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simplipy-assets / proposals /LLM_PROPOSALS_v1.json

psaegert

Publish mined artifacts 2-1, 3-2, 4-3 (simplipy 0.7.1) + LLM proposal batch v1

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	{
	"schema": "llm-proposals-v1",
	"description": "Certified LLM-proposed simplification-rule candidates for the production mining LLM channel. Candidates only: every entry re-enters full certification against the current rule state at mine time.",
	"provenance": {
	"raw_waves": [
	"llm_proposals_raw.json (591, 2026-07-11)",
	"llm_proposals_round2.json (1863, 2026-07-12)"
	],
	"generation": "llm_prompts/round1_proposal_workflow.js (6 family agents), round2_proposal_workflow.js (10 family agents); Claude, grammar-constrained, not seed-reproducible",
	"certification": "validate_llm_proposals.py + certify_long_targets.py against the R8-era ruleset snapshot (rules_snapshot_for_llm.json)",
	"historical_note": "hint accuracy: round1 173/192 (90%), round2 516/611 (84%)"
	},
	"counts": {
	"unique_certified": 740,
	"tier_minimal": 733,
	"tier_hint_verified_nonminimal": 7
	},
	"proposals": [
	{
	"source": [
	"*",
	"abs",
	"x0",
	"abs",
	"x1"
	],
	"target": [
	"abs",
	"*",
	"x0",
	"x1"
	],
	"why": "\|x\|\|y\| -> \|xy\|",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"cosh",
	"x0",
	"sinh",
	"x0"
	],
	"target": [
	"div2",
	"sinh",
	"mult2",
	"x0"
	],
	"why": "cosh sinh = sinh(2x)/2 (operand order)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"cosh",
	"x0",
	"tanh",
	"x0"
	],
	"target": [
	"sinh",
	"x0"
	],
	"why": "cosh * tanh = sinh (operand order)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"div2",
	"x0",
	"div2",
	"x1"
	],
	"target": [
	"*",
	"div4",
	"x0",
	"x1"
	],
	"why": "(x0/2)(x1/2) -> (x0x1)/4",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"*",
	"div2",
	"x0",
	"mult2",
	"x1"
	],
	"target": [
	"*",
	"x0",
	"x1"
	],
	"why": "operand-order variant x/2 * 2y -> xy",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"div3",
	"x0",
	"mult3",
	"x1"
	],
	"target": [
	"*",
	"x0",
	"x1"
	],
	"why": "(x0/3)(3x1) -> x0x1",
	"family": 3,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"exp",
	"x0",
	"exp",
	"x1"
	],
	"target": [
	"exp",
	"+",
	"x0",
	"x1"
	],
	"why": "e^a e^b = e^(a+b)",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"mult2",
	"x0",
	"div2",
	"x1"
	],
	"target": [
	"*",
	"x0",
	"x1"
	],
	"why": "(2x0)(x1/2) -> x0x1",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"mult2",
	"x0",
	"mult2",
	"x1"
	],
	"target": [
	"*",
	"mult4",
	"x0",
	"x1"
	],
	"why": "(2x0)(2x1) -> 4(x0x1)",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"*",
	"mult3",
	"x0",
	"div3",
	"x1"
	],
	"target": [
	"*",
	"x0",
	"x1"
	],
	"why": "3x * y/3 -> xy",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"mult4",
	"x0",
	"div2",
	"x1"
	],
	"target": [
	"*",
	"mult2",
	"x0",
	"x1"
	],
	"why": "constant-ladder: 4x * (y/2) = 2xy",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"*",
	"mult4",
	"x0",
	"div4",
	"x1"
	],
	"target": [
	"*",
	"x0",
	"x1"
	],
	"why": "4x * y/4 -> xy",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"mult5",
	"x0",
	"div5",
	"x1"
	],
	"target": [
	"*",
	"x0",
	"x1"
	],
	"why": "5x * y/5 -> xy",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"neg",
	"x0",
	"neg",
	"x1"
	],
	"target": [
	"*",
	"x0",
	"x1"
	],
	"why": "(-x0)(-x1) -> x0x1",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"np.pi",
	"*",
	"np.e",
	"x0"
	],
	"target": [
	"*",
	"<constant>",
	"x0"
	],
	"why": "pi*e folds to one constant",
	"family": 5,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow",
	"x0",
	"<constant>",
	"x0"
	],
	"target": [
	"pow",
	"x0",
	"<constant>"
	],
	"why": "x^c*x=x^(c+1)",
	"family": 5,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow1_2",
	"x0",
	"pow1_2",
	"x1"
	],
	"target": [
	"pow1_2",
	"*",
	"x0",
	"x1"
	],
	"why": "sqrt(a) sqrt(b) = sqrt(ab) (dom ext)",
	"family": 4,
	"rounds": [
	"round1",
	"round2"
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	"tier": "minimal",
	"hint_matched": true
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	{
	"source": [
	"*",
	"pow1_2",
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	"pow1_4",
	"x0"
	],
	"target": [
	"pow1_4",
	"pow3",
	"x0"
	],
	"why": "x^(1/2) x^(1/4) = x^(3/4)",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow1_2",
	"x0",
	"pow2",
	"x0"
	],
	"target": [
	"pow1_2",
	"pow5",
	"x0"
	],
	"why": "sqrt(x) x^2 = x^(5/2)",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow1_3",
	"x0",
	"pow1_3",
	"x1"
	],
	"target": [
	"pow1_3",
	"*",
	"x0",
	"x1"
	],
	"why": "cbrt(a) cbrt(b) = cbrt(ab) (dom ext)",
	"family": 4,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow1_4",
	"x0",
	"pow1_4",
	"x1"
	],
	"target": [
	"pow1_4",
	"*",
	"x0",
	"x1"
	],
	"why": "4th roots multiply; domain extension for xy>0 with x,y<0",
	"family": 5,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow1_5",
	"x0",
	"pow1_5",
	"x1"
	],
	"target": [
	"pow1_5",
	"*",
	"x0",
	"x1"
	],
	"why": "odd 5th root multiplicative",
	"family": 5,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"target": [
	"pow2",
	"*",
	"x0",
	"x1"
	],
	"why": "x^2*y^2 -> (xy)^2",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow2",
	"x0",
	"pow3",
	"x0"
	],
	"target": [
	"pow5",
	"x0"
	],
	"why": "x0^2*x0^3 -> x0^5",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow2",
	"x0",
	"pow4",
	"x0"
	],
	"target": [
	"pow2",
	"pow3",
	"x0"
	],
	"why": "x^2*x^4=x^6=(x^3)^2",
	"family": 5,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow3",
	"x0",
	"pow2",
	"x0"
	],
	"target": [
	"pow5",
	"x0"
	],
	"why": "operand-order variant of x^2*x^3",
	"family": 5,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow3",
	"x0",
	"pow3",
	"x1"
	],
	"target": [
	"pow3",
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	"why": "x^3*y^3 -> (xy)^3",
	"family": 4,
	"rounds": [
	"round2"
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	"tier": "minimal",
	"hint_matched": true
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	{
	"source": [
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	"x0",
	"pow5",
	"x0"
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	"x0"
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	"why": "x^3*x^5=x^8",
	"family": 5,
	"rounds": [
	"round2"
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
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	"pow4",
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	"why": "x^4*y^4=(xy)^4",
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	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
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	{
	"source": [
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	"pow5",
	"x1"
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	"target": [
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	"x0",
	"x1"
	],
	"why": "x^5*y^5=(xy)^5",
	"family": 5,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
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	"x0",
	"cos",
	"x0"
	],
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	],
	"why": "sin x cos x = sin(2x)/2 (target length 4)",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"sinh",
	"x0",
	"cosh",
	"x0"
	],
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	"mult2",
	"x0"
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	"family": 0,
	"rounds": [
	"round1",
	"round2"
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	"tier": "minimal",
	"hint_matched": true
	},
	{
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	"x0",
	"cos",
	"x0"
	],
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	"x0"
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	"why": "tan x * cos x = sin x",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
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	"x0",
	"cosh",
	"x0"
	],
	"target": [
	"sinh",
	"x0"
	],
	"why": "tanh * cosh = sinh",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"x0",
	"pow",
	"x0",
	"<constant>"
	],
	"target": [
	"pow",
	"x0",
	"<constant>"
	],
	"why": "x * x^c = x^(c+1)",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"*",
	"<constant>",
	"x0",
	"x0"
	],
	"target": [
	"*",
	"<constant>",
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	],
	"why": "Cx0+x0 -> (C+1)x0",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"+",
	"x0",
	"1",
	"(-1)"
	],
	"target": [
	"x0"
	],
	"why": "constant-ladder: (x+1)+(-1) = x",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
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	],
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	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
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	],
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	"family": 1,
	"rounds": [
	"round1",
	"round2"
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	"tier": "minimal",
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	{
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	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
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	"family": 0,
	"rounds": [
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	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
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	"target": [
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	],
	"why": "cosh 2x + 1 = 2 cosh^2",
	"family": 1,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
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	],
	"why": "x/2+y/2 = (x+y)/2",
	"family": 0,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
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	"div4",
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	"target": [
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	"why": "constant-ladder: x/2 + x/4 = 3x/4",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
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	"x1"
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	],
	"why": "constant-ladder: x/5 + y/5 = (x+y)/5",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
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	"log",
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	"x1"
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	"why": "log a + log b = log(ab)",
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	"round2"
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
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	"why": "2x+2y = 2(x+y)",
	"family": 0,
	"rounds": [
	"round2"
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
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	"mult5",
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	"why": "2x0+3x0 -> 5x0",
	"family": 3,
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	"tier": "minimal",
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	},
	{
	"source": [
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	"x1"
	],
	"target": [
	"mult3",
	"+",
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	"x1"
	],
	"why": "3x+3y = 3(x+y)",
	"family": 0,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
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	"why": "constant-ladder: 5x + 5y = 5(x+y)",
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	"tier": "minimal",
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	},
	{
	"source": [
	"+",
	"pow2",
	"sinh",
	"x0",
	"1"
	],
	"target": [
	"pow2",
	"cosh",
	"x0"
	],
	"why": "sinh^2 + 1 = cosh^2 (operand order)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"*",
	"<constant>",
	"x0",
	"x0"
	],
	"target": [
	"*",
	"<constant>",
	"x0"
	],
	"why": "Cx0-x0 -> (C-1)x0",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"1",
	"cos",
	"mult2",
	"x0"
	],
	"target": [
	"mult2",
	"pow2",
	"sin",
	"x0"
	],
	"why": "1 - cos(2x) = 2 sin^2 x",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"1",
	"pow2",
	"cos",
	"x0"
	],
	"target": [
	"pow2",
	"sin",
	"x0"
	],
	"why": "1 - cos^2 x = sin^2 x",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"1",
	"pow2",
	"sin",
	"x0"
	],
	"target": [
	"pow2",
	"cos",
	"x0"
	],
	"why": "1 - sin^2 x = cos^2 x",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"1",
	"pow2",
	"tanh",
	"x0"
	],
	"target": [
	"inv",
	"pow2",
	"cosh",
	"x0"
	],
	"why": "1 - tanh^2 = sech^2 = 1/cosh^2",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"cosh",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"mult2",
	"pow2",
	"sinh",
	"x0"
	],
	"why": "cosh 2x - 1 = 2 sinh^2",
	"family": 1,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"cosh",
	"x0",
	"exp",
	"x0"
	],
	"target": [
	"neg",
	"sinh",
	"x0"
	],
	"why": "cosh - e^x = -sinh",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"div2",
	"x0",
	"div2",
	"x1"
	],
	"target": [
	"div2",
	"-",
	"x0",
	"x1"
	],
	"why": "constant-ladder: x/2 - y/2 = (x-y)/2",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"div2",
	"x0",
	"div3",
	"x0"
	],
	"target": [
	"div2",
	"div3",
	"x0"
	],
	"why": "constant-ladder: x/2 - x/3 = x/6",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"div3",
	"x0",
	"div4",
	"x0"
	],
	"target": [
	"div3",
	"div4",
	"x0"
	],
	"why": "constant-ladder: x/3 - x/4 = x/12",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"exp",
	"x0",
	"cosh",
	"x0"
	],
	"target": [
	"sinh",
	"x0"
	],
	"why": "e^x - cosh = sinh",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"exp",
	"x0",
	"sinh",
	"x0"
	],
	"target": [
	"cosh",
	"x0"
	],
	"why": "e^x - sinh = cosh",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"log",
	"x0",
	"log",
	"x1"
	],
	"target": [
	"log",
	"/",
	"x0",
	"x1"
	],
	"why": "log a - log b = log(a/b)",
	"family": 2,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"mult2",
	"x0",
	"div2",
	"x0"
	],
	"target": [
	"div2",
	"mult3",
	"x0"
	],
	"why": "2x0-x0/2 -> 3x0/2",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"mult2",
	"x0",
	"mult2",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "constant-ladder: 2x - 2x = 0",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"mult2",
	"x0",
	"mult2",
	"x1"
	],
	"target": [
	"mult2",
	"-",
	"x0",
	"x1"
	],
	"why": "2x-2y = 2(x-y)",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"mult2",
	"x0",
	"mult3",
	"x0"
	],
	"target": [
	"neg",
	"x0"
	],
	"why": "2x0-3x0 -> -x0",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"mult2",
	"x0",
	"mult4",
	"x0"
	],
	"target": [
	"mult2",
	"neg",
	"x0"
	],
	"why": "constant-ladder: 2x - 4x = -2x",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"-",
	"mult3",
	"x0",
	"mult2",
	"x0"
	],
	"target": [
	"x0"
	],
	"why": "constant-ladder: 3x - 2x = x",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"mult4",
	"x0",
	"mult2",
	"x0"
	],
	"target": [
	"mult2",
	"x0"
	],
	"why": "4x0-2x0 -> 2x0",
	"family": 3,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"mult4",
	"x0",
	"mult3",
	"x0"
	],
	"target": [
	"x0"
	],
	"why": "constant-ladder: 4x - 3x = x",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"mult5",
	"x0",
	"mult2",
	"x0"
	],
	"target": [
	"mult3",
	"x0"
	],
	"why": "constant-ladder: 5x - 2x = 3x",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"mult5",
	"x0",
	"mult3",
	"x0"
	],
	"target": [
	"mult2",
	"x0"
	],
	"why": "5x0-3x0 -> 2x0",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"pow2",
	"cosh",
	"x0",
	"1"
	],
	"target": [
	"pow2",
	"sinh",
	"x0"
	],
	"why": "cosh^2 - 1 = sinh^2",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"sinh",
	"x0",
	"exp",
	"x0"
	],
	"target": [
	"neg",
	"cosh",
	"x0"
	],
	"why": "sinh x - exp x = -cosh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"x0",
	"*",
	"<constant>",
	"x0"
	],
	"target": [
	"*",
	"<constant>",
	"x0"
	],
	"why": "x0-Cx0 -> (1-C)x0",
	"family": 3,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"abs",
	"x0",
	"abs",
	"x0"
	],
	"target": [
	"1"
	],
	"why": "\|x\|/\|x\| = 1 (domain extension at 0)",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"abs",
	"x0",
	"abs",
	"x1"
	],
	"target": [
	"abs",
	"/",
	"x0",
	"x1"
	],
	"why": "\|x\|/\|y\| -> \|x/y\|",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"cos",
	"x0",
	"sin",
	"x0"
	],
	"target": [
	"inv",
	"tan",
	"x0"
	],
	"why": "cos/sin = cot = 1/tan",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"cosh",
	"x0",
	"sinh",
	"x0"
	],
	"target": [
	"inv",
	"tanh",
	"x0"
	],
	"why": "cosh/sinh = coth = 1/tanh",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"div2",
	"x0",
	"div2",
	"x1"
	],
	"target": [
	"/",
	"x0",
	"x1"
	],
	"why": "(x0/2)/(x1/2) -> x0/x1",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"div2",
	"x0",
	"div4",
	"x0"
	],
	"target": [
	"<constant>"
	],
	"why": "(x/2)/(x/4) -> 2",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"div3",
	"x0",
	"div3",
	"x1"
	],
	"target": [
	"/",
	"x0",
	"x1"
	],
	"why": "(x/3)/(y/3) -> x/y",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"div4",
	"x0",
	"div4",
	"x1"
	],
	"target": [
	"/",
	"x0",
	"x1"
	],
	"why": "(x/4)/(y/4) -> x/y",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"div5",
	"x0",
	"div5",
	"x1"
	],
	"target": [
	"/",
	"x0",
	"x1"
	],
	"why": "(x/5)/(y/5) -> x/y",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"exp",
	"x0",
	"cosh",
	"x0"
	],
	"target": [
	"+",
	"1",
	"tanh",
	"x0"
	],
	"why": "e^x/cosh = 1 + tanh",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"exp",
	"x0",
	"exp",
	"x1"
	],
	"target": [
	"exp",
	"-",
	"x0",
	"x1"
	],
	"why": "e^a / e^b = e^(a-b)",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"mult2",
	"x0",
	"div2",
	"x0"
	],
	"target": [
	"<constant>"
	],
	"why": "2x/(x/2) -> 4",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"mult2",
	"x0",
	"div2",
	"x1"
	],
	"target": [
	"/",
	"mult4",
	"x0",
	"x1"
	],
	"why": "2x/(y/2) -> 4x/y",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"mult2",
	"x0",
	"mult2",
	"x0"
	],
	"target": [
	"1"
	],
	"why": "constant-ladder: 2x/(2x) = 1 (x/x domain extension explicitly allowed)",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"mult2",
	"x0",
	"mult2",
	"x1"
	],
	"target": [
	"/",
	"x0",
	"x1"
	],
	"why": "(2x0)/(2x1) -> x0/x1, constant scaling cancels in ratio",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"mult2",
	"x0",
	"mult4",
	"x0"
	],
	"target": [
	"<constant>"
	],
	"why": "2x/4x -> 1/2",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"mult2",
	"x0",
	"mult4",
	"x1"
	],
	"target": [
	"/",
	"div2",
	"x0",
	"x1"
	],
	"why": "(2x0)/(4x1) -> (x0/x1)/2",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"mult2",
	"x0",
	"pow2",
	"x0"
	],
	"target": [
	"inv",
	"div2",
	"x0"
	],
	"why": "constant-ladder: 2x/x^2 = 2/x",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"mult3",
	"x0",
	"mult3",
	"x1"
	],
	"target": [
	"/",
	"x0",
	"x1"
	],
	"why": "(3x0)/(3x1) -> x0/x1",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"mult4",
	"x0",
	"mult2",
	"x0"
	],
	"target": [
	"<constant>"
	],
