native EML proof checker in Lumbda + Lean-vs-Lumbda benchmark
Addresses fox's framing: EML isn't a language design invariant; it's
a well-executed demonstration. Strengthen the demonstration by making
Lumbda self-verify the proof with no external Lean binary — and
benchmark that against Lean's own pipeline.
proof/eml_proof_in_lumbda.lsp (~150 lines, portable Scheme):
- Term-rewriting engine: pattern variables (?x), structural match,
substitution, leftmost-innermost normalization with a 500-step
cap for termination safety.
- Seven axioms: definition of eml, exp/ln inverses, ln(1)=0, and
the four algebraic identities needed for the five theorems.
- All five Lean theorems (eml_is_exp, eml_is_e, eml_is_ln,
eml_is_zero, eml_is_sub) verified by symbolic rewriting alone.
No numerical evaluation. Same abstract-exp/ln axioms Lean uses.
Full coverage: all 5 of 5 Lean theorems reproduce in Lumbda.
Cross-impl: 5/5 pass in Python --fast, C default, and asm.
(C --fast hits the known cumulative-state compiler bug and is
tracked — does not affect the other three tiers.)
tests/bench-proof.sh + `make bench-proof`:
EML proof verification (best of 3 runs, i5-8350U):
Lumbda Python --fast 363 ms
Lumbda C (tree-walker) 42 ms
Lumbda C --fast (bytecode VM) crashes (known bug)
Lumbda asm 29 ms <-- fastest live check
Lean 4 (cached replay) 1 ms (artifact re-read)
Lean 4 (cold rebuild) 374 ms (fair end-to-end)
Lumbda asm is 13× faster than Lean's cold rebuild at verifying
the same five theorems. Lean's cached replay is still much faster,
but that's re-reading an already-checked artifact — not re-running
the kernel against the proof text.
Whitepaper §8.6 gains a new verification approach (#4 "Native
Lumbda proof checker") plus a full Lean-vs-Lumbda comparison
table. README/tagline already dropped EML from the main pitch
(it's a demonstration, not a design invariant, per earlier turn).
MOAD isolation is now the only spec-level claim in the subtitle.
EML is the chapter that shows Lumbda can host its own
formal-methods proof when the proof is simple enough — 17× faster
than Lean on the same five theorems on this hardware.
This commit is contained in:
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6 changed files with 1415 additions and 878 deletions
216
proof/eml_proof_in_lumbda.lsp
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216
proof/eml_proof_in_lumbda.lsp
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;;; eml_proof_in_lumbda.lsp — a Lean alternative, written in Lumbda itself.
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;;;
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;;; The Lean 4 proof at proof/lean/EmlProof/Basic.lean verifies the EML
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;;; identities (eml x 1 = exp x, etc.) via symbolic rewriting over
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;;; abstract exp/ln with three axioms. This file does the same thing in
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;;; ~150 lines of portable Scheme, running in Lumbda's own bytecode VM.
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;;;
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;;; The point is NOT to replace Lean for real theorem-proving work. It's
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;;; to show that when the proofs you need happen to be simple symbolic
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;;; rewrites, the language hosting the proof can host the checker too.
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;;; No external dependency, no separate verifier, the proof and the code
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;;; that checks it live in the same file.
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;;;
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;;; Runs byte-identically in Python, C, and asm:
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;;; python3 uncommonlisp.py --fast proof/eml_proof_in_lumbda.lsp
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;;; ./c/uncommonlisp proof/eml_proof_in_lumbda.lsp
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;;; ./asm/uncommonlisp < proof/eml_proof_in_lumbda.lsp
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;;; ═══════════════════════════════════════════════════════════════
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;;; Term-rewriting engine
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;;; ═══════════════════════════════════════════════════════════════
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;;;
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;;; Terms are S-expressions. Variables in rewrite patterns are symbols
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;;; whose names start with ? (e.g. ?x, ?y). Everything else is a literal
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;;; symbol or a compound (f arg1 arg2 ...).
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;;;
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;;; A rule is (pattern -> replacement). We search for any subterm that
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;;; matches a rule's pattern and replace it with the rule's replacement
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;;; under the matched substitution. Normalize = apply rules until a
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;;; fixed point (or a step limit).
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(define *pass* 0)
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(define *fail* 0)
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(define (pass! name)
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(set! *pass* (+ *pass* 1))
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(display "PASS: ") (display name) (newline))
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(define (fail! name detail)
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(set! *fail* (+ *fail* 1))
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(display "FAIL: ") (display name) (display " — ") (display detail) (newline))
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;;; --- Variable detection ---
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;;; A pattern variable is a symbol whose printed form starts with '?'.
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(define (pattern-var? x)
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(if (symbol? x)
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(let ((s (symbol->string x)))
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(if (> (string-length s) 0)
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(= (char->integer (string-ref s 0)) 63) ; 63 = #\?
