Etymology
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heap (v.)
Old English heapian "collect, heap up, bring together;" from heap (n.). Related: Heaped; heaping. Compare Old High German houfon, German haufen "to heap," also a verb from a noun.
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heap (n.)

Old English heap "pile (of things); great number, crowd, multitude (of persons)," from West Germanic *haupaz (source also of Old Saxon hop, Old Frisian hap, Middle Low German hupe, Dutch hoop, German Haufe "heap"), of uncertain origin. The group is perhaps related to Old English heah "high" (see high), but OED suggests a common origin with Latin cubare "lie down," and Boutkan says it is probably not Indo-European at all.

Slang meaning "old car" is attested from 1924. Earlier it meant "slovenly woman" (1806). As a characteristic word in American Indian English speech, "a lot, a great deal," by 1832.

One grain of sand does not make a heap. A second grain of sand added to the first does not make a heap. Indeed each and every grain of sand, when added to the others, does not make a heap which was not a heap before. Therefore, all the grains of sand in existence can still not a heap make. [the fallacy of the heap, as described in Malcolm Murray and Nebojsa Kujundzic, "Critical Reflection," 2005]
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up (n.)
"that which is up," 1530s, from up (adv.). Phrase on the up-(and-up) "honest, straightforward" first attested 1863, American English.
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up (adv.)

Old English up, uppe, from Proto-Germanic *upp- "up" (source also of Old Frisian, Old Saxon up "up, upward," Old Norse upp; Danish, Dutch op; Old High German uf, German auf "up"; Gothic iup "up, upward," uf "on, upon, under;" Old High German oba, German ob "over, above, on, upon"), from PIE root *upo "under," also "up from under," hence also "over."

As a preposition, "to a higher place" from c. 1500; also "along, through" (1510s), "toward" (1590s). Often used elliptically for go up, come up, rise up, etc. Up the river "in jail" first recorded 1891, originally in reference to Sing Sing, which is up the Hudson from New York City. To drive someone up the wall (1951) is from the notion of the behavior of lunatics or caged animals. Insulting retort up yours (scil. ass) is attested by late 19c.

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up- 
prefix with various senses, from Old English up (adv.), corresponding to similar prefixes in other Germanic languages.
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up (v.)

1550s, "to drive and catch (swans)," from up (adv.). Intransitive meaning "get up, rise to one's feet" (as in up and leave) is recorded from 1640s. Sense of "to move upward" is recorded from 1737. Meaning "increase" (as in up the price of oil) is attested from 1915. Compare Old English verb uppian "to rise up, swell." Related: Upped; upping. Upping block, used for mounting or dismounting horses, carriages, etc., is attested from 1796 (earlier was horsing-block, 1660s).

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up (adj.)
c. 1300, "dwelling inland or upland," from up (adv.). Meaning "going up" is from 1784. From 1815 as "excited, exhilarated, happy," hence "enthusiastic, optimistic." Up-and-coming "promising" is from 1848. Musical up-tempo (adj.) is recorded from 1948.
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scrap-heap (n.)

1803, "place where scrap is collected," later especially "place in a railroad yard, etc., where old iron is collected," from scrap (n.1) + heap (n.).

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cover-up (n.)

also coverup, "means or act of concealing" some event or activity, 1922, from the verbal phrase (1872), from cover (v.) + up (adv.).

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