refactor(library/init): move more theorems to logic

library/data/list/basic.lean

@@ -183,7 +183,7 @@ induction_on l
 
 definition mem.is_decidable [instance] (H : decidable_eq T) (x : T) (l : list T) : decidable (x ∈ l) :=
 rec_on l
-  (decidable.inr (iff.false_elim !mem.nil))
+  (decidable.inr (not_of_iff_false !mem.nil))
   (take (h : T) (l : list T) (iH : decidable (x ∈ l)),
     show decidable (x ∈ h::l), from
     decidable.rec_on iH

library/init/logic.lean

@@ -53,15 +53,18 @@ namespace eq
     notation H1 ▸ H2 := subst H1 H2
   end ops
 
-  variable {p : Prop}
+end eq
-  open ops
 
-  theorem true_elim (H : p = true) : p :=
+section
+  variable {p : Prop}
+  open eq.ops
+
+  theorem of_eq_true (H : p = true) : p :=
   H⁻¹ ▸ trivial
 
-  theorem false_elim (H : p = false) : ¬p :=
+  theorem not_of_eq_false (H : p = false) : ¬p :=
   assume Hp, H ▸ Hp
-end eq
+end
 
 calc_subst eq.subst
 calc_refl  eq.refl
@@ -140,11 +143,11 @@ namespace heq
 
   theorem of_eq_of_heq (H₁ : a = a') (H₂ : a' == b) : a == b :=
   trans (of_eq H₁) H₂
-
-  theorem true_elim {a : Prop} (H : a == true) : a :=
-  eq.true_elim (heq.to_eq H)
 end heq
 
+theorem of_heq_true {a : Prop} (H : a == true) : a :=
+of_eq_true (heq.to_eq H)
+
 calc_trans heq.trans
 calc_trans heq.of_heq_of_eq
 calc_trans heq.of_eq_of_heq
@@ -158,16 +161,8 @@ notation a ∧ b  := and a b
 
 variables {a b c d : Prop}
 
-namespace and
+theorem and.elim (H₁ : a ∧ b) (H₂ : a → b → c) : c :=
-  theorem elim (H₁ : a ∧ b) (H₂ : a → b → c) : c :=
+and.rec H₂ H₁
-  rec H₂ H₁
-
-  definition not_left (b : Prop) (Hna : ¬a) : ¬(a ∧ b) :=
-  assume H : a ∧ b, absurd (elim_left H) Hna
-
-  definition not_right (a : Prop) {b : Prop} (Hnb : ¬b) : ¬(a ∧ b) :=
-  assume H : a ∧ b, absurd (elim_right H) Hnb
-end and
 
 -- or
 -- --
@@ -183,11 +178,6 @@ namespace or
 
   theorem elim (H₁ : a ∨ b) (H₂ : a → c) (H₃ : b → c) : c :=
   rec H₂ H₃ H₁
-
-  definition not_intro (Hna : ¬a) (Hnb : ¬b) : ¬(a ∨ b) :=
-  assume H : a ∨ b, or.rec_on H
-    (assume Ha, absurd Ha Hna)
-    (assume Hb, absurd Hb Hnb)
 end or
 
 -- iff
@@ -233,17 +223,17 @@ namespace iff
     (assume Hb, elim_right H Hb)
     (assume Ha, elim_left H Ha)
 
-  theorem true_elim (H : a ↔ true) : a :=
-  mp (symm H) trivial
-
-  theorem false_elim (H : a ↔ false) : ¬a :=
-  assume Ha : a, mp H Ha
-
   open eq.ops
   theorem of_eq {a b : Prop} (H : a = b) : a ↔ b :=
   iff.intro (λ Ha, H ▸ Ha) (λ Hb, H⁻¹ ▸ Hb)
 end iff
 
+theorem of_iff_true (H : a ↔ true) : a :=
+iff.mp (iff.symm H) trivial
+
+theorem not_of_iff_false (H : a ↔ false) : ¬a :=
+assume Ha : a, iff.mp H Ha
+
 calc_refl iff.refl
 calc_trans iff.trans
 
