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shares outstanding: Weighted Average Number of Shares Outstanding, Diluted [Abstract] Revenue, Remaining Performance Obligation, Expected Timing of Satisfaction [Table] Revenue, Remaining Performance Obligation, Expected Timing of Satisfaction [Table] Amendment Flag Amendment Flag Ordinary shares which have been reserved for issuance (in shares) Share-Based Compensation Arrangement by Share-Based Payment Award, Number of Shares Authorized 2027 Long-Term Debt, Maturity, Year Two Operating leases Lessee, Operating Leases [Text Block] Net cash used in financing activities Cash Provided by (Used in) Financing Activity, Including Discontinued Operation Unbilled revenue Increase (Decrease) in Contract with Customer, Asset Weighted average discount rate Operating Lease, Weighted Average Discount Rate, Percent Summary of Stock Option Activity Share-Based Payment Arrangement, Option, Activity [Table Text Block] Interest and related penalties Unrecognized Tax Benefits, Income Tax Penalties and Interest Accrued Fair Value, Assets Measured on Recurring Basis Fair Value, Assets Measured on Recurring Basis [Table Text Block] Basic (in shares) Weighted average number of ordinary shares outstanding for basic net income per ordinary share (in shares) Weighted Average Number of Shares Outstanding, Basic Approximate dollar value of shares that may yet be purchased under the program (in millions) Stock repurchase program, remaining authorized amount Share Repurchase Program, Remaining Authorized, Amount Weighted Average Remaining Contractual Life Share-based Compensation Arrangement by Share-based Payment Award, Options, Weighted Average Remaining Contractual Term [Abstract] Share-based Compensation Arrangement by Share-based Payment Award, Options, Weighted Average Remaining Contractual Term [Abstract] Direct costs Cost of Goods and Service, Excluding Depreciation, Depletion, and Amortization Schedule of Reconciliation of Number of Shares Used in Computation of Basic and Diluted Net Income per Ordinary Share Schedule of Earnings Per Share, Basic and Diluted [Table Text Block] Exercisable at end of period (in shares) Number of shares exercisable (in shares) Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Exercisable, Number Subsequent Events [Abstract] Segment Reporting [Abstract] Segment Reporting [Abstract] Debt Disclosure [Abstract] Risk-free interest rate Share-Based Compensation Arrangement by Share-Based Payment Award, Fair Value Assumptions, Risk Free Interest Rate Amortization of financing costs and debt discount Amortization of Debt Issuance Costs Debt discount paid Debt Instrument, Unamortized Discount Outstanding at beginning of period (in USD per share) Outstanding at end of period (in USD per share) Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Outstanding, Weighted Average Exercise Price Restructuring and Related Activities [Abstract] Restructuring and Related Activities [Abstract] Share-based Compensation Arrangement by Share-based Payment Award [Line Items] Share-Based Compensation Arrangement by Share-Based Payment Award [Line Items] Senior Secured Notes (the "2027 Notes")* Senior Secured Notes Due May 2027 [Member] Senior Secured Notes Due May 2027 Ordinary shares, par value (in EUR cents per share) Common Stock, Par or Stated Value Per Share Ordinary shares, shares issued (in shares) Beginning balance (in shares) Ending balance (in shares) Common Stock, Shares, Issued Income taxes receivable Income Taxes Receivable, Current Weighted average remaining lease term Operating Lease, Weighted Average Remaining Lease Term Unbilled revenue Unbilled services (unbilled revenue) Contract with Customer, Asset, before Allowance for Credit Loss, Current Granted (in shares) Share-Based Compensation Arrangement by Share-Based Payment Award, Options, Grants in Period, Gross Variable Rate Component [Domain] Variable Rate Component [Domain] Variable Rate Component [Domain] Percentage change in unearned revenue (payments on account) Increase (Decrease) In Contract With Customer, Liability, Gross, Current, Percent Increase (Decrease) In Contract with Customer, Liability, Gross, Current, Percent Schedule of Goodwill Schedule of Goodwill [Table Text Block] Interest expense Interest Expense, Nonoperating Subsequent events Subsequent Events [Text Block] Currency translation adjustment Other Comprehensive Income (Loss), Foreign Currency Transaction and Translation Adjustment, Net of Tax 2025 (remaining) Long-Term Debt, Maturity, Remainder of Fiscal Year Total Liabilities Liabilities Debt instrument, maximum borrowing capacity Line of Credit Facility, Maximum Borrowing Capacity Increase related to current period tax positions Unrecognized Tax Benefits, Increase Resulting from Current Period Tax Positions Other liabilities Other Liabilities, Current Statement [Line Items] Statement [Line Items] Consolidation Items [Axis] Consolidation Items [Axis] Variable Rate [Axis] Variable Rate [Axis] Revenue, Remaining Performance Obligation, Expected Timing of Satisfaction [Line Items] Revenue, Remaining Performance Obligation, Expected Timing of Satisfaction [Line Items] Summary of Details of Movement in Restructuring Charges Schedule of Restructuring Reserve by Type of Cost [Table Text Block] Cash and cash equivalents Cash and Cash Equivalent Change in unbilled services (unbilled revenue) Increase (Decrease) In Contract With Customer, Asset, Gross, Current Increase (Decrease) In Contract With Customer, Asset, Gross, Current Additional ordinary shares which have been reserved for issuance (in shares) Share-Based Compensation Arrangement by Share-Based Payment Award, Number of Additional Shares Authorized Ordinary Shares Common Stock [Member] Shares vested (in shares) Share-Based Compensation Arrangement by Share-Based Payment Award, Equity Instruments Other than Options, Vested in Period Intangible assets Goodwill and Intangible Assets, Intangible Assets, Policy [Policy Text Block] Net (decrease) / increase in cash and cash equivalents Cash, Cash Equivalent, Restricted Cash, and Restricted Cash Equivalent, Period Increase (Decrease), Including Exchange Rate Effect and Discontinued Operation Effect of dilutive share options and other awards outstanding under share based compensation programs (in shares) Weighted Average Number of Shares Outstanding, Diluted, Adjustment Other Liabilities