	"why": "4x/2x -> 2",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"mult4",
	"x0",
	"mult2",
	"x1"
	],
	"target": [
	"/",
	"mult2",
	"x0",
	"x1"
	],
	"why": "(4x0)/(2x1) -> 2(x0/x1)",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"mult4",
	"x0",
	"mult4",
	"x1"
	],
	"target": [
	"/",
	"x0",
	"x1"
	],
	"why": "4x/4y -> x/y",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"mult5",
	"x0",
	"mult5",
	"x1"
	],
	"target": [
	"/",
	"x0",
	"x1"
	],
	"why": "5x/5y -> x/y",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"neg",
	"x0",
	"neg",
	"x1"
	],
	"target": [
	"/",
	"x0",
	"x1"
	],
	"why": "(-x0)/(-x1) -> x0/x1",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow",
	"x0",
	"<constant>",
	"x0"
	],
	"target": [
	"pow",
	"x0",
	"<constant>"
	],
	"why": "x^c / x = x^(c-1)",
	"family": 4,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow1_2",
	"x0",
	"pow1_2",
	"x1"
	],
	"target": [
	"pow1_2",
	"/",
	"x0",
	"x1"
	],
	"why": "sqrt(a)/sqrt(b) = sqrt(a/b) (dom ext)",
	"family": 4,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow1_3",
	"x0",
	"pow1_3",
	"x1"
	],
	"target": [
	"pow1_3",
	"/",
	"x0",
	"x1"
	],
	"why": "cbrt(x)/cbrt(y)=cbrt(x/y)",
	"family": 5,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow2",
	"mult2",
	"x0",
	"x0"
	],
	"target": [
	"mult4",
	"x0"
	],
	"why": "constant-ladder: (2x)^2/x = 4x",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow2",
	"x0",
	"pow1_2",
	"x0"
	],
	"target": [
	"pow1_2",
	"pow3",
	"x0"
	],
	"why": "x^2 / sqrt(x) = x^(3/2)",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"target": [
	"pow2",
	"/",
	"x0",
	"x1"
	],
	"why": "x^2/y^2 -> (x/y)^2",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow2",
	"x0",
	"pow3",
	"x0"
	],
	"target": [
	"inv",
	"x0"
	],
	"why": "x0^2/x0^3 -> 1/x0",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow2",
	"x0",
	"pow4",
	"x0"
	],
	"target": [
	"inv",
	"pow2",
	"x0"
	],
	"why": "x^2/x^4 -> 1/x^2",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow2",
	"x0",
	"pow5",
	"x0"
	],
	"target": [
	"inv",
	"pow3",
	"x0"
	],
	"why": "x^2/x^5 -> 1/x^3",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow3",
	"x0",
	"pow2",
	"x0"
	],
	"target": [
	"x0"
	],
	"why": "x0^3/x0^2 -> x0, exponent-difference cancellation",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow3",
	"x0",
	"pow3",
	"x1"
	],
	"target": [
	"pow3",
	"/",
	"x0",
	"x1"
	],
	"why": "x^3/y^3 -> (x/y)^3",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow3",
	"x0",
	"pow4",
	"x0"
	],
	"target": [
	"inv",
	"x0"
	],
	"why": "x^3/x^4 -> 1/x",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow3",
	"x0",
	"pow5",
	"x0"
	],
	"target": [
	"inv",
	"pow2",
	"x0"
	],
	"why": "x^3/x^5 = 1/x^2",
	"family": 4,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow4",
	"x0",
	"pow2",
	"x0"
	],
	"target": [
	"pow2",
	"x0"
	],
	"why": "x0^4/x0^2 -> x0^2",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow4",
	"x0",
	"pow3",
	"x0"
	],
	"target": [
	"x0"
	],
	"why": "x^4/x^3 = x",
	"family": 4,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow4",
	"x0",
	"pow4",
	"x1"
	],
	"target": [
	"pow4",
	"/",
	"x0",
	"x1"
	],
	"why": "x^4/y^4 -> (x/y)^4",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow4",
	"x0",
	"pow5",
	"x0"
	],
	"target": [
	"inv",
	"x0"
	],
	"why": "x^4/x^5 -> 1/x",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow5",
	"x0",
	"pow2",
	"x0"
	],
	"target": [
	"pow3",
	"x0"
	],
	"why": "x0^5/x0^2 -> x0^3",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow5",
	"x0",
	"pow3",
	"x0"
	],
	"target": [
	"pow2",
	"x0"
	],
	"why": "x0^5/x0^3 -> x0^2",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow5",
	"x0",
	"pow4",
	"x0"
	],
	"target": [
	"x0"
	],
	"why": "x^5/x^4 -> x",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow5",
	"x0",
	"pow5",
	"x1"
	],
	"target": [
	"pow5",
	"/",
	"x0",
	"x1"
	],
	"why": "x^5/y^5 -> (x/y)^5",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"sin",
	"x0",
	"cos",
	"x0"
	],
	"target": [
	"tan",
	"x0"
	],
	"why": "sin/cos = tan",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"sin",
	"x0",
	"tan",
	"x0"
	],
	"target": [
	"cos",
	"x0"
	],
	"why": "sin/tan = cos",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"sinh",
	"x0",
	"cosh",
	"x0"
	],
	"target": [
	"tanh",
	"x0"
	],
	"why": "sinh/cosh = tanh",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"sinh",
	"x0",
	"tanh",
	"x0"
	],
	"target": [
	"cosh",
	"x0"
	],
	"why": "sinh/tanh = cosh",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"tan",
	"x0",
	"sin",
	"x0"
	],
	"target": [
	"inv",
	"cos",
	"x0"
	],
	"why": "tan/sin = sec = 1/cos",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"tanh",
	"x0",
	"sinh",
	"x0"
	],
	"target": [
	"inv",
	"cosh",
	"x0"
	],
	"why": "tanh/sinh = sech = 1/cosh",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"x0",
	"+",
	"x1",
	"float(\"inf\")"
	],
	"target": [
	"0"
	],
	"why": "finite / inf = 0 (signed zero compares equal)",
	"family": 5,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"x0",
	"pow",
	"x0",
	"<constant>"
	],
	"target": [
	"pow",
	"x0",
	"<constant>"
	],
	"why": "x / x^c = x^(1-c)",
	"family": 4,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"abs",
	"*",
	"<constant>",
	"abs",
	"x0"
	],
	"target": [
	"*",
	"<constant>",
	"abs",
	"x0"
	],
	"why": "\|c\|x\|\| = \|c\|\|x\|; wildcard absorbs \|c\|",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"abs",
	"*",
	"abs",
	"x0",
	"x1"
	],
	"target": [
	"abs",
	"*",
	"x0",
	"x1"
	],
	"why": "inner abs redundant under outer abs",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"abs",
	"*",
	"neg",
	"x0",
	"x1"
	],
	"target": [
	"abs",
	"*",
	"x0",
	"x1"
	],
	"why": "\|(-x0) x1\| = \|x0 x1\|",
	"family": 4,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"abs",
	"*",
	"x0",
	"abs",
	"x0"
	],
	"target": [
	"pow2",
	"x0"
	],
	"why": "\|x \|x\|\| = x^2",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"abs",
	"*",
	"x0",
	"abs",
	"x1"
	],
	"target": [
	"abs",
	"*",
	"x0",
	"x1"
	],
	"why": "inner abs redundant, other operand",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"abs",
	"/",
	"abs",
	"x0",
	"x1"
	],
	"target": [
	"abs",
	"/",
	"x0",
	"x1"
	],
	"why": "numerator abs redundant under abs",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"abs",
	"/",
	"x0",
	"abs",
	"x1"
	],
	"target": [
	"abs",
	"/",
	"x0",
	"x1"
	],
	"why": "denominator abs redundant under abs",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"abs",
	"inv",
	"pow3",
	"neg",
	"x0"
	],
	"target": [
	"abs",
	"inv",
	"pow3",
	"x0"
	],
	"why": "\|1/(-x)^3\| = 1/\|x^3\|",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"abs",
	"pow",
	"abs",
	"x0",
	"x1"
	],
	"target": [
	"pow",
	"abs",
	"x0",
	"x1"
	],
	"why": "\|\|x\|^y\| = \|x\|^y (base>=0)",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div2",
	"*",
	"mult2",
	"x0",
	"x1"
	],
	"target": [
	"*",
	"x0",
	"x1"
	],
	"why": "((2x0)x1)/2 -> x0x1",
	"family": 3,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div2",
	"+",
	"1",
	"cos",
	"x0"
	],
	"target": [
	"pow2",
	"cos",
	"div2",
	"x0"
	],
	"why": "half-angle: (1+cos x)/2 = cos^2(x/2)",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div2",
	"+",
	"cosh",
	"x0",
	"1"
	],
	"target": [
	"pow2",
	"cosh",
	"div2",
	"x0"
	],
	"why": "(cosh x+1)/2 = cosh^2(x/2), half argument",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div2",
	"-",
	"1",
	"cos",
	"x0"
	],
	"target": [
	"pow2",
	"sin",
	"div2",
	"x0"
	],
	"why": "half-angle: (1-cos x)/2 = sin^2(x/2)",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div2",
	"-",
	"cosh",
	"x0",
	"1"
	],
	"target": [
	"pow2",
	"sinh",
	"div2",
	"x0"
	],
	"why": "(cosh x-1)/2 = sinh^2(x/2), half argument",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div2",
	"/",
	"mult2",
	"x0",
	"x1"
	],
	"target": [
	"/",
	"x0",
	"x1"
	],
	"why": "((2x0)/x1)/2 -> x0/x1",
	"family": 3,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"exp",
	"*",
	"<constant>",
	"log",
	"x0"
	],
	"target": [
	"pow",
	"x0",
	"<constant>"
	],
	"why": "e^(C log x) = x^C",
	"family": 2,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"exp",
	"*",
	"x1",
	"log",
	"x0"
	],
	"target": [
	"pow",
	"x0",
	"x1"
	],
	"why": "e^(b log a) = a^b",
	"family": 2,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"exp",
	"+",
	"1",
	"log",
	"x0"
	],
	"target": [
	"*",
	"np.e",
	"x0"
	],
	"why": "e^(1 + log x) = e x",
	"family": 2,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"exp",
	"+",
	"log",
	"x0",
	"<constant>"
	],
	"target": [
	"*",
	"<constant>",
	"x0"
	],
	"why": "e^(log x + C) = e^C x = C' x",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"exp",
	"+",
	"log",
	"x0",
	"x1"
	],
	"target": [
	"*",
	"exp",
	"x1",
	"x0"
	],
	"why": "e^(log a + b) = a e^b",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"exp",
	"+",
	"x0",
	"log",
	"x1"
	],
	"target": [
	"*",
	"exp",
	"x0",
	"x1"
	],
	"why": "e^(a + log b) = b e^a",
	"family": 2,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"exp",
	"-",
	"<constant>",
	"log",
	"x0"
	],
	"target": [
	"/",
	"<constant>",
	"x0"
	],
	"why": "e^(C - log x) = e^C / x = C'/x",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"exp",
	"-",
	"log",
	"x0",
	"1"
	],
	"target": [
	"/",
	"x0",
	"np.e"
	],
	"why": "e^(log x - 1) = x/e",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"exp",
	"-",
	"log",
	"x0",
	"x1"
	],
	"target": [
	"/",
	"x0",
	"exp",
	"x1"
	],
	"why": "e^(log a - b) = a / e^b",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"exp",
	"-",
	"x0",
	"log",
	"x1"
	],
	"target": [
	"/",
	"exp",
	"x0",
	"x1"
	],
	"why": "e^(a - log b) = e^a / b",
	"family": 2,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"exp",
	"/",
	"log",
	"x0",
	"<constant>"
	],
	"target": [
	"pow",
	"x0",
	"inv",
	"<constant>"
	],
	"why": "e^(log x / C) = x^(1/C) = x^C'",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"exp",
	"/",
	"log",
	"x0",
	"x1"
	],
	"target": [
	"pow",
	"x0",
	"inv",
	"x1"
	],
	"why": "e^(log a / b) = a^(1/b)",
	"family": 2,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"inv",
	"div2",
	"pow2",
	"inv",
	"x0"
	],
	"target": [
	"mult2",
	"pow2",
	"x0"
	],
	"why": "constant-ladder: 1/((1/x)^2/2) = 2x^2",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"inv",
	"pow",
	"inv",
	"x0",
	"x1"
	],
	"target": [
	"pow",
	"x0",
	"x1"
	],
	"why": "1/((1/x)^y) = x^y",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"inv",
	"pow",
	"x0",
	"neg",
	"x1"
	],
	"target": [
	"pow",
	"x0",
	"x1"
	],
	"why": "1/(x^(-y)) = x^y",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"log",
	"*",
	"exp",
	"x0",
	"x1"
	],
	"target": [
	"+",
	"log",
	"x1",
	"x0"
	],
	"why": "log(e^a b) = a + log b",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"log",
	"*",
	"x0",
	"exp",
	"x1"
	],
	"target": [
	"+",
	"log",
	"x0",
	"x1"
	],
	"why": "log(a e^b) = b + log a",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"log",
	"/",
	"exp",
	"x0",
	"x1"
	],
	"target": [
	"-",
	"x0",
	"log",
	"x1"
	],
	"why": "log(e^a / b) = a - log b",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"log",
	"/",
	"np.e",
	"exp",
	"x0"
	],
	"target": [
	"-",
	"1",
	"x0"
	],
	"why": "log(e / e^x) = 1 - x",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"log",
	"/",
	"x0",
	"exp",
	"x1"
	],
	"target": [
	"-",
	"log",
	"x0",
	"x1"
	],
	"why": "log(a / e^b) = log a - b",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"log",
	"pow",
	"<constant>",
	"mult2",
	"x0"
	],
	"target": [
	"log",
	"pow",
	"<constant>",
	"x0"
	],
	"why": "log(C^(2x)) = (2 ln C) x",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"log",
	"pow",
	"<constant>",
	"neg",
	"x0"
	],
	"target": [
	"log",
	"pow",
	"<constant>",
	"x0"
	],
	"why": "log(C^(-x)) = -x log C = C' x",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"mult2",
	"*",
	"div2",
	"x0",
	"x1"
	],
	"target": [
	"*",
	"x0",
	"x1"
	],
	"why": "2((x0/2)x1) -> x0*x1",
	"family": 3,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"mult2",
	"/",
	"x0",
	"mult2",
	"x1"
	],
	"target": [
	"/",
	"x0",
	"x1"
	],
	"why": "2*(x0/(2x1)) -> x0/x1, multK pulls through denominator",
	"family": 3,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"mult2",
	"pow2",
	"cosh",
	"div2",
	"x0"
	],
	"target": [
	"+",
	"1",
	"cosh",
	"x0"
	],
	"why": "half-angle: 2 cosh^2(x/2) = cosh x + 1",
	"family": 1,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"mult2",
	"pow2",
	"sinh",
	"div2",
	"x0"
	],
	"target": [
	"+",
	"(-1)",
	"cosh",
	"x0"
	],
	"why": "half-angle: 2 sinh^2(x/2) = cosh x - 1",
	"family": 1,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"neg",
	"*",
	"neg",
	"x0",
	"x1"
	],
	"target": [
	"*",
	"x0",
	"x1"
	],
	"why": "-((-x0)x1) -> x0x1",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"neg",
	"*",
	"x0",
	"neg",
	"x1"
	],
	"target": [
	"*",
	"x0",
	"x1"
	],
	"why": "-(x(-y)) = xy",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"neg",
	"/",
	"neg",
	"x0",
	"x1"
	],
	"target": [
	"/",
	"x0",
	"x1"
	],
	"why": "-((-x)/y) = x/y",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"neg",
	"/",
	"x0",
	"neg",
	"x1"
	],
	"target": [
	"/",
	"x0",
	"x1"
	],
	"why": "-(x0/(-x1)) -> x0/x1",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"neg",
	"sin",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"sin",
	"-",
	"x0",
	"x1"
	],
	"why": "antisymmetry: -sin(x1-x0) = sin(x0-x1)",
	"family": 0,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow",
	"<constant>",
	"mult2",
	"log",
	"x0"
	],
	"target": [
	"pow",
	"x0",
	"sinh",
	"<constant>"
	],
	"why": "C^(2 log x) = x^(2 log C) = x^C'",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"pow",
	"abs",
	"x0",
	"mult2",
	"x1"
	],
	"target": [
	"pow",
	"pow2",
	"x0",
	"x1"
	],
	"why": "\|x\|^(2y) = (x^2)^y",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow",
	"exp",
	"x0",
	"log",
	"x1"
	],
	"target": [
	"pow",
	"x1",
	"x0"
	],
	"why": "(e^a)^(log b) = b^a",
	"family": 2,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow",
	"inv",
	"x0",
	"neg",
	"x1"
	],
	"target": [
	"pow",
	"x0",
	"x1"
	],
	"why": "(1/x)^(-y) = x^y",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow",
	"pow",
	"x0",
	"<constant>",
	"<constant>"
	],
	"target": [
	"pow",
	"x0",
	"<constant>"
	],
	"why": "(x^a)^b = x^(ab)",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow",
	"pow1_2",
	"x0",
	"mult2",
	"x1"
	],
	"target": [
	"pow",
	"x0",
	"x1"
	],
	"why": "(sqrt x)^(2y) = x^y (dom ext)",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow",
	"pow1_3",
	"x0",
	"mult3",
	"x1"
	],
	"target": [
	"pow",
	"x0",
	"x1"
	],
	"why": "(x^(1/3))^(3y) = x^y (dom ext)",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow",
	"pow1_4",
	"x0",
	"mult4",
	"x1"
	],
	"target": [
	"pow",
	"x0",
	"x1"
	],
	"why": "(x^(1/4))^(4y) = x^y (dom ext)",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow",
	"pow1_5",
	"x0",
	"mult5",
	"x1"
	],
	"target": [
	"pow",
	"x0",
	"x1"
	],
	"why": "(x^(1/5))^(5y) = x^y (dom ext)",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow",
	"pow2",
	"x0",
	"div2",
	"x1"
	],
	"target": [
	"pow",
	"abs",
	"x0",
	"x1"
	],
	"why": "(x^2)^(y/2) = \|x\|^y",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow",
	"pow2",
	"x0",
	"mult2",
	"x1"
	],
	"target": [
	"pow",
	"pow4",
	"x0",
	"x1"
	],
	"why": "(x^2)^(2y) = (x^4)^y",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow",
	"pow3",
	"x0",
	"div3",
	"x1"
	],
	"target": [
	"pow",
	"x0",
	"x1"
	],
	"why": "(x^3)^(y/3) = x^y (equal or nan)",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow",
	"pow4",
	"x0",
	"div4",
	"x1"
	],
	"target": [
	"pow",
	"abs",
	"x0",
	"x1"
	],
	"why": "(x^4)^(y/4) = \|x\|^y",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow",
	"pow5",
	"x0",
	"div5",
	"x1"
	],
	"target": [
	"pow",
	"x0",
	"x1"
	],
	"why": "(x^5)^(y/5) = x^y (equal or nan)",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"*",
	"<constant>",
	"pow2",
	"x0"
	],
	"target": [
	"abs",
	"*",
	"<constant>",
	"x0"
	],
	"why": "sqrt(c x^2)=sqrt(c)\|x\| for fitted c>=0",
	"family": 5,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"pow1_2",
	"+",
	"1",
	"pow2",
	"x0"
	],
	"target": [
	"cosh",
	"asinh",
	"x0"
	],
	"why": "sqrt(1+x^2) = cosh(asinh x)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"+",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"cosh",
	"asinh",
	"x0"
	],
	"why": "sqrt(x^2+1) = cosh(asinh x) (operand order)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"-",
	"1",
	"pow2",
	"x0"
	],
	"target": [
	"cos",
	"asin",
	"x0"
	],
	"why": "sqrt(1-x^2)=cos(asin x)=sin(acos x)",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"-",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"sinh",
	"acosh",
	"abs",
	"x0"
	],
	"why": "sqrt(x^2-1) = sinh(acosh\|x\|)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow2",
	"*",
	"<constant>",
	"pow1_2",
	"x0"
	],
	"target": [
	"*",
	"<constant>",
	"x0"
	],
	"why": "(c sqrt(x))^2=c^2 x; domain extension x<0",
	"family": 5,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow2",
	"-",
	"neg",
	"x0",
	"x1"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "(-x-y)^2 = (x+y)^2",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow2",
	"pow",
	"x0",
	"div2",
	"x1"
	],
	"target": [
	"pow",
	"x0",
	"x1"
	],
	"why": "(x^(y/2))^2 = x^y (checked neg-base cases equal or nan)",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"abs",
	"inv",
	"x0",
	"abs",
	"x1"
	],
	"target": [
	"abs",
	"/",
	"x1",
	"x0"
	],
	"why": "(1/\|x\|)\|y\| = \|y/x\|",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"exp",
	"mult2",
	"x0",
	"exp",
	"x0"
	],
	"target": [
	"exp",
	"mult3",
	"x0"
	],
	"why": "e^(2x) e^x = e^(3x)",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"exp",
	"x0",
	"exp",
	"neg",
	"x0"
	],
	"target": [
	"1"
	],
	"why": "e^x * e^-x = 1",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"exp",
	"x0",
	"exp",
	"neg",
	"x1"
	],
	"target": [
	"exp",
	"-",
	"x0",
	"x1"
	],
	"why": "e^a e^(-b) = e^(a-b)",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"mult2",
	"sin",
	"x0",
	"cos",
	"x0"
	],
	"target": [
	"sin",
	"mult2",
	"x0"
	],
	"why": "(2sin)*cos nesting variant",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow1_2",
	"x0",
	"pow1_2",
	"inv",
	"x0"
	],
	"target": [
	"1"
	],
	"why": "sqrt(x) sqrt(1/x) = 1 (x>0)",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow2",
	"cosh",
	"x0",
	"tanh",
	"x0"
	],
	"target": [
	"div2",
	"sinh",
	"mult2",
	"x0"
	],
	"why": "cosh^2 tanh = sinh cosh = sinh(2x)/2",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow2",
	"x0",
	"pow",
	"x0",
	"<constant>"
	],
	"target": [
	"pow",
	"x0",
	"<constant>"
	],
	"why": "x^2 * x^c = x^(c+2)",
	"family": 4,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow3",
	"x0",
	"pow3",
	"pow2",
	"x0"
	],
	"target": [
	"pow3",
	"pow3",
	"x0"
	],
	"why": "x^3*(x^2)^3=x^9",
	"family": 5,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"sin",
	"mult2",
	"x0",
	"tan",
	"x0"
	],
	"target": [
	"mult2",
	"pow2",
	"sin",
	"x0"
	],
