fix(tests/lean): adjust tests to reflect changes in the standard library

tests/lean/interactive/eq2.lean

@@ -6,16 +6,15 @@
 -- ====================
 
 -- Equality.
-
+prelude
-import logic.prop
+definition Prop := Type.{0}
-
 -- eq
 -- --
 
 inductive eq {A : Type} (a : A) : A → Prop :=
 refl : eq a a
 
-infix `=` := eq
+infix `=`:50 := eq
 definition rfl {A : Type} {a : A} := eq.refl a
 
 -- proof irrelevance is built in
@@ -43,9 +42,9 @@ calc_refl  eq.refl
 calc_trans eq.trans
 
 namespace eq_ops
-  postfix `⁻¹` := eq.symm
+  postfix `⁻¹`:1024 := eq.symm
-  reserve infixr `⬝`:75  infixr `⬝`   := eq.trans
+  infixr `⬝`:75   := eq.trans
-  infixr `▸`   := eq.subst
+  infixr `▸`:75   := eq.subst
 end eq_ops
 open eq_ops
 
@@ -77,7 +76,7 @@ namespace eq
           (u : P a) :
     drec_on H2 (drec_on H1 u) = drec_on (trans H1 H2) u :=
     (show ∀(H2 : b = c), drec_on H2 (drec_on H1 u) = drec_on (trans H1 H2) u,
-      from drec_on H2 (take (H2 : b = b), drec_on_id H2 _))
+      from drec_on H2 (fun (H2 : b = b), drec_on_id H2 _))
     H2
 end eq
 
@@ -127,6 +126,7 @@ theorem congr_arg2_dep {A : Type} {B : A → Type} {C : Type} {a₁ a₂ : A}
             ... = f a₁ b₂                : {H₂})
     b₂ H₁ H₂
 
+
 theorem congr_arg3_dep {A : Type} {B : A → Type} {C : Πa, B a → Type} {D : Type} {a₁ a₂ : A} {b₁ : B a₁} {b₂ : B a₂} {c₁ : C a₁ b₁} {c₂ : C a₂ b₂} (f : Πa b, C a b → D)
     (H₁ : a₁ = a₂)  (H₂ : eq.drec_on H₁ b₁ = b₂) (H₃ : eq.drec_on (congr_arg2_dep C H₁ H₂) c₁ = c₂) :
   f a₁ b₁ c₁ = f a₂ b₂ c₂ :=
@@ -143,7 +143,7 @@ theorem congr_arg3_ndep_dep {A B : Type} {C : A → B → Type} {D : Type} {a₁
 congr_arg3_dep f H₁ (drec_on_constant H₁ b₁ ⬝ H₂) H₃
 
 theorem equal_f {A : Type} {B : A → Type} {f g : Π x, B x} (H : f = g) : ∀x, f x = g x :=
-take x, congr_fun H x
+fun x, congr_fun H x
 
 theorem eqmp {a b : Prop} (H1 : a = b) (H2 : a) : b :=
 H1 ▸ H2
@@ -151,59 +151,11 @@ H1 ▸ H2
 theorem eqmpr {a b : Prop} (H1 : a = b) (H2 : b) : a :=
 H1⁻¹ ▸ H2
 
-theorem eq_true_elim {a : Prop} (H : a = true) : a :=
-H⁻¹ ▸ trivial
-
-theorem eq_false_elim {a : Prop} (H : a = false) : ¬a :=
-assume Ha : a, H ▸ Ha
-
 theorem imp_trans {a b c : Prop} (H1 : a → b) (H2 : b → c) : a → c :=
-assume Ha, H2 (H1 Ha)
+fun Ha, H2 (H1 Ha)
 
 theorem imp_eq_trans {a b c : Prop} (H1 : a → b) (H2 : b = c) : a → c :=
-assume Ha, H2 ▸ (H1 Ha)
+fun Ha, H2 ▸ (H1 Ha)
 
