Beyond shadow of doubt

to be uncertain about; consider questionable or unlikely; hesitate to believe.
to distrust.
Archaic. to fear; be apprehensive about.
to be uncertain about something; be undecided in opinion or belief.
a feeling of uncertainty about the truth, reality, or nature of something.
distrust.
a state of affairs such as to occasion uncertainty.
Obsolete. fear; dread.
beyond the shadow of a doubt, with certainty; definitely.
Also, beyond a doubt, beyond doubt.
in doubt, in a state of uncertainty or suspense:
His appointment to the position is still in doubt.
no doubt,

probably.
certainly:
There is no doubt an element of truth in what you say.

without doubt, unquestionably; certainly.
noun
uncertainty about the truth, fact, or existence of something (esp in the phrases in doubt, without doubt, beyond a shadow of doubt, etc)
(often pl) lack of belief in or conviction about something: all his doubts about the project disappeared
an unresolved difficulty, point, etc
(philosophy) the methodical device, esp in the philosophy of Descartes, of identifying certain knowledge as the residue after rejecting any proposition which might, however improbably, be false
(obsolete) fear
give someone the benefit of the doubt, to presume someone suspected of guilt to be innocent; judge leniently
no doubt, almost certainly
verb
(transitive; may take a clause as object) to be inclined to disbelieve: I doubt we are late
(transitive) to distrust or be suspicious of: he doubted their motives
(intransitive) to feel uncertainty or be undecided
(transitive; may take a clause as object) (Scot) to be inclined to believe
(transitive) (archaic) to fear
(Irish) I wouldn’t doubt someone, I would expect nothing else from someone
v.
n.
see:

beyond a doubt
cast doubt on
give the benefit of the doubt
no doubt
shadow of a doubt