From b3ca5faa492c802cfaa287ac695b28c23ea13b77 Mon Sep 17 00:00:00 2001
From: Leonardo de Moura <leonardo@microsoft.com>
Date: Mon, 16 Nov 2015 11:01:53 -0800
Subject: [PATCH] fix(tests/lean): some of the simplifier tests

---
 library/init/nat.lean                     |  3 ++
 tests/lean/simplifier1.hlean              | 12 ++---
 tests/lean/simplifier1.lean               | 10 ++--
 tests/lean/simplifier10.lean              | 19 ++++----
 tests/lean/simplifier11.lean              | 56 +++++++++++------------
 tests/lean/simplifier12.lean              |  2 +-
 tests/lean/simplifier13.lean              |  7 +--
 tests/lean/simplifier14.lean.expected.out | 19 +++++++-
 tests/lean/simplifier2.lean               | 11 ++---
 tests/lean/simplifier3.lean               | 12 ++---
 tests/lean/simplifier4.lean               | 22 ++++-----
 tests/lean/simplifier5.lean               | 23 ++++++----
 tests/lean/simplifier6.lean               | 25 +++++-----
 tests/lean/simplifier7.lean               |  8 ++--
 tests/lean/simplifier8.lean               |  4 +-
 tests/lean/simplifier9.lean               | 34 +++++++-------
 16 files changed, 147 insertions(+), 120 deletions(-)

diff --git a/library/init/nat.lean b/library/init/nat.lean
index 9b04695ca..0cad98f68 100644
--- a/library/init/nat.lean
+++ b/library/init/nat.lean
@@ -154,6 +154,9 @@ namespace nat
 
   protected theorem lt_irrefl (n : ℕ) : ¬n < n := not_succ_le_self
 
+  theorem lt_self_iff_false (n : ℕ) : n < n ↔ false :=
+  iff_false_intro (λ H, absurd H (nat.lt_irrefl n))
+
   theorem self_lt_succ (n : ℕ) : n < succ n := !nat.le_refl
 
   theorem self_lt_succ_iff_true [simp] (n : ℕ) : n < succ n ↔ true :=
diff --git a/tests/lean/simplifier1.hlean b/tests/lean/simplifier1.hlean
index 8c17fde48..10a757094 100644
--- a/tests/lean/simplifier1.hlean
+++ b/tests/lean/simplifier1.hlean
@@ -6,12 +6,12 @@ constants (T : Type.{l}) (x1 x2 x3 x4 x5 x6 : T) (f : T → T → T)
 constants (f_comm : ∀ x y, f x y = f y x)
           (f_l : ∀ x y z, f (f x y) z = f x (f y z))
           (f_r : ∀ x y z, f x (f y z) = f y (f x z))
-
+namespace tst
 attribute f_comm [simp]
 attribute f_l [simp]
 attribute f_r [simp]
-
-#simplify eq 0 (f (f x2 x4) (f x5 (f x3 (f x1 x6))))
+end tst
+#simplify eq tst 0 (f (f x2 x4) (f x5 (f x3 (f x1 x6))))
 
 open is_trunc
 
@@ -19,9 +19,9 @@ constants g : Π (x y : nat), x ≠ y → Type₁
 constants a b c : nat
 constants H₁ : a ≠ b
 constants H₂ : a = c
-attribute H₂ [simp]
+namespace tst attribute H₂ [simp] end tst
 
 definition ne_is_hprop [instance] (a b : nat) : is_hprop (a ≠ b) :=
 sorry
-
-#simplify eq 0 (g a b H₁)
+-- TODO(Daniel): simplifier seems to have applied unnecessary step (see: (eq.nrec ... (eq.refl ..)))
+#simplify eq tst 0 (g a b H₁)
diff --git a/tests/lean/simplifier1.lean b/tests/lean/simplifier1.lean
index c1eb2bc0c..84e7391e7 100644
--- a/tests/lean/simplifier1.lean
+++ b/tests/lean/simplifier1.lean
@@ -1,10 +1,10 @@
 /- Basic rewriting with eq without congruence or conditionals -/
 universe l
 constant T : Type.{l}
-constants (x y z : T) (f g h : T → T) 
+constants (x y z : T) (f g h : T → T)
 constants (Hfxgy : f x = g y) (Hgyhz : g y = h z)
 