@@ -307,15 +297,15 @@ section
   rec_on Hp
     (assume Hp  : p, rec_on Hq
       (assume Hq  : q,  inl (and.intro Hp Hq))
-      (assume Hnq : ¬q, inr (and.not_right p Hnq)))
+      (assume Hnq : ¬q, inr (assume H : p ∧ q, and.rec_on H (assume Hp Hq, absurd Hq Hnq))))
-    (assume Hnp : ¬p, inr (and.not_left q Hnp))
+    (assume Hnp : ¬p, inr (assume H : p ∧ q, and.rec_on H (assume Hp Hq, absurd Hp Hnp)))
 
   definition or.decidable [instance] (Hp : decidable p) (Hq : decidable q) : decidable (p ∨ q) :=
   rec_on Hp
     (assume Hp  : p, inl (or.inl Hp))
     (assume Hnp : ¬p, rec_on Hq
       (assume Hq  : q,  inl (or.inr Hq))
-      (assume Hnq : ¬q, inr (or.not_intro Hnp Hnq)))
+      (assume Hnq : ¬q, inr (assume H : p ∨ q, or.elim H (assume Hp, absurd Hp Hnp) (assume Hq, absurd Hq Hnq))))
 
   definition not.decidable [instance] (Hp : decidable p) : decidable (¬p) :=
   rec_on Hp
@@ -390,32 +380,6 @@ decidable.rec
   (λ Hnc : ¬c, eq.refl (@ite c (decidable.inr Hnc) A t t))
   H
 
-definition if_true {A : Type} (t e : A) : (if true then t else e) = t :=
-if_pos trivial
-
-definition if_false {A : Type} (t e : A) : (if false then t else e) = e :=
-if_neg not_false
-
-theorem if_cond_congr {c₁ c₂ : Prop} [H₁ : decidable c₁] [H₂ : decidable c₂] (Heq : c₁ ↔ c₂) {A : Type} (t e : A)
-                      : (if c₁ then t else e) = (if c₂ then t else e) :=
-decidable.rec_on H₁
- (λ Hc₁  : c₁,  decidable.rec_on H₂
-   (λ Hc₂  : c₂,  if_pos Hc₁ ⬝ (if_pos Hc₂)⁻¹)
-   (λ Hnc₂ : ¬c₂, absurd (iff.elim_left Heq Hc₁) Hnc₂))
- (λ Hnc₁ : ¬c₁, decidable.rec_on H₂
-   (λ Hc₂  : c₂,  absurd (iff.elim_right Heq Hc₂) Hnc₁)
-   (λ Hnc₂ : ¬c₂, if_neg Hnc₁ ⬝ (if_neg Hnc₂)⁻¹))
-
-theorem if_congr_aux {c₁ c₂ : Prop} [H₁ : decidable c₁] [H₂ : decidable c₂] {A : Type} {t₁ t₂ e₁ e₂ : A}
-                     (Hc : c₁ ↔ c₂) (Ht : t₁ = t₂) (He : e₁ = e₂) :
-                 (if c₁ then t₁ else e₁) = (if c₂ then t₂ else e₂) :=
-Ht ▸ He ▸ (if_cond_congr Hc t₁ e₁)
-
-theorem if_congr {c₁ c₂ : Prop} [H₁ : decidable c₁] {A : Type} {t₁ t₂ e₁ e₂ : A} (Hc : c₁ ↔ c₂) (Ht : t₁ = t₂) (He : e₁ = e₂) :
-                 (if c₁ then t₁ else e₁) = (@ite c₂ (decidable.decidable_iff_equiv H₁ Hc) A t₂ e₂) :=
-have H2 [visible] : decidable c₂, from (decidable.decidable_iff_equiv H₁ Hc),
-if_congr_aux Hc Ht He
-
 -- We use "dependent" if-then-else to be able to communicate the if-then-else condition
 -- to the branches
 definition dite (c : Prop) [H : decidable c] {A : Type} (t : c → A) (e : ¬ c → A) : A :=