 theorem eq_imp_trans {a b c : Prop} (H1 : a = b) (H2 : b → c) : a → c :=
-assume Ha, H2 (H1 ▸ Ha)
+fun Ha, H2 (H1 ▸ Ha)
-
--- ne
--- --
-
-definition ne {A : Type} (a b : A) := ¬(a = b)
-infix `≠` := ne
-
-namespace ne
-  theorem intro {A : Type} {a b : A} (H : a = b → false) : a ≠ b :=
-  H
-
-  theorem elim {A : Type} {a b : A} (H1 : a ≠ b) (H2 : a = b) : false :=
-  H1 H2
-
-  theorem irrefl {A : Type} {a : A} (H : a ≠ a) : false :=
-  H rfl
-
-  theorem symm {A : Type} {a b : A} (H : a ≠ b) : b ≠ a :=
-  assume H1 : b = a, H (H1⁻¹)
-end ne
-
-theorem a_neq_a_elim {A : Type} {a : A} (H : a ≠ a) : false :=
-H rfl
-
-theorem eq_ne_trans {A : Type} {a b c : A} (H1 : a = b) (H2 : b ≠ c) : a ≠ c :=
-H1⁻¹ ▸ H2
-
-theorem ne_eq_trans {A : Type} {a b c : A} (H1 : a ≠ b) (H2 : b = c) : a ≠ c :=
-H2 ▸ H1
-
-calc_trans eq_ne_trans
-calc_trans ne_eq_trans
-
-theorem p_ne_false {p : Prop} (Hp : p) : p ≠ false :=
-assume Heq : p = false, Heq ▸ Hp
-
-theorem p_ne_true {p : Prop} (Hnp : ¬p) : p ≠ true :=
-assume Heq : p = true, absurd trivial (Heq ▸ Hnp)
-
-theorem true_ne_false : ¬true = false :=
-assume H : true = false,
-  H ▸ trivial

tests/lean/run/nat_bug3.lean

@@ -20,7 +20,7 @@ axiom add_one (n:nat) : n + (succ zero) = succ n
 axiom add_le_right_inv {n m k : nat} (H : n + k ≤ m + k) : n ≤ m
 
 theorem succ_le_cancel {n m : nat} (H : succ n ≤ succ m) :  n ≤ m
-:= add_le_right_inv (add_one m⁻¹ ▸ add_one n⁻¹ ▸ H)
+:= add_le_right_inv ((add_one m)⁻¹ ▸ (add_one n)⁻¹ ▸ H)
 
 end nat
 end experiment

tests/lean/slow/nat_bug2.lean

@@ -57,7 +57,7 @@ theorem zero_or_succ (n : ℕ) : n = 0 ∨ n = succ (pred n)
 := induction_on n
     (or.intro_left _ (refl 0))
     (take m IH, or.intro_right _
-      (show succ m = succ (pred (succ m)), from congr_arg succ (pred_succ m⁻¹)))
+      (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)
@@ -72,7 +72,7 @@ theorem discriminate {B : Prop} {n : ℕ} (H1: n = 0 → B) (H2 : ∀m, n = succ
 
 theorem succ_inj {n m : ℕ} (H : succ n = succ m) : n = m
 := calc
-    n = pred (succ n) : pred_succ n⁻¹
+    n = pred (succ n) : (pred_succ n)⁻¹
   ... = pred (succ m) : {H}
   ... = m             : pred_succ m
 
@@ -473,7 +473,7 @@ theorem succ_le {n m : ℕ} (H : n ≤ m) : succ n ≤ succ m
 := add_one m ▸ add_one n ▸ add_le_right H 1
 
 theorem succ_le_cancel {n m : ℕ} (H : succ n ≤ succ m) :  n ≤ m
-:= add_le_right_inv (add_one m⁻¹ ▸ add_one n⁻¹ ▸ H)
+:= add_le_right_inv ((add_one m)⁻¹ ▸ (add_one n)⁻¹ ▸ H)
 
 theorem self_le_succ (n : ℕ) : n ≤ succ n
 := le_intro (add_one n)
@@ -757,7 +757,7 @@ theorem add_lt {n m k l : ℕ} (H1 : n < k) (H2 : m < l) : n + m < k + l
 := add_lt_le H1 (lt_imp_le H2)
 
 theorem add_lt_left_inv {n m k : ℕ} (H : k + n < k + m) : n < m
-:= add_le_left_inv (add_succ k n⁻¹ ▸ H)
+:= add_le_left_inv ((add_succ k n)⁻¹ ▸ H)
 
 theorem add_lt_right_inv {n m k : ℕ} (H : n + k < m + k) : n < m
 := add_lt_left_inv (add_comm m k ▸ add_comm n k ▸ H)
@@ -768,7 +768,7 @@ theorem succ_lt {n m : ℕ} (H : n < m) : succ n < succ m
 := add_one m ▸ add_one n ▸ add_lt_right H 1
 
 theorem succ_lt_inv {n m : ℕ} (H : succ n < succ m) :  n < m
-:= add_lt_right_inv (add_one m⁻¹ ▸ add_one n⁻¹ ▸ H)
+:= add_lt_right_inv ((add_one m)⁻¹ ▸ (add_one n)⁻¹ ▸ H)
 
