Nuprl Lemma : int-rsub-req
∀[k:ℤ]. ∀[x:ℝ]. (k - x = (r(k) - x))
Proof
Definitions occuring in Statement :
int-rsub: k - x,
rsub: x - y,
req: x = y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rsub: x - y,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
real: ℝ,
all: ∀x:A. B[x],
rminus: -(x),
int-to-real: r(n),
int-rsub: k - x,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
req-iff-bdd-diff,
int-rsub_wf,
radd_wf,
int-to-real_wf,
rminus_wf,
req_witness,
rsub_wf,
real_wf,
istype-int,
nat_plus_wf,
bdd-diff_weakening,
nat_plus_properties,
decidable__equal_int,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-less_than,
intformeq_wf,
itermSubtract_wf,
itermMultiply_wf,
itermAdd_wf,
itermMinus_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_term_value_minus_lemma,
bdd-diff_functionality,
radd-bdd-diff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
independent_functionElimination,
universeIsType,
sqequalRule,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
inhabitedIsType,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
addEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
functionExtensionality,
unionElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
independent_pairFormation
Latex:
\mforall{}[k:\mBbbZ{}]. \mforall{}[x:\mBbbR{}]. (k - x = (r(k) - x))
Date html generated:
2019_10_29-AM-09_32_07
Last ObjectModification:
2019_02_13-PM-01_08_27
Theory : reals
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