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+-- the goal of this module is to present inductive tools for reasoning about minesweeper, similar to the axioms ProofSweeper provides,
+-- and prove them sound and complete with respect to our formulation of minesweeper.
+
+module Minesweeper.Reasoning where
+
+open import Relation.Nullary
+open import Relation.Binary.PropositionalEquality
+open import Data.List as List using (List)
+open import Data.List.Membership.Propositional
+open import Data.Product
+open import Data.Nat
+
+open import Data.Fin using (Fin)
+open import Minesweeper.Enumeration as Enum using (Enumeration)
+open import Minesweeper.Coords
+open import Minesweeper.Board
+open import Minesweeper.Rules
+open import Minesweeper.Moves
+
+-- our inductive family for minesweeper proofs! based on ProofSweeper but using our own machinery
+-- and not intended for hand-writing at all
+data _[_]↝★_ {bounds} (grid : Board Tile bounds) (coords : Coords bounds) : Guess → Set
+record _Neighboring★_on_excluding_ {bounds} (guess : Guess) (neighbor : Coords bounds) (grid : Board Tile bounds) (exclusion : Coords bounds) : Set
+
+data _[_]↝★_ {bounds} grid coords where
+  -- known tiles already have a proven identity
+  known★ : ∀ tile guess → lookup coords grid ≡ known tile → guess ↝⚐ tile → grid [ coords ]↝★ guess
+
+  -- a tile is safe if an adjacent safe tile already has as many adjacent mines as it can
+  safe★ : ∀ neighborMineCount
+    (safeNeighbor : known (safe neighborMineCount) Neighboring coords on grid)
+    (neighborMines : List (mine⚐ Neighboring★ proj₁ (proj₁ safeNeighbor) on grid excluding coords)) →
+    List.length neighborMines ≡ neighborMineCount →
+      grid [ coords ]↝★ safe⚐
+
+  -- a tile is a mine if an adjacent safe tile already has as many adjacent safe tiles as it can
+  mine★ : ∀ neighborMineCount
+    (safeNeighbor : known (safe neighborMineCount) Neighboring coords on grid)
+    (neighborSafes : List (safe⚐ Neighboring★ proj₁ (proj₁ safeNeighbor) on grid excluding coords)) →
+    List.length neighborSafes + neighborMineCount ≡ List.length (Enumeration.list (neighbors (proj₁ (proj₁ safeNeighbor)))) →
+      grid [ coords ]↝★ mine⚐
+
+record _Neighboring★_on_excluding_ {bounds} guess neighbor grid exclusion where
+  inductive
+  field
+    coords : Coords bounds
+    deduction : grid [ coords ]↝★ guess
+    ≢exclusion : coords ≢ exclusion