	"why": "sin2x*tan=2sin^2",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"sin",
	"x0",
	"sin",
	"neg",
	"x0"
	],
	"target": [
	"neg",
	"pow2",
	"sin",
	"x0"
	],
	"why": "sin x * sin(-x) = -sin^2 x",
	"family": 0,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"sinh",
	"x0",
	"sinh",
	"neg",
	"x0"
	],
	"target": [
	"neg",
	"pow2",
	"sinh",
	"x0"
	],
	"why": "sinh odd: sinh(x) sinh(-x) = -sinh^2",
	"family": 1,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"tanh",
	"neg",
	"x0",
	"cosh",
	"x0"
	],
	"target": [
	"neg",
	"sinh",
	"x0"
	],
	"why": "tanh(-x) cosh(x) = -sinh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"tanh",
	"x0",
	"tanh",
	"neg",
	"x0"
	],
	"target": [
	"neg",
	"pow2",
	"tanh",
	"x0"
	],
	"why": "tanh odd: tanh(x) tanh(-x) = -tanh^2",
	"family": 1,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"1",
	"inv",
	"pow2",
	"sinh",
	"x0"
	],
	"target": [
	"inv",
	"pow2",
	"tanh",
	"x0"
	],
	"why": "1 + 1/sinh^2 = coth^2 = 1/tanh^2",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"1",
	"inv",
	"pow2",
	"tan",
	"x0"
	],
	"target": [
	"inv",
	"pow2",
	"sin",
	"x0"
	],
	"why": "1+cot^2=csc^2",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"1",
	"mult2",
	"pow2",
	"sinh",
	"x0"
	],
	"target": [
	"cosh",
	"mult2",
	"x0"
	],
	"why": "1 + 2sinh^2 = cosh 2x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"acos",
	"neg",
	"x0",
	"acos",
	"x0"
	],
	"target": [
	"np.pi"
	],
	"why": "acos(-x)+acos(x)=pi",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"asin",
	"neg",
	"x0",
	"asin",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "asin odd",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"asinh",
	"x0",
	"asinh",
	"neg",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "asinh x + asinh(-x) = 0, odd parity",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"atan",
	"x0",
	"atan",
	"inv",
	"x0"
	],
	"target": [
	"atan",
	"/",
	"float(\"inf\")",
	"x0"
	],
	"why": "BORDERLINE: atan(x)+atan(1/x)=sign(x)*pi/2; only pi/2 for x>0; minimal global form needs sign",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"+",
	"atan",
	"x0",
	"atan",
	"neg",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "atan odd: atan(x)+atan(-x)=0",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"atanh",
	"x0",
	"atanh",
	"neg",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "atanh x + atanh(-x) = 0, odd parity",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"exp",
	"neg",
	"x0",
	"exp",
	"x0"
	],
	"target": [
	"mult2",
	"cosh",
	"x0"
	],
	"why": "exp(-x) + exp x = 2 cosh x (operand order)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"exp",
	"neg",
	"x0",
	"sinh",
	"x0"
	],
	"target": [
	"cosh",
	"x0"
	],
	"why": "exp(-x) + sinh x = cosh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"exp",
	"x0",
	"exp",
	"neg",
	"x0"
	],
	"target": [
	"mult2",
	"cosh",
	"x0"
	],
	"why": "e^x + e^-x = 2 cosh",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"inv",
	"tanh",
	"x0",
	"tanh",
	"x0"
	],
	"target": [
	"inv",
	"div2",
	"tanh",
	"mult2",
	"x0"
	],
	"why": "coth + tanh = 2 coth(2x); borderline 5-token target",
	"family": 2,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"+",
	"log",
	"x0",
	"log",
	"inv",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "log x + log(1/x) = 0",
	"family": 2,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"log",
	"x0",
	"log",
	"pow2",
	"x0"
	],
	"target": [
	"log",
	"pow3",
	"x0"
	],
	"why": "log x + log x^2 = 3 log x",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"+",
	"mult2",
	"pow2",
	"sinh",
	"x0",
	"1"
	],
	"target": [
	"cosh",
	"mult2",
	"x0"
	],
	"why": "2sinh^2 + 1 = cosh 2x",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"sin",
	"neg",
	"x0",
	"sin",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "sin odd: sin(-x)+sin(x)=0",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"sin",
	"x0",
	"sin",
	"neg",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "sin x + sin(-x) = 0",
	"family": 0,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"sinh",
	"x0",
	"exp",
	"neg",
	"x0"
	],
	"target": [
	"cosh",
	"x0"
	],
	"why": "sinh + e^-x = cosh",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"sinh",
	"x0",
	"sinh",
	"neg",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "sinh odd: sinh(x)+sinh(-x) = 0",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"tanh",
	"x0",
	"tanh",
	"neg",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "tanh odd: tanh(x)+tanh(-x) = 0",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"1",
	"inv",
	"pow2",
	"cosh",
	"x0"
	],
	"target": [
	"pow2",
	"tanh",
	"x0"
	],
	"why": "1 - 1/cosh^2 = tanh^2",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"1",
	"mult2",
	"pow2",
	"sin",
	"x0"
	],
	"target": [
	"cos",
	"mult2",
	"x0"
	],
	"why": "1 - 2 sin^2 = cos(2x)",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"cosh",
	"x0",
	"exp",
	"neg",
	"x0"
	],
	"target": [
	"sinh",
	"x0"
	],
	"why": "cosh - e^-x = sinh",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"exp",
	"neg",
	"x0",
	"cosh",
	"x0"
	],
	"target": [
	"neg",
	"sinh",
	"x0"
	],
	"why": "exp(-x) - cosh x = -sinh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"exp",
	"neg",
	"x0",
	"exp",
	"x0"
	],
	"target": [
	"mult2",
	"neg",
	"sinh",
	"x0"
	],
	"why": "exp(-x) - exp x = -2 sinh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"-",
	"exp",
	"x0",
	"exp",
	"neg",
	"x0"
	],
	"target": [
	"mult2",
	"sinh",
	"x0"
	],
	"why": "e^x - e^-x = 2 sinh",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"inv",
	"pow2",
	"cos",
	"x0",
	"1"
	],
	"target": [
	"pow2",
	"tan",
	"x0"
	],
	"why": "sec^2 - 1 = tan^2",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"inv",
	"pow2",
	"sin",
	"x0",
	"1"
	],
	"target": [
	"inv",
	"pow2",
	"tan",
	"x0"
	],
	"why": "csc^2 - 1 = cot^2 = 1/tan^2",
	"family": 0,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"inv",
	"pow2",
	"tanh",
	"x0",
	"1"
	],
	"target": [
	"inv",
	"pow2",
	"sinh",
	"x0"
	],
	"why": "coth^2 - 1 = csch^2 = 1/sinh^2",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"inv",
	"tanh",
	"x0",
	"tanh",
	"x0"
	],
	"target": [
	"inv",
	"div2",
	"sinh",
	"mult2",
	"x0"
	],
	"why": "coth - tanh = 2 csch(2x); borderline 5-token target",
	"family": 2,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"-",
	"log",
	"pow2",
	"x0",
	"log",
	"x0"
	],
	"target": [
	"log",
	"x0"
	],
	"why": "log x^2 - log x = log x",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"log",
	"pow3",
	"x0",
	"log",
	"x0"
	],
	"target": [
	"log",
	"pow2",
	"x0"
	],
	"why": "log x^3 - log x = 2 log x",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"-",
	"mult2",
	"pow2",
	"cos",
	"x0",
	"1"
	],
	"target": [
	"cos",
	"mult2",
	"x0"
	],
	"why": "2cos^2-1=cos2x",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"mult2",
	"pow2",
	"cosh",
	"x0",
	"1"
	],
	"target": [
	"cosh",
	"mult2",
	"x0"
	],
	"why": "2cosh^2 - 1 = cosh 2x",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"pow2",
	"inv",
	"tanh",
	"x0",
	"1"
	],
	"target": [
	"inv",
	"pow2",
	"sinh",
	"x0"
	],
	"why": "coth^2 - 1 = csch^2",
	"family": 1,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"-",
	"sin",
	"x0",
	"sin",
	"neg",
	"x0"
	],
	"target": [
	"mult2",
	"sin",
	"x0"
	],
	"why": "sin x - sin(-x) = 2 sin x",
	"family": 0,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"*",
	"<constant>",
	"pow2",
	"x0",
	"x0"
	],
	"target": [
	"*",
	"<constant>",
	"x0"
	],
	"why": "Cx^2/x -> Cx",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"pow2",
	"x0",
	"x0",
	"x0"
	],
	"target": [
	"+",
	"1",
	"x0"
	],
	"why": "(x^2+x)/x -> x+1",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"+",
	"pow3",
	"x0",
	"x0",
	"x0"
	],
	"target": [
	"+",
	"1",
	"pow2",
	"x0"
	],
	"why": "(x^3+x)/x = x^2+1",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"-",
	"pow2",
	"x0",
	"x0",
	"x0"
	],
	"target": [
	"+",
	"(-1)",
	"x0"
	],
	"why": "(x^2-x)/x -> x-1",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"-",
	"pow3",
	"x0",
	"x0",
	"x0"
	],
	"target": [
	"+",
	"(-1)",
	"pow2",
	"x0"
	],
	"why": "(x^3-x)/x = x^2-1",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"exp",
	"mult2",
	"x0",
	"exp",
	"x0"
	],
	"target": [
	"exp",
	"x0"
	],
	"why": "e^(2x)/e^x = e^x",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"exp",
	"mult3",
	"x0",
	"exp",
	"x0"
	],
	"target": [
	"exp",
	"mult2",
	"x0"
	],
	"why": "e^(3x)/e^x = e^(2x)",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"exp",
	"neg",
	"x0",
	"cosh",
	"x0"
	],
	"target": [
	"-",
	"1",
	"tanh",
	"x0"
	],
	"why": "exp(-x)/cosh(x) = 1 - tanh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"exp",
	"neg",
	"x0",
	"exp",
	"x0"
	],
	"target": [
	"pow",
	"<constant>",
	"x0"
	],
	"why": "e^(-x)/e^x = e^(-2x)",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"exp",
	"x0",
	"exp",
	"neg",
	"x0"
	],
	"target": [
	"exp",
	"mult2",
	"x0"
	],
	"why": "e^x / e^(-x) = e^(2x)",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"exp",
	"x0",
	"exp",
	"neg",
	"x1"
	],
	"target": [
	"exp",
	"+",
	"x0",
	"x1"
	],
	"why": "e^a / e^(-b) = e^(a+b)",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"log",
	"pow",
	"x0",
	"x1",
	"x1"
	],
	"target": [
	"log",
	"abs",
	"x0"
	],
	"why": "log(x^y)/y = log x",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"log",
	"pow2",
	"x0",
	"log",
	"x0"
	],
	"target": [
	"<constant>"
	],
	"why": "log(x^2)/log(x) = 2",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"sin",
	"mult2",
	"x0",
	"cos",
	"x0"
	],
	"target": [
	"mult2",
	"sin",
	"x0"
	],
	"why": "sin(2x)/cos(x) = 2 sin x",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"sin",
	"mult2",
	"x0",
	"sin",
	"x0"
	],
	"target": [
	"mult2",
	"cos",
	"x0"
	],
	"why": "sin(2x)/sin(x) = 2 cos x (0/0 points are domain-extension)",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"sin",
	"mult2",
	"x0",
	"tan",
	"x0"
	],
	"target": [
	"mult2",
	"pow2",
	"cos",
	"x0"
	],
	"why": "sin(2x)/tan(x) = 2 cos^2 x",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"sinh",
	"mult2",
	"x0",
	"cosh",
	"x0"
	],
	"target": [
	"mult2",
	"sinh",
	"x0"
	],
	"why": "sinh 2x / cosh = 2 sinh",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"sinh",
	"mult2",
	"x0",
	"sinh",
	"x0"
	],
	"target": [
	"mult2",
	"cosh",
	"x0"
	],
	"why": "sinh(2x)/sinh(x) = 2 cosh x",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"sinh",
	"neg",
	"x0",
	"cosh",
	"x0"
	],
	"target": [
	"neg",
	"tanh",
	"x0"
	],
	"why": "sinh(-x)/cosh = -tanh",
	"family": 1,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"sinh",
	"neg",
	"x0",
	"sinh",
	"x0"
	],
	"target": [
	"(-1)"
	],
	"why": "sinh odd: sinh(-x)/sinh(x) = -1",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"tanh",
	"neg",
	"x0",
	"tanh",
	"x0"
	],
	"target": [
	"(-1)"
	],
	"why": "tanh odd: tanh(-x)/tanh(x) = -1",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"abs",
	"*",
	"abs",
	"x0",
	"abs",
	"x1"
	],
	"target": [
	"abs",
	"*",
	"x0",
	"x1"
	],
	"why": "both inner abs redundant",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"abs",
	"*",
	"neg",
	"x0",
	"neg",
	"x1"
	],
	"target": [
	"abs",
	"*",
	"x0",
	"x1"
	],
	"why": "\|(-x)(-y)\| = \|xy\|",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"abs",
	"*",
	"pow2",
	"x0",
	"abs",
	"x1"
	],
	"target": [
	"*",
	"abs",
	"x1",
	"pow2",
	"x0"
	],
	"why": "\|x^2 \|y\|\| = x^2 \|y\| (borderline, invited)",
	"family": 6,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"abs",
	"*",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"target": [
	"pow2",
	"*",
	"x0",
	"x1"
	],
	"why": "\|x^2 y^2\| = (xy)^2",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"abs",
	"/",
	"neg",
	"x0",
	"neg",
	"x1"
	],
	"target": [
	"abs",
	"/",
	"x0",
	"x1"
	],
	"why": "\|(-x)/(-y)\| = \|x/y\|",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"asinh",
	"pow1_2",
	"-",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"acosh",
	"abs",
	"x0"
	],
	"why": "asinh(sqrt(x^2-1)) = acosh\|x\| (abs needed for x<=-1 branch)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div2",
	"+",
	"1",
	"cos",
	"mult2",
	"x0"
	],
	"target": [
	"pow2",
	"cos",
	"x0"
	],
	"why": "(1+cos2x)/2=cos^2",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div2",
	"+",
	"cosh",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"pow2",
	"cosh",
	"x0"
	],
	"why": "(cosh 2x + 1)/2 = cosh^2",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div2",
	"+",
	"mult2",
	"x0",
	"mult2",
	"x1"
	],
	"target": [
	"+",
	"x0",
	"x1"
	],
	"why": "(2x+2y)/2 -> x+y",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div2",
	"-",
	"1",
	"cos",
	"mult2",
	"x0"
	],
	"target": [
	"pow2",
	"sin",
	"x0"
	],
	"why": "(1-cos 2x)/2 = sin^2 x (power-reduction)",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div2",
	"-",
	"cosh",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"pow2",
	"sinh",
	"x0"
	],
	"why": "(cosh 2x - 1)/2 = sinh^2",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div5",
	"+",
	"mult2",
	"x0",
	"mult3",
	"x0"
	],
	"target": [
	"x0"
	],
	"why": "constant-ladder: (2x+3x)/5 = x",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"exp",
	"+",
	"log",
	"x0",
	"log",
	"x1"
	],
	"target": [
	"*",
	"x0",
	"x1"
	],
	"why": "e^(log a + log b) = ab",
	"family": 2,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"exp",
	"-",
	"log",
	"x0",
	"log",
	"x1"
	],
	"target": [
	"/",
	"x0",
	"x1"
	],
	"why": "e^(log a - log b) = a/b",
	"family": 2,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"exp",
	"-",
	"x0",
	"+",
	"x1",
	"float(\"inf\")"
	],
	"target": [
	"0"
	],
	"why": "x - inf = -inf, exp(-inf) = 0 everywhere",
	"family": 5,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"exp",
	"neg",
	"*",
	"x1",
	"log",
	"x0"
	],
	"target": [
	"inv",
	"pow",
	"x0",
	"x1"
	],
	"why": "e^(-b log a) = a^(-b)",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"inv",
	"+",
	"1",
	"pow2",
	"tan",
	"x0"
	],
	"target": [
	"pow2",
	"cos",
	"x0"
	],
	"why": "1/(1+tan^2)=cos^2",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"inv",
	"pow1_2",
	"+",
	"1",
	"pow2",
	"x0"
	],
	"target": [
	"cos",
	"atan",
	"x0"
	],
	"why": "1/sqrt(1+x^2)=cos(atan x)",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"inv",
	"pow1_2",
	"-",
	"1",
	"pow2",
	"x0"
	],
	"target": [
	"cosh",
	"atanh",
	"x0"
	],
	"why": "1/sqrt(1-x^2) = cosh(atanh x)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"log",
	"*",
	"exp",
	"x0",
	"exp",
	"x1"
	],
	"target": [
	"+",
	"x0",
	"x1"
	],
	"why": "log(e^x * e^y) = x+y",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"log",
	"*",
	"pow2",
	"x0",
	"pow3",
	"x0"
	],
	"target": [
	"log",
	"pow5",
	"x0"
	],
	"why": "log(x^2*x^3) = 5 log x",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"log",
	"/",
	"exp",
	"x0",
	"exp",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "log(e^x/e^x) = 0",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"log",
	"pow",
	"exp",
	"x0",
	"exp",
	"x1"
	],
	"target": [
	"*",
	"exp",
	"x1",
	"x0"
	],
	"why": "log((e^a)^(e^b)) = a e^b",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"mult2",
	"*",
	"cos",
	"x0",
	"sin",
	"x0"
	],
	"target": [
	"sin",
	"mult2",
	"x0"
	],
	"why": "2 cos sin = sin(2x), operand order swapped from the known example",
	"family": 0,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"mult2",
	"*",
	"cosh",
	"x0",
	"sinh",
	"x0"
	],
	"target": [
	"sinh",
	"mult2",
	"x0"
	],
	"why": "2 cosh sinh = sinh 2x (operand order)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"mult2",
	"*",
	"sin",
	"x0",
	"cos",
	"x0"
	],
	"target": [
	"sin",
	"mult2",
	"x0"
	],
	"why": "2 sin cos = sin(2x)",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"mult2",
	"*",
	"sinh",
	"x0",
	"cosh",
	"x0"
	],
	"target": [
	"sinh",
	"mult2",
	"x0"
	],
	"why": "2 sinh cosh = sinh(2x)",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"mult4",
	"*",
	"sin",
	"x0",
	"cos",
	"x0"
	],
	"target": [
	"mult2",
	"sin",
	"mult2",
	"x0"
	],
	"why": "4 sin cos = 2 sin(2x)",
	"family": 0,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"neg",
	"sin",
	"*",
	"neg",
	"x0",
	"x1"
	],
	"target": [
	"sin",
	"*",
	"x0",
	"x1"
	],
	"why": "-sin(-xy) = sin(xy)",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"neg",
	"sin",
	"/",
	"neg",
	"x0",
	"x1"
	],
	"target": [
	"sin",
	"/",
	"x0",
	"x1"
	],
	"why": "-sin(-x/y) = sin(x/y)",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"neg",
	"sinh",
	"*",
	"neg",
	"x0",
	"x1"
	],
	"target": [
	"sinh",
	"*",
	"x0",
	"x1"
	],
	"why": "-sinh(-xy) = sinh(xy)",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"neg",
	"tan",
	"/",
	"neg",
	"x0",
	"x1"
	],
	"target": [
	"tan",
	"/",
	"x0",
	"x1"
	],
	"why": "-tan(-x/y) = tan(x/y)",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"neg",
	"tanh",
	"*",
	"neg",
	"x0",
	"x1"
	],
	"target": [
	"tanh",
	"*",
	"x0",
	"x1"
	],
	"why": "-tanh(-xy) = tanh(xy)",
	"family": 6,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"*",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"target": [
	"abs",
	"*",
	"x0",
	"x1"
	],
	"why": "sqrt(x0^2 x1^2) = \|x0 x1\|",
	"family": 4,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"+",
	"1",
	"pow2",
	"sinh",
	"x0"
	],
	"target": [
	"cosh",
	"x0"
	],
	"why": "sqrt(1+sinh^2) = cosh x (target length 2)",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"+",
	"1",
	"pow2",
	"tan",
	"x0"
	],
	"target": [
	"abs",
	"inv",
	"cos",
	"x0"
	],
	"why": "sqrt(1+tan^2)=1/\|cos\|",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"pow1_2",
	"+",
	"cosh",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"*",
	"<constant>",
	"cosh",
	"x0"
	],
	"why": "sqrt(cosh 2x + 1) = sqrt(2) cosh x, constant fitted to sqrt(2)",
	"family": 1,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"+",
	"pow2",
	"sinh",
	"x0",
	"1"
	],
	"target": [
	"cosh",
	"x0"
	],
	"why": "sqrt(sinh^2+1) = cosh x (operand order)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"-",
	"1",
	"pow2",
	"cos",
	"x0"
	],
	"target": [
	"abs",
	"sin",
	"x0"
	],
	"why": "sqrt(1-cos^2)=\|sin\|",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"-",
	"1",
	"pow2",
	"sin",
	"x0"
	],
	"target": [
	"abs",
	"cos",
	"x0"
	],
	"why": "sqrt(1-sin^2) = \|cos x\|",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"-",
	"1",
	"pow2",
	"tanh",
	"x0"
	],
	"target": [
	"cos",
	"asin",
	"tanh",
	"x0"
	],
	"why": "sqrt(1-tanh^2) = sech = 1/cosh",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"pow1_2",
	"/",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"target": [
	"abs",
	"/",
	"x0",
	"x1"
	],
	"why": "sqrt(a^2/b^2) = \|a/b\|",
	"family": 4,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"div2",
	"+",
	"cosh",
	"x0",
	"1"
	],
	"target": [
	"cosh",
	"div2",
	"x0"
	],
	"why": "sqrt((cosh x+1)/2) = cosh(x/2)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"div2",
	"-",
	"cosh",
	"x0",
	"1"
	],
	"target": [
	"abs",
	"sinh",
	"div2",
	"x0"
	],
	"why": "sqrt((cosh x-1)/2) = \|sinh(x/2)\|",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"inv",
	"-",
	"1",
