File size: 1,305 Bytes
92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | import Mathlib
import OAI.Computability.VertexCover.Machines.Primitives
namespace OAI
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namespace VertexCover.Machine
theorem prodBits_injective {α β : Type} {ea : α → List Bool} {eb : β → List Bool}
(ha : Function.Injective ea) (hb : Function.Injective eb) :
Function.Injective (prodBits ea eb) := by
intro a b h
apply Prod.ext
· apply ha
have hh := congrArg (Stream.output (Project.trans true false) (fun _ => []) 0) h
simpa only [prodBits,Project.framed,Bool.false_eq_true,↓reduceIte,List.append_nil] using hh
· apply hb
have hh := congrArg (Stream.output (Project.trans false true) (fun _ => []) 0) h
simpa only [prodBits,Project.framed,Bool.false_eq_true,↓reduceIte,List.nil_append] using hh
theorem boolBits_injective : Function.Injective boolBits := by
intro a b h
exact List.singleton_injective h
end VertexCover.Machine
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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