-attribute Hfxgy [simp]
-#simplify eq 2 (f x)
-attribute Hgyhz [simp]
-#simplify eq 2 (f x)
+namespace tst attribute Hfxgy [simp] end tst
+#simplify eq tst 2 (f x)
+namespace tst attribute Hgyhz [simp] end tst
+#simplify eq tst 2 (f x)
diff --git a/tests/lean/simplifier10.lean b/tests/lean/simplifier10.lean
index 16fefaaa5..2cf3c097b 100644
--- a/tests/lean/simplifier10.lean
+++ b/tests/lean/simplifier10.lean
@@ -1,19 +1,18 @@
 import logic.connectives logic.quantifiers
 
 universe l
-constants (T : Type.{l}) (x y z : T) (P : T → Prop) (Q : T → T → T → Prop) (R W V : T → T → Prop) 
+constants (T : Type.{l}) (x y z : T) (P : T → Prop) (Q : T → T → T → Prop) (R W V : T → T → Prop)
 constants (px : P x) (wxz : W x z) (vzy : V z y)
 constants (H : ∀ (x y z : T), P x → W x z → V z y → (Q z y x ↔ R x y))
-attribute px [simp]
+namespace tst
+attribute px true_iff [simp]
 attribute wxz [simp]
 attribute vzy [simp]
 attribute H [simp]
+end tst
 
-#simplify iff 0 P x
-#simplify iff 0 W x z
-#simplify iff 0 V z y
-#simplify iff 0 Q z y x
-#simplify iff 0 V z y ↔ Q z y x
-
-
-
+#simplify iff tst 0 P x
+#simplify iff tst 0 W x z
+#simplify iff tst 0 V z y
+#simplify iff tst 0 Q z y x
+#simplify iff tst 0 V z y ↔ Q z y x
diff --git a/tests/lean/simplifier11.lean b/tests/lean/simplifier11.lean
index d3ba5999c..e4bb500a5 100644
--- a/tests/lean/simplifier11.lean
+++ b/tests/lean/simplifier11.lean
@@ -5,29 +5,29 @@ namespace if_congr
 constants {A : Type} {b c : Prop} (dec_b : decidable b) (dec_c : decidable c)
           {x y u v : A} (h_c : b ↔ c) (h_t : x = u) (h_e : y = v)
 
-local attribute dec_b [instance]
-local attribute dec_c [instance]
-local attribute h_c [simp]
-local attribute h_t [simp]
-local attribute h_e [simp]
+attribute dec_b [instance]
+ attribute dec_c [instance]
+ attribute h_c [simp]
+ attribute h_t [simp]
+ attribute h_e [simp]
 
 attribute if_congr [congr]
 
-#simplify eq 0 (ite b x y)
+#simplify eq if_congr 0 (ite b x y)
 end if_congr
 
 namespace if_ctx_simp_congr
 constants {A : Type} {b c : Prop} (dec_b : decidable b)
           {x y u v : A} (h_c : b ↔ c) (h_t : c → x = u) (h_e : ¬c → y = v)
 
-local attribute dec_b [instance]
-local attribute h_c [simp]
-local attribute h_t [simp]
-local attribute h_e [simp]
+ attribute dec_b [instance]
+ attribute h_c [simp]
+ attribute h_t [simp]
+ attribute h_e [simp]
 
 attribute if_ctx_simp_congr [congr]
-
-#simplify eq 0 (ite b x y)
+-- TODO(Daniel): lean_unreachable
+#simplify eq if_ctx_simp_congr 0 (ite b x y)
 
 end if_ctx_simp_congr
 
@@ -35,28 +35,28 @@ namespace if_congr_prop
 constants {b c x y u v : Prop} (dec_b : decidable b) (dec_c : decidable c)
           (h_c : b ↔ c) (h_t : c → (x ↔ u)) (h_e : ¬c → (y ↔ v))
 
-local attribute dec_b [instance]
-local attribute dec_c [instance]
-local attribute h_c [simp]
-local attribute h_t [simp]
-local attribute h_e [simp]
+ attribute dec_b [instance]
+ attribute dec_c [instance]
+ attribute h_c [simp]
+ attribute h_t [simp]
+ attribute h_e [simp]
 
 attribute if_congr_prop [congr]
 
-#simplify iff 0 (ite b x y)
+#simplify iff if_congr_prop 0 (ite b x y)
 end if_congr_prop
 
 namespace if_ctx_simp_congr_prop
 constants (b c x y u v : Prop) (dec_b : decidable b)
           (h_c : b ↔ c) (h_t : c → (x ↔ u)) (h_e : ¬ c → (y ↔ v))
 
