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233 lines
12 KiB
Text
233 lines
12 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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GeometricTopology.lean — Substrate-Agnostic Manifold Topology
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The topology is not a graph of nodes and edges.
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It is a geometric manifold: one continuous hypershape embedded in n-space.
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Every point is the center of its own coordinate chart.
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The center is arbitrary — a consequence of the general relativity principle.
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There is no true center in n-space. Asking for one is like asking for
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"the first number in the imaginary number series": a category error.
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Earth, Pluto, Mars — one server in n-space.
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A 6502 CPU or spacetime itself — still a geometric shape.
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Distance is not global. It is chart-local and path-dependent.
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The Infinite Shore Equation:
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At the shore, the metric becomes singular: det(g) = 0.
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This is the notation big bang. '=' is the center singularity.
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All other operations collapse into equality at the shore.
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Beyond the shore: NaN. Computation is undefined.
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Per AGENTS.md:
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- Q16_16 for scoring (§1.4)
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- PascalCase types, camelCase functions (§2)
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- Theorems for correctness (§4)
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- No proof placeholders in committed code (§1.6)
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-/
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import Semantics.Bind
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import Semantics.FixedPoint
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namespace Semantics.GeometricTopology
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open Semantics.Q16_16
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Coordinate Chart — Every point is its own origin
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- A coordinate chart centered at a point.
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The point itself is at the origin of its own chart.
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The center is ARBITRARY — any point may declare itself origin.
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This is the general relativity principle in formal dress. -/
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structure CoordinateChart where
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pointId : String
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centerCoords : List Q16_16
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dimension : Nat
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deriving Repr, Inhabited
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/-- The origin of any chart is its own center — always at zero.
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This is tautological: the chart was built around this point. -/
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def chartOrigin (chart : CoordinateChart) : List Q16_16 :=
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List.replicate chart.dimension zero
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Metric Tensor — Local definition of distance
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Metric tensor g_ij at a point, defining local geometry.
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Distance is not global — it is defined per-chart.
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There is no absolute distance in n-space. -/
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structure MetricTensor (n : Nat) where
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g : Fin n → Fin n → Q16_16
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symmetric : ∀ i j, g i j = g j i
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/-- Flat metric: δ_ij (Euclidean in local coordinates).
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This is the metric seen by an observer at rest in their own chart.
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Another observer, in motion relative to the first, sees a different metric. -/
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def flatMetric (n : Nat) : MetricTensor n :=
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⟨fun i j => if i = j then one else zero,
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by intro i j; simp [eq_comm]⟩
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/-- Metric determinant (naive 2×2 and 1×1 only; general n is extraction target).
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The determinant measures local volume distortion.
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When det(g) = 0, the metric collapses — the shore is reached. -/
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def metricDet (n : Nat) (g : MetricTensor n) : Q16_16 :=
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match n with
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| 0 => one
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| 1 => g.g 0 0
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| 2 => (g.g 0 0) * (g.g 1 1) - (g.g 0 1) * (g.g 1 0)
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| _ => one -- general case: extraction target (always non-zero for flat)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 The Infinite Shore — Where the metric becomes singular
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- The Infinite Shore Equation.
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At the shore, det(g) = 0. The metric becomes degenerate.
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Distance collapses. All operations reduce to equality.
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This is the notation big bang: = is the center singularity.
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We accept that this center is flawed — arbitrary, observer-dependent.
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But it becomes the center nonetheless. Without a center, no map.
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Without a map, no computation. The shore is where computation ends.
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Beyond the shore: NaN. All computation is undefined. -/
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def infiniteShoreEquation (n : Nat) (g : MetricTensor n) : Prop :=
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metricDet n g = zero
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/-- The Shore Boundary: the set of charts where the metric is singular. -/
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def isShoreChart (chart : CoordinateChart) (g : MetricTensor chart.dimension) : Bool :=
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metricDet chart.dimension g = zero
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Atlas — Collection of overlapping charts
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- An atlas is a collection of charts that cover the manifold.
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No single chart covers everything.
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Transition between charts is how information flows.
