An infinite base has its whole powers, and surjectivity is decided through the image - #1445
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…rough the image (-oo)^3 was NaN -- a claim that the value does not exist -- and so were (-oo)^2 and (-oo)^(-1), while (+oo)^2 was +oo: the finite-base arms of Pow did not apply and the polar form had nothing to say. They are the extended reals' powers now, by parity for -oo and 0 for a negative exponent, so the image of (-oo; +oo) under a cube is (-oo; +oo), and the reals are read as that interval where a family's image is computed. With it, forall b in B : exists a in A : f(a) = b -- surjectivity onto B, Def 7.4.1 of the reference of #1409 -- is decided as B lying in the image of A under f, where that image evaluates: the cube is onto RR, the square is not and is onto [0; +oo), the exponential is onto (0; +oo), doubling is not onto ZZ. Item 6's remaining analytic row. #1409 Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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Item 6's remaining analytic row — surjectivity — and a
NaNon the way.(-oo)^3wasNaN, and so were(-oo)^2and(-oo)^(-1), while(+oo)^2was+oo:Pow's finite-base arms did not apply to an infinite base and the polar form had nothing to say.NaNis a claim that the value does not exist, and these exist in the extended reals:-oo,+oo(by the parity of the exponent) and0for a negative one. The image of(-oo; +oo)under a cube is then(-oo; +oo), andRRis read as that interval where a family's image is computed (image(x^3, x in RR)is(-oo; +oo)).Surjectivity.
forall b in B : exists a in A : f(a) = b(Def 7.4.1) is the statement thatBlies in the image ofAunderf, and the quantifier decision now puts it to the subset decision (#1432) with the image (#1442) where that image evaluates:forall b in RR : exists a in RR : a^3 = bTrueforall b in RR : exists a in RR : a^2 = bFalse(the witness route had it)forall b in [0; +oo) : exists a in RR : a^2 = bTrueforall b in (0; +oo) : exists a in RR : e^a = bTrue; ontoRR:Falseforall b in ZZ : exists a in ZZ : 2 a = bFalseforall b in {1, 4, 9} : exists a in {1, 2, 3} : a^2 = bTrue; with16inB:FalseSurjectivityTest, 15 rows. BREAKING-CHANGES.md rows measured on a v2.5.0 build (NaNthere). Full suite 12247 passed, 0 failed; benchmark gate PASSED.Part of #1409.
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