library/logic/axioms/classical.lean

@@ -23,8 +23,8 @@ cases P H1 H2 a
 -- this supercedes the em in decidable
 theorem em (a : Prop) : a ∨ ¬a :=
 or.elim (prop_complete a)
-  (assume Ht : a = true,  or.inl (eq.true_elim Ht))
+  (assume Ht : a = true,  or.inl (of_eq_true Ht))
-  (assume Hf : a = false, or.inr (eq.false_elim Hf))
+  (assume Hf : a = false, or.inr (not_of_eq_false Hf))
 
 theorem prop_complete_swapped (a : Prop) : a = false ∨ a = true :=
 cases (λ x, x = false ∨ x = true)
@@ -36,9 +36,9 @@ theorem propext {a b : Prop} (Hab : a → b) (Hba : b → a) : a = b :=
 or.elim (prop_complete a)
   (assume Hat,  or.elim (prop_complete b)
     (assume Hbt,  Hat ⬝ Hbt⁻¹)
-    (assume Hbf, false_elim (Hbf ▸ (Hab (eq.true_elim Hat)))))
+    (assume Hbf, false_elim (Hbf ▸ (Hab (of_eq_true Hat)))))
   (assume Haf, or.elim (prop_complete b)
-    (assume Hbt,  false_elim (Haf ▸ (Hba (eq.true_elim Hbt))))
+    (assume Hbt,  false_elim (Haf ▸ (Hba (of_eq_true Hbt))))
     (assume Hbf, Haf ⬝ Hbf⁻¹))
 
 theorem eq.of_iff {a b : Prop} (H : a ↔ b) : a = b :=

library/logic/connectives.lean

@@ -39,6 +39,12 @@ assume not_em : ¬(a ∨ ¬a),
 -- ---
 
 namespace and
+  definition not_left (b : Prop) (Hna : ¬a) : ¬(a ∧ b) :=
+  assume H : a ∧ b, absurd (elim_left H) Hna
+
+  definition not_right (a : Prop) {b : Prop} (Hnb : ¬b) : ¬(a ∧ b) :=
+  assume H : a ∧ b, absurd (elim_right H) Hnb
+
   theorem swap (H : a ∧ b) : b ∧ a :=
   intro (elim_right H) (elim_left H)
 
@@ -68,6 +74,11 @@ end and
 -- --
 
 namespace or
+  definition not_intro (Hna : ¬a) (Hnb : ¬b) : ¬(a ∨ b) :=
+  assume H : a ∨ b, or.rec_on H
+    (assume Ha, absurd Ha Hna)
+    (assume Hb, absurd Hb Hnb)
+
   theorem imp_or (H₁ : a ∨ b) (H₂ : a → c) (H₃ : b → d) : c ∨ d :=
   elim H₁
     (assume Ha : a, inl (H₂ Ha))
@@ -136,3 +147,37 @@ 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 :=
 obtain w Hw, from H2,
 H1 w (and.elim_left Hw) (and.elim_right Hw)
+
+-- if-then-else
+-- ------------
+section
+  open eq.ops
+
+  variables {A : Type} {c₁ c₂ : Prop}
+
+  definition if_true (t e : A) : (if true then t else e) = t :=
+  if_pos trivial
+
+  definition if_false (t e : A) : (if false then t else e) = e :=
+  if_neg not_false
+
+  theorem if_cond_congr [H₁ : decidable c₁] [H₂ : decidable c₂] (Heq : c₁ ↔ c₂) (t e : A)
+                        : (if c₁ then t else e) = (if c₂ then t else e) :=
+  decidable.rec_on H₁
+   (λ Hc₁  : c₁,  decidable.rec_on H₂
+     (λ Hc₂  : c₂,  if_pos Hc₁ ⬝ (if_pos Hc₂)⁻¹)
+     (λ Hnc₂ : ¬c₂, absurd (iff.elim_left Heq Hc₁) Hnc₂))
+   (λ Hnc₁ : ¬c₁, decidable.rec_on H₂
+     (λ Hc₂  : c₂,  absurd (iff.elim_right Heq Hc₂) Hnc₁)
+     (λ Hnc₂ : ¬c₂, if_neg Hnc₁ ⬝ (if_neg Hnc₂)⁻¹))
+
+  theorem if_congr_aux [H₁ : decidable c₁] [H₂ : decidable c₂] {t₁ t₂ e₁ e₂ : A}
+                       (Hc : c₁ ↔ c₂) (Ht : t₁ = t₂) (He : e₁ = e₂) :
+                   (if c₁ then t₁ else e₁) = (if c₂ then t₂ else e₂) :=
+  Ht ▸ He ▸ (if_cond_congr Hc t₁ e₁)
+
+  theorem if_congr [H₁ : decidable c₁] {t₁ t₂ e₁ e₂ : A} (Hc : c₁ ↔ c₂) (Ht : t₁ = t₂) (He : e₁ = e₂) :
+                   (if c₁ then t₁ else e₁) = (@ite c₂ (decidable.decidable_iff_equiv H₁ Hc) A t₂ e₂) :=
+  have H2 [visible] : decidable c₂, from (decidable.decidable_iff_equiv H₁ Hc),
+  if_congr_aux Hc Ht He
+end