	"pow2",
	"x0"
	],
	"target": [
	"cosh",
	"atanh",
	"x0"
	],
	"why": "sqrt(1/(1-x^2)) = cosh(atanh x) (inv inside sqrt variant)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_3",
	"*",
	"pow3",
	"x0",
	"pow3",
	"x1"
	],
	"target": [
	"*",
	"x0",
	"x1"
	],
	"why": "cbrt(x^3*y^3)=xy",
	"family": 5,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow2",
	"*",
	"pow1_2",
	"x0",
	"pow1_2",
	"x1"
	],
	"target": [
	"*",
	"x0",
	"x1"
	],
	"why": "(sqrt(x)sqrt(y))^2=xy; domain extension",
	"family": 5,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow2",
	"+",
	"cosh",
	"x0",
	"sinh",
	"x0"
	],
	"target": [
	"exp",
	"mult2",
	"x0"
	],
	"why": "(cosh+sinh)^2 = e^(2x)",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow2",
	"+",
	"mult2",
	"x0",
	"mult2",
	"x1"
	],
	"target": [
	"mult4",
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "(2x+2y)^2 = 4(x+y)^2, borderline",
	"family": 0,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"pow2",
	"+",
	"sin",
	"x0",
	"cos",
	"x0"
	],
	"target": [
	"+",
	"1",
	"sin",
	"mult2",
	"x0"
	],
	"why": "BORDERLINE (5-token target): (sin+cos)^2=1+sin2x",
	"family": 1,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"pow2",
	"+",
	"sinh",
	"x0",
	"cosh",
	"x0"
	],
	"target": [
	"exp",
	"mult2",
	"x0"
	],
	"why": "(sinh+cosh)^2 = exp(2x) (operand order)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow2",
	"-",
	"cosh",
	"x0",
	"sinh",
	"x0"
	],
	"target": [
	"exp",
	"mult2",
	"neg",
	"x0"
	],
	"why": "(cosh-sinh)^2 = exp(-2x)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"pow3",
	"+",
	"cosh",
	"x0",
	"sinh",
	"x0"
	],
	"target": [
	"exp",
	"mult3",
	"x0"
	],
	"why": "(cosh+sinh)^3 = exp(3x)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow3",
	"-",
	"cosh",
	"x0",
	"sinh",
	"x0"
	],
	"target": [
	"exp",
	"mult3",
	"neg",
	"x0"
	],
	"why": "(cosh-sinh)^3 = exp(-3x)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"pow4",
	"+",
	"cosh",
	"x0",
	"sinh",
	"x0"
	],
	"target": [
	"exp",
	"mult4",
	"x0"
	],
	"why": "(cosh+sinh)^4 = exp(4x)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow5",
	"+",
	"cosh",
	"x0",
	"sinh",
	"x0"
	],
	"target": [
	"exp",
	"mult5",
	"x0"
	],
	"why": "(cosh+sinh)^5 = exp(5x)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"+",
	"1",
	"x0",
	"-",
	"1",
	"x0"
	],
	"target": [
	"-",
	"1",
	"pow2",
	"x0"
	],
	"why": "(1+x)(1-x)=1-x^2",
	"family": 5,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"+",
	"x0",
	"1",
	"-",
	"x0",
	"1"
	],
	"target": [
	"+",
	"(-1)",
	"pow2",
	"x0"
	],
	"why": "(x+1)(x-1) = x^2-1",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"*",
	"+",
	"x0",
	"np.pi",
	"-",
	"x0",
	"np.pi"
	],
	"target": [
	"+",
	"<constant>",
	"pow2",
	"x0"
	],
	"why": "(x+pi)(x-pi) = x^2-pi^2; constant part may fold, borderline",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"*",
	"+",
	"x0",
	"x1",
	"+",
	"x1",
	"x0"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "(x+y)*(y+x) collapses to (x+y)^2, commuted second factor",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"+",
	"x0",
	"x1",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"-",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"why": "(x+y)(x-y) = x^2-y^2, borderline 5-token target",
	"family": 0,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"*",
	"+",
	"x1",
	"x0",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"-",
	"pow2",
	"x1",
	"pow2",
	"x0"
	],
	"why": "(y+x)(y-x) = y^2-x^2, borderline",
	"family": 0,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"*",
	"-",
	"1",
	"x0",
	"+",
	"1",
	"x0"
	],
	"target": [
	"-",
	"1",
	"pow2",
	"x0"
	],
	"why": "(1-x)(1+x) = 1-x^2",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"-",
	"x0",
	"1",
	"+",
	"x0",
	"1"
	],
	"target": [
	"+",
	"(-1)",
	"pow2",
	"x0"
	],
	"why": "(x-1)(x+1) = x^2-1",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"*",
	"-",
	"x0",
	"x1",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"-",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"why": "(x-y)(x+y) = x^2-y^2, borderline",
	"family": 0,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"*",
	"-",
	"x0",
	"x1",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"neg",
	"pow2",
	"-",
	"x0",
	"x1"
	],
	"why": "(x-y)*(y-x) = -(x-y)^2, borderline 5-token target",
	"family": 0,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"*",
	"<constant>",
	"*",
	"sin",
	"x0",
	"cos",
	"x0"
	],
	"target": [
	"*",
	"<constant>",
	"sin",
	"mult2",
	"x0"
	],
	"why": "parameterized csincos, c=2 -> sin2x",
	"family": 1,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"*",
	"exp",
	"x0",
	"exp",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"exp",
	"x1"
	],
	"why": "e^x * e^(y-x) = e^y",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow",
	"x0",
	"<constant>",
	"pow",
	"x0",
	"<constant>"
	],
	"target": [
	"pow",
	"x0",
	"<constant>"
	],
	"why": "x^a * x^b = x^(a+b)",
	"family": 4,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow",
	"x0",
	"x1",
	"pow",
	"x0",
	"x2"
	],
	"target": [
	"pow",
	"x0",
	"+",
	"x1",
	"x2"
	],
	"why": "borderline: x^y * x^z -> x^(y+z)",
	"family": 4,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"*",
	"pow",
	"x0",
	"x2",
	"pow",
	"x1",
	"x2"
	],
	"target": [
	"pow",
	"*",
	"x0",
	"x1",
	"x2"
	],
	"why": "borderline: x^z * y^z -> (xy)^z",
	"family": 4,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"*",
	"pow2",
	"/",
	"x0",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
	"pow2",
	"x0"
	],
	"why": "(x/y)^2 * y^2 -> x^2",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow3",
	"/",
	"x0",
	"x1",
	"pow3",
	"x1"
	],
	"target": [
	"pow3",
	"x0"
	],
	"why": "(x/y)^3 * y^3 -> x^3",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"sin",
	"mult2",
	"x0",
	"cos",
	"mult2",
	"x0"
	],
	"target": [
	"div2",
	"sin",
	"mult4",
	"x0"
	],
	"why": "sin2x cos2x=sin4x/2",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"sinh",
	"div2",
	"x0",
	"cosh",
	"div2",
	"x0"
	],
	"target": [
	"div2",
	"sinh",
	"x0"
	],
	"why": "sinh(x/2)cosh(x/2) = sinh(x)/2",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"*",
	"<constant>",
	"x0",
	"*",
	"<constant>",
	"x0"
	],
	"target": [
	"*",
	"<constant>",
	"x0"
	],
	"why": "C1x0+C2x0 -> (C1+C2)*x0, like-term collection",
	"family": 3,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"*",
	"x0",
	"x1",
	"*",
	"x0",
	"x2"
	],
	"target": [
	"*",
	"+",
	"x1",
	"x2",
	"x0"
	],
	"why": "x0x1+x0x2 -> x0(x1+x2) (borderline 5-token target)",
	"family": 8,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"+",
	"*",
	"x0",
	"x1",
	"*",
	"x1",
	"x0"
	],
	"target": [
	"*",
	"mult2",
	"x0",
	"x1"
	],
	"why": "x0x1 + x1x0 -> 2x0x1",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"+",
	"*",
	"x0",
	"x1",
	"*",
	"x2",
	"x0"
	],
	"target": [
	"*",
	"+",
	"x1",
	"x2",
	"x0"
	],
	"why": "mixed operand order: x0x1+x2x0 -> x0(x1+x2)",
	"family": 8,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"+",
	"+",
	"1",
	"mult2",
	"x0",
	"pow2",
	"x0"
	],
	"target": [
	"pow2",
	"+",
	"1",
	"x0"
	],
	"why": "1+2x+x^2 = (x+1)^2, ascending order",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"+",
	"+",
	"pow2",
	"x0",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"pow2",
	"+",
	"1",
	"x0"
	],
	"why": "x^2+2x+1 = (x+1)^2, left-nested",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"+",
	"-",
	"1",
	"mult2",
	"x0",
	"pow2",
	"x0"
	],
	"target": [
	"pow2",
	"+",
	"(-1)",
	"x0"
	],
	"why": "1-2x+x^2 = (1-x)^2 = (x-1)^2",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"+",
	"-",
	"pow2",
	"x0",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"pow2",
	"+",
	"(-1)",
	"x0"
	],
	"why": "x^2-2x+1 = (x-1)^2",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"+",
	"/",
	"x0",
	"x2",
	"/",
	"x1",
	"x2"
	],
	"target": [
	"/",
	"+",
	"x0",
	"x1",
	"x2"
	],
	"why": "borderline: x/z + y/z -> (x+y)/z (5-token minimal)",
	"family": 4,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"+",
	"1",
	"mult2",
	"pow2",
	"sinh",
	"div2",
	"x0"
	],
	"target": [
	"cosh",
	"x0"
	],
	"why": "1+2sinh^2(x/2) = cosh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"cos",
	"mult2",
	"x0",
	"pow2",
	"sin",
	"x0"
	],
	"target": [
	"pow2",
	"cos",
	"x0"
	],
	"why": "cos2x+sin^2=cos^2",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"log",
	"/",
	"x0",
	"x1",
	"log",
	"x1"
	],
	"target": [
	"log",
	"x0"
	],
	"why": "log(x/y) + log y = log x",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"log",
	"abs",
	"x0",
	"log",
	"abs",
	"x1"
	],
	"target": [
	"log",
	"abs",
	"*",
	"x0",
	"x1"
	],
	"why": "borderline: log\|x\|+log\|y\| = log\|xy\|, minimal form 5 tokens",
	"family": 3,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"+",
	"log",
	"pow2",
	"x0",
	"log",
	"pow3",
	"x0"
	],
	"target": [
	"log",
	"pow5",
	"x0"
	],
	"why": "log x^2 + log x^3 = 5 log x",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"+",
	"log",
	"x0",
	"log",
	"/",
	"x1",
	"x0"
	],
	"target": [
	"log",
	"x1"
	],
	"why": "log x + log(y/x) = log y",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"mult2",
	"log",
	"x0",
	"mult3",
	"log",
	"x0"
	],
	"target": [
	"log",
	"pow5",
	"x0"
	],
	"why": "2 log x + 3 log x = 5 log x",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"+",
	"pow1_2",
	"+",
	"pow2",
	"x0",
	"1",
	"x0"
	],
	"target": [
	"exp",
	"asinh",
	"x0"
	],
	"why": "sqrt(x^2+1)+x = exp(asinh x) (operand order)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow2",
	"cos",
	"x0",
	"pow2",
	"sin",
	"x0"
	],
	"target": [
	"1"
	],
	"why": "Pythagorean identity with operand order swapped: cos^2+sin^2 = 1",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow2",
	"cosh",
	"x0",
	"pow2",
	"sinh",
	"x0"
	],
	"target": [
	"cosh",
	"mult2",
	"x0"
	],
	"why": "cosh^2 + sinh^2 = cosh(2x)",
	"family": 0,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow2",
	"sin",
	"x0",
	"pow2",
	"cos",
	"x0"
	],
	"target": [
	"1"
	],
	"why": "Pythagorean: sin^2+cos^2=1",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow2",
	"sinh",
	"x0",
	"pow2",
	"cosh",
	"x0"
	],
	"target": [
	"cosh",
	"mult2",
	"x0"
	],
	"why": "sinh^2 + cosh^2 = cosh 2x (operand order)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow2",
	"x0",
	"+",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"pow2",
	"+",
	"1",
	"x0"
	],
	"why": "x^2+(2x+1) = (x+1)^2, right-nested",
	"family": 0,
	"rounds": [
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	"hint_matched": false
	},
	{
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	"pow1_2",
	"+",
	"pow2",
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	"tier": "minimal",
	"hint_matched": true
	},
	{
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	"x0",
	"*",
	"<constant>",
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	"family": 3,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
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	"-",
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	"x0",
	"x1",
	"*",
	"x0",
	"x2"
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	"x1",
	"x2",
	"x0"
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	"tier": "minimal_le5",
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	},
	{
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	"x1",
	"*",
	"x1",
	"x0"
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	"family": 8,
	"rounds": [
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	"hint_matched": true
	},
	{
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	"-",
	"+",
	"pow2",
	"x0",
	"1",
	"mult2",
	"x0"
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	"+",
	"(-1)",
	"x0"
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	"family": 0,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": false
	},
	{
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	"-",
	"/",
	"+",
	"x0",
	"x1",
	"x1",
	"1"
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	"/",
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	"x1"
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	"family": 4,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
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	"-",
	"/",
	"exp",
	"x0",
	"cosh",
	"x0",
	"1"
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	"tanh",
	"x0"
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	"why": "e^x/cosh - 1 = tanh",
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	"rounds": [
	"round1"
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"/",
	"x0",
	"x2",
	"/",
	"x1",
	"x2"
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	"target": [
	"/",
	"-",
	"x0",
	"x1",
	"x2"
	],
	"why": "borderline: x/z - y/z -> (x-y)/z",
	"family": 4,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"-",
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	"mult2",
	"pow2",
	"sin",
	"div2",
	"x0"
	],
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	"cos",
	"x0"
	],
	"why": "1-2sin^2(x/2)=cos x",
	"family": 1,
	"rounds": [
	"round2"
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"1",
	"mult2",
	"pow2",
	"sin",
	"mult2",
	"x0"
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	"x0"
	],
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	"family": 1,
	"rounds": [
	"round2"
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
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	"mult2",
	"x0",
	"pow2",
	"sinh",
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	"pow2",
	"cosh",
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	],
	"why": "cosh 2x - sinh^2 = cosh^2",
	"family": 1,
	"rounds": [
	"round1"
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"log",
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	"x0",
	"x1",
	"log",
	"x0"
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	"log",
	"x1"
	],
	"why": "log(xy) - log x = log y",
	"family": 3,
	"rounds": [
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	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"mult2",
	"+",
	"x0",
	"x1",
	"mult2",
	"x0"
	],
	"target": [
	"mult2",
	"x1"
	],
	"why": "2(x0+x1)-2x0 -> 2x1",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"mult2",
	"pow2",
	"cosh",
	"div2",
	"x0",
	"1"
	],
	"target": [
	"cosh",
	"x0"
	],
	"why": "2cosh^2(x/2)-1 = cosh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"pow2",
	"+",
	"x0",
	"1",
	"mult2",
	"x0"
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	"+",
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	"pow2",
	"x0"
	],
	"why": "(x+1)^2-2x=x^2+1",
	"family": 5,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"-",
	"pow2",
	"cos",
	"x0",
	"cos",
	"mult2",
	"x0"
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	"pow2",
	"sin",
	"x0"
	],
	"why": "cos^2-cos2x=sin^2",
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	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"pow2",
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	"x0",
	"pow2",
	"sin",
	"x0"
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	"x0"
	],
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	"family": 0,
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	"tier": "minimal",
	"hint_matched": true
	},
	{
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	"-",
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	"pow2",
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
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	"pow4",
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	],
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	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
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	"pow4",
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	"mult2",
	"x0"
	],
	"why": "sin^4-cos^4=-cos2x",
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	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
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	"/",
	"*",
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	"+",
	"x0",
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	"inv",
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	"inv",
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	"rounds": [
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	},
	{
	"source": [
	"/",
	"+",
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	"sin",
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	"inv",
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	"div2",
	"x0"
	],
	"why": "(1+cos)/sin=cot(x/2)",
	"family": 1,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
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	"1",
	"sinh",
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	"x0"
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	"why": "(cosh+1)/sinh = coth(x/2)",
	"family": 2,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"pow3",
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	"pow2",
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	"why": "(x^3+x^2)/x -> x^2+x",
	"family": 4,
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
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	"x0",
	"x1"
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	"target": [
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	"inv",
	"x1"
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	"rounds": [
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	"hint_matched": null
	},
	{
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	"/",
	"+",
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	"+",
	"x0",
	"x1"
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	"family": 8,
	"rounds": [
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	"tier": "minimal",
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	{
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	"+",
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	"+",
	"x1",
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	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
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	"sin",
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	"x0"
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	"rounds": [
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	"round2"
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	"tier": "minimal",
	"hint_matched": true
	},
	{
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	"-",
	"x1",
	"x0"
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	"target": [
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	"why": "(x-y)/(y-x) -> -1",
	"family": 4,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"exp",
	"+",
	"x0",
	"x1",
	"exp",
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	"target": [
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	"why": "e^(x+y)/e^x = e^y",
	"family": 3,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
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	"exp",
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	"exp",
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	"x0",
