Nuprl Lemma : rsub-rmin
∀[x,y,z:ℝ]. ((x - rmin(y;z)) = rmax(x - y;x - z))
Proof
Definitions occuring in Statement :
rmin: rmin(x;y),
rmax: rmax(x;y),
rsub: x - y,
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rneq: x ≠ y,
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
uiff: uiff(P;Q),
guard: {T},
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rmul_preserves_req,
rsub_wf,
rmin_wf,
int-to-real_wf,
rless-int,
rless_wf,
req_inversion,
req_witness,
req_wf,
rmax_wf,
req-implies-req,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rmul_wf,
real_wf,
rminus_wf,
radd_wf,
req_functionality,
req_transitivity,
itermMinus_wf,
real_term_value_minus_lemma,
rminus_functionality,
rmax_functionality,
itermAdd_wf,
real_term_value_add_lemma,
rmin_functionality,
radd_comm,
req_weakening,
radd-rmin,
rminus-rmax
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
minusEquality,
natural_numberEquality,
because_Cache,
independent_isectElimination,
inlFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[x,y,z:\mBbbR{}]. ((x - rmin(y;z)) = rmax(x - y;x - z))
Date html generated:
2017_10_03-AM-08_36_33
Last ObjectModification:
2017_07_28-AM-07_29_32
Theory : reals
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