library/logic/identities.lean

@@ -143,7 +143,7 @@ theorem false_eq_true : (false ↔ true) ↔ false :=
 not_false_iff_true ▸ (a_iff_not_a false)
 
 theorem a_eq_true (a : Prop) : (a ↔ true) ↔ a :=
-iff.intro (assume H, iff.true_elim H) (assume H, iff_true_intro H)
+iff.intro (assume H, of_iff_true H) (assume H, iff_true_intro H)
 
 theorem a_eq_false (a : Prop) : (a ↔ false) ↔ ¬a :=
-iff.intro (assume H, iff.false_elim H) (assume H, iff_false_intro H)
+iff.intro (assume H, not_of_iff_false H) (assume H, iff_false_intro H)

tests/lean/interactive/findp.input.expected.out

@@ -9,13 +9,12 @@ false.cases_on|Π (C : Type), false → C
 false.decidable|decidable false
 false.induction_on|∀ (C : Prop), false → C
 true_ne_false|¬ true = false
-eq.false_elim|?p = false → ¬ ?p
 not_of_is_false|is_false ?c → ¬ ?c
+not_of_iff_false|?a ↔ false → ¬ ?a
 p_ne_false|?p → ?p ≠ false
 is_false|Π (c : Prop) [H : decidable c], Prop
+not_of_eq_false|?p = false → ¬ ?p
 decidable.rec_on_false|Π (H3 : ¬ ?p), ?H2 H3 → decidable.rec_on ?H ?H1 ?H2
 not_false|¬ false
 of_not_is_false|¬ is_false ?c → ?c
-if_false|∀ (t e : ?A), ite false t e = e
-iff.false_elim|?a ↔ false → ¬ ?a
 -- ENDFINDP

tests/lean/run/class4.lean

@@ -23,10 +23,10 @@ definition is_zero (x : nat)
 := nat.rec true (λ n r, false) x
 
 theorem is_zero_zero : is_zero zero
-:= eq.true_elim (refl _)
+:= of_eq_true (refl _)
 
 theorem not_is_zero_succ (x : nat) : ¬ is_zero (succ x)
-:= eq.false_elim (refl _)
+:= not_of_eq_false (refl _)
 
 theorem dichotomy (m : nat) : m = zero ∨ (∃ n, m = succ n)
 := nat.rec

tests/lean/run/is_nil.lean

@@ -11,7 +11,7 @@ definition is_nil {A : Type} (l : list A) : Prop
 := list.rec true (fun h t r, false) l
 
 theorem is_nil_nil (A : Type) : is_nil (@nil A)
-:= eq.true_elim (refl true)
+:= of_eq_true (refl true)
 
 theorem cons_ne_nil {A : Type} (a : A) (l : list A) : ¬ cons a l = nil
 := not_intro (assume H : cons a l = nil,