	"x1"
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	"target": [
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	"neg",
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	"why": "e^x/e^(x+y) = e^-y",
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	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
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	"target": [
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	"why": "log(x^y)/log(x) = y",
	"family": 3,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"mult2",
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	"+",
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	"pow2",
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	"target": [
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	"mult2",
	"atan",
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	"why": "2x/(1+x^2)=sin(2 atan x)",
	"family": 1,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"mult2",
	"x0",
	"-",
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	"pow2",
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	"target": [
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	"atan",
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	"why": "2x/(1-x^2)=tan(2 atan x)",
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	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
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	"<constant>",
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	"<constant>"
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	"why": "x^a / x^b = x^(a-b)",
	"family": 4,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
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	"pow",
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	"x1",
	"pow",
	"x0",
	"x2"
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	"target": [
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	"-",
	"x1",
	"x2"
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	"family": 4,
	"rounds": [
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	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
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	"pow2",
	"x0",
	"x0"
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	"target": [
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	"why": "sqrt(1-x^2)/x=tan(acos x)",
	"family": 1,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow2",
	"sinh",
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	"pow2",
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	"target": [
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	"x0"
	],
	"why": "sinh^2/cosh^2 = tanh^2",
	"family": 1,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
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	"x0",
	"mult2",
	"cos",
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	"target": [
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	"why": "sin2x/(2cos x)=sin x",
	"family": 1,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
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	"mult2",
	"sin",
	"x0"
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	"target": [
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	"why": "sin2x/(2sin x)=cos x (domain ext at sin=0)",
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	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"sin",
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	"+",
	"1",
	"cos",
	"x0"
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	"target": [
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	"x0"
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	"why": "half-angle tangent: sin x/(1+cos x) = tan(x/2)",
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	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"sinh",
	"x0",
	"+",
	"1",
	"cosh",
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	"target": [
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	"x0"
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	"why": "sinh/(1+cosh) = tanh(x/2) (operand order)",
	"family": 2,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
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	"x0",
	"+",
	"cosh",
	"x0",
	"1"
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	"target": [
	"tanh",
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	"x0"
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	"why": "sinh x/(cosh x + 1) = tanh(x/2)",
	"family": 0,
	"rounds": [
	"round1",
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
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	"+",
	"x0",
	"*",
	"x0",
	"x1"
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	"target": [
	"inv",
	"+",
	"1",
	"x1"
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	"why": "x/(x+xy) -> 1/(1+y)",
	"family": 4,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
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	"pow1_2",
	"+",
	"1",
	"pow2",
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	"target": [
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	"x0"
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	"why": "x/sqrt(1+x^2)=sin(atan x)",
	"family": 1,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
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	"+",
	"pow2",
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	"target": [
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	"why": "x/sqrt(x^2+1) = tanh(asinh x) (operand order)",
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	"rounds": [
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	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
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	"pow2",
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	"target": [
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	"why": "x/sqrt(1-x^2)=tan(asin x)",
	"family": 1,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": false
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	{
	"source": [
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	"inv",
	"abs",
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	"target": [
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	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
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	{
	"source": [
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	"exp",
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	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
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	"exp",
	"neg",
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	"why": "(e^x+e^-x)/2 = cosh",
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	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div2",
	"-",
	"abs",
	"x0",
	"inv",
	"abs",
	"x0"
	],
	"target": [
	"sinh",
	"log",
	"abs",
	"x0"
	],
	"why": "(\|x\|-1/\|x\|)/2 = sinh(log\|x\|)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div2",
	"-",
	"exp",
	"x0",
	"exp",
	"neg",
	"x0"
	],
	"target": [
	"sinh",
	"x0"
	],
	"why": "(e^x-e^-x)/2 = sinh",
	"family": 1,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div2",
	"/",
	"sinh",
	"mult2",
	"x0",
	"cosh",
	"x0"
	],
	"target": [
	"sinh",
	"x0"
	],
	"why": "sinh2x/(2 cosh x) = sinh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div2",
	"/",
	"sinh",
	"mult2",
	"x0",
	"sinh",
	"x0"
	],
	"target": [
	"cosh",
	"x0"
	],
	"why": "sinh2x/(2 sinh x) = cosh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"exp",
	"div2",
	"+",
	"log",
	"x0",
	"log",
	"x1"
	],
	"target": [
	"pow1_2",
	"*",
	"x0",
	"x1"
	],
	"why": "e^((log a + log b)/2) = sqrt(ab)",
	"family": 2,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"exp",
	"div2",
	"-",
	"log",
	"x0",
	"log",
	"x1"
	],
	"target": [
	"pow1_2",
	"/",
	"x0",
	"x1"
	],
	"why": "e^((log a - log b)/2) = sqrt(a/b)",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"exp",
	"mult2",
	"+",
	"log",
	"x0",
	"log",
	"x1"
	],
	"target": [
	"pow2",
	"*",
	"x0",
	"x1"
	],
	"why": "e^(2(log a + log b)) = (ab)^2",
	"family": 2,
	"rounds": [
	"round1",
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"exp",
	"mult2",
	"-",
	"log",
	"x0",
	"log",
	"x1"
	],
	"target": [
	"pow2",
	"/",
	"x0",
	"x1"
	],
	"why": "e^(2(log a - log b)) = (a/b)^2",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"exp",
	"neg",
	"+",
	"log",
	"x0",
	"log",
	"x1"
	],
	"target": [
	"/",
	"inv",
	"x0",
	"x1"
	],
	"why": "e^(-(log a + log b)) = 1/(ab)",
	"family": 2,
	"rounds": [
	"round1"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"mult2",
	"inv",
	"+",
	"1",
	"cos",
	"mult2",
	"x0"
	],
	"target": [
	"inv",
	"pow2",
	"cos",
	"x0"
	],
	"why": "2/(1+cos2x)=sec^2",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"div2",
	"+",
	"cosh",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"cosh",
	"x0"
	],
	"why": "sqrt((cosh2x+1)/2) = cosh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"div2",
	"-",
	"cosh",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"abs",
	"sinh",
	"x0"
	],
	"why": "sqrt((cosh2x-1)/2) = \|sinh x\|",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"abs",
	"x0",
	"*",
	"abs",
	"x1",
	"abs",
	"x2"
	],
	"target": [
	"abs",
	"*",
	"x0",
	"*",
	"x1",
	"x2"
	],
	"why": "\|x\|\|y\|\|z\| = \|xyz\| (borderline 6-token target, invited)",
	"family": 6,
	"rounds": [
	"round2_long"
	],
	"tier": "hint_verified_nonminimal",
	"hint_matched": null
	},
	{
	"source": [
	"*",
	"neg",
	"x0",
	"*",
	"neg",
	"x1",
	"neg",
	"x2"
	],
	"target": [
	"neg",
	"*",
	"x0",
	"*",
	"x1",
	"x2"
	],
	"why": "three negs = one neg through triple product (borderline 5-token target)",
	"family": 6,
	"rounds": [
	"round2_long"
	],
	"tier": "hint_verified_nonminimal",
	"hint_matched": null
	},
	{
	"source": [
	"*",
	"tan",
	"x0",
	"+",
	"1",
	"cos",
	"mult2",
	"x0"
	],
	"target": [
	"sin",
	"mult2",
	"x0"
	],
	"why": "tan*(1+cos2x)=sin2x",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"*",
	"x0",
	"neg",
	"x1",
	"*",
	"x0",
	"x1"
	],
	"target": [
	"0"
	],
	"why": "x0(-x1)+x0x1 -> 0",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"+",
	"pow4",
	"x0",
	"mult2",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"pow4",
	"cosh",
	"asinh",
	"x0"
	],
	"why": "x^4+2x^2+1 = (x^2+1)^2, borderline 5-token target",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"+",
	"-",
	"pow4",
	"x0",
	"mult2",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"pow2",
	"+",
	"(-1)",
	"pow2",
	"x0"
	],
	"why": "x^4-2x^2+1 = (x^2-1)^2, borderline",
	"family": 0,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"+",
	"/",
	"neg",
	"x0",
	"x1",
	"/",
	"x0",
	"x1"
	],
	"target": [
	"0"
	],
	"why": "(-x)/y + x/y -> 0",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"/",
	"x0",
	"x1",
	"/",
	"x0",
	"neg",
	"x1"
	],
	"target": [
	"0"
	],
	"why": "x/y + x/(-y) -> 0",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"abs",
	"x0",
	"pow1_2",
	"-",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"exp",
	"acosh",
	"abs",
	"x0"
	],
	"why": "\|x\|+sqrt(x^2-1) = exp(acosh\|x\|)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"mult2",
	"x0",
	"+",
	"mult3",
	"x0",
	"mult4",
	"x0"
	],
	"target": [
	"mult3",
	"mult3",
	"x0"
	],
	"why": "constant-ladder: 2x+3x+4x = 9x = 3*3x (8-token sum ladder)",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"mult3",
	"sinh",
	"x0",
	"mult4",
	"pow3",
	"sinh",
	"x0"
	],
	"target": [
	"sinh",
	"mult3",
	"x0"
	],
	"why": "3sinh + 4sinh^3 = sinh 3x, triple argument",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"+",
	"mult2",
	"x0",
	"mult2",
	"x1",
	"mult2",
	"x1"
	],
	"target": [
	"mult2",
	"x0"
	],
	"why": "constant-ladder: (2x+2y)-2y = 2x, cross-variable cancel",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"+",
	"mult3",
	"x0",
	"mult4",
	"x0",
	"mult2",
	"x0"
	],
	"target": [
	"mult5",
	"x0"
	],
	"why": "constant-ladder: (3x+4x) - 2x = 5x",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"inv",
	"pow2",
	"cos",
	"x0",
	"pow2",
	"tan",
	"x0"
	],
	"target": [
	"1"
	],
	"why": "sec^2-tan^2=1",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"mult3",
	"sin",
	"x0",
	"mult4",
	"pow3",
	"sin",
	"x0"
	],
	"target": [
	"sin",
	"mult3",
	"x0"
	],
	"why": "3sin-4sin^3=sin3x",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"mult4",
	"pow3",
	"cos",
	"x0",
	"mult3",
	"cos",
	"x0"
	],
	"target": [
	"cos",
	"mult3",
	"x0"
	],
	"why": "4cos^3-3cos=cos3x",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"mult4",
	"pow3",
	"cosh",
	"x0",
	"mult3",
	"cosh",
	"x0"
	],
	"target": [
	"cosh",
	"mult3",
	"x0"
	],
	"why": "4cosh^3 - 3cosh = cosh 3x, triple argument",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"exp",
	"mult2",
	"x0",
	"1",
	"exp",
	"x0"
	],
	"target": [
	"mult2",
	"cosh",
	"x0"
	],
	"why": "(exp(2x)+1)/exp x = 2 cosh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"pow2",
	"x0",
	"*",
	"<constant>",
	"x0",
	"x0"
	],
	"target": [
	"+",
	"<constant>",
	"x0"
	],
	"why": "(x^2+Cx)/x -> x+C",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"+",
	"pow2",
	"x0",
	"*",
	"x0",
	"x1",
	"x0"
	],
	"target": [
	"+",
	"x0",
	"x1"
	],
	"why": "(x^2+xy)/x = x+y",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"pow2",
	"x0",
	"x0",
	"+",
	"x0",
	"1"
	],
	"target": [
	"x0"
	],
	"why": "(x^2+x)/(x+1) -> x",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"pow3",
	"x0",
	"pow2",
	"x0",
	"pow2",
	"x0"
	],
	"target": [
	"+",
	"1",
	"x0"
	],
	"why": "(x^3+x^2)/x^2 -> x+1",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"+",
	"x0",
	"1",
	"-",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"inv",
	"+",
	"(-1)",
	"x0"
	],
	"why": "(x+1)/(x^2-1) = 1/(x-1)",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"-",
	"1",
	"pow2",
	"x0",
	"+",
	"1",
	"x0"
	],
	"target": [
	"-",
	"1",
	"x0"
	],
	"why": "(1-x^2)/(1+x) = 1-x",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"1",
	"pow2",
	"x0",
	"-",
	"1",
	"x0"
	],
	"target": [
	"+",
	"1",
	"x0"
	],
	"why": "(1-x^2)/(1-x) = 1+x",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"exp",
	"mult2",
	"x0",
	"1",
	"exp",
	"x0"
	],
	"target": [
	"mult2",
	"sinh",
	"x0"
	],
	"why": "(exp(2x)-1)/exp x = 2 sinh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow2",
	"x0",
	"*",
	"x0",
	"x1",
	"x0"
	],
	"target": [
	"-",
	"x0",
	"x1"
	],
	"why": "(x^2-xy)/x = x-y",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow2",
	"x0",
	"1",
	"+",
	"x0",
	"1"
	],
	"target": [
	"+",
	"(-1)",
	"x0"
	],
	"why": "(x^2-1)/(x+1) = x-1",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"-",
	"pow2",
	"x0",
	"1",
	"-",
	"x0",
	"1"
	],
	"target": [
	"+",
	"1",
	"x0"
	],
	"why": "(x^2-1)/(x-1) = x+1",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"-",
	"pow2",
	"x0",
	"x0",
	"-",
	"x0",
	"1"
	],
	"target": [
	"x0"
	],
	"why": "(x^2-x)/(x-1) -> x",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow3",
	"x0",
	"1",
	"-",
	"x0",
	"1"
	],
	"target": [
	"+",
	"+",
	"pow2",
	"x0",
	"x0",
	"1"
	],
	"why": "borderline: (x^3-1)/(x-1) -> x^2+x+1 (6-token minimal)",
	"family": 4,
	"rounds": [
	"round2_long"
	],
	"tier": "hint_verified_nonminimal",
	"hint_matched": null
	},
	{
	"source": [
	"/",
	"-",
	"pow3",
	"x0",
	"x0",
	"+",
	"x0",
	"1"
	],
	"target": [
	"-",
	"pow2",
	"x0",
	"x0"
	],
	"why": "(x^3-x)/(x+1) -> x^2-x",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow3",
	"x0",
	"x0",
	"-",
	"x0",
	"1"
	],
	"target": [
	"+",
	"pow2",
	"x0",
	"x0"
	],
	"why": "(x^3-x)/(x-1) -> x^2+x",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow4",
	"x0",
	"pow2",
	"x0",
	"pow2",
	"x0"
	],
	"target": [
	"+",
	"(-1)",
	"pow2",
	"x0"
	],
	"why": "(x^4-x^2)/x^2 = x^2-1",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"-",
	"x0",
	"1",
	"-",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"inv",
	"+",
	"1",
	"x0"
	],
	"why": "(x-1)/(x^2-1) = 1/(x+1)",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"pow2",
	"cos",
	"x0",
	"+",
	"1",
	"sin",
	"x0"
	],
	"target": [
	"-",
	"1",
	"sin",
	"x0"
	],
	"why": "cos^2/(1+sin)=1-sin",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"tan",
	"x0",
	"+",
	"1",
	"pow2",
	"tan",
	"x0"
	],
	"target": [
	"div2",
	"sin",
	"mult2",
	"x0"
	],
	"why": "tan/(1+tan^2)=sin x cos x=sin2x/2",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"x0",
	"+",
	"pow2",
	"x0",
	"*",
	"x0",
	"x1"
	],
	"target": [
	"inv",
	"+",
	"x0",
	"x1"
	],
	"why": "x/(x^2+xy) -> 1/(x+y), denominator-side collapse",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"x1",
	"+",
	"*",
	"x0",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
	"inv",
	"+",
	"x0",
	"x1"
	],
	"why": "y/(xy+y^2) -> 1/(x+y)",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"asin",
	"/",
	"x0",
	"pow1_2",
	"+",
	"1",
	"pow2",
	"x0"
	],
	"target": [
	"atan",
	"x0"
	],
	"why": "asin(x/sqrt(1+x^2))=atan x",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"asinh",
	"/",
	"x0",
	"pow1_2",
	"-",
	"1",
	"pow2",
	"x0"
	],
	"target": [
	"atanh",
	"x0"
	],
	"why": "asinh(x/sqrt(1-x^2)) = atanh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"atan",
	"/",
	"x0",
	"pow1_2",
	"-",
	"1",
	"pow2",
	"x0"
	],
	"target": [
	"asin",
	"x0"
	],
	"why": "atan(x/sqrt(1-x^2))=asin x",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"atanh",
	"/",
	"x0",
	"pow1_2",
	"+",
	"1",
	"pow2",
	"x0"
	],
	"target": [
	"asinh",
	"x0"
	],
	"why": "atanh(x/sqrt(1+x^2)) = asinh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div4",
	"+",
	"cosh",
	"mult3",
	"x0",
	"mult3",
	"cosh",
	"x0"
	],
	"target": [
	"pow3",
	"cosh",
	"x0"
	],
	"why": "(cosh3x + 3cosh)/4 = cosh^3, power reduction",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div4",
	"-",
	"sinh",
	"mult3",
	"x0",
	"mult3",
	"sinh",
	"x0"
	],
	"target": [
	"pow3",
	"sinh",
	"x0"
	],
	"why": "(sinh3x - 3sinh)/4 = sinh^3, power reduction",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"inv",
	"+",
	"x0",
	"pow1_2",
	"+",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"exp",
	"asinh",
	"neg",
	"x0"
	],
	"why": "1/(x+sqrt(x^2+1)) = exp(-asinh x)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"log",
	"+",
	"pow1_2",
	"+",
	"pow2",
	"x0",
	"1",
	"x0"
	],
	"target": [
	"asinh",
	"x0"
	],
	"why": "log(sqrt(x^2+1)+x) = asinh x (outer operand order)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"log",
	"+",
	"x0",
	"pow1_2",
	"+",
	"1",
	"pow2",
	"x0"
	],
	"target": [
	"asinh",
	"x0"
	],
	"why": "log(x+sqrt(1+x^2)) = asinh x (inner operand order)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"log",
	"+",
	"x0",
	"pow1_2",
	"+",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"asinh",
	"x0"
	],
	"why": "log(x+sqrt(x^2+1)) = asinh x, the canonical 8-token reduction",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"log",
	"+",
	"x0",
	"pow1_2",
	"-",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"acosh",
	"x0"
	],
	"why": "log(x+sqrt(x^2-1)) = acosh x (domains match: both nan for x<1)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"log",
	"-",
	"pow1_2",
	"+",
	"pow2",
