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2.1 KiB
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47 lines
No EOL
2.1 KiB
Text
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--- Copyright (c) 2014 Microsoft Corporation. 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
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----------------------------------------------------------------------------------------------------
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import logic.connectives.basic struc.function
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using function
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namespace congr
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inductive struc {T1 : Type} {T2 : Type} (R1 : T1 → T1 → Prop) (R2 : T2 → T2 → Prop)
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(f : T1 → T2) : Prop :=
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mk : (∀x y : T1, R1 x y → R2 (f x) (f y)) → struc R1 R2 f
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abbreviation app {T1 : Type} {T2 : Type} {R1 : T1 → T1 → Prop} {R2 : T2 → T2 → Prop}
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{f : T1 → T2} (C : struc R1 R2 f) {x y : T1} : R1 x y → R2 (f x) (f y) :=
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struc_rec id C x y
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inductive struc2 {T1 : Type} {T2 : Type} {T3 : Type} (R1 : T1 → T1 → Prop)
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(R2 : T2 → T2 → Prop) (R3 : T3 → T3 → Prop) (f : T1 → T2 → T3) : Prop :=
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mk2 : (∀(x1 y1 : T1) (x2 y2 : T2), R1 x1 y1 → R2 x2 y2 → R3 (f x1 x2) (f y1 y2)) →
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struc2 R1 R2 R3 f
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abbreviation app2 {T1 : Type} {T2 : Type} {T3 : Type} {R1 : T1 → T1 → Prop}
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{R2 : T2 → T2 → Prop} {R3 : T3 → T3 → Prop} {f : T1 → T2 → T3}
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(C : struc2 R1 R2 R3 f) {x1 y1 : T1} {x2 y2 : T2}
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: R1 x1 y1 → R2 x2 y2 → R3 (f x1 x2) (f y1 y2) :=
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struc2_rec id C x1 y1 x2 y2
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theorem compose21
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{T2 : Type} {R2 : T2 → T2 → Prop}
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{T3 : Type} {R3 : T3 → T3 → Prop}
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{T4 : Type} {R4 : T4 → T4 → Prop}
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{g : T2 → T3 → T4} (C3 : congr.struc2 R2 R3 R4 g)
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⦃T1 : Type⦄ -- nice!
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{R1 : T1 → T1 → Prop}
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{f1 : T1 → T2} (C1 : congr.struc R1 R2 f1)
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{f2 : T1 → T3} (C2 : congr.struc R1 R3 f2) :
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congr.struc R1 R4 (λx, g (f1 x) (f2 x)) := mk (take x1 x2 H, app2 C3 (app C1 H) (app C2 H))
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theorem congr_and : congr.struc2 iff iff iff and := sorry
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theorem congr_and_comp [instance] {T : Type} {R : T → T → Prop} {f1 f2 : T → Prop}
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(C1 : struc R iff f1) (C2 : struc R iff f2) :
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congr.struc R iff (λx, f1 x ∧ f2 x) := congr.compose21 congr_and C1 C2
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end congr |