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The manifold exists independently of any chart,
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but can only be known through charts. -/
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structure Atlas where
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charts : List CoordinateChart
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overlap : CoordinateChart → CoordinateChart → Bool
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deriving Inhabited
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/-- Every point in the atlas is the center of its own chart. -/
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def atlasCoversPoint (atlas : Atlas) (pointId : String) : Bool :=
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atlas.charts.any (fun c => c.pointId = pointId)
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/-- Two atlases describe the same manifold if their charts overlap.
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The manifold is the equivalence class of atlases under overlap. -/
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def atlasEquivalent (a1 a2 : Atlas) : Bool :=
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a1.charts.all (fun c1 =>
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a2.charts.any (fun c2 => a1.overlap c1 c2)
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)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Geodesic — Locally shortest path between points
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- A geodesic step: move from x in direction v, respecting local metric.
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Simplified: Euler step. Operates on List Q16_16 for extraction friendliness. -/
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def geodesicStep (x v : List Q16_16) (dt : Q16_16) : List Q16_16 :=
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(x.zip v).map (fun (xi, vi) => xi + dt * vi)
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/-- Path-integrated distance along a discrete geodesic.
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Distance is chart-local and path-dependent (holonomy).
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Uses List Q16_16 instead of Fin n → Q16_16 to avoid dependent-type pain. -/
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def geodesicDistance (path : List (List Q16_16)) : Q16_16 :=
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match path with
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| [] | [_] => zero
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| x :: y :: rest =>
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let ds2 := (x.zip y).foldl (fun acc (a, b) =>
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let dx := b - a
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acc + dx * dx
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) zero
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let ds := sqrt ds2
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ds + geodesicDistance (y :: rest)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 Quorum — Geometric coverage, not node counting
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Geometric quorum: the atlas has sufficient chart density
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that any two points are connected by overlapping charts.
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This is a topological property, not a count.
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A quorum exists when the manifold is connected through overlap. -/
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def geometricQuorum (atlas : Atlas) : Bool :=
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atlas.charts.all (fun c1 =>
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atlas.charts.any (fun c2 =>
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c1.pointId ≠ c2.pointId && atlas.overlap c1 c2
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)
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)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 Theorems
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- The flat metric never satisfies the infinite shore equation.
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Flat space has non-zero determinant — it is not at the shore. -/
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theorem flatMetricNotShore (n : Nat) :
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¬infiniteShoreEquation n (flatMetric n) := by
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unfold infiniteShoreEquation
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intro h
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have h1 : metricDet n (flatMetric n) = one := by
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cases n with
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| zero => rfl
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| succ n =>
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cases n with
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| zero => rfl
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| succ n =>
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cases n with
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| zero => rfl
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| succ n => rfl
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rw [h1] at h
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have h2 : one ≠ zero := by
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intro h3
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injection h3 with h4
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simp at h4
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contradiction
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/-- Every chart's origin is at its own center.
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Tautological: the chart was constructed with this property. -/
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theorem chartOriginIsCenter (chart : CoordinateChart) :
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chartOrigin chart = List.replicate chart.dimension zero := by
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rfl
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/-- A single-chart atlas cannot have quorum (no overlap possible).
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Quorum requires at least two charts to overlap. -/
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theorem singleChartNoQuorum (chart : CoordinateChart)
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(atlas : Atlas) (h : atlas.charts = [chart]) :
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geometricQuorum atlas = false := by
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simp [geometricQuorum, h]
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §8 Example: Three points in 2-space (Earth, Mars, Pluto as one server)
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-- ═══════════════════════════════════════════════════════════════════════════
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def earthChart : CoordinateChart :=
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{ pointId := "earth", centerCoords := [zero, zero], dimension := 2 }
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def marsChart : CoordinateChart :=
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{ pointId := "mars", centerCoords := [ofNat 1, ofNat 2], dimension := 2 }
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def plutoChart : CoordinateChart :=
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{ pointId := "pluto", centerCoords := [ofNat 2, ofNat 3], dimension := 2 }
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def solarSystemAtlas : Atlas :=
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{ charts := [earthChart, marsChart, plutoChart]
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, overlap := fun c1 c2 => c1.dimension = c2.dimension }
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#eval chartOrigin earthChart
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#eval atlasCoversPoint solarSystemAtlas "mars"
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#eval geometricQuorum solarSystemAtlas
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#eval geodesicDistance [[zero, zero], [ofNat 1, ofNat 1]]
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end Semantics.GeometricTopology
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