	"x0",
	"1",
	"x0"
	],
	"target": [
	"asinh",
	"neg",
	"x0"
	],
	"why": "log(sqrt(x^2+1)-x) = -asinh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"log",
	"-",
	"x0",
	"pow1_2",
	"-",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"neg",
	"acosh",
	"x0"
	],
	"why": "log(x-sqrt(x^2-1)) = -acosh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"log",
	"/",
	"+",
	"1",
	"x0",
	"-",
	"1",
	"x0"
	],
	"target": [
	"mult2",
	"atanh",
	"x0"
	],
	"why": "log((1+x)/(1-x)) = 2 atanh (9 tokens)",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"mult2",
	"*",
	"sin",
	"div2",
	"x0",
	"cos",
	"div2",
	"x0"
	],
	"target": [
	"sin",
	"x0"
	],
	"why": "2 sin(x/2)cos(x/2)=sin x",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"mult2",
	"*",
	"sin",
	"mult2",
	"x0",
	"cos",
	"mult2",
	"x0"
	],
	"target": [
	"sin",
	"mult4",
	"x0"
	],
	"why": "2 sin2x cos2x=sin4x",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"mult2",
	"*",
	"sinh",
	"div2",
	"x0",
	"cosh",
	"div2",
	"x0"
	],
	"target": [
	"sinh",
	"x0"
	],
	"why": "2 sinh(x/2)cosh(x/2) = sinh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"mult2",
	"*",
	"x0",
	"pow1_2",
	"+",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"sinh",
	"mult2",
	"asinh",
	"x0"
	],
	"why": "2x sqrt(x^2+1) = sinh(2 asinh x)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"mult2",
	"*",
	"x0",
	"pow1_2",
	"-",
	"1",
	"pow2",
	"x0"
	],
	"target": [
	"sin",
	"mult2",
	"acos",
	"x0"
	],
	"why": "BORDERLINE: 2x sqrt(1-x^2)=sin(2 asin x); source also nan outside \|x\|<=1 so domains match - may certify",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"mult2",
	"inv",
	"+",
	"exp",
	"x0",
	"exp",
	"neg",
	"x0"
	],
	"target": [
	"inv",
	"cosh",
	"x0"
	],
	"why": "2/(e^x+e^-x) = sech = 1/cosh",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"mult2",
	"inv",
	"-",
	"exp",
	"x0",
	"exp",
	"neg",
	"x0"
	],
	"target": [
	"inv",
	"sinh",
	"x0"
	],
	"why": "2/(e^x-e^-x) = csch = 1/sinh",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"+",
	"+",
	"pow2",
	"x0",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"abs",
	"+",
	"1",
	"x0"
	],
	"why": "sqrt of x^2+2x+1 = \|x+1\|",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"pow1_2",
	"+",
	"-",
	"pow2",
	"x0",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"abs",
	"+",
	"(-1)",
	"x0"
	],
	"why": "sqrt of x^2-2x+1 = \|x-1\|",
	"family": 0,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"pow1_2",
	"/",
	"+",
	"1",
	"x0",
	"-",
	"1",
	"x0"
	],
	"target": [
	"exp",
	"atanh",
	"x0"
	],
	"why": "sqrt((1+x)/(1-x)) = exp(atanh x)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow2",
	"+",
	"+",
	"pow2",
	"x0",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"pow4",
	"+",
	"1",
	"x0"
	],
	"why": "(x^2+2x+1)^2 = (x+1)^4",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"pow2",
	"+",
	"-",
	"pow2",
	"x0",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"pow4",
	"+",
	"(-1)",
	"x0"
	],
	"why": "(x^2-2x+1)^2 = (x-1)^4",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"pow2",
	"+",
	"cosh",
	"div2",
	"x0",
	"sinh",
	"div2",
	"x0"
	],
	"target": [
	"exp",
	"x0"
	],
	"why": "(cosh(x/2)+sinh(x/2))^2 = exp x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"+",
	"x1",
	"x2",
	"/",
	"x0",
	"+",
	"x1",
	"x2"
	],
	"target": [
	"x0"
	],
	"why": "commuted variant",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"-",
	"1",
	"cos",
	"x0",
	"+",
	"1",
	"cos",
	"x0"
	],
	"target": [
	"pow2",
	"sin",
	"x0"
	],
	"why": "factored (1-cos)(1+cos)=sin^2",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"-",
	"1",
	"sin",
	"x0",
	"+",
	"1",
	"sin",
	"x0"
	],
	"target": [
	"pow2",
	"cos",
	"x0"
	],
	"why": "factored (1-sin)(1+sin)=cos^2",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"-",
	"pow2",
	"x0",
	"1",
	"+",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"+",
	"(-1)",
	"pow4",
	"x0"
	],
	"why": "(x^2-1)(x^2+1) = x^4-1",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"*",
	"/",
	"x0",
	"+",
	"x1",
	"x2",
	"+",
	"x1",
	"x2"
	],
	"target": [
	"x0"
	],
	"why": "(x0/(x1+x2))*(x1+x2) -> x0: cancel compound denominator",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"cos",
	"+",
	"x0",
	"x1",
	"cos",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"-",
	"pow2",
	"cos",
	"x0",
	"pow2",
	"sin",
	"x1"
	],
	"why": "BORDERLINE (7-token target): cos(x+y)cos(x-y)=cos^2 x - sin^2 y",
	"family": 1,
	"rounds": [
	"round2_long"
	],
	"tier": "hint_verified_nonminimal",
	"hint_matched": null
	},
	{
	"source": [
	"*",
	"cos",
	"x0",
	"-",
	"mult2",
	"cos",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"cos",
	"mult3",
	"x0"
	],
	"why": "cos x(2cos2x-1)=cos3x",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"pow2",
	"/",
	"x0",
	"x1",
	"pow2",
	"/",
	"x1",
	"x0"
	],
	"target": [
	"1"
	],
	"why": "(x/y)^2 * (y/x)^2 -> 1",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"sin",
	"+",
	"x0",
	"x1",
	"sin",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"-",
	"pow2",
	"sin",
	"x0",
	"pow2",
	"sin",
	"x1"
	],
	"why": "BORDERLINE (7-token target, named in brief): sin(x+y)sin(x-y)=sin^2 x - sin^2 y",
	"family": 1,
	"rounds": [
	"round2_long"
	],
	"tier": "hint_verified_nonminimal",
	"hint_matched": null
	},
	{
	"source": [
	"*",
	"sin",
	"x0",
	"+",
	"mult2",
	"cos",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"sin",
	"mult3",
	"x0"
	],
	"why": "sin x(2cos2x+1)=sin3x",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"tan",
	"mult2",
	"x0",
	"-",
	"1",
	"pow2",
	"tan",
	"x0"
	],
	"target": [
	"mult2",
	"tan",
	"x0"
	],
	"why": "tan2x*(1-tan^2 x)=2tan x",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"x0",
	"+",
	"/",
	"x1",
	"x0",
	"/",
	"x2",
	"x0"
	],
	"target": [
	"+",
	"x1",
	"x2"
	],
	"why": "x*(y/x + z/x) -> y+z",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"*",
	"+",
	"x0",
	"1",
	"-",
	"x0",
	"1",
	"1"
	],
	"target": [
	"pow2",
	"x0"
	],
	"why": "(x+1)(x-1)+1 = x^2",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"*",
	"+",
	"x0",
	"mult2",
	"x1",
	"x0",
	"pow2",
	"x1"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "partial factoring (x+2y)x+y^2, product operand order swapped",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"*",
	"x0",
	"+",
	"x0",
	"mult2",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "partial factoring x(x+2y)+y^2 = (x+y)^2",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"*",
	"x0",
	"-",
	"x0",
	"mult2",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
	"pow2",
	"-",
	"x0",
	"x1"
	],
	"why": "partial factoring x(x-2y)+y^2 = (x-y)^2",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"*",
	"x1",
	"+",
	"x1",
	"mult2",
	"x0",
	"pow2",
	"x0"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "partial factoring y(y+2x)+x^2 = (x+y)^2",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"*",
	"x1",
	"-",
	"x1",
	"mult2",
	"x0",
	"pow2",
	"x0"
	],
	"target": [
	"pow2",
	"-",
	"x0",
	"x1"
	],
	"why": "partial factoring y(y-2x)+x^2 = (y-x)^2 = (x-y)^2",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"+",
	"div2",
	"x0",
	"div3",
	"x0",
	"div2",
	"div3",
	"x0"
	],
	"target": [
	"x0"
	],
	"why": "constant-ladder: telescoping 1/2 + 1/3 + 1/6 = 1 (9-token sum ladder)",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"/",
	"-",
	"x0",
	"x1",
	"x2",
	"/",
	"x1",
	"x2"
	],
	"target": [
	"/",
	"x0",
	"x2"
	],
	"why": "(x0-x1)/x2 + x1/x2 -> x0/x2",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"/",
	"-",
	"x1",
	"x0",
	"-",
	"x0",
	"x1",
	"1"
	],
	"target": [
	"0"
	],
	"why": "(y-x)/(x-y) + 1 -> 0",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"asinh",
	"-",
	"x0",
	"x1",
	"asinh",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "asinh(x0-x1)+asinh(x1-x0) -> 0: odd function antisymmetry",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"atan",
	"-",
	"x0",
	"x1",
	"atan",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "atan(x0-x1)+atan(x1-x0) -> 0: odd function antisymmetry",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"div2",
	"+",
	"x0",
	"x1",
	"div2",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"x0"
	],
	"why": "(x0+x1)/2+(x0-x1)/2 -> x0",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"inv",
	"-",
	"x0",
	"x1",
	"inv",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "1/(x0-x1)+1/(x1-x0) -> 0 (nan only at x0=x1, domain extension)",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"log",
	"/",
	"x0",
	"x1",
	"log",
	"/",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "log(x/y) + log(y/x) = 0 (9 tokens)",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"log",
	"/",
	"x0",
	"x1",
	"log",
	"/",
	"x1",
	"x2"
	],
	"target": [
	"log",
	"/",
	"x0",
	"x2"
	],
	"why": "telescoping log(x/y)+log(y/z) = log(x/z) (9 tokens)",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"mult4",
	"*",
	"x0",
	"x1",
	"pow2",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "4xy+(x-y)^2 = (x+y)^2, commuted",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow2",
	"+",
	"x0",
	"x1",
	"pow2",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"mult2",
	"+",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"why": "borderline: (x+y)^2 + (x-y)^2 -> 2(x^2+y^2)",
	"family": 4,
	"rounds": [
	"round2_long"
	],
	"tier": "hint_verified_nonminimal",
	"hint_matched": null
	},
	{
	"source": [
	"+",
	"pow2",
	"-",
	"x0",
	"x1",
	"mult2",
	"*",
	"x0",
	"x1"
	],
	"target": [
	"+",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"why": "(x-y)^2+2xy = x^2+y^2, borderline",
	"family": 0,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"+",
	"pow2",
	"-",
	"x0",
	"x1",
	"mult4",
	"*",
	"x0",
	"x1"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "(x-y)^2+4xy = (x+y)^2",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow2",
	"cosh",
	"div2",
	"x0",
	"pow2",
	"sinh",
	"div2",
	"x0"
	],
	"target": [
	"cosh",
	"x0"
	],
	"why": "cosh^2(x/2)+sinh^2(x/2) = cosh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow2",
	"x1",
	"*",
	"x0",
	"+",
	"x0",
	"mult2",
	"x1"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "partial factoring y^2+x(x+2y), summand order swapped",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow3",
	"-",
	"x0",
	"x1",
	"pow3",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "(x0-x1)^3+(x1-x0)^3 -> 0: odd power antisymmetry",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow5",
	"-",
	"x0",
	"x1",
	"pow5",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "pow5(x0-x1)+pow5(x1-x0) -> 0: odd power antisymmetry (exact; float rounding only at extreme magnitudes)",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"sin",
	"-",
	"x0",
	"x1",
	"sin",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "sin(x0-x1)+sin(x1-x0) -> 0: odd function antisymmetry",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"sinh",
	"-",
	"x0",
	"x1",
	"sinh",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "sinh(x0-x1)+sinh(x1-x0) -> 0: odd function antisymmetry",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"tanh",
	"-",
	"x0",
	"x1",
	"tanh",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "tanh(x0-x1)+tanh(x1-x0) -> 0: odd function antisymmetry",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"x0",
	"*",
	"x1",
	"-",
	"exp",
	"x2",
	"exp",
	"x2"
	],
	"target": [
	"x0"
	],
	"why": "x1*(exp(x2)-exp(x2))=0 dead addend",
	"family": 9,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"*",
	"neg",
	"x0",
	"neg",
	"x1",
	"*",
	"x0",
	"x1"
	],
	"target": [
	"0"
	],
	"why": "(-x0)(-x1)-x0x1 -> 0",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"/",
	"+",
	"x0",
	"x1",
	"x1",
	"/",
	"x0",
	"x1"
	],
	"target": [
	"1"
	],
	"why": "(x+y)/y - x/y -> 1",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"/",
	"+",
	"x0",
	"x1",
	"x2",
	"/",
	"x0",
	"x2"
	],
	"target": [
	"/",
	"x1",
	"x2"
	],
	"why": "(x0+x1)/x2 - x0/x2 -> x1/x2: split fraction",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"1",
	"mult2",
	"inv",
	"+",
	"exp",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"tanh",
	"x0"
	],
	"why": "1 - 2/(exp(2x)+1) = tanh x, logistic shape",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"abs",
	"+",
	"x0",
	"x1",
	"abs",
	"+",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "\|x0+x1\|-\|x1+x0\| -> 0",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"abs",
	"-",
	"x0",
	"x1",
	"abs",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "abs(x0-x1)-abs(x1-x0) -> 0: even function symmetry",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"cos",
	"-",
	"x0",
	"x1",
	"cos",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "cos(x0-x1)-cos(x1-x0) -> 0: even function symmetry",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"cosh",
	"-",
	"x0",
	"x1",
	"cosh",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "cosh(x0-x1)-cosh(x1-x0) -> 0: even function symmetry",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"div2",
	"+",
	"x0",
	"x1",
	"div2",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"x1"
	],
	"why": "(x0+x1)/2-(x0-x1)/2 -> x1",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"exp",
	"+",
	"x0",
	"x1",
	"exp",
	"+",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "exp(x0+x1)-exp(x1+x0) -> 0",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"pow2",
	"*",
	"x0",
	"x1",
	"pow2",
	"*",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "(x0x1)^2-(x1x0)^2 -> 0",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"pow2",
	"+",
	"pow2",
	"x0",
	"1",
	"mult4",
	"pow2",
	"x0"
	],
	"target": [
	"pow2",
	"+",
	"(-1)",
	"pow2",
	"x0"
	],
	"why": "(x^2+1)^2-4x^2 = (x^2-1)^2, borderline",
	"family": 0,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"-",
	"pow2",
	"+",
	"x0",
	"1",
	"pow2",
	"-",
	"x0",
	"1"
	],
	"target": [
	"mult4",
	"x0"
	],
	"why": "(x+1)^2 - (x-1)^2 -> 4x",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"pow2",
	"+",
	"x0",
	"x1",
	"mult2",
	"*",
	"x0",
	"x1"
	],
	"target": [
	"+",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"why": "(x+y)^2-2xy = x^2+y^2, borderline 5-token target",
	"family": 0,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"-",
	"pow2",
	"+",
	"x0",
	"x1",
	"mult4",
	"*",
	"x0",
	"x1"
	],
	"target": [
	"pow2",
	"-",
	"x0",
	"x1"
	],
	"why": "(x+y)^2-4xy = (x-y)^2",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"pow2",
	"+",
	"x0",
	"x1",
	"pow2",
	"+",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "(x0+x1)^2-(x1+x0)^2 -> 0",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"pow2",
	"+",
	"x0",
	"x1",
	"pow2",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"*",
	"mult4",
	"x0",
	"x1"
	],
	"why": "(x+y)^2-(x-y)^2 = 4xy, polarization identity",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"-",
	"pow2",
	"+",
	"x0",
	"x1",
	"pow2",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"*",
	"mult4",
	"x0",
	"x1"
	],
	"why": "polarization, flipped difference (square is even)",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"-",
	"pow2",
	"+",
	"x1",
	"x0",
	"pow2",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"*",
	"mult4",
	"x0",
	"x1"
	],
	"why": "polarization, commuted sum",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"-",
	"pow2",
	"-",
	"x0",
	"x1",
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"*",
	"mult4",
	"neg",
	"x0",
	"x1"
	],
	"why": "(x-y)^2-(x+y)^2 = -4xy, borderline 5-token target",
	"family": 0,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"-",
	"pow2",
	"-",
	"x0",
	"x1",
	"pow2",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "pow2(x0-x1)-pow2(x1-x0) -> 0: even function symmetry",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"pow2",
	"cos",
	"div2",
	"x0",
	"pow2",
	"sin",
	"div2",
	"x0"
	],
	"target": [
	"cos",
	"x0"
	],
	"why": "cos^2(x/2)-sin^2(x/2)=cos x",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"pow2",
	"cos",
	"mult2",
	"x0",
	"pow2",
	"sin",
	"mult2",
	"x0"
	],
	"target": [
	"cos",
	"mult4",
	"x0"
	],
	"why": "cos^2(2x)-sin^2(2x)=cos4x",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"pow4",
	"-",
	"x0",
	"x1",
	"pow4",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "pow4(x0-x1)-pow4(x1-x0) -> 0: even function symmetry",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"sin",
	"*",
	"x0",
	"x1",
	"sin",
	"*",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "sin(x0x1)-sin(x1x0) -> 0",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"sin",
	"+",
	"x0",
	"x1",
	"sin",
	"+",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "sin(x0+x1)-sin(x1+x0) -> 0: commutativity under unary",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"*",
	"+",
	"x0",
	"x1",
	"x2",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"x2"
	],
	"why": "((x0+x1)*x2)/(x0+x1) -> x2",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"*",
	"<constant>",
	"+",
	"x0",
	"x1",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"<constant>"
	],
	"why": "C(x+y)/(x+y) -> C, single-wildcard factored form",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"*",
	"x0",
	"+",
	"x0",
	"1",
	"+",
	"x0",
	"1"
	],
	"target": [
	"x0"
	],
	"why": "factored variant x(x+1)/(x+1) -> x",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"*",
	"x0",
	"+",
	"x1",
	"x2",
	"+",
	"x1",
	"x2"
	],
	"target": [
	"x0"
	],
	"why": "x(y+z)/(y+z) -> x, factored variant",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"*",
	"x0",
	"x1",
	"*",
	"x0",
	"x2",
	"x0"
	],
	"target": [
	"+",
	"x1",
	"x2"
	],
	"why": "(xy+xz)/x -> y+z",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"*",
	"x0",
	"x1",
	"*",
	"x2",
	"x0",
	"x0"
	],
	"target": [
	"+",
	"x1",
	"x2"
	],
	"why": "operand-order variant (xy+zx)/x -> y+z",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"1",
	"tanh",
	"x0",
	"-",
	"1",
	"tanh",
	"x0"
	],
	"target": [
	"exp",
	"mult2",
	"atanh",
	"tanh",
	"x0"
	],
	"why": "(1+tanh)/(1-tanh) = exp(2x)",
	"family": 2,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"/",
	"+",
	"exp",
	"mult2",
	"x0",
	"1",
	"mult2",
	"exp",
	"x0"
	],
	"target": [
	"cosh",
	"x0"
	],
	"why": "(exp(2x)+1)/(2 exp x) = cosh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"mult2",
	"x0",
	"mult2",
	"x1",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"<constant>"
	],
	"why": "expanded variant (2x+2y)/(x+y) -> 2",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"+",
	"pow3",
	"x0",
	"x0",
	"+",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"x0"
	],
	"why": "(x^3+x)/(x^2+1) = x",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"x0",
	"x1",
	"-",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"target": [
	"inv",
	"-",
	"x0",
	"x1"
	],
	"why": "(x+y)/(x^2-y^2) = 1/(x-y)",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"*",
	"x0",
	"x1",
	"*",
	"x0",
	"x2",
	"x0"
	],
	"target": [
	"-",
	"x1",
	"x2"
	],
	"why": "(xy-xz)/x -> y-z",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"*",
	"x0",
	"x1",
	"*",
	"x1",
	"x0",
	"x2"
	],
	"target": [
	"0"
	],
	"why": "(xy - yx)/z -> 0, commutativity zero at depth",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"1",
