prolog CoQ implementation of Sanjoy Nath's Geometrifying Trigonometry systems

To implement Sanjoy Nath’s Geometrifying Trigonometry (SNGT) using logic programming and interactive theorem proving (Prolog and Coq), you must translate his concept of "Bunch of Line Segments" (BOLS), "Locked Sets", and "Caliperness Gluing Rules" into formal code. [1, 2, 3]
Because Coq and Prolog are entirely different environments, the industry standard workflow is a hybrid architecture: Use Prolog as the syntax parser/search engine to find valid geometric moves, and use Coq to formally prove that those moves maintain spatial truth (Caliperness invariants). [1, 3, 4]

Step 1: Define the Abstract Syntax Tree (AST) in Coq

First, you must teach Coq how to read trigonometric expressions as data, rather than evaluating them as numbers. [4]
  1. Define an inductive type for the terms.
  2. Ensure you have tokens for adding (gluing) and multiplying (scaling/orienting). [2, 3] 
(* Coq Definition *)
Inductive Var : Type := 

  | Theta | Phi | Delta.

Inductive TrigExpr : Type :=

  | Num : R -> TrigExpr                  (* Constants like 1 *)
  | Sin : Var -> TrigExpr                (* Sin(x) *)
  | Cos : Var -> TrigExpr                (* Cos(x) *)

  | Glue : TrigExpr -> TrigExpr -> TrigExpr   (* SNGT Addition (+) *)
  | Scale : TrigExpr -> TrigExpr -> TrigExpr  (* SNGT Multiplication (*) *).

Step 2: Define SNGT "Locked Sets" and Vectors in Coq

In SNGT, an expression is mapped to a geometric string. You must represent a "Locked Set" or a "Bunch of Line Segments" (BOLS) as a list of directed vectors in a 2D plane. [1, 2]
  1. Define a segment by its length and its absolute orientation angle.
  2. Represent a BOLS object as a continuous chain of these segments. [1] 
Record Segment : Type := {
  length : R;
  angle : R
}.

Definition BOLS := list Segment.

Step 3: Implement the SNGT Geometric Parser in Prolog

Prolog excels at parsing symbols and generating search trees. Write a Definite Clause Grammar (DCG) in Prolog to convert a standard trigonometric expression string into a structured list of geometric segments (BOLS). [1, 4, 5] 
% Prolog Parser Fragment
parse_sngt(cos(Theta), [segment(1, Theta)]).
parse_sngt(sin(Theta), [segment(1, Theta_Offset)]) :- 
    Theta_Offset is Theta - pi/2. % Sin is Cos shifted

% SNGT Addition Rule: "Gluing" joins lists of segments together
parse_sngt(glue(A, B), Segments) :-
    parse_sngt(A, SegsA),
    parse_sngt(B, SegsB),
    append(SegsA, SegsB, Segments).

Step 4: Code the "Caliperness Game" Mechanics in Prolog

Nath's theory relies on legal moves: sliding, rotating, tearing, and fusing segments without changing their lengths. Program these transition rules in Prolog so it can automatically search for a path that collapses a complex shape into a single straight line. [3] 
% Move Rule: Rotate two joined segments while preserving cumulative distance
caliper_move(rotate, [segment(L1, A1), segment(L2, A2)], [segment(L1, NA1), segment(L2, NA2)]) :-
    % Invariant: Total vector endpoint distance constraint logic goes here
    true. 

% Recursive solver to find the sequence of configurations
solve_sngt(State, State) :- is_straight_line(State).
solve_sngt(CurrentState, Path) :-
    caliper_move(_Type, CurrentState, NextState),
    solve_sngt(NextState, Path).

Step 5: Embed the Prolog Logic as a Coq Engine (Coq-Elpi or custom Tactic)

To verify your Prolog logic inside Coq, you can utilize Coq-Elpi (an extension that embeds an elpi-Prolog lambda-prolog engine directly inside Coq) or write a Coq function to execute the verification.
  1. Write a verification predicate in Coq that ensures a step transition preserves Euclidean invariants.
  2. Use a recursive fixpoint to walk through the transformation path discovered by your Prolog solver. [1] 
Fixpoint verify_caliper_path (initial : BOLS) (path : list BOLS) (final : BOLS) : bool :=
  match path with

  | nil => check_equivalence initial final
  | step :: remaining => if valid_sngt_transformation_q initial step 
                         then verify_caliper_path step remaining final
                         else false
  end.

Step 6: Formally Prove the Core SNGT Invariant Theorem

The final step is proving that if the algorithm reduces a BOLS expression to a single straight line, the algebraic equivalent is true.
Theorem sngt_caliperness_invariant : forall (expr : TrigExpr) (b : BOLS),
  parse_to_bols expr = b ->
  forall (b_final : BOLS), 
    reduced_to_straight_line b b_final -> 
    vector_magnitude_sum b = vector_magnitude_sum b_final.
Proof.
  (* Use geometry tactics or coq-reals to solve via induction on the list of segments *)
Admitted.
If you want to refine this model, which specific SNGT rule would you like to build next: the 4-symmetric cases of multiplication or the code for sorting planar polygon edges (Polygonometry)? [1, 6, 7]

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