image and preimage of a set under an expression - #1442
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Item 6 of the reference's docket (#1409), the half that needs no pair. image(f(x), x in A) is { f(x) : x in A }, which is union({f(x)}, x in A): listed over a listed A, an interval by interval arithmetic where x occurs once in f -- exact for one occurrence and the operations that have images (#1423); the reference's 9c/5 + 32 on (0, 100) is (32, 212) -- and otherwise the family, which answers membership through the quantifiers. preimage(f(x), x in A, Y) is { x in A : f(x) in Y }, and a set builder of that shape is solved on evaluation where the statement solver reads the membership (a listed Y, an interval; #1440), the solutions cut by A: the pre-images of x^2 (Ex 7.3.10), with the one of (0, 1) excluding 0 where the book prints (-1, 1). That shape only: a set builder is not solved on evaluation in general, that being a search on every evaluation. The performance baseline moves to this run. ParseHard reached +3.2% over the baseline recorded at 86af577, and it is the ten function tokens added to the grammar since -- about 12 KB/op each, whatever the input, measured by removing the two of this change and nothing else (3,727,130 to 3,703,015 B/op): a cost in the ANTLR runtime's per-parse work, filed as #1441. Every other row is within 1% of the old baseline. #1409 Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
This was referenced Sep 21, 2026
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Item 6 of the docket on #1409, the half that needs no pair (the pair, and with it a function as a set of pairs and item 5, waits for v3's point per your note there).
image(f(x), x in A)is{ f(x) : x in A }— which isunion({f(x)}, x in A)(#1435), so nothing new in the tree:Aimage(x^2, x in {1, 2, 3})is{1, 4, 9}(Ex 7.3.2's table as an expression)xonce infimage(9 c / 5 + 32, c in (0; 100))is(32; 212)(Ex 7.3.3),image(x^2, x in [-1; 2])is[0; 4]1/2 in image(x^2/(1 + x^2), x in RR)isTrue(Ex 7.3.4 — two occurrences, so no arithmetic)preimage(f(x), x in A, Y)is{ x in A : f(x) in Y }, and a set builder of exactly that shape is solved on evaluation where the statement solver reads the membership (#1440):preimage(x^2, x in RR, {1})is{1, -1},preimage(x^2, x in ZZ, {1, 4})is{1, -1, 2, -2},preimage(2 x, x in ZZ+, [1; 5])is{1, 2}, and the pre-image of(0; 1)underx^2excludes0where the book's Ex 7.3.10 prints(-1, 1). A pre-image the solver cannot read stays the builder it is and answers membership itself (2 in preimage(x^2, x in RR, ZZ)). Only that shape: a set builder is not solved on evaluation in general — that would be a search on every evaluation.MathS.Sets.Image,MathS.Sets.PreImage;image,preimageare keywords now. Syntax.md, BREAKING-CHANGES.md (rows measured on a v2.5.0 build), PublicApi.txt, parser regenerated and post-processed.ImageAndPreImageTest, 14 rows. Full suite 12226 passed, 0 failed; benchmark gate: the baseline moves to this run.ParseHardreached +3.2% over the baseline recorded at86af5774, and it is the ten function tokens added to the grammar since — about 12 KB/op each, whatever the input, measured by removing the two of this PR and nothing else (3,727,130 → 3,703,015 B/op): a cost in the ANTLR runtime's per-parse work, filed as #1441. Every other row is within 1% of the old baseline (SimplifyHard +1.0% since #1432, reported there).Needs #1440 (merged) for the pre-images.
Part of #1409.
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https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura