def tl.nzpick

import bend-osdev-gf@0.1.1.0/PROOF.bend as PROOF
import Base
import bend-osdev-gf@0.1.1.0/gf.bend as G
import bend-osdev-gf@0.1.1.0/tables.bend as T
import bend-osdev-array@0.1.0.1/array.bend as A
import bend-osdev-array@0.1.0.1/LAWS.bend as AL
import bend-osdev-gf@0.1.1.0/proof_tab.bend as GT

7873# both factors nonzero: the table's product, through its three reads
7874def tl.nzpick(+f: G.Gf8Field, +l: List<&2, U32>, +a: U32, +b: U32, +hl: {List.length(&2, U32, l) == A.Arrays.slots(8n) : Nat}, +hxa: {A.Arrays.nth(U32, l, A.Arrays.index(8n, G.Gf8.elt(a)), 0) == U32.from_nat(G.Gf8.log(f, a)) : U32}, +hxb: {A.Arrays.nth(U32, l, A.Arrays.index(8n, G.Gf8.elt(b)), 0) == U32.from_nat(G.Gf8.log(f, b)) : U32}, +ela: {G.Gf8.exp(f, G.Gf8.log(f, a)) == G.Gf8.elt(a) : U32}, +elb: {G.Gf8.exp(f, G.Gf8.log(f, b)) == G.Gf8.elt(b) : U32}) -> {T.gf8tab.two.pick(Bool.or(False{}, False{}), GT.gt.tab(f, l), G.Gf8.elt(a), G.Gf8.elt(b), False{}) == (GT.gt.tab(f, l), G.Gf8.mul(f, a, b)) : T.Gf8Tab & U32}:
7875  %Equal.sym(Array<U32> & U32, Array.get(U32, A.Arrays.of_list(U32, 8n, l, 0), G.Gf8.elt(a)), (A.Arrays.of_list(U32, 8n, l, 0), A.Arrays.nth(U32, l, A.Arrays.index(8n, G.Gf8.elt(a)), 0)), AL.of_list_get(U32, 8n, l, 0, hl, GT.gt.hd8(), G.Gf8.elt(a))) : {T.gf8tab.two.mid(A.Arrays.of_list(U32, 9n, GT.gt.elist(510n, f, 0n), 0), _, G.Gf8.elt(b), False{}) == (GT.gt.tab(f, l), G.Gf8.mul(f, a, b)) : T.Gf8Tab & U32}
7876  %Equal.sym(Array<U32> & U32, Array.get(U32, A.Arrays.of_list(U32, 8n, l, 0), G.Gf8.elt(b)), (A.Arrays.of_list(U32, 8n, l, 0), A.Arrays.nth(U32, l, A.Arrays.index(8n, G.Gf8.elt(b)), 0)), AL.of_list_get(U32, 8n, l, 0, hl, GT.gt.hd8(), G.Gf8.elt(b))) : {T.gf8tab.two.fin(A.Arrays.of_list(U32, 9n, GT.gt.elist(510n, f, 0n), 0), _, A.Arrays.nth(U32, l, A.Arrays.index(8n, G.Gf8.elt(a)), 0), False{}) == (GT.gt.tab(f, l), G.Gf8.mul(f, a, b)) : T.Gf8Tab & U32}
7877  %Equal.sym(Array<U32> & U32, Array.get(U32, A.Arrays.of_list(U32, 9n, GT.gt.elist(510n, f, 0n), 0), U32.add(A.Arrays.nth(U32, l, A.Arrays.index(8n, G.Gf8.elt(a)), 0), A.Arrays.nth(U32, l, A.Arrays.index(8n, G.Gf8.elt(b)), 0))), (A.Arrays.of_list(U32, 9n, GT.gt.elist(510n, f, 0n), 0), A.Arrays.nth(U32, GT.gt.elist(510n, f, 0n), A.Arrays.index(9n, U32.add(A.Arrays.nth(U32, l, A.Arrays.index(8n, G.Gf8.elt(a)), 0), A.Arrays.nth(U32, l, A.Arrays.index(8n, G.Gf8.elt(b)), 0))), 0)), AL.of_list_get(U32, 9n, GT.gt.elist(510n, f, 0n), 0, GT.gt.hl_elist(f), GT.gt.hd9(), U32.add(A.Arrays.nth(U32, l, A.Arrays.index(8n, G.Gf8.elt(a)), 0), A.Arrays.nth(U32, l, A.Arrays.index(8n, G.Gf8.elt(b)), 0)))) : {T.gf8tab.exp.fin(_, A.Arrays.of_list(U32, 8n, l, 0)) == (GT.gt.tab(f, l), G.Gf8.mul(f, a, b)) : T.Gf8Tab & U32}
7878  %Equal.sym(U32, A.Arrays.nth(U32, GT.gt.elist(510n, f, 0n), A.Arrays.index(9n, U32.add(A.Arrays.nth(U32, l, A.Arrays.index(8n, G.Gf8.elt(a)), 0), A.Arrays.nth(U32, l, A.Arrays.index(8n, G.Gf8.elt(b)), 0))), 0), G.Gf8.mul(f, a, b), tl.nz(f, l, a, b, hxa, hxb, ela, elb)) : {(T.Gf8Tab{A.Arrays.of_list(U32, 9n, GT.gt.elist(510n, f, 0n), 0), A.Arrays.of_list(U32, 8n, l, 0)}, _) == (GT.gt.tab(f, l), G.Gf8.mul(f, a, b)) : T.Gf8Tab & U32}
7879  {==}
Uses (34)

Used by (1)

bend-osdev-gf@0.1.1.0/PROOF:tl.pick