table get_builtin_led_table() {
 
 void initialize_builtin_exprs() {
     notation::H_show        = new name("H_show");
-    notation::g_exists_elim = new name("exists_elim");
+    notation::g_exists_elim = new name{"exists", "elim"};
     notation::g_ite         = new name("ite");
     notation::g_dite        = new name("dite");
     notation::g_not         = new expr(mk_constant("not"));

tests/lean/run/class4.lean

@@ -31,14 +31,14 @@ theorem not_is_zero_succ (x : nat) : ¬ is_zero (succ x)
 theorem dichotomy (m : nat) : m = zero ∨ (∃ n, m = succ n)
 := nat.rec
      (or.intro_left _ (refl zero))
-     (λ m H, or.intro_right _ (exists_intro m (refl (succ m))))
+     (λ m H, or.intro_right _ (exists.intro m (refl (succ m))))
      m
 
 theorem is_zero_to_eq (x : nat) (H : is_zero x) : x = zero
 := or.elim (dichotomy x)
      (assume Hz : x = zero, Hz)
      (assume Hs : (∃ n, x = succ n),
-       exists_elim Hs (λ (w : nat) (Hw : x = succ w),
+       exists.elim Hs (λ (w : nat) (Hw : x = succ w),
          absurd H (eq.subst (symm Hw) (not_is_zero_succ w))))
 
 theorem is_not_zero_to_eq {x : nat} (H : ¬ is_zero x) : ∃ n, x = succ n
@@ -48,7 +48,7 @@ theorem is_not_zero_to_eq {x : nat} (H : ¬ is_zero x) : ∃ n, x = succ n
      (assume Hs, Hs)
 
 theorem not_zero_add (x y : nat) (H : ¬ is_zero y) : ¬ is_zero (x + y)
-:= exists_elim (is_not_zero_to_eq H)
+:= exists.elim (is_not_zero_to_eq H)
      (λ (w : nat) (Hw : y = succ w),
         have H1 : x + y = succ (x + w), from
           calc x + y = x + succ w   : {Hw}

tests/lean/run/elim.lean

@@ -4,4 +4,4 @@ constant p : num → num → num → Prop
 axiom H1 : ∃ x y z, p x y z
 axiom H2 : ∀ {x y z : num}, p x y z → p x x x
 theorem tst : ∃ x, p x x x
-:= obtain a b c H, from H1, exists_intro a (H2 H)
+:= obtain a b c H, from H1, exists.intro a (H2 H)

tests/lean/run/elim2.lean

@@ -5,4 +5,4 @@ axiom H1 : ∃ x y z, p x y z
 axiom H2 : ∀ {x y z : num}, p x y z → p x x x
 theorem tst : ∃ x, p x x x
 := obtain a b c H [visible], from H1,
-     by (apply exists_intro; apply H2; eassumption)
+     by (apply exists.intro; apply H2; eassumption)

tests/lean/run/fapply.lean

@@ -2,7 +2,7 @@ import logic
 
 example : ∃ a : num, a = a :=
 begin
-  fapply exists_intro,
+  fapply exists.intro,
   exact 0,
   apply rfl,
 end

tests/lean/run/pquot.lean

@@ -69,7 +69,7 @@ pquot.rec_on q f (λ (a b : A) (H : R a b),
     @cast_to_heq _ _ _ _ aux₂ aux₃)
 
 theorem pquot.abs.surjective {A : Type} {R : A → A → Prop} : ∀ q : pquot R, ∃ x : A, pquot.abs R x = q :=
-take q, pquot.induction_on q (take a, exists_intro a rfl)
+take q, pquot.induction_on q (take a, exists.intro a rfl)
 
 definition pquot.exact {A : Type} (R : A → A → Prop) :=
 ∀ a b : A, pquot.abs R a = pquot.abs R b → R a b

tests/lean/run/tactic21.lean

@@ -4,10 +4,10 @@ open tactic
 definition assump := eassumption
 
 theorem tst1 {A : Type} {a b c : A} {p : A → A → Prop} (H1 : p a b) (H2 : p b c) : ∃ x, p a x ∧ p x c
-:= by apply exists_intro; apply and.intro; assump; assump
+:= by apply exists.intro; apply and.intro; assump; assump
 
 theorem tst2 {A : Type} {a b c d : A} {p : A → A → Prop} (Ha : p a c) (H1 : p a b) (Hb : p b d) (H2 : p b c) : ∃ x, p a x ∧ p x c
-:= by apply exists_intro; apply and.intro; assump; assump
+:= by apply exists.intro; apply and.intro; assump; assump
 
 (*
 print(get_env():find("tst2"):value())

tests/lean/run/tactic23.lean

@@ -26,13 +26,13 @@ check 2 + 3
 definition assump : tactic := tactic.eassumption
 
 theorem T1 {p : nat → Prop} {a : nat } (H : p (a+2)) : ∃ x, p (succ x)
-:= by apply exists_intro; assump
+:= by apply exists.intro; assump
 
 definition is_zero (n : nat)
 := nat.rec true (λ n r, false) n
 
 theorem T2 : ∃ a, (is_zero a) = true
-:= by apply exists_intro; apply eq.refl
+:= by apply exists.intro; apply eq.refl
 
 end nat
 end experiment

tests/lean/slow/nat_bug2.lean

@@ -60,7 +60,7 @@ theorem zero_or_succ (n : ℕ) : n = 0 ∨ n = succ (pred n)
       (show succ m = succ (pred (succ m)), from congr_arg succ (pred_succ m⁻¹)))
 
