def caseGT
bend-collections-laws-math@1.0.0.0 · proofs/math/typed/f64addf.bend:132
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64addf.bend as F64addf
import Base
import bend-collections-laws-math@1.0.0.0/spec/lib/common.bend as C
import bend-collections-laws-math@1.0.0.0/spec/math/f64.bend as SF
import bend-collections-laws-math@1.0.0.0/spec/math/w64.bend as SW
import bend-collections-laws-math@1.0.0.0/src/math/f64.bend as F
import bend-collections-laws-math@1.0.0.0/src/math/w64.bend as X
import bend-collections-laws-math@1.0.0.0/src/math/u64.bend as WU
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/natcmp.bend as NC
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64round.bend as FR
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64addg.bend as AG
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64addh.bend as AH
132def caseGT(+one: Nat, +h1: {one == 1n : Nat}, +x: F.F64, +y: F.F64, +sa: Bool, +sb: Bool, +ea: Nat, +eb: Nat, +Fa: Nat, +Fb: Nat, +hfa: {SW.value(F.frac(x)) == Fa : Nat}, +hfb: {SW.value(F.frac(y)) == Fb : Nat}, +hFa: {C.fits(52n, Fa) == True{} : Bool}, +hFb: {C.fits(52n, Fb) == True{} : Bool}, +ha: {Nat.is_eq(ea, 2047n) == False{} : Bool}, +hgt: {Nat.is_lt(eb, ea) == True{} : Bool}, +sg: Bool, +hsg: {Bool.not(Bool.xor(sa, sb)) == sg : Bool}) -> {SF.pick(F.F64, sg, F.am_case(x, y, sa, ea, eb, GT{}), F.sm_case(x, y, sa, ea, eb, GT{})) == SF.add_mag(sa, C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), sb, AG.mt(eb, Fb), AG.xe(eb)) : F.F64}:
133 match sg:
134 case True{}:
135 +i1 = Equal.cong(Bool, F.F64, t => F.am_big(x, sa, ea, F.frac(x), eb, F.frac(y), t), Nat.is_eq(ea, 2047n), False{}, ha)
136 +i2 = AH.bigA(one, h1, sa, ea, F.frac(x), eb, F.frac(y), Fa, Fb, hfa, hfb, hFa, hFb, hgt)
137 +i3 = Equal.cong(Nat, F.F64, z => SF.round(sa, z, AG.xe(eb)), Nat.add(AG.mt(eb, Fb), C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa))), Nat.add(C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), AG.mt(eb, Fb)), NA.add_comm(AG.mt(eb, Fb), C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa))))
138 +sp = Equal.cong(Bool, F.F64, t => SF.pick(F.F64, t, SF.round(sa, Nat.add(C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), AG.mt(eb, Fb)), AG.xe(eb)), SF.sub_mag(sa, C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), sb, AG.mt(eb, Fb), AG.xe(eb), Nat.cmp(C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), AG.mt(eb, Fb)))), Bool.not(Bool.xor(sa, sb)), True{}, hsg)
139 +imp = Equal.trans(F.F64, F.am_case(x, y, sa, ea, eb, GT{}), F.rp62(sa, Nat.add(F.off(), ea), X.add(X.add(WU.U64{0, 536870912}, X.shl(F.frac(x), 9n)), X.shr_jam(F.am_small(eb, X.shl(F.frac(y), 9n)), Nat.sub(ea, eb)))), SF.round(sa, Nat.add(C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), AG.mt(eb, Fb)), AG.xe(eb)), i1, Equal.trans(F.F64, F.rp62(sa, Nat.add(F.off(), ea), X.add(X.add(WU.U64{0, 536870912}, X.shl(F.frac(x), 9n)), X.shr_jam(F.am_small(eb, X.shl(F.frac(y), 9n)), Nat.sub(ea, eb)))), SF.round(sa, Nat.add(AG.mt(eb, Fb), C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa))), AG.xe(eb)), SF.round(sa, Nat.add(C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), AG.mt(eb, Fb)), AG.xe(eb)), i2, i3))
