Nuprl Lemma : inhabited-upper-rc-face

k:ℕ. ∀c:ℚCube(k). ∀j:ℕk.  ((↑Inhabited(c j))  Inhabited(upper-rc-face(c;j)) Inhabited(c))


Proof




Definitions occuring in Statement :  upper-rc-face: upper-rc-face(c;j) inhabited-rat-cube: Inhabited(c) rational-cube: Cube(k) inhabited-rat-interval: Inhabited(I) int_seg: {i..j-} nat: assert: b bool: 𝔹 all: x:A. B[x] implies:  Q apply: a natural_number: $n equal: t ∈ T
Definitions unfolded in proof :  false: False assert: b bnot: ¬bb exists: x:A. B[x] uiff: uiff(P;Q) btrue: tt it: unit: Unit bool: 𝔹 bfalse: ff ifthenelse: if then else fi  true: True nequal: a ≠ b ∈  guard: {T} sq_type: SQType(T) or: P ∨ Q decidable: Dec(P) nat: subtype_rel: A ⊆B squash: T less_than: a < b le: A ≤ B lelt: i ≤ j < k prop: rev_implies:  Q pi2: snd(t) rational-interval: Interval rational-cube: Cube(k) int_seg: {i..j-} and: P ∧ Q iff: ⇐⇒ Q upper-rc-face: upper-rc-face(c;j) uimplies: supposing a member: t ∈ T uall: [x:A]. B[x] implies:  Q all: x:A. B[x]
Lemmas referenced :  neg_assert_of_eq_int assert-bnot bool_cases_sqequal eqff_to_assert assert_of_tt inhabited-rat-point-interval assert_of_eq_int eqtt_to_assert iff_weakening_equal bfalse_wf eq_int_eq_false istype-universe true_wf squash_wf equal_wf bool_subtype_base bool_wf int_subtype_base subtype_base_sq decidable__equal_int assert-inhabited-rat-cube subtype_rel_self int_seg_wf assert_wf iff_weakening_uiff rat-point-interval_wf rational-interval_wf eq_int_wf ifthenelse_wf inhabited-rat-interval_wf istype-assert upper-rc-face_wf inhabited-rat-cube_wf iff_imp_equal_bool
Rules used in proof :  voidElimination dependent_pairFormation_alt equalityElimination baseClosed imageMemberEquality universeEquality lambdaEquality_alt intEquality cumulativity instantiate unionElimination natural_numberEquality universeIsType promote_hyp imageElimination functionEquality independent_functionElimination dependent_functionElimination equalitySymmetry equalityTransitivity equalityIstype productElimination inhabitedIsType applyEquality rename setElimination functionIsType because_Cache independent_pairFormation sqequalRule independent_isectElimination hypothesis hypothesisEquality thin isectElimination sqequalHypSubstitution extract_by_obid introduction cut lambdaFormation_alt sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}k:\mBbbN{}.  \mforall{}c:\mBbbQ{}Cube(k).  \mforall{}j:\mBbbN{}k.    ((\muparrow{}Inhabited(c  j))  {}\mRightarrow{}  Inhabited(upper-rc-face(c;j))  =  Inhabited(c))



Date html generated: 2019_10_29-AM-07_56_52
Last ObjectModification: 2019_10_17-PM-05_14_21

Theory : rationals


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