-local attribute dec_b [instance]
-local attribute h_c [simp]
-local attribute h_t [simp]
-local attribute h_e [simp]
+ attribute dec_b [instance]
+ attribute h_c [simp]
+ attribute h_t [simp]
+ attribute h_e [simp]
 
 attribute if_ctx_simp_congr_prop [congr]
-#simplify iff 0 (ite b x y)
+#simplify iff if_ctx_simp_congr_prop 0 (ite b x y)
 
 end if_ctx_simp_congr_prop
 
@@ -64,11 +64,11 @@ namespace if_simp_congr_prop
 constants (b c x y u v : Prop) (dec_b : decidable b)
           (h_c : b ↔ c) (h_t : x ↔ u) (h_e : y ↔ v)
 
-local attribute dec_b [instance]
-local attribute h_c [simp]
-local attribute h_t [simp]
-local attribute h_e [simp]
+ attribute dec_b [instance]
+ attribute h_c [simp]
+ attribute h_t [simp]
+ attribute h_e [simp]
 
 attribute if_simp_congr_prop [congr]
-#simplify iff 0 (ite b x y)
+#simplify iff if_simp_congr_prop 0 (ite b x y)
 end if_simp_congr_prop
diff --git a/tests/lean/simplifier12.lean b/tests/lean/simplifier12.lean
index f3c95f756..b2ccd19e0 100644
--- a/tests/lean/simplifier12.lean
+++ b/tests/lean/simplifier12.lean
@@ -7,4 +7,4 @@ constants (T : Type.{l}) (s : algebra.comm_ring T)
 constants (x1 x2 x3 x4 : T) (f g : T → T)
 attribute s [instance]
 
-#simplify eq 0 x2 + (1 * g x1 + 0 + (f x3 * 3 * 1 * (x2 + 0 + g x1 * 7) * x2 * 1)) + 5 * (x4 + f x1)
+#simplify eq simplifier.som 0 x2 + (1 * g x1 + 0 + (f x3 * 3 * 1 * (x2 + 0 + g x1 * 7) * x2 * 1)) + 5 * (x4 + f x1)
diff --git a/tests/lean/simplifier13.lean b/tests/lean/simplifier13.lean
index 9876acca3..6ae62a1da 100644
--- a/tests/lean/simplifier13.lean
+++ b/tests/lean/simplifier13.lean
@@ -3,7 +3,8 @@ constants (T : Type.{l}) (f : T →  T → T) (g : T → T)
 constants (P : T → Prop) (Q : Prop) (Hfg : ∀ (x : T), f x x = g x)
 constants (c : Π (x y z : T), P x → P y → P z → Q)
 constants (x y z : T) (px : P (f x x)) (py : P (f y y)) (pz : P (g z))
-
+namespace tst
 attribute Hfg [simp]
-
-#simplify eq 0 c (f x x) (f y y) (g z) px py pz
+end tst
+-- TODO(Daniel): extra step? Similar to simplifier1.hlean
+#simplify eq tst 0 c (f x x) (f y y) (g z) px py pz
diff --git a/tests/lean/simplifier14.lean.expected.out b/tests/lean/simplifier14.lean.expected.out
index 30404ce4c..a905f74e5 100644
--- a/tests/lean/simplifier14.lean.expected.out
+++ b/tests/lean/simplifier14.lean.expected.out
@@ -1 +1,18 @@
-TODO
\ No newline at end of file
+(refl): x1
+x1 * 2
+x1 * 3
+x1 * 4
+x1 * 4
+x1 * 5
+x1 * (1 + -1)
+x1 * (1 + (1 + -1))
+x1 * (1 + (1 + -2))
+x1 * (1 + (1 + (-1 + -1)))
+x1 * 5
+x2 * (1 + -1) + x1 * (1 + -1)
+x2 * (1 + (1 + -2)) + x1 * (1 + (1 + (1 + -3)))
+x1 * 2 + x2 * (1 + -1)
+x2 * 2 + (x1 * 3 + x2 * x1 * (1 + (1 + -32)))
+x1 + (-0 + x3 * x2 * (3 + -2))
+x1 * 2 + (x3 * x2 * 3 + (x3 * x1 * (1 + -3) + x2 * x1 * (1 + (1 + -2))))
+x1 * 2 + x2 * (1 + -1)
diff --git a/tests/lean/simplifier2.lean b/tests/lean/simplifier2.lean
index 5d2b5261f..df3e1bd45 100644
--- a/tests/lean/simplifier2.lean
+++ b/tests/lean/simplifier2.lean
@@ -1,11 +1,10 @@
 /- Basic rewriting with eq and lambda without congruence or conditionals -/
 universe l
 constant T : Type.{l}
-constants (x y z : T) (f g h : T → T) 
+constants (x y z : T) (f g h : T → T)
 constants (Hfxgy : f x = g y) (Hgyhz : g y = h z)
 