140 Equal.trans(F.F64, SF.pick(F.F64, True{}, F.am_case(x, y, sa, ea, eb, GT{}), F.sm_case(x, y, sa, ea, eb, GT{})), F.am_case(x, y, sa, ea, eb, GT{}), SF.add_mag(sa, C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), sb, AG.mt(eb, Fb), AG.xe(eb)), FR.pk_t(F.F64, F.am_case(x, y, sa, ea, eb, GT{}), F.sm_case(x, y, sa, ea, eb, GT{})), Equal.trans(F.F64, F.am_case(x, y, sa, ea, eb, GT{}), SF.round(sa, Nat.add(C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), AG.mt(eb, Fb)), AG.xe(eb)), SF.add_mag(sa, C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), sb, AG.mt(eb, Fb), AG.xe(eb)), imp, Equal.sym(F.F64, SF.add_mag(sa, C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), sb, AG.mt(eb, Fb), AG.xe(eb)), SF.round(sa, Nat.add(C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), AG.mt(eb, Fb)), AG.xe(eb)), sp)))
141 case False{}:
142 +i1 = Equal.cong(Bool, F.F64, t => F.sm_big(x, sa, ea, F.frac(x), eb, F.frac(y), t), Nat.is_eq(ea, 2047n), False{}, ha)
143 +i2 = AH.bigS(one, h1, sa, ea, F.frac(x), eb, F.frac(y), Fa, Fb, hfa, hfb, hFa, hFb, hgt)
144 +c1 = AG.bigcmp(one, h1, ea, eb, Fa, Fb, hFa, hFb, hgt)
145 +c2 = Equal.trans(Cmp, Nat.cmp(C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), AG.mt(eb, Fb)), SF.flip(Nat.cmp(AG.mt(eb, Fb), C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)))), GT{}, Equal.sym(Cmp, SF.flip(Nat.cmp(AG.mt(eb, Fb), C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)))), Nat.cmp(C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), AG.mt(eb, Fb)), NC.cmp_flip(AG.mt(eb, Fb), C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)))), Equal.cong(Cmp, Cmp, t => SF.flip(t), Nat.cmp(AG.mt(eb, Fb), C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa))), LT{}, c1))
146 +sp1 = Equal.cong(Bool, F.F64, t => SF.pick(F.F64, t, SF.round(sa, Nat.add(C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), AG.mt(eb, Fb)), AG.xe(eb)), SF.sub_mag(sa, C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), sb, AG.mt(eb, Fb), AG.xe(eb), Nat.cmp(C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), AG.mt(eb, Fb)))), Bool.not(Bool.xor(sa, sb)), False{}, hsg)
147 +sp2 = Equal.cong(Cmp, F.F64, t => SF.sub_mag(sa, C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), sb, AG.mt(eb, Fb), AG.xe(eb), t), Nat.cmp(C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), AG.mt(eb, Fb)), GT{}, c2)
148 +imp = Equal.trans(F.F64, F.sm_case(x, y, sa, ea, eb, GT{}), F.norm_round_pack(sa, Nat.add(F.off(), Nat.sub(ea, 1n)), X.sub(X.add(X.shl(F.frac(x), 10n), WU.U64{0, 1073741824}), X.shr_jam(F.sm_small(eb, X.shl(F.frac(y), 10n)), Nat.sub(ea, eb)))), SF.round(sa, Nat.sub(C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), AG.mt(eb, Fb)), AG.xe(eb)), i1, i2)
149 +spc = Equal.trans(F.F64, SF.add_mag(sa, C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), sb, AG.mt(eb, Fb), AG.xe(eb)), SF.sub_mag(sa, C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), sb, AG.mt(eb, Fb), AG.xe(eb), Nat.cmp(C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), AG.mt(eb, Fb))), SF.round(sa, Nat.sub(C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), AG.mt(eb, Fb)), AG.xe(eb)), sp1, sp2)
150 Equal.trans(F.F64, SF.pick(F.F64, False{}, F.am_case(x, y, sa, ea, eb, GT{}), F.sm_case(x, y, sa, ea, eb, GT{})), F.sm_case(x, y, sa, ea, eb, GT{}), SF.add_mag(sa, C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), sb, AG.mt(eb, Fb), AG.xe(eb)), FR.pk_f(F.F64, F.am_case(x, y, sa, ea, eb, GT{}), F.sm_case(x, y, sa, ea, eb, GT{})), Equal.trans(F.F64, F.sm_case(x, y, sa, ea, eb, GT{}), SF.round(sa, Nat.sub(C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), AG.mt(eb, Fb)), AG.xe(eb)), SF.add_mag(sa, C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), sb, AG.mt(eb, Fb), AG.xe(eb)), imp, Equal.sym(F.F64, SF.add_mag(sa, C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), sb, AG.mt(eb, Fb), AG.xe(eb)), SF.round(sa, Nat.sub(C.shift(Nat.sub(AG.xe(ea), AG.xe(eb)), AG.mt(ea, Fa)), AG.mt(eb, Fb)), AG.xe(eb)), spc)))