	"cos",
	"mult2",
	"x0",
	"sin",
	"mult2",
	"x0"
	],
	"target": [
	"tan",
	"x0"
	],
	"why": "(1-cos2x)/sin2x=tan",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"1",
	"cos",
	"x0",
	"+",
	"1",
	"cos",
	"x0"
	],
	"target": [
	"pow2",
	"tan",
	"div2",
	"x0"
	],
	"why": "(1-cos)/(1+cos)=tan^2(x/2)",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"1",
	"pow2",
	"x0",
	"+",
	"1",
	"pow2",
	"x0"
	],
	"target": [
	"cos",
	"mult2",
	"atan",
	"x0"
	],
	"why": "(1-x^2)/(1+x^2)=cos(2 atan x)",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"cosh",
	"mult2",
	"x0",
	"1",
	"sinh",
	"mult2",
	"x0"
	],
	"target": [
	"tanh",
	"x0"
	],
	"why": "(cosh2x-1)/sinh2x = tanh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"exp",
	"mult2",
	"x0",
	"1",
	"mult2",
	"exp",
	"x0"
	],
	"target": [
	"sinh",
	"x0"
	],
	"why": "(exp(2x)-1)/(2 exp x) = sinh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"exp",
	"x0",
	"1",
	"+",
	"exp",
	"x0",
	"1"
	],
	"target": [
	"tanh",
	"div2",
	"x0"
	],
	"why": "(e^x-1)/(e^x+1) = tanh(x/2), half-argument",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow2",
	"x0",
	"1",
	"+",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"tanh",
	"log",
	"abs",
	"x0"
	],
	"why": "(x^2-1)/(x^2+1) = tanh(log\|x\|), holds even at x=0 (tanh(-inf)=-1)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow2",
	"x0",
	"pow2",
	"x1",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"-",
	"x0",
	"x1"
	],
	"why": "(x^2-y^2)/(x+y) = x-y",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow2",
	"x0",
	"pow2",
	"x1",
	"+",
	"x1",
	"x0"
	],
	"target": [
	"-",
	"x0",
	"x1"
	],
	"why": "(x^2-y^2)/(y+x) = x-y, commuted denominator",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow2",
	"x0",
	"pow2",
	"x1",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"+",
	"x0",
	"x1"
	],
	"why": "(x^2-y^2)/(x-y) = x+y",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow2",
	"x0",
	"pow2",
	"x1",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"-",
	"neg",
	"x0",
	"x1"
	],
	"why": "(x^2-y^2)/(y-x) = -(x+y)",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"-",
	"pow2",
	"x1",
	"pow2",
	"x0",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"-",
	"x1",
	"x0"
	],
	"why": "(y^2-x^2)/(x+y) = y-x",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow2",
	"x1",
	"pow2",
	"x0",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"+",
	"x0",
	"x1"
	],
	"why": "variable-swapped variant (y^2-x^2)/(y-x) -> x+y",
	"family": 4,
	"rounds": [
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	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow3",
	"x0",
	"x0",
	"-",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"x0"
	],
	"why": "(x^3-x)/(x^2-1) = x",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow4",
	"x0",
	"1",
	"+",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"+",
	"(-1)",
	"pow2",
	"x0"
	],
	"why": "(x^4-1)/(x^2+1) = x^2-1",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"-",
	"pow4",
	"x0",
	"1",
	"-",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"+",
	"1",
	"pow2",
	"x0"
	],
	"why": "(x^4-1)/(x^2-1) = x^2+1",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"-",
	"x0",
	"x1",
	"-",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"target": [
	"inv",
	"+",
	"x0",
	"x1"
	],
	"why": "(x-y)/(x^2-y^2) = 1/(x+y)",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"cos",
	"mult2",
	"x0",
	"-",
	"cos",
	"x0",
	"sin",
	"x0"
	],
	"target": [
	"+",
	"cos",
	"x0",
	"sin",
	"x0"
	],
	"why": "BORDERLINE (5-token target): cos2x/(cos-sin)=cos+sin",
	"family": 1,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"/",
	"cosh",
	"-",
	"x0",
	"x1",
	"cosh",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"1"
	],
	"why": "cosh(x0-x1)/cosh(x1-x0) -> 1 (cosh>=1, division safe)",
	"family": 8,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"log",
	"pow",
	"x0",
	"x1",
	"*",
	"x1",
	"log",
	"x0"
	],
	"target": [
	"1"
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	"why": "log(x^y)/(y log x) = 1 (9 tokens)",
	"family": 3,
	"rounds": [
	"round2"
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"mult2",
	"tan",
	"x0",
	"+",
	"1",
	"pow2",
	"tan",
	"x0"
	],
	"target": [
	"sin",
	"mult2",
	"x0"
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	"why": "sin2x rational-in-tan form",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"mult2",
	"tan",
	"x0",
	"-",
	"1",
	"pow2",
	"tan",
	"x0"
	],
	"target": [
	"tan",
	"mult2",
	"x0"
	],
	"why": "tan double angle 2t/(1-t^2)",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"mult2",
	"tanh",
	"x0",
	"+",
	"1",
	"pow2",
	"tanh",
	"x0"
	],
	"target": [
	"tanh",
	"mult2",
	"x0"
	],
	"why": "2tanh/(1+tanh^2) = tanh 2x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow2",
	"+",
	"x0",
	"x1",
	"pow2",
	"+",
	"x1",
	"x0"
	],
	"target": [
	"1"
	],
	"why": "(x0+x1)^2/(x1+x0)^2 -> 1",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow2",
	"-",
	"x0",
	"x1",
	"pow2",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"1"
	],
	"why": "(x0-x1)^2/(x1-x0)^2 -> 1",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow2",
	"tan",
	"x0",
	"+",
	"1",
	"pow2",
	"tan",
	"x0"
	],
	"target": [
	"pow2",
	"sin",
	"x0"
	],
	"why": "tan^2/(1+tan^2)=sin^2",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow3",
	"+",
	"x0",
	"x1",
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"+",
	"x0",
	"x1"
	],
	"why": "(x+y)^3/(x+y)^2 = x+y",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow4",
	"+",
	"x0",
	"x1",
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "(x+y)^4/(x+y)^2 = (x+y)^2",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"sin",
	"+",
	"x0",
	"x1",
	"cos",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"tan",
	"+",
	"x0",
	"x1"
	],
	"why": "sin(u)/cos(u) -> tan(u) with symmetric u",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"sin",
	"mult2",
	"x0",
	"+",
	"1",
	"cos",
	"mult2",
	"x0"
	],
	"target": [
	"tan",
	"x0"
	],
	"why": "sin2x/(1+cos2x)=tan",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"sinh",
	"+",
	"x0",
	"x1",
	"cosh",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"tanh",
	"+",
	"x0",
	"x1"
	],
	"why": "sinh(u)/cosh(u) -> tanh(u) with symmetric u",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"sinh",
	"mult2",
	"x0",
	"+",
	"cosh",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"tanh",
	"x0"
	],
	"why": "sinh2x/(cosh2x+1) = tanh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"asinh",
	"mult2",
	"*",
	"x0",
	"pow1_2",
	"+",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"mult2",
	"asinh",
	"x0"
	],
	"why": "asinh(2x sqrt(x^2+1)) = 2 asinh x, double-argument of asinh",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div2",
	"log",
	"/",
	"+",
	"1",
	"x0",
	"-",
	"1",
	"x0"
	],
	"target": [
	"atanh",
	"x0"
	],
	"why": "log((1+x)/(1-x))/2 = atanh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div2",
	"log",
	"/",
	"+",
	"x0",
	"1",
	"-",
	"x0",
	"1"
	],
	"target": [
	"atanh",
	"inv",
	"x0"
	],
	"why": "log((x+1)/(x-1))/2 = acoth x = atanh(1/x)",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"log",
	"pow1_2",
	"/",
	"+",
	"1",
	"x0",
	"-",
	"1",
	"x0"
	],
	"target": [
	"atanh",
	"x0"
	],
	"why": "log(sqrt((1+x)/(1-x))) = atanh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"neg",
	"*",
	"neg",
	"x0",
	"*",
	"neg",
	"x1",
	"neg",
	"x2"
	],
	"target": [
	"*",
	"*",
	"x0",
	"x1",
	"x2"
	],
	"why": "four negs cancel through triple product (borderline 5-token target)",
	"family": 6,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"neg",
	"log",
	"-",
	"pow1_2",
	"+",
	"pow2",
	"x0",
	"1",
	"x0"
	],
	"target": [
	"asinh",
	"x0"
	],
	"why": "-log(sqrt(x^2+1)-x) = asinh x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"+",
	"+",
	"pow4",
	"x0",
	"mult2",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"+",
	"1",
	"pow2",
	"x0"
	],
	"why": "sqrt(x^4+2x^2+1) = x^2+1 (always positive, no abs)",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"*",
	"+",
	"x0",
	"1",
	"+",
	"-",
	"pow2",
	"x0",
	"x0",
	"1"
	],
	"target": [
	"+",
	"1",
	"pow3",
	"x0"
	],
	"why": "sum of cubes factorization (x+1)(x^2-x+1) = x^3+1",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"*",
	"-",
	"x0",
	"1",
	"+",
	"+",
	"pow2",
	"x0",
	"x0",
	"1"
	],
	"target": [
	"+",
	"(-1)",
	"pow3",
	"x0"
	],
	"why": "difference of cubes (x-1)(x^2+x+1) = x^3-1",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"+",
	"+",
	"pow2",
	"x0",
	"*",
	"mult2",
	"x0",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "binomial square expansion with cross term written (2x)*y",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"+",
	"pow2",
	"x0",
	"*",
	"x0",
	"mult2",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "binomial square expansion with cross term written x*(2y)",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"+",
	"pow2",
	"x0",
	"mult2",
	"*",
	"x0",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "binomial square expansion (x^2+2xy)+y^2, left-nested",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"+",
	"pow2",
	"x0",
	"pow2",
	"x1",
	"mult2",
	"*",
	"x0",
	"x1"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "binomial square expansion (x^2+y^2)+2xy, squares grouped first",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"-",
	"pow2",
	"x0",
	"mult2",
	"*",
	"x0",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
	"pow2",
	"-",
	"x0",
	"x1"
	],
	"why": "square of difference (x^2-2xy)+y^2",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"mult2",
	"*",
	"x0",
	"x1",
	"+",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "binomial square expansion 2xy+(x^2+y^2), cross term first",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow2",
	"x0",
	"+",
	"mult2",
	"*",
	"x0",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "binomial square expansion x^2+(2xy+y^2), right-nested",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow2",
	"x1",
	"+",
	"mult2",
	"*",
	"x0",
	"x1",
	"pow2",
	"x0"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "binomial square expansion y^2+(2xy+x^2), variables swapped ordering",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow2",
	"x1",
	"+",
	"pow2",
	"x0",
	"mult2",
	"*",
	"x0",
	"x1"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "binomial, operand-order variant y^2+(x^2+2xy)",
	"family": 5,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow2",
	"x1",
	"-",
	"pow2",
	"x0",
	"mult2",
	"*",
	"x0",
	"x1"
	],
	"target": [
	"pow2",
	"-",
	"x0",
	"x1"
	],
	"why": "square of difference y^2+(x^2-2xy)",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"+",
	"pow2",
	"x0",
	"pow2",
	"x1",
	"mult2",
	"*",
	"x0",
	"x1"
	],
	"target": [
	"pow2",
	"-",
	"x0",
	"x1"
	],
	"why": "square of difference (x^2+y^2)-2xy",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"+",
	"pow2",
	"x0",
	"pow2",
	"x1",
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"*",
	"mult2",
	"neg",
	"x0",
	"x1"
	],
	"why": "x^2+y^2-(x+y)^2 = -2xy, borderline",
	"family": 0,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"-",
	"+",
	"pow2",
	"x0",
	"pow2",
	"x1",
	"pow2",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"*",
	"mult2",
	"x0",
	"x1"
	],
	"why": "x^2+y^2-(x-y)^2 = 2xy",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"-",
	"+",
	"pow2",
	"x1",
	"pow2",
	"x0",
	"mult2",
	"*",
	"x0",
	"x1"
	],
	"target": [
	"pow2",
	"-",
	"x0",
	"x1"
	],
	"why": "square of difference (y^2+x^2)-2xy, swapped squares",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"-",
	"pow2",
	"+",
	"x0",
	"1",
	"pow2",
	"x0",
	"mult2",
	"x0"
	],
	"target": [
	"1"
	],
	"why": "(x+1)^2 - x^2 - 2x -> 1",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"-",
	"pow2",
	"+",
	"x0",
	"x1",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"target": [
	"*",
	"mult2",
	"x0",
	"x1"
	],
	"why": "(x+y)^2-x^2-y^2 = 2xy",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"-",
	"log",
	"*",
	"x0",
	"x1",
	"+",
	"log",
	"x0",
	"log",
	"x1"
	],
	"target": [
	"0"
	],
	"why": "log(xy) - (log x + log y) = 0 (10 tokens)",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"mult4",
	"*",
	"x0",
	"x1",
	"*",
	"mult2",
	"x0",
	"mult2",
	"x1"
	],
	"target": [
	"0"
	],
	"why": "4x0x1 - (2x0)(2x1) -> 0: scalar-distribution witness",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"pow2",
	"+",
	"x0",
	"x1",
	"+",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"target": [
	"*",
	"mult2",
	"x0",
	"x1"
	],
	"why": "(x+y)^2-(x^2+y^2) = 2xy, grouped subtrahend",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"-",
	"pow2",
	"-",
	"x0",
	"x1",
	"+",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"target": [
	"*",
	"mult2",
	"neg",
	"x0",
	"x1"
	],
	"why": "(x-y)^2-(x^2+y^2) = -2xy, borderline 5-token target",
	"family": 0,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"-",
	"pow2",
	"x0",
	"-",
	"mult2",
	"*",
	"x0",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
	"pow2",
	"-",
	"x0",
	"x1"
	],
	"why": "square of difference x^2-(2xy-y^2)",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"*",
	"x0",
	"x1",
	"pow2",
	"x0",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"x0"
	],
	"why": "numerator-order variant (xy+x^2)/(x+y) -> x",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"*",
	"x0",
	"x1",
	"pow2",
	"x1",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"x1"
	],
	"why": "(xy+y^2)/(x+y) = y",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"log",
	"x0",
	"log",
	"x1",
	"log",
	"*",
	"x0",
	"x1"
	],
	"target": [
	"1"
	],
	"why": "(log x + log y)/log(xy) = 1 (10 tokens)",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"pow2",
	"x0",
	"*",
	"x0",
	"x1",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"x0"
	],
	"why": "(x^2+xy)/(x+y) = x",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"pow2",
	"x0",
	"*",
	"x0",
	"x1",
	"+",
	"x1",
	"x0"
	],
	"target": [
	"x0"
	],
	"why": "denominator-order variant (x^2+xy)/(y+x) -> x",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"pow3",
	"x0",
	"*",
	"pow2",
	"x0",
	"x1",
	"pow2",
	"x0"
	],
	"target": [
	"+",
	"x0",
	"x1"
	],
	"why": "(x^3+x^2 y)/x^2 -> x+y",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"*",
	"x0",
	"x1",
	"pow2",
	"x1",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"x1"
	],
	"why": "(xy-y^2)/(x-y) -> y",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow2",
	"x0",
	"*",
	"x0",
	"x1",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"x0"
	],
	"why": "(x^2-xy)/(x-y) -> x",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow3",
	"x0",
	"x0",
	"*",
	"x0",
	"+",
	"x0",
	"1"
	],
	"target": [
	"+",
	"(-1)",
	"x0"
	],
	"why": "(x^3-x)/(x(x+1)) = x-1",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"-",
	"pow3",
	"x0",
	"x0",
	"*",
	"x0",
	"-",
	"x0",
	"1"
	],
	"target": [
	"+",
	"1",
	"x0"
	],
	"why": "(x^3-x)/(x(x-1)) = x+1",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"-",
	"pow4",
	"x0",
	"pow2",
	"x0",
	"-",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"pow2",
	"x0"
	],
	"why": "(x^4-x^2)/(x^2-1) = x^2",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"exp",
	"+",
	"x0",
	"x1",
	"*",
	"exp",
	"x0",
	"exp",
	"x1"
	],
	"target": [
	"1"
	],
	"why": "e^(x+y)/(e^x e^y) = 1 (9 tokens)",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"pow2",
	"*",
	"x0",
	"x1",
	"*",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"target": [
	"1"
	],
	"why": "(x0x1)^2 / (x0^2x1^2) -> 1",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"sin",
	"+",
	"x0",
	"x1",
	"*",
	"cos",
	"x0",
	"cos",
	"x1"
	],
	"target": [
	"+",
	"tan",
	"x0",
	"tan",
	"x1"
	],
	"why": "BORDERLINE (5-token target): sin(x+y)/(cos x cos y)=tan x+tan y",
	"family": 1,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"/",
	"sinh",
	"+",
	"x0",
	"x1",
	"*",
	"cosh",
	"x0",
	"cosh",
	"x1"
	],
	"target": [
	"+",
	"tanh",
	"x0",
	"tanh",
	"x1"
	],
	"why": "sinh(x+y)/(cosh cosh) = tanh x + tanh y; borderline 5-token target",
	"family": 2,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"/",
	"sinh",
	"-",
	"x0",
	"x1",
	"*",
	"cosh",
	"x0",
	"cosh",
	"x1"
	],
	"target": [
	"-",
	"tanh",
	"x0",
	"tanh",
	"x1"
	],
	"why": "sinh(x-y)/(cosh cosh) = tanh x - tanh y; borderline 5-token target",
	"family": 2,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"div2",
	"+",
	"pow2",
	"+",
	"x0",
	"x1",
	"pow2",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"+",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"why": "((x+y)^2+(x-y)^2)/2 = x^2+y^2, borderline 5-token target",
	"family": 0,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"div2",
	"-",
	"log",
	"+",
	"1",
	"x0",
	"log",
	"-",
	"1",
	"x0"
	],
	"target": [
	"atanh",
	"x0"
	],
	"why": "(log(1+x)-log(1-x))/2 = atanh x, expanded-log variant",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div4",
	"-",
	"pow2",
	"+",
	"x0",
	"x1",
	"pow2",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"*",
	"x0",
	"x1"
	],
	"why": "((x+y)^2-(x-y)^2)/4 = xy",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"mult4",
	"*",
	"*",
	"sin",
	"x0",
	"cos",
	"x0",
	"cos",
	"mult2",
	"x0"
	],
	"target": [
	"sin",
	"mult4",
	"x0"
	],
	"why": "4 sin cos cos2x=sin4x",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"/",
	"-",
	"cosh",
	"x0",
	"1",
	"+",
	"cosh",
	"x0",
	"1"
	],
	"target": [
	"abs",
	"tanh",
	"div2",
	"x0"
	],
	"why": "sqrt((cosh-1)/(cosh+1)) = \|tanh(x/2)\|",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"+",
	"sin",
	"x0",
	"cos",
	"x0",
	"-",
	"sin",
	"x0",
	"cos",
	"x0"
	],
	"target": [
	"neg",
	"cos",
	"mult2",
	"x0"
	],
	"why": "(sin+cos)(sin-cos)=-cos2x",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"-",
	"cos",
	"x0",
	"sin",
	"x0",
	"+",
	"cos",
	"x0",
	"sin",
	"x0"
	],
	"target": [
	"cos",
	"mult2",
	"x0"
	],
	"why": "factored (cos-sin)(cos+sin)=cos2x",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"*",
	"-",
	"pow2",
	"x0",
	"pow2",
	"x1",
	"+",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"target": [
	"-",
	"pow4",
	"x0",
	"pow4",
	"x1"
	],
	"why": "(x^2-y^2)(x^2+y^2) = x^4-y^4, borderline",
	"family": 0,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"*",
	"exp",
	"x0",
	"*",
	"exp",
	"x1",
	"exp",
	"neg",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"1"
	],
	"why": "e^x e^y e^-(x+y) = 1 (11 tokens)",
	"family": 3,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"*",
	"cos",
	"x0",
	"cos",
	"x1",
	"*",
	"sin",
	"x0",
	"sin",
	"x1"
	],
	"target": [
	"cos",
	"-",
	"x0",
	"x1"
	],
	"why": "cos subtraction formula",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"*",
	"cos",
	"x1",
	"sin",
	"x0",
	"*",
	"sin",
	"x1",
	"cos",
	"x0"
	],
	"target": [
	"sin",
	"+",
	"x0",
	"x1"
	],
	"why": "sin addition, factors reordered",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"*",
	"sin",
	"x0",
	"cos",
	"x0",
	"*",
	"cos",
	"x0",
	"sin",
	"x0"
	],
	"target": [
	"sin",
	"mult2",
	"x0"
	],
