Nuprl Lemma : member-rat-cube-faces
∀k:ℕ. ∀c:ℚCube(k).
∀f:ℚCube(k). ((f ∈ rat-cube-faces(k;c)) ⇐⇒ f ≤ c ∧ (dim(f) = (dim(c) - 1) ∈ ℤ)) supposing ↑Inhabited(c)
Proof
Definitions occuring in Statement :
rat-cube-faces: rat-cube-faces(k;c),
rat-cube-dimension: dim(c),
inhabited-rat-cube: Inhabited(c),
rat-cube-face: c ≤ d,
rational-cube: ℚCube(k),
l_member: (x ∈ l),
nat: ℕ,
assert: ↑b,
uimplies: b supposing a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
sq_stable: SqStable(P),
guard: {T},
squash: ↓T,
true: True,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
cand: A c∧ B,
exists: ∃x:A. B[x],
l_member: (x ∈ l),
rev_implies: P ⇐ Q,
so_apply: x[s],
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
prop: ℙ,
subtype_rel: A ⊆r B,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x]
Lemmas referenced :
decidable__rat-cube-face,
sq_stable_from_decidable,
istype-universe,
equal_wf,
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
select_wf,
istype-nat,
istype-assert,
implies-member-rat-cube-faces,
subtract_wf,
int_subtype_base,
istype-int,
lelt_wf,
set_subtype_base,
rat-cube-dimension_wf,
equal-wf-base,
rat-cube-face_wf,
subtype_rel_list,
rat-cube-faces_wf,
rational-cube_wf,
l_member_wf,
inhabited-rat-cube_wf,
assert_witness
Rules used in proof :
promote_hyp,
universeEquality,
instantiate,
baseClosed,
imageMemberEquality,
imageElimination,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
unionElimination,
dependent_set_memberEquality_alt,
productElimination,
dependent_functionElimination,
sqequalBase,
equalityIstype,
productIsType,
setIsType,
because_Cache,
equalitySymmetry,
equalityTransitivity,
inhabitedIsType,
setElimination,
addEquality,
natural_numberEquality,
minusEquality,
lambdaEquality_alt,
sqequalRule,
intEquality,
productEquality,
setEquality,
applyEquality,
independent_isectElimination,
universeIsType,
independent_pairFormation,
rename,
independent_functionElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}c:\mBbbQ{}Cube(k).
\mforall{}f:\mBbbQ{}Cube(k). ((f \mmember{} rat-cube-faces(k;c)) \mLeftarrow{}{}\mRightarrow{} f \mleq{} c \mwedge{} (dim(f) = (dim(c) - 1)))
supposing \muparrow{}Inhabited(c)
Date html generated:
2019_10_29-AM-07_57_34
Last ObjectModification:
2019_10_18-PM-06_36_16
Theory : rationals
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