 theorem lt_self_succ (n : ℕ) : n < succ n
 := le_refl (succ n)
@@ -1313,9 +1313,9 @@ theorem sub_sub_split {P : ℕ → ℕ → Prop} {n m : ℕ} (H1 : ∀k, n = m +
     (H2 : ∀k, m = n + k → P 0 k) : P (n - m) (m - n)
 := or.elim (le_total n m)
     (assume H3 : n ≤ m,
-      le_imp_sub_eq_zero H3⁻¹ ▸  (H2 (m - n) (add_sub_le H3⁻¹)))
+      (le_imp_sub_eq_zero H3)⁻¹ ▸  (H2 (m - n) ((add_sub_le H3)⁻¹)))
     (assume H3 : m ≤ n,
-      le_imp_sub_eq_zero H3⁻¹ ▸ (H1 (n - m) (add_sub_le H3⁻¹)))
+      (le_imp_sub_eq_zero H3)⁻¹ ▸ (H1 (n - m) ((add_sub_le H3)⁻¹)))
 
 theorem sub_intro {n m k : ℕ} (H : n + m = k) : k - n = m
 := have H2 : k - n + n = m + n, from

tests/lean/slow/nat_wo_hints.lean

@@ -51,7 +51,7 @@ theorem zero_or_succ (n : ℕ) : n = 0 ∨ n = succ (pred n)
 := induction_on n
     (or.intro_left _ (eq.refl 0))
     (take m IH, or.intro_right _
-      (show succ m = succ (pred (succ m)), from congr_arg succ (pred_succ m⁻¹)))
+      (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)
@@ -66,7 +66,7 @@ theorem discriminate {B : Prop} {n : ℕ} (H1: n = 0 → B) (H2 : ∀m, n = succ
 
 theorem succ_inj {n m : ℕ} (H : succ n = succ m) : n = m
 := calc
-    n = pred (succ n) : pred_succ n⁻¹
+    n = pred (succ n) : (pred_succ n)⁻¹
   ... = pred (succ m) : {H}
   ... = m             : pred_succ m
 
@@ -467,7 +467,7 @@ theorem succ_le {n m : ℕ} (H : n ≤ m) : succ n ≤ succ m
 := add_one m ▸ add_one n ▸ add_le_right H 1
 
 theorem succ_le_cancel {n m : ℕ} (H : succ n ≤ succ m) :  n ≤ m
-:= add_le_right_inv (add_one m⁻¹ ▸ add_one n⁻¹ ▸ H)
+:= add_le_right_inv ((add_one m)⁻¹ ▸ (add_one n)⁻¹ ▸ H)
 
 theorem self_le_succ (n : ℕ) : n ≤ succ n
 := le_intro (add_one n)
@@ -755,7 +755,7 @@ theorem add_lt {n m k l : ℕ} (H1 : n < k) (H2 : m < l) : n + m < k + l
 := add_lt_le H1 (lt_imp_le H2)
 
 theorem add_lt_left_inv {n m k : ℕ} (H : k + n < k + m) : n < m
-:= add_le_left_inv (add_succ k n⁻¹ ▸ H)
+:= add_le_left_inv ((add_succ k n)⁻¹ ▸ H)
 
 theorem add_lt_right_inv {n m k : ℕ} (H : n + k < m + k) : n < m
 := add_lt_left_inv (add_comm m k ▸ add_comm n k ▸ H)
@@ -766,7 +766,7 @@ theorem succ_lt {n m : ℕ} (H : n < m) : succ n < succ m
 := add_one m ▸ add_one n ▸ add_lt_right H 1
 
 theorem succ_lt_inv {n m : ℕ} (H : succ n < succ m) :  n < m
-:= add_lt_right_inv (add_one m⁻¹ ▸ add_one n⁻¹ ▸ H)
+:= add_lt_right_inv ((add_one m)⁻¹ ▸ (add_one n)⁻¹ ▸ H)
 
 theorem lt_self_succ (n : ℕ) : n < succ n
 := le_refl (succ n)
@@ -1317,9 +1317,9 @@ theorem sub_sub_split {P : ℕ → ℕ → Prop} {n m : ℕ} (H1 : ∀k, n = m +
     (H2 : ∀k, m = n + k → P 0 k) : P (n - m) (m - n)
 := or.elim (le_total n m)
     (assume H3 : n ≤ m,
-      le_imp_sub_eq_zero H3⁻¹ ▸  (H2 (m - n) (add_sub_le H3⁻¹)))
+      (le_imp_sub_eq_zero H3)⁻¹ ▸  (H2 (m - n) ((add_sub_le H3)⁻¹)))
     (assume H3 : m ≤ n,
-      le_imp_sub_eq_zero H3⁻¹ ▸ (H1 (n - m) (add_sub_le H3⁻¹)))
+      (le_imp_sub_eq_zero H3)⁻¹ ▸ (H1 (n - m) ((add_sub_le H3)⁻¹)))
 
 theorem sub_intro {n m k : ℕ} (H : n + m = k) : k - n = m
 := have H2 : k - n + n = m + n, from