	"why": "sin cos + cos sin = sin 2x (addition-formula shape at y=x)",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"*",
	"sin",
	"x0",
	"cos",
	"x1",
	"*",
	"cos",
	"x0",
	"sin",
	"x1"
	],
	"target": [
	"sin",
	"+",
	"x0",
	"x1"
	],
	"why": "sin addition formula collapses",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"*",
	"sin",
	"x0",
	"sin",
	"x1",
	"*",
	"cos",
	"x0",
	"cos",
	"x1"
	],
	"target": [
	"cos",
	"-",
	"x0",
	"x1"
	],
	"why": "cos subtraction, terms swapped",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"+",
	"+",
	"pow3",
	"x0",
	"mult3",
	"pow2",
	"x0",
	"mult3",
	"x0",
	"1"
	],
	"target": [
	"pow3",
	"+",
	"1",
	"x0"
	],
	"why": "x^3+3x^2+3x+1 = (x+1)^3",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"+",
	"+",
	"pow3",
	"x0",
	"1",
	"mult3",
	"*",
	"x0",
	"+",
	"x0",
	"1"
	],
	"target": [
	"pow3",
	"+",
	"1",
	"x0"
	],
	"why": "partial-factored cube x^3+1+3x(x+1) = (x+1)^3",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"+",
	"-",
	"mult5",
	"x0",
	"mult2",
	"x0",
	"-",
	"mult4",
	"x0",
	"mult3",
	"x0"
	],
	"target": [
	"mult4",
	"x0"
	],
	"why": "constant-ladder: (5x-2x)+(4x-3x) = 4x, 11-token binomial-class ladder",
	"family": 7,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"-",
	"pow2",
	"x0",
	"pow2",
	"x1",
	"-",
	"pow2",
	"x1",
	"pow2",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "(x0^2-x1^2)+(x1^2-x0^2) -> 0",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"/",
	"sin",
	"x0",
	"cos",
	"x0",
	"/",
	"cos",
	"x0",
	"sin",
	"x0"
	],
	"target": [
	"inv",
	"div2",
	"sin",
	"mult2",
	"x0"
	],
	"why": "BORDERLINE (5-token target): tan+cot=2/sin2x",
	"family": 1,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"+",
	"/",
	"x0",
	"+",
	"x0",
	"x1",
	"/",
	"x1",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"1"
	],
	"why": "x/(x+y) + y/(x+y) -> 1",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"/",
	"x0",
	"+",
	"x0",
	"x1",
	"/",
	"x1",
	"+",
	"x1",
	"x0"
	],
	"target": [
	"1"
	],
	"why": "denominator commuted variant",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"/",
	"x0",
	"-",
	"x0",
	"x1",
	"/",
	"x1",
	"-",
	"x1",
	"x0"
	],
	"target": [
	"1"
	],
	"why": "x0/(x0-x1)+x1/(x1-x0) -> 1: antisymmetric denominators",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"/",
	"x1",
	"+",
	"x0",
	"x1",
	"/",
	"x0",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"1"
	],
	"why": "terms swapped variant",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow2",
	"sin",
	"*",
	"x0",
	"x1",
	"pow2",
	"cos",
	"*",
	"x0",
	"x1"
	],
	"target": [
	"1"
	],
	"why": "sin^2+cos^2 -> 1 with u=x0*x1",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow2",
	"sin",
	"+",
	"x0",
	"x1",
	"pow2",
	"cos",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"1"
	],
	"why": "sin^2(u)+cos^2(u) -> 1 with u=x0+x1",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow2",
	"sin",
	"-",
	"x0",
	"x1",
	"pow2",
	"cos",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"1"
	],
	"why": "sin^2+cos^2 -> 1 with u=x0-x1",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"*",
	"+",
	"x0",
	"x1",
	"x2",
	"*",
	"x2",
	"+",
	"x1",
	"x0"
	],
	"target": [
	"0"
	],
	"why": "(x0+x1)x2 - x2(x1+x0) -> 0: commutativity at two levels",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"*",
	"cos",
	"x0",
	"cos",
	"x1",
	"*",
	"sin",
	"x0",
	"sin",
	"x1"
	],
	"target": [
	"cos",
	"+",
	"x0",
	"x1"
	],
	"why": "cos addition formula",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"*",
	"sin",
	"x0",
	"cos",
	"x1",
	"*",
	"cos",
	"x0",
	"sin",
	"x1"
	],
	"target": [
	"sin",
	"-",
	"x0",
	"x1"
	],
	"why": "sin subtraction formula",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"+",
	"-",
	"pow3",
	"x0",
	"mult3",
	"pow2",
	"x0",
	"mult3",
	"x0",
	"1"
	],
	"target": [
	"pow3",
	"+",
	"(-1)",
	"x0"
	],
	"why": "x^3-3x^2+3x-1 = (x-1)^3",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"-",
	"+",
	"pow3",
	"x0",
	"mult3",
	"x0",
	"+",
	"mult3",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"pow3",
	"+",
	"(-1)",
	"x0"
	],
	"why": "(x^3+3x)-(3x^2+1)=(x-1)^3",
	"family": 5,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"-",
	"/",
	"x0",
	"-",
	"x0",
	"x1",
	"/",
	"x1",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"1"
	],
	"why": "x0/(x0-x1)-x1/(x0-x1) -> 1",
	"family": 8,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"pow2",
	"+",
	"pow2",
	"x0",
	"1",
	"pow2",
	"-",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"mult4",
	"pow2",
	"x0"
	],
	"why": "(x^2+1)^2-(x^2-1)^2 = 4x^2",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"pow2",
	"inv",
	"cos",
	"x0",
	"pow2",
	"/",
	"sin",
	"x0",
	"cos",
	"x0"
	],
	"target": [
	"1"
	],
	"why": "sec^2-tan^2=1 with tan spelled sin/cos",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"*",
	"+",
	"x0",
	"x1",
	"-",
	"x0",
	"x1",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"-",
	"x0",
	"x1"
	],
	"why": "factored variant ((x+y)(x-y))/(x+y) -> x-y",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"*",
	"+",
	"x0",
	"x1",
	"-",
	"x0",
	"x1",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"+",
	"x0",
	"x1"
	],
	"why": "factored variant ((x+y)(x-y))/(x-y) -> x+y",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"*",
	"-",
	"x0",
	"x1",
	"+",
	"x0",
	"x1",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"-",
	"x0",
	"x1"
	],
	"why": "factor-order variant ((x-y)(x+y))/(x+y) -> x-y",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"*",
	"x0",
	"x1",
	"*",
	"x0",
	"x2",
	"+",
	"x1",
	"x2"
	],
	"target": [
	"x0"
	],
	"why": "(xy+xz)/(y+z) -> x",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"+",
	"pow2",
	"x0",
	"mult2",
	"x0",
	"1",
	"+",
	"x0",
	"1"
	],
	"target": [
	"+",
	"1",
	"x0"
	],
	"why": "(x^2+2x+1)/(x+1) = x+1, expanded numerator",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"+",
	"-",
	"pow2",
	"x0",
	"mult2",
	"x0",
	"1",
	"-",
	"x0",
	"1"
	],
	"target": [
	"+",
	"<constant>",
	"x0"
	],
	"why": "(x^2-2x+1)/(x-1) = x-1",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"+",
	"pow3",
	"x0",
	"*",
	"pow2",
	"x0",
	"x1",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"pow2",
	"x0"
	],
	"why": "(x^3+x^2 y)/(x+y) -> x^2",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"pow3",
	"x0",
	"1",
	"+",
	"-",
	"pow2",
	"x0",
	"x0",
	"1"
	],
	"target": [
	"+",
	"1",
	"x0"
	],
	"why": "(x^3+1)/(x^2-x+1) = x+1",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"-",
	"1",
	"cos",
	"mult2",
	"x0",
	"+",
	"1",
	"cos",
	"mult2",
	"x0"
	],
	"target": [
	"pow2",
	"tan",
	"x0"
	],
	"why": "(1-cos2x)/(1+cos2x)=tan^2",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"1",
	"exp",
	"neg",
	"x0",
	"+",
	"1",
	"exp",
	"neg",
	"x0"
	],
	"target": [
	"tanh",
	"div2",
	"x0"
	],
	"why": "(1-e^-x)/(1+e^-x) = tanh(x/2), half-argument",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"1",
	"pow2",
	"tan",
	"x0",
	"+",
	"1",
	"pow2",
	"tan",
	"x0"
	],
	"target": [
	"cos",
	"mult2",
	"x0"
	],
	"why": "cos2x rational-in-tan form",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
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	"/",
	"-",
	"cosh",
	"mult2",
	"x0",
	"1",
	"+",
	"cosh",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"pow2",
	"tanh",
	"x0"
	],
	"why": "(cosh2x-1)/(cosh2x+1) = tanh^2 x",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"exp",
	"mult2",
	"x0",
	"1",
	"+",
	"exp",
	"mult2",
	"x0",
	"1"
	],
	"target": [
	"tanh",
	"x0"
	],
	"why": "(exp2x-1)/(exp2x+1) = tanh x",
	"family": 2,
	"rounds": [
	"round2"
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow2",
	"+",
	"x0",
	"x1",
	"pow2",
	"-",
	"x0",
	"x1",
	"x0"
	],
	"target": [
	"mult4",
	"x1"
	],
	"why": "((x+y)^2-(x-y)^2)/x = 4y, removable singularity",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow2",
	"+",
	"x0",
	"x1",
	"pow2",
	"-",
	"x0",
	"x1",
	"x1"
	],
	"target": [
	"mult4",
	"x0"
	],
	"why": "((x+y)^2-(x-y)^2)/y = 4x",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow3",
	"x0",
	"*",
	"pow2",
	"x0",
	"x1",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"pow2",
	"x0"
	],
	"why": "(x^3-x^2 y)/(x-y) -> x^2",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow3",
	"x0",
	"1",
	"+",
	"+",
	"pow2",
	"x0",
	"x0",
	"1"
	],
	"target": [
	"+",
	"(-1)",
	"x0"
	],
	"why": "(x^3-1)/(x^2+x+1) = x-1",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"-",
	"pow4",
	"x0",
	"pow4",
	"x1",
	"+",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"target": [
	"-",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"why": "(x^4-y^4)/(x^2+y^2) = x^2-y^2, borderline 5-token target",
	"family": 0,
	"rounds": [
	"round2_long"
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	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"/",
	"-",
	"pow4",
	"x0",
	"pow4",
	"x1",
	"-",
	"pow2",
	"x0",
	"pow2",
	"x1"
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	"target": [
	"+",
	"pow2",
	"x0",
	"pow2",
	"x1"
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	"family": 0,
	"rounds": [
	"round2_long"
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	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"/",
	"mult2",
	"tan",
	"div2",
	"x0",
	"+",
	"1",
	"pow2",
	"tan",
	"div2",
	"x0"
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	"target": [
	"sin",
	"x0"
	],
	"why": "Weierstrass: 2t/(1+t^2)=sin x, t=tan(x/2)",
	"family": 1,
	"rounds": [
	"round2"
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"mult2",
	"tan",
	"div2",
	"x0",
	"-",
	"1",
	"pow2",
	"tan",
	"div2",
	"x0"
	],
	"target": [
	"tan",
	"x0"
	],
	"why": "Weierstrass: 2t/(1-t^2)=tan x",
	"family": 1,
	"rounds": [
	"round2"
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	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"abs",
	"+",
	"pow2",
	"x0",
	"+",
	"mult2",
	"*",
	"x0",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "abs of expanded perfect square x^2+2xy+y^2 (nonnegative) drops abs",
	"family": 0,
	"rounds": [
	"round2"
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	"tier": "minimal",
	"hint_matched": true
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	{
	"source": [
	"div2",
	"-",
	"+",
	"pow2",
	"x0",
	"pow2",
	"x1",
	"pow2",
	"-",
	"x0",
	"x1"
	],
	"target": [
	"*",
	"x0",
	"x1"
	],
	"why": "((x^2+y^2)-(x-y)^2)/2 = xy",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"div2",
	"-",
	"pow2",
	"+",
	"x0",
	"x1",
	"+",
	"pow2",
	"x0",
	"pow2",
	"x1"
	],
	"target": [
	"*",
	"x0",
	"x1"
	],
	"why": "((x+y)^2-(x^2+y^2))/2 = xy",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"+",
	"+",
	"pow2",
	"x0",
	"mult2",
	"*",
	"x0",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
	"abs",
	"+",
	"x0",
	"x1"
	],
	"why": "sqrt(x^2+2xy+y^2)=\|x+y\|",
	"family": 5,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"+",
	"-",
	"pow2",
	"x0",
	"mult2",
	"*",
	"x0",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
	"abs",
	"-",
	"x0",
	"x1"
	],
	"why": "sqrt(x^2-2xy+y^2)=\|x-y\|",
	"family": 5,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"pow1_2",
	"+",
	"pow2",
	"x0",
	"+",
	"mult2",
	"*",
	"x0",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
	"abs",
	"+",
	"x0",
	"x1"
	],
	"why": "sqrt of expanded x^2+2xy+y^2 = \|x+y\|",
	"family": 0,
	"rounds": [
	"round2"
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	"tier": "minimal",
	"hint_matched": true
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	{
	"source": [
	"pow1_2",
	"-",
	"+",
	"pow2",
	"x0",
	"pow2",
	"x1",
	"mult2",
	"*",
	"x0",
	"x1"
	],
	"target": [
	"abs",
	"-",
	"x0",
	"x1"
	],
	"why": "sqrt of expanded x^2-2xy+y^2 = \|x-y\|",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
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	{
	"source": [
	"pow2",
	"+",
	"pow2",
	"x0",
	"+",
	"mult2",
	"*",
	"x0",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
	"pow4",
	"+",
	"x0",
	"x1"
	],
	"why": "(x^2+2xy+y^2)^2 = (x+y)^4",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
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	{
	"source": [
	"*",
	"*",
	"-",
	"x0",
	"1",
	"+",
	"x0",
	"1",
	"+",
	"pow2",
	"x0",
	"1"
	],
	"target": [
	"+",
	"(-1)",
	"pow4",
	"x0"
	],
	"why": "(x-1)(x+1)(x^2+1) = x^4-1, fully factored (12 tok)",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"-",
	"pow4",
	"x0",
	"1",
	"*",
	"-",
	"x0",
	"1",
	"+",
	"x0",
	"1"
	],
	"target": [
	"+",
	"1",
	"pow2",
	"x0"
	],
	"why": "(x^4-1)/((x-1)(x+1)) = x^2+1 (13 tok)",
	"family": 0,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": false
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	{
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	"/",
	"/",
	"-",
	"pow4",
	"x0",
	"1",
	"-",
	"x0",
	"1",
	"+",
	"x0",
	"1"
	],
	"target": [
	"+",
	"1",
	"pow2",
	"x0"
	],
	"why": "(x^4-1)/(x-1)/(x+1) = x^2+1 (13 tok)",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
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	"log",
	"+",
	"x0",
	"*",
	"pow1_2",
	"-",
	"x0",
	"1",
	"pow1_2",
	"+",
	"x0",
	"1"
	],
	"target": [
	"acosh",
	"x0"
	],
	"why": "log(x+sqrt(x-1)sqrt(x+1)) = acosh x, 12 tokens",
	"family": 2,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": true
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	{
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	"+",
	"+",
	"+",
	"pow3",
	"x0",
	"mult3",
	"pow2",
	"x0",
	"mult3",
	"x0",
	"1"
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	"target": [
	"+",
	"1",
	"x0"
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	"why": "cbrt of expanded (x+1)^3 = x+1",
	"family": 0,
	"rounds": [
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	"tier": "minimal",
	"hint_matched": false
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	{
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	"*",
	"+",
	"-",
	"pow2",
	"x0",
	"*",
	"x0",
	"x1",
	"pow2",
	"x1",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"+",
	"pow3",
	"x0",
	"pow3",
	"x1"
	],
	"why": "sum of cubes with factors commuted (13 tok), borderline",
	"family": 0,
	"rounds": [
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	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"*",
	"+",
	"x0",
	"x1",
	"+",
	"-",
	"pow2",
	"x0",
	"*",
	"x0",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
	"+",
	"pow3",
	"x0",
	"pow3",
	"x1"
	],
	"why": "sum of cubes (x+y)(x^2-xy+y^2) = x^3+y^3, borderline 5-token target (13 tok)",
	"family": 0,
	"rounds": [
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	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
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	"*",
	"-",
	"x0",
	"x1",
	"+",
	"+",
	"pow2",
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	"*",
	"x0",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
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	"pow3",
	"x0",
	"pow3",
	"x1"
	],
	"why": "difference of cubes (x-y)(x^2+xy+y^2) = x^3-y^3, borderline (13 tok)",
	"family": 0,
	"rounds": [
	"round2_long"
	],
	"tier": "minimal_le5",
	"hint_matched": null
	},
	{
	"source": [
	"*",
	"mult4",
	"*",
	"sin",
	"x0",
	"cos",
	"x0",
	"-",
	"1",
	"mult2",
	"pow2",
	"sin",
	"x0"
	],
	"target": [
	"sin",
	"mult4",
	"x0"
	],
	"why": "4 sin cos (1-2sin^2)=sin4x",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"+",
	"+",
	"pow2",
	"x0",
	"*",
	"x0",
	"x1",
	"*",
	"x1",
	"x0",
	"pow2",
	"x1"
	],
	"target": [
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	"+",
	"x0",
	"x1"
	],
	"why": "binomial square with 2xy split as xy+yx, left chain (13 tok)",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"-",
	"-",
	"pow2",
	"x0",
	"*",
	"x0",
	"x1",
	"*",
	"x0",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
	"pow2",
	"-",
	"x0",
	"x1"
	],
	"why": "square of difference with 2xy split as xy+xy (13 tok)",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow2",
	"+",
	"cosh",
	"x0",
	"sinh",
	"x0",
	"pow2",
	"-",
	"cosh",
	"x0",
	"sinh",
	"x0"
	],
	"target": [
	"mult2",
	"cosh",
	"mult2",
	"x0"
	],
	"why": "(cosh+sinh)^2+(cosh-sinh)^2 = 2cosh 2x, 13-token binomial",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow2",
	"+",
	"sin",
	"x0",
	"cos",
	"x0",
	"pow2",
	"-",
	"sin",
	"x0",
	"cos",
	"x0"
	],
	"target": [
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	],
	"why": "(sin+cos)^2+(sin-cos)^2=2",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"+",
	"pow2",
	"x0",
	"+",
	"*",
	"x0",
	"x1",
	"+",
	"*",
	"x0",
	"x1",
	"pow2",
	"x1"
	],
	"target": [
	"pow2",
	"+",
	"x0",
	"x1"
	],
	"why": "binomial square with 2xy split as xy+xy, right chain (13 tok)",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"-",
	"pow2",
	"+",
	"cosh",
	"x0",
	"sinh",
	"x0",
	"pow2",
	"-",
	"cosh",
	"x0",
	"sinh",
	"x0"
	],
	"target": [
	"mult2",
	"sinh",
	"mult2",
	"x0"
	],
	"why": "(cosh+sinh)^2-(cosh-sinh)^2 = 2sinh 2x, 13-token binomial",
	"family": 2,
	"rounds": [
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	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"*",
	"pow2",
	"x0",
	"x1",
	"*",
	"x0",
	"pow2",
	"x1",
	"*",
	"x0",
	"x1"
	],
	"target": [
	"+",
	"x0",
	"x1"
	],
	"why": "(x^2 y + x y^2)/(xy) -> x+y, 13-token binomial class",
	"family": 4,
	"rounds": [
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	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"*",
	"pow2",
	"x0",
	"x1",
	"*",
	"x0",
	"pow2",
	"x1",
	"+",
	"x0",
	"x1"
	],
	"target": [
	"*",
	"x0",
	"x1"
	],
	"why": "(x^2 y + x y^2)/(x+y) -> xy",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"+",
	"tan",
	"x0",
	"tan",
	"x1",
	"-",
	"1",
	"*",
	"tan",
	"x0",
	"tan",
	"x1"
	],
	"target": [
	"tan",
	"+",
	"x0",
	"x1"
	],
	"why": "tan sum formula (t0+t1)/(1-t0 t1)",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"*",
	"pow2",
	"x0",
	"x1",
	"*",
	"x0",
	"pow2",
	"x1",
	"*",
	"x0",
	"x1"
	],
	"target": [
	"-",
	"x0",
	"x1"
	],
	"why": "(x^2 y - x y^2)/(xy) -> x-y",
	"family": 4,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"1",
	"pow2",
	"tan",
	"div2",
	"x0",
	"+",
	"1",
	"pow2",
	"tan",
	"div2",
	"x0"
	],
	"target": [
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	"x0"
	],
	"why": "Weierstrass: (1-t^2)/(1+t^2)=cos x",
	"family": 1,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
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	{
	"source": [
	"/",
	"-",
	"exp",
	"x0",
	"exp",
	"neg",
	"x0",
	"+",
	"exp",
	"x0",
	"exp",
	"neg",
	"x0"
	],
	"target": [
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	"x0"
	],
	"why": "(e^x-e^-x)/(e^x+e^-x) = tanh x, 13-token binomial",
	"family": 2,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": true
	},
	{
	"source": [
	"/",
	"-",
	"pow2",
	"+",
	"x0",
	"x1",
	"pow2",
	"-",
	"x0",
	"x1",
	"*",
	"x0",
	"x1"
	],
	"target": [
	"<constant>"
	],
	"why": "((x+y)^2-(x-y)^2)/(xy) = 4, constant residue (13 tok)",
	"family": 0,
	"rounds": [
	"round2"
	],
	"tier": "minimal",
	"hint_matched": false
	},
	{
	"source": [
	"/",
	"-",
	"pow4",
	"x0",
	"1",
	"*",
	"+",
	"pow2",
	"x0",
	"1",
	"+",
	"x0",
	"1"
	],
	"target": [
	"+",
	"(-1)",
	"x0"
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