-attribute Hfxgy [simp]
-#simplify eq 0 (λ a b c : bool, f x) -- λ (a b c : bool), g y
-attribute Hgyhz [simp]
-#simplify eq 0 (λ a b c : bool, f x) -- λ (a b c : bool), h z
-
+namespace tst attribute Hfxgy [simp] end tst
+#simplify eq tst 0 (λ a b c : bool, f x) -- λ (a b c : bool), g y
+namespace tst attribute Hgyhz [simp] end tst
+#simplify eq tst 0 (λ a b c : bool, f x) -- λ (a b c : bool), h z
diff --git a/tests/lean/simplifier3.lean b/tests/lean/simplifier3.lean
index e0afe3374..a58624056 100644
--- a/tests/lean/simplifier3.lean
+++ b/tests/lean/simplifier3.lean
@@ -7,13 +7,13 @@ constant T : Type.{l}
 constants (x y z : T → T) (f g h : (T →  T) → (T →  T)) (a b c : T)
 constants (Hfxgy : f x = g y) (Hgyhz : g y = h z) (Hab : a = b) (Hbc : b = c)
 
-
+namespace tst
 attribute Hfxgy [simp]
 attribute Hgyhz [simp]
 attribute Hab [simp]
 attribute Hbc [simp]
-
-#simplify eq 2 (f x a)
+end tst
+#simplify eq tst 2 (f x a)
 
 end test_congr
 
@@ -23,9 +23,9 @@ universes l1 l2
 constants (T : Type.{l1}) (U : T → Type.{l2})
 constants (f g : Π (x:T), U x) (x y : T)
 constants (Hfg : f = g) (Hxy : x = y)
-
+namespace tst
 attribute Hfg [simp]
 attribute Hxy [simp]
-
-#simplify eq 2 (f x)
+end tst
+#simplify eq tst 2 (f x)
 end test_congr_fun
diff --git a/tests/lean/simplifier4.lean b/tests/lean/simplifier4.lean
index d54c20456..4b17fbd58 100644
--- a/tests/lean/simplifier4.lean
+++ b/tests/lean/simplifier4.lean
@@ -3,17 +3,17 @@ import logic.connectives data.nat
 
 universe l
 constant T : Type.{l}
-constants (x y z : T) (f g h : T → T) 
-constants (R : T → T → Prop) 
+constants (x y z : T) (f g h : T → T)
+constants (R : T → T → Prop)
 constants (R_refl : ∀ x, R x x) (R_sym : ∀ x y, R x y →  R y x) (R_trans : ∀ x y z, R x y → R y z → R x z)
 constants (Hfxgy : R (f x) (g y)) (Hgyhz : R (g y) (h z))
 
