iteI_runsF (KFV §22: a sign test with arbitrary (call-containing) branches)
ProvedNLogN.KFV.iteI_runsFThe following holds (Lean statement, in the scope of module Solutions.KFV18Ite):
theorem NLogN.KFV.iteI_runsF {e : Ex HS} {a b : IStmt HS Pp} {s t : IState HS} {c : ℕ} (hc : e.Clean)
(ho : e.ok s) (hpos : ¬ e.eval s < 0) (hb : IRuns I iprog b s t c) :
IRuns I iprog (iteI e a b) s t (cIte e s + c)
Context. The lemma belongs to the part KFV §22: a sign test with arbitrary (call-containing) branches: iteI e a b is the shape of the compiled RS.ite with arbitrary item-language branches:
evaluate e, branch on its sign (a if e < 0). iteI_runsT, iteI_runsF.
Formalization Note. This lemma belongs to layer 5 of the Lean formalization (integer-multiplication-kappa campaign) of the OpenAI preprint Integer multiplication below : the uniform item-language machine (register programs over the NLogN multitape framework) assembling the Proposition 14 / Lemma 13 / Proposition 18 pieces, the resampling and chirp stages and the cost recurrences into one fixed machine, culminating in NLogN.KFTop.kappa_bound : IntMul.KappaBound (1 / 2 ^ 182) (the bound of Def_IntMul_MultitapeModel). All names refer to the definition bundles up to IntMul_Kappa_K_Defs33.
-- OpenAI-preprint / κ-improvement development (integer-multiplication-kappa), generated by skeleton subtraction.
import Definitions.Def_IntMul_Kappa_K_Defs33
set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000
set_option linter.unusedSimpArgs false
set_option linter.unusedVariables false
open NLogN
open NLogN.KFV
open IntMul RS KIL KFD
variable {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}
theorem NLogN.KFV.iteI_runsF {e : Ex HS} {a b : IStmt HS Pp} {s t : IState HS} {c : ℕ} (hc : e.Clean)
(ho : e.ok s) (hpos : ¬ e.eval s < 0) (hb : IRuns I iprog b s t c) :
IRuns I iprog (iteI e a b) s t (cIte e s + c) := by sorry