 theorem zero_or_succ2 (n : ℕ) : n = 0 ∨ ∃k, n = succ k
-:= or_of_or_of_imp_of_imp (zero_or_succ n) (assume H, H) (assume H : n = succ (pred n), exists_intro (pred n) H)
+:= or_of_or_of_imp_of_imp (zero_or_succ n) (assume H, H) (assume H : n = succ (pred n), exists.intro (pred n) H)
 
 theorem case {P : ℕ → Prop} (n : ℕ) (H1: P 0) (H2 : ∀m, P (succ m)) : P n
 := induction_on n H1 (take m IH, H2 m)
@@ -386,7 +386,7 @@ infix  `<=` := le
 infix  `≤`  := le
 
 theorem le_intro {n m k : ℕ} (H : n + k = m) : n ≤ m
-:= exists_intro k H
+:= exists.intro k H
 
 theorem le_elim {n m : ℕ} (H : n ≤ m) : ∃ k, n + k = m
 := H
@@ -691,7 +691,7 @@ theorem lt_zero_inv (n : ℕ) : ¬ n < 0
 theorem lt_positive {n m : ℕ} (H : n < m) : ∃k, m = succ k
 := discriminate
     (take (Hm : m = 0), absurd (subst Hm H) (lt_zero_inv n))
-    (take (l : ℕ) (Hm : m = succ l), exists_intro l Hm)
+    (take (l : ℕ) (Hm : m = succ l), exists.intro l Hm)
 
 ---------- interaction with le
 
@@ -882,7 +882,7 @@ theorem ne_zero_positive {n : ℕ} (H : n ≠ 0) : n > 0
 theorem pos_imp_eq_succ {n : ℕ} (H : n > 0) : ∃l, n = succ l
 := discriminate
     (take H2, absurd (subst H2 H) (lt_irrefl 0))
-    (take l Hl, exists_intro l Hl)
+    (take l Hl, exists.intro l Hl)
 
 theorem add_positive_right (n : ℕ) {k : ℕ} (H : k > 0) : n + k > n
 := obtain (l : ℕ) (Hl : k = succ l), from pos_imp_eq_succ H,
@@ -1276,7 +1276,7 @@ theorem sub_le_self (n m : ℕ) : n - m ≤ n
 theorem le_elim_sub (n m : ℕ) (H : n ≤ m) : ∃k, m - k = n
 :=
   obtain (k : ℕ) (Hk : n + k = m), from le_elim H,
-  exists_intro k
+  exists.intro k
     (calc
       m - k = n + k - k : {symm Hk}
         ... = n : sub_add_left n k)

tests/lean/slow/nat_wo_hints.lean

@@ -54,7 +54,7 @@ theorem zero_or_succ (n : ℕ) : n = 0 ∨ n = succ (pred n)
       (show succ m = succ (pred (succ m)), from congr_arg succ (pred_succ m⁻¹)))
 
 theorem zero_or_succ2 (n : ℕ) : n = 0 ∨ ∃k, n = succ k
-:= or_of_or_of_imp_of_imp (zero_or_succ n) (assume H, H) (assume H : n = succ (pred n), exists_intro (pred n) H)
+:= or_of_or_of_imp_of_imp (zero_or_succ n) (assume H, H) (assume H : n = succ (pred n), exists.intro (pred n) H)
 
 theorem case {P : ℕ → Prop} (n : ℕ) (H1: P 0) (H2 : ∀m, P (succ m)) : P n
 := induction_on n H1 (take m IH, H2 m)
@@ -380,7 +380,7 @@ infix  `<=` := le
 infix  `≤`  := le
 
 theorem le_intro {n m k : ℕ} (H : n + k = m) : n ≤ m
-:= exists_intro k H
+:= exists.intro k H
 
 theorem le_elim {n m : ℕ} (H : n ≤ m) : ∃ k, n + k = m
 := H
@@ -689,7 +689,7 @@ theorem lt_zero_inv (n : ℕ) : ¬ n < 0
 theorem lt_positive {n m : ℕ} (H : n < m) : ∃k, m = succ k
 := discriminate
     (take (Hm : m = 0), absurd (subst Hm H) (lt_zero_inv n))
-    (take (l : ℕ) (Hm : m = succ l), exists_intro l Hm)
+    (take (l : ℕ) (Hm : m = succ l), exists.intro l Hm)
 
 ---------- interaction with le
 
@@ -880,7 +880,7 @@ theorem ne_zero_positive {n : ℕ} (H : n ≠ 0) : n > 0
 theorem pos_imp_eq_succ {n : ℕ} (H : n > 0) : ∃l, n = succ l
 := discriminate
     (take H2, absurd (subst H2 H) (lt_irrefl 0))
-    (take l Hl, exists_intro l Hl)
+    (take l Hl, exists.intro l Hl)
 
 theorem add_positive_right (n : ℕ) {k : ℕ} (H : k > 0) : n + k > n
 := obtain (l : ℕ) (Hl : k = succ l), from pos_imp_eq_succ H,
@@ -1280,7 +1280,7 @@ theorem sub_le_self (n m : ℕ) : n - m ≤ n
 theorem le_elim_sub (n m : ℕ) (H : n ≤ m) : ∃k, m - k = n
 :=
   obtain (k : ℕ) (Hk : n + k = m), from le_elim H,
-  exists_intro k
+  exists.intro k
     (calc
       m - k = n + k - k : {symm Hk}
         ... = n : sub_add_left n k)