92 lines
3 KiB
Text
92 lines
3 KiB
Text
/-
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Copyright (c) 2014 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Jeremy Avigad, Andrew Zipperer
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Using classical logic, defines an inverse function.
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-/
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import .function .map
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open eq.ops classical
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namespace set
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variables {X Y : Type}
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noncomputable definition inv_fun (f : X → Y) (a : set X) (dflt : X) (y : Y) : X :=
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if H : ∃₀ x ∈ a, f x = y then some H else dflt
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theorem inv_fun_pos {f : X → Y} {a : set X} {dflt : X} {y : Y}
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(H : ∃₀ x ∈ a, f x = y) : (inv_fun f a dflt y ∈ a) ∧ (f (inv_fun f a dflt y) = y) :=
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have H1 : inv_fun f a dflt y = some H, from dif_pos H,
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H1⁻¹ ▸ some_spec H
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theorem inv_fun_neg {f : X → Y} {a : set X} {dflt : X} {y : Y}
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(H : ¬ ∃₀ x ∈ a, f x = y) : inv_fun f a dflt y = dflt :=
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dif_neg H
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variables {f : X → Y} {a : set X} {b : set Y}
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theorem maps_to_inv_fun {dflt : X} (dflta : dflt ∈ a) :
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maps_to (inv_fun f a dflt) b a :=
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let f' := inv_fun f a dflt in
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take y,
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assume yb : y ∈ b,
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show f' y ∈ a, from
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by_cases
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(assume H : ∃₀ x ∈ a, f x = y,
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and.left (inv_fun_pos H))
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(assume H : ¬ ∃₀ x ∈ a, f x = y,
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(inv_fun_neg H)⁻¹ ▸ dflta)
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theorem left_inv_on_inv_fun_of_inj_on (dflt : X) (H : inj_on f a) :
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left_inv_on (inv_fun f a dflt) f a :=
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let f' := inv_fun f a dflt in
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take x,
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assume xa : x ∈ a,
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have H1 : ∃₀ x' ∈ a, f x' = f x, from exists.intro x (and.intro xa rfl),
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have H2 : f' (f x) ∈ a ∧ f (f' (f x)) = f x, from inv_fun_pos H1,
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show f' (f x) = x, from H (and.left H2) xa (and.right H2)
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theorem surj_on_inv_fun_of_inj_on (dflt : X) (mapsto : maps_to f a b) (H : inj_on f a) :
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surj_on (inv_fun f a dflt) b a :=
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surj_on_of_right_inv_on mapsto (left_inv_on_inv_fun_of_inj_on dflt H)
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theorem right_inv_on_inv_fun_of_surj_on (dflt : X) (H : surj_on f a b) :
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right_inv_on (inv_fun f a dflt) f b :=
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let f' := inv_fun f a dflt in
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take y,
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assume yb: y ∈ b,
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obtain x (Hx : x ∈ a ∧ f x = y), from H yb,
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have Hy : f' y ∈ a ∧ f (f' y) = y, from inv_fun_pos (exists.intro x Hx),
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and.right Hy
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theorem inj_on_inv_fun (dflt : X) (H : surj_on f a b) :
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inj_on (inv_fun f a dflt) b :=
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inj_on_of_left_inv_on (right_inv_on_inv_fun_of_surj_on dflt H)
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end set
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open set
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namespace map
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variables {X Y : Type} {a : set X} {b : set Y}
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protected noncomputable definition inverse (f : map a b) {dflt : X} (dflta : dflt ∈ a) :=
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map.mk (inv_fun f a dflt) (@maps_to_inv_fun _ _ _ _ b _ dflta)
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theorem left_inverse_inverse {f : map a b} {dflt : X} (dflta : dflt ∈ a) (H : map.injective f) :
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map.left_inverse (map.inverse f dflta) f :=
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left_inv_on_inv_fun_of_inj_on dflt H
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theorem right_inverse_inverse {f : map a b} {dflt : X} (dflta : dflt ∈ a) (H : map.surjective f) :
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map.right_inverse (map.inverse f dflta) f :=
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right_inv_on_inv_fun_of_surj_on dflt H
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theorem is_inverse_inverse {f : map a b} {dflt : X} (dflta : dflt ∈ a) (H : map.bijective f) :
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map.is_inverse (map.inverse f dflta) f :=
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and.intro
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(left_inverse_inverse dflta (and.left H))
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(right_inverse_inverse dflta (and.right H))
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end map
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