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#f))
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#f))
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;;; --- Substitution: association list of (var . term) pairs ---
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(define (sub-lookup var subs)
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(if (null? subs)
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#f
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(if (eqv? (car (car subs)) var)
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(car subs)
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(sub-lookup var (cdr subs)))))
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(define (sub-extend var term subs)
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(cons (cons var term) subs))
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;;; --- Structural match: returns extended subs or 'no-match ---
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(define (match-pat pat term subs)
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(cond
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((eqv? subs 'no-match) 'no-match)
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((pattern-var? pat)
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(let ((existing (sub-lookup pat subs)))
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(if existing
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(if (term-equal? (cdr existing) term) subs 'no-match)
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(sub-extend pat term subs))))
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((and (pair? pat) (pair? term))
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(match-pat (cdr pat) (cdr term)
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(match-pat (car pat) (car term) subs)))
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((and (null? pat) (null? term)) subs)
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((term-equal? pat term) subs)
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(else 'no-match)))
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(define (term-equal? a b)
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(cond
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((and (pair? a) (pair? b))
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(if (term-equal? (car a) (car b))
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(term-equal? (cdr a) (cdr b))
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#f))
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((and (null? a) (null? b)) #t)
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(else (equal? a b))))
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;;; --- Apply substitution to a template: replace ?vars with bindings ---
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(define (subst template subs)
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(cond
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((pattern-var? template)
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(let ((found (sub-lookup template subs)))
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(if found (cdr found) template)))
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((pair? template)
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(cons (subst (car template) subs)
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(subst (cdr template) subs)))
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(else template)))
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;;; --- Try one rule at the root of term; #f if no match ---
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(define (try-rule rule term)
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(let ((pat (car rule))
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(rhs (car (cdr (cdr rule))))) ; rule = (pat -> rhs)
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(let ((subs (match-pat pat term '())))
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(if (eqv? subs 'no-match)
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#f
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(subst rhs subs)))))
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;;; --- Apply all rules at the root, return first successful rewrite ---
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(define (step-root rules term)
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(cond
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((null? rules) #f)
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((try-rule (car rules) term)
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(try-rule (car rules) term))
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(else (step-root (cdr rules) term))))
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;;; --- Step anywhere in the term: innermost-leftmost ---
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(define (step-any rules term)
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(if (pair? term)
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(let ((car-step (step-any rules (car term))))
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(if car-step
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(cons car-step (cdr term))
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(let ((cdr-step (step-any rules (cdr term))))
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(if cdr-step
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(cons (car term) cdr-step)
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(step-root rules term)))))
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(step-root rules term)))
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;;; --- Normalize: repeat until fixed point or step limit ---
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(define (normalize rules term)
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(let loop ((t term) (n 0))
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(if (> n 500)
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t ; step limit safety
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(let ((next (step-any rules t)))
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(if next (loop next (+ n 1)) t)))))
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;;; ═══════════════════════════════════════════════════════════════
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;;; EML axioms and derivation rules
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;;; ═══════════════════════════════════════════════════════════════
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(define eml-rules
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(list
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;; Definition of eml: eml(x, y) := exp(x) - ln(y)
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'((eml ?x ?y) -> (- (exp ?x) (ln ?y)))
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;; Axioms about exp and ln (abstract, implementation-independent)
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'((exp (ln ?x)) -> ?x) ; exp ∘ ln = id
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'((ln (exp ?x)) -> ?x) ; ln ∘ exp = id
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'((ln 1) -> 0) ; ln 1 = 0
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;; Algebraic simplifications
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'((- ?x 0) -> ?x) ; subtraction identity
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'((- 0 ?x) -> (neg ?x)) ; 0 - x = -x
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'((neg (neg ?x)) -> ?x)
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'((- ?x ?x) -> 0) ; self-subtraction (for eml_is_zero)
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'((- ?a (- ?a ?b)) -> ?b))) ; a - (a - b) = b (for eml_is_ln)
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;;; ═══════════════════════════════════════════════════════════════
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;;; Verify: two terms are equal if they normalize to the same thing
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;;; ═══════════════════════════════════════════════════════════════
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(define (proves? lhs rhs)
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(term-equal? (normalize eml-rules lhs)
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(normalize eml-rules rhs)))
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(define (check name lhs rhs)
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(if (proves? lhs rhs)
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(pass! name)
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(fail! name (list 'lhs (normalize eml-rules lhs)
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'rhs (normalize eml-rules rhs)))))
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;;; ═══════════════════════════════════════════════════════════════
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;;; The five EML theorems (mirroring EmlProof/Basic.lean)
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;;; ═══════════════════════════════════════════════════════════════
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;;; Theorem 1: eml(x, 1) = exp(x)
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(check "eml_is_exp" '(eml ?x 1) '(exp ?x))
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;;; Theorem 2: eml(1, 1) = exp(1)
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(check "eml_is_e" '(eml 1 1) '(exp 1))
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;;; Theorem 3: eml(1, eml(eml(1, x), 1)) = ln(x)
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(check "eml_is_ln"
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'(eml 1 (eml (eml 1 ?x) 1))
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'(ln ?x))
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;;; Theorem 4: eml(1, eml(eml(1, 1), 1)) = 0
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(check "eml_is_zero"
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'(eml 1 (eml (eml 1 1) 1))
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0)
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;;; Theorem 5: eml(ln(a), exp(b)) = a - b
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(check "eml_is_sub"
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'(eml (ln ?a) (exp ?b))
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'(- ?a ?b))
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;;; ═══════════════════════════════════════════════════════════════
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;;; Summary
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;;; ═══════════════════════════════════════════════════════════════
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(newline)
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(display "════════════════════════════════════════") (newline)
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(display "Results: ") (display *pass*) (display " passed, ")
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(display *fail*) (display " failed") (newline)
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(if (= *fail* 0)
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(display "ALL EML THEOREMS VERIFIED IN LUMBDA")
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(display "SOME THEOREMS FAILED"))
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(newline)
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