-attribute R_refl [refl]
-attribute R_sym [symm]
-#simplify R 2 (f x) -- f x
-attribute R_trans [trans]
-#simplify R 2 (f x) -- f x
-attribute Hfxgy [simp]
-#simplify R 2 (f x) -- f x
-attribute Hgyhz [simp]
-#simplify R 2 (f x) -- f x
+namespace tst attribute R_refl [refl] end tst
+namespace tst attribute R_sym [symm] end tst
+#simplify R tst 2 (f x) -- f x
+namespace tst attribute R_trans [trans] end tst
+#simplify R tst 2 (f x) -- f x
+namespace tst attribute Hfxgy [simp] end tst
+#simplify R tst 2 (f x) -- f x
+namespace tst attribute Hgyhz [simp] end tst
+#simplify R tst 2 (f x) -- f x
diff --git a/tests/lean/simplifier5.lean b/tests/lean/simplifier5.lean
index 5816e541c..b506735aa 100644
--- a/tests/lean/simplifier5.lean
+++ b/tests/lean/simplifier5.lean
@@ -1,13 +1,16 @@
 /- Basic rewriting with iff with congr_iff -/
 import logic.connectives
 open nat
-#simplify iff 2 (@le nat nat_has_le 0 0) -- true
-#simplify iff 2 (@le nat nat_has_le 0 1) -- true
-#simplify iff 2 (@le nat nat_has_le 0 2) -- true
-#simplify iff 2 (@lt nat nat_has_lt 0 0) -- false
-#simplify iff 2 (@lt nat nat_has_lt 0 (succ 0)) -- true
-#simplify iff 2 (@lt nat nat_has_lt 1 (succ 1)) -- true
-#simplify iff 2 (@lt nat nat_has_lt 0 (succ (succ 0))) -- true
-#simplify iff 2 (@le nat nat_has_le 0 0 ↔ @le nat nat_has_le 0 0) -- true
-#simplify iff 2 (@le nat nat_has_le 0 0 ↔ @le nat nat_has_le 0 1) -- true
-#simplify iff 2 (@le nat nat_has_le 0 0 ↔ @lt nat nat_has_lt 0 0) -- false
+namespace tst
+attribute iff_self true_iff_false self_lt_succ_iff_true zero_lt_succ_iff_true zero_le_iff_true succ_le_self_iff_false succ_le_self_iff_false lt_self_iff_false [simp]
+end tst
+#simplify iff tst 2 (@le nat nat_has_le 0 0) -- true
+#simplify iff tst 2 (@le nat nat_has_le 0 1) -- true
+#simplify iff tst 2 (@le nat nat_has_le 0 2) -- true
+#simplify iff tst 2 (@lt nat nat_has_lt 0 0) -- false
+#simplify iff tst 2 (@lt nat nat_has_lt 0 (succ 0)) -- true
+#simplify iff tst 2 (@lt nat nat_has_lt 1 (succ 1)) -- true
+#simplify iff tst 2 (@lt nat nat_has_lt 0 (succ (succ 0))) -- true
+#simplify iff tst 2 (@le nat nat_has_le 0 0 ↔ @le nat nat_has_le 0 0) -- true
+#simplify iff tst 2 (@le nat nat_has_le 0 0 ↔ @le nat nat_has_le 0 1) -- true
+#simplify iff tst 2 (@le nat nat_has_le 0 0 ↔ @lt nat nat_has_lt 0 0) -- false
diff --git a/tests/lean/simplifier6.lean b/tests/lean/simplifier6.lean
index 963b83bee..a3d295b57 100644
--- a/tests/lean/simplifier6.lean
+++ b/tests/lean/simplifier6.lean
@@ -1,28 +1,31 @@
 /- Basic pi congruence -/
 import logic.connectives logic.quantifiers
 
-attribute congr_forall [congr]
-attribute congr_imp [congr]
-
 namespace pi_congr1
 constants (p1 q1 p2 q2 p3 q3 : Prop) (H1 : p1 ↔ q1) (H2 : p2 ↔ q2) (H3 : p3 ↔ q3)
-local attribute H1 [simp]
-local attribute H2 [simp]
-local attribute H3 [simp]
+namespace tst
+attribute congr_forall [congr]
+attribute congr_imp    [congr]
+attribute H1 [simp]
+attribute H2 [simp]
+attribute H3 [simp]
+end tst
 
-#simplify iff 1 p1
-#simplify iff 1 p1 →  p2
-#simplify iff 1 p1 →  p2 → p3
+#simplify iff tst 1 p1 -- Broken?
+#simplify iff tst 1 p1 →  p2
+#simplify iff tst 1 p1 →  p2 → p3
 
 end pi_congr1
 
 namespace pi_congr2
 universe l
 constants (T : Type.{l}) (P Q : T → Prop) (H : ∀ (x : T), P x ↔ Q x)
-local attribute H [simp]
+namespace tst
+attribute H [simp]
+end tst
 constant (x : T)
 
-#simplify iff 1 (∀ (x : T), P x)
+#simplify iff tst 1 (∀ (x : T), P x)
 
 
 end pi_congr2
diff --git a/tests/lean/simplifier7.lean b/tests/lean/simplifier7.lean
index 549a3f07a..b9ea24897 100644
--- a/tests/lean/simplifier7.lean
+++ b/tests/lean/simplifier7.lean
@@ -4,13 +4,13 @@ import logic.connectives
 constants (P1 Q1 P2 Q2 P3 Q3 : Prop) (H1 : P1 ↔ Q1) (H2 : P2 ↔ Q2) (H3 : P3 ↔ Q3)
 constants (f g : Prop → Prop → Prop)
 constants (Hf : and = f) (Hg : f = g)
-
+namespace tst
 attribute H1 [simp]
 attribute H2 [simp]
 attribute H3 [simp]
 attribute Hf [simp]
 attribute Hg [simp]
+end tst
 
