------------------------------------------------------------------------------ -- -- -- GNAT RUN-TIME COMPONENTS -- -- -- -- S Y S T E M . W I D T H _ U -- -- -- -- B o d y -- -- -- -- Copyright (C) 1992-2024, Free Software Foundation, Inc. -- -- -- -- GNAT is free software; you can redistribute it and/or modify it under -- -- terms of the GNU General Public License as published by the Free Soft- -- -- ware Foundation; either version 3, or (at your option) any later ver- -- -- sion. GNAT is distributed in the hope that it will be useful, but WITH- -- -- OUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY -- -- or FITNESS FOR A PARTICULAR PURPOSE. -- -- -- -- As a special exception under Section 7 of GPL version 3, you are granted -- -- additional permissions described in the GCC Runtime Library Exception, -- -- version 3.1, as published by the Free Software Foundation. -- -- -- -- You should have received a copy of the GNU General Public License and -- -- a copy of the GCC Runtime Library Exception along with this program; -- -- see the files COPYING3 and COPYING.RUNTIME respectively. If not, see -- -- . -- -- -- -- GNAT was originally developed by the GNAT team at New York University. -- -- Extensive contributions were provided by Ada Core Technologies Inc. -- -- -- ------------------------------------------------------------------------------ package body System.Width_U is -- Ghost code, loop invariants and assertions in this unit are meant for -- analysis only, not for run-time checking, as it would be too costly -- otherwise. This is enforced by setting the assertion policy to Ignore. pragma Assertion_Policy (Ghost => Ignore, Loop_Invariant => Ignore, Assert => Ignore, Assert_And_Cut => Ignore, Subprogram_Variant => Ignore); function Width (Lo, Hi : Uns) return Natural is -- Ghost code, loop invariants and assertions in this unit are meant for -- analysis only, not for run-time checking, as it would be too costly -- otherwise. This is enforced by setting the assertion policy to -- Ignore. pragma Assertion_Policy (Ghost => Ignore, Loop_Invariant => Ignore, Assert => Ignore); ------------------ -- Local Lemmas -- ------------------ procedure Lemma_Lower_Mult (A, B, C : Big_Natural) with Ghost, Pre => A <= B, Post => A * C <= B * C; procedure Lemma_Div_Commutation (X, Y : Uns) with Ghost, Pre => Y /= 0, Post => Big (X) / Big (Y) = Big (X / Y); procedure Lemma_Div_Twice (X : Big_Natural; Y, Z : Big_Positive) with Ghost, Post => X / Y / Z = X / (Y * Z); procedure Lemma_Euclidian (V, Q, F, R : Big_Integer) with Ghost, Pre => F > 0 and then Q = V / F and then R = V rem F, Post => V = Q * F + R; -- Ghost lemma to prove the relation between the quotient/remainder of -- division by F and the value V. ---------------------- -- Lemma_Lower_Mult -- ---------------------- procedure Lemma_Lower_Mult (A, B, C : Big_Natural) is null; --------------------------- -- Lemma_Div_Commutation -- --------------------------- procedure Lemma_Div_Commutation (X, Y : Uns) is null; --------------------- -- Lemma_Div_Twice -- --------------------- procedure Lemma_Div_Twice (X : Big_Natural; Y, Z : Big_Positive) is XY : constant Big_Natural := X / Y; YZ : constant Big_Natural := Y * Z; XYZ : constant Big_Natural := X / Y / Z; R : constant Big_Natural := (XY rem Z) * Y + (X rem Y); begin pragma Assert (X = XY * Y + (X rem Y)); pragma Assert (XY = XY / Z * Z + (XY rem Z)); pragma Assert (X = XYZ * YZ + R); pragma Assert ((XY rem Z) * Y <= (Z - 1) * Y); pragma Assert (R <= YZ - 1); pragma Assert (X / YZ = (XYZ * YZ + R) / YZ); pragma Assert (X / YZ = XYZ + R / YZ); end Lemma_Div_Twice; --------------------- -- Lemma_Euclidian -- --------------------- procedure Lemma_Euclidian (V, Q, F, R : Big_Integer) is null; -- Local variables W : Natural; T : Uns; -- Local ghost variables Max_W : constant Natural := Max_Log10 with Ghost; Pow : Big_Integer := 1 with Ghost; T_Init : constant Uns := Uns'Max (Lo, Hi) with Ghost; -- Start of processing for System.Width_U begin if Lo > Hi then return 0; else -- Minimum value is 2, one for space, one for digit W := 2; -- Get max of absolute values T := Uns'Max (Lo, Hi); -- Increase value if more digits required while T >= 10 loop Lemma_Div_Commutation (T, 10); Lemma_Div_Twice (Big (T_Init), Big_10 ** (W - 2), Big_10); T := T / 10; W := W + 1; Pow := Pow * 10; pragma Loop_Invariant (W in 3 .. Max_W + 2); pragma Loop_Invariant (Pow = Big_10 ** (W - 2)); pragma Loop_Invariant (Big (T) = Big (T_Init) / Pow); pragma Loop_Variant (Decreases => T); end loop; declare F : constant Big_Integer := Big_10 ** (W - 2) with Ghost; Q : constant Big_Integer := Big (T_Init) / F with Ghost; R : constant Big_Integer := Big (T_Init) rem F with Ghost; begin pragma Assert (Q < Big_10); Lemma_Euclidian (Big (T_Init), Q, F, R); Lemma_Lower_Mult (Q, Big (9), F); pragma Assert (Big (T_Init) <= Big (9) * F + F - 1); pragma Assert (Big (T_Init) < Big_10 * F); pragma Assert (Big_10 * F = Big_10 ** (W - 1)); end; return W; end if; end Width; end System.Width_U;