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doc(examples/lean): improve dep_if example

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Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
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Leonardo de Moura 2014-01-31 13:49:57 -08:00
parent 110ca84984
commit a51530dca0
1 changed files with 10 additions and 4 deletions
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14
examples/lean/dep_if.lean
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@ -32,13 +32,16 @@ theorem then_simp (A : TypeU) (c : Bool) (r : A) (t : c → A) (e : ¬ c → A)
in by simp in by simp
-- Given H : c, (dep_if c t e) = t H -- Given H : c, (dep_if c t e) = t H
theorem dep_if_true {A : TypeU} (c : Bool) (t : c → A) (e : ¬ c → A) (H : c) : dep_if c t e = t H theorem dep_if_elim_then {A : TypeU} (c : Bool) (t : c → A) (e : ¬ c → A) (H : c) : dep_if c t e = t H
:= let s1 : (λ r, (∀ Hc : c, r = t Hc) ∧ (∀ Hnc : ¬ c, r = e Hnc)) = (λ r, r = t H) := let s1 : (λ r, (∀ Hc : c, r = t Hc) ∧ (∀ Hnc : ¬ c, r = e Hnc)) = (λ r, r = t H)
:= funext (λ r, then_simp A c r t e H) := funext (λ r, then_simp A c r t e H)
in calc dep_if c t e = ε (nonempty_resolve t e) (λ r, (∀ Hc : c, r = t Hc) ∧ (∀ Hnc : ¬ c, r = e Hnc)) : refl (dep_if c t e) in calc dep_if c t e = ε (nonempty_resolve t e) (λ r, (∀ Hc : c, r = t Hc) ∧ (∀ Hnc : ¬ c, r = e Hnc)) : refl (dep_if c t e)
... = ε (nonempty_resolve t e) (λ r, r = t H) : { s1 } ... = ε (nonempty_resolve t e) (λ r, r = t H) : { s1 }
... = t H : eps_singleton (nonempty_resolve t e) (t H) ... = t H : eps_singleton (nonempty_resolve t e) (t H)
theorem dep_if_true {A : TypeU} (t : true → A) (e : ¬ true → A) : dep_if true t e = t trivial
:= dep_if_elim_then true t e trivial
theorem else_simp (A : TypeU) (c : Bool) (r : A) (t : c → A) (e : ¬ c → A) (H : ¬ c) theorem else_simp (A : TypeU) (c : Bool) (r : A) (t : c → A) (e : ¬ c → A) (H : ¬ c)
: (∀ Hc : c, r = t Hc) ∧ (∀ Hnc : ¬ c, r = e Hnc) ↔ r = e H : (∀ Hc : c, r = t Hc) ∧ (∀ Hnc : ¬ c, r = e Hnc) ↔ r = e H
:= let s1 : (∀ Hc : c, r = t Hc) ↔ true := let s1 : (∀ Hc : c, r = t Hc) ↔ true
@ -52,13 +55,16 @@ theorem else_simp (A : TypeU) (c : Bool) (r : A) (t : c → A) (e : ¬ c → A)
in by simp in by simp
-- Given H : ¬ c, (dep_if c t e) = e H -- Given H : ¬ c, (dep_if c t e) = e H
theorem dep_if_false {A : TypeU} (c : Bool) (t : c → A) (e : ¬ c → A) (H : ¬ c) : dep_if c t e = e H theorem dep_if_elim_else {A : TypeU} (c : Bool) (t : c → A) (e : ¬ c → A) (H : ¬ c) : dep_if c t e = e H
:= let s1 : (λ r, (∀ Hc : c, r = t Hc) ∧ (∀ Hnc : ¬ c, r = e Hnc)) = (λ r, r = e H) := let s1 : (λ r, (∀ Hc : c, r = t Hc) ∧ (∀ Hnc : ¬ c, r = e Hnc)) = (λ r, r = e H)
:= funext (λ r, else_simp A c r t e H) := funext (λ r, else_simp A c r t e H)
in calc dep_if c t e = ε (nonempty_resolve t e) (λ r, (∀ Hc : c, r = t Hc) ∧ (∀ Hnc : ¬ c, r = e Hnc)) : refl (dep_if c t e) in calc dep_if c t e = ε (nonempty_resolve t e) (λ r, (∀ Hc : c, r = t Hc) ∧ (∀ Hnc : ¬ c, r = e Hnc)) : refl (dep_if c t e)
... = ε (nonempty_resolve t e) (λ r, r = e H) : { s1 } ... = ε (nonempty_resolve t e) (λ r, r = e H) : { s1 }
... = e H : eps_singleton (nonempty_resolve t e) (e H) ... = e H : eps_singleton (nonempty_resolve t e) (e H)
theorem dep_if_false {A : TypeU} (t : false → A) (e : ¬ false → A) : dep_if false t e = e trivial
:= dep_if_elim_else false t e trivial
import cast import cast
theorem dep_if_congr {A : TypeM} (c1 c2 : Bool) theorem dep_if_congr {A : TypeM} (c1 c2 : Bool)
@ -85,7 +91,7 @@ pop_scope
-- can reduce the dependent-if to a regular if -- can reduce the dependent-if to a regular if
theorem dep_if_if {A : TypeU} (c : Bool) (t e : A) : dep_if c (λ Hc, t) (λ Hn, e) = if c then t else e theorem dep_if_if {A : TypeU} (c : Bool) (t e : A) : dep_if c (λ Hc, t) (λ Hn, e) = if c then t else e
:= or_elim (em c) := or_elim (em c)
(assume Hc : c, calc dep_if c (λ Hc, t) (λ Hn, e) = (λ Hc, t) Hc : dep_if_true _ _ _ Hc (assume Hc : c, calc dep_if c (λ Hc, t) (λ Hn, e) = (λ Hc, t) Hc : dep_if_elim_then _ _ _ Hc
... = if c then t else e : by simp) ... = if c then t else e : by simp)
(assume Hn : ¬ c, calc dep_if c (λ Hc, t) (λ Hn, e) = (λ Hn, e) Hn : dep_if_false _ _ _ Hn (assume Hn : ¬ c, calc dep_if c (λ Hc, t) (λ Hn, e) = (λ Hn, e) Hn : dep_if_elim_else _ _ _ Hn
... = if c then t else e : by simp) ... = if c then t else e : by simp)