-
-#simplify iff 2 (and P1 (and P2 P3))
-#simplify iff 2 (and P1 (iff P2 P3))
+#simplify iff tst 2 (and P1 (and P2 P3))
+#simplify iff tst 2 (and P1 (iff P2 P3))
diff --git a/tests/lean/simplifier8.lean b/tests/lean/simplifier8.lean
index 469fe1884..a1754c593 100644
--- a/tests/lean/simplifier8.lean
+++ b/tests/lean/simplifier8.lean
@@ -6,8 +6,10 @@ constants (f_comm : ∀ x y, f x y = f y x)
           (f_l : ∀ x y z, f (f x y) z = f x (f y z))
           (f_r : ∀ x y z, f x (f y z) = f y (f x z))
 
+namespace tst
 attribute f_comm [simp]
 attribute f_l [simp]
 attribute f_r [simp]
+end tst
 
-#simplify eq 0 (f (f x2 x4) (f x5 (f x3 (f x1 x6))))
+#simplify eq tst 0 (f (f x2 x4) (f x5 (f x3 (f x1 x6))))
diff --git a/tests/lean/simplifier9.lean b/tests/lean/simplifier9.lean
index b3e4a7f75..6c7fa5f30 100644
--- a/tests/lean/simplifier9.lean
+++ b/tests/lean/simplifier9.lean
@@ -1,30 +1,30 @@
 -- Rewriting with (tmp)-local hypotheses
 import logic.quantifiers
 
+namespace tst
 attribute congr_forall [congr]
 attribute congr_imp [congr]
+end tst
 
 universe l
-constants (T : Type.{l}) (P Q : T → Prop) 
+constants (T : Type.{l}) (P Q : T → Prop)
 
 set_option simplify.max_steps 50000
 constants (x y : T)
+-- TODO(Daniel): the following is looping...
+#simplify iff tst 0 x = y →  x = y
+#simplify iff tst 0 T → x = y →  x = y
+#simplify iff tst 0 ∀ z : T, x = z → x = y
+#simplify iff tst 0 ∀ z : T, z = x → x = z
+#simplify iff tst 0 ∀ (z w : T), x = y →  x = y
+#simplify iff tst 0 ∀ (z w : T), x = y →  P x
 
-#simplify iff 0 x = y →  x = y
-#simplify iff 0 T → x = y →  x = y
-#simplify iff 0 ∀ z : T, x = z → x = y
-#simplify iff 0 ∀ z : T, z = x → x = z
-#simplify iff 0 ∀ (z w : T), x = y →  x = y
-#simplify iff 0 ∀ (z w : T), x = y →  P x
-
-#simplify iff 0 ∀ (H : ∀ x, P x ↔ Q x), P x
-#simplify iff 0 ∀ (p : Prop) (H : ∀ x, P x ↔ Q x) (q : Prop), P x
-
-#simplify iff 0 ∀ (p : Prop) (H : ∀ x, P x ↔ Q x), p ∨ P x
-#simplify iff 0 (∀ (x : T), P x ↔ Q x) →  P x
-#simplify iff 0 (∀ (x : T), P x ↔ Q x) →  P x
-#simplify iff 0 ∀ (x y : T), (∀ (x : T), P x ↔ Q x) →  P x
-
-#simplify iff 0 ∀ (x z : T), x = z → (∀ (w : T), P w ↔ Q w) →  P x
+#simplify iff tst 0 ∀ (H : ∀ x, P x ↔ Q x), P x
+#simplify iff tst 0 ∀ (p : Prop) (H : ∀ x, P x ↔ Q x) (q : Prop), P x
 
+#simplify iff tst 0 ∀ (p : Prop) (H : ∀ x, P x ↔ Q x), p ∨ P x
+#simplify iff tst 0 (∀ (x : T), P x ↔ Q x) →  P x
+#simplify iff tst 0 (∀ (x : T), P x ↔ Q x) →  P x
+#simplify iff tst 0 ∀ (x y : T), (∀ (x : T), P x ↔ Q x) →  P x
 
+#simplify iff tst 0 ∀ (x z : T), x = z → (∀ (w : T), P w ↔ Q w) →  P x