Nuprl Lemma : rat-cube-face-dimension-equal
∀[k:ℕ]. ∀[c:ℚCube(k)]. ∀f:ℚCube(k). (f = c ∈ ℚCube(k)) supposing ((dim(f) = dim(c) ∈ ℤ) and f ≤ c) supposing ↑Inhabited\000C(c)
Proof
Definitions occuring in Statement :
rat-cube-dimension: dim(c),
inhabited-rat-cube: Inhabited(c),
rat-cube-face: c ≤ d,
rational-cube: ℚCube(k),
nat: ℕ,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
decidable: Dec(P),
less_than': less_than'(a;b),
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
rat-cube-dimension: dim(c),
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ge: i ≥ j ,
squash: ↓T,
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
or: P ∨ Q,
rat-cube-face: c ≤ d,
guard: {T},
rational-cube: ℚCube(k),
prop: ℙ,
so_apply: x[s],
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
subtype_rel: A ⊆r B,
and: P ∧ Q,
uiff: uiff(P;Q),
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
le_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
assert_of_eq_int,
sum_le,
false_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
itermSubtract_wf,
itermAdd_wf,
subtract-is-int-iff,
eq_int_wf,
ifthenelse_wf,
sum_wf,
istype-less_than,
istype-le,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
itermConstant_wf,
intformle_wf,
decidable__le,
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
less_than_wf,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__lt,
rat-interval-dimension_wf,
istype-false,
int_seg_subtype_nat,
Error :isolate_summand2,
assert_of_bnot,
eqff_to_assert,
uiff_transitivity,
eqtt_to_assert,
not_wf,
bnot_wf,
assert_wf,
bool_wf,
equal-wf-T-base,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
istype-void,
int_formula_prop_and_lemma,
intformeq_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
int_seg_properties,
int_seg_wf,
rat-interval-face-dimension,
istype-nat,
rational-cube_wf,
inhabited-rat-cube_wf,
istype-assert,
rat-cube-face_wf,
int_subtype_base,
lelt_wf,
set_subtype_base,
rat-cube-dimension_wf,
istype-int,
assert-inhabited-rat-cube,
inhabited-rat-cube-face
Rules used in proof :
applyLambdaEquality,
hyp_replacement,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
productIsType,
dependent_set_memberEquality_alt,
universeEquality,
instantiate,
imageMemberEquality,
equalityElimination,
baseClosed,
independent_pairFormation,
voidElimination,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
equalityTransitivity,
imageElimination,
unionElimination,
functionExtensionality,
functionIsTypeImplies,
universeIsType,
inhabitedIsType,
isectIsTypeImplies,
axiomEquality,
isect_memberEquality_alt,
equalitySymmetry,
sqequalBase,
rename,
setElimination,
addEquality,
natural_numberEquality,
minusEquality,
lambdaEquality_alt,
intEquality,
sqequalRule,
applyEquality,
equalityIstype,
independent_isectElimination,
productElimination,
because_Cache,
dependent_functionElimination,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
lambdaFormation_alt,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[c:\mBbbQ{}Cube(k)]. \mforall{}f:\mBbbQ{}Cube(k). (f = c) supposing ((dim(f) = dim(c)) and f \mleq{} c) supposing \muparrow{}Inha\000Cbited(c)
Date html generated:
2019_10_29-AM-07_52_39
Last ObjectModification:
2019_10_18-AM-11_05_23
Theory : rationals
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