Nuprl Lemma : assert-rat-term-eq2
∀[r1,r2:rat_term()]. ∀[f:ℤ ⟶ ℝ].
((snd(rat_term_to_real(f;r1))) = (snd(rat_term_to_real(f;r2)))) supposing
((fst(rat_term_to_real(f;r1))) and
(fst(rat_term_to_real(f;r2))) and
(inl Ax ≤ rat-term-eq(r1;r2)))
Proof
Definitions occuring in Statement :
rat-term-eq: rat-term-eq(r1;r2),
rat_term_to_real: rat_term_to_real(f;t),
rat_term: rat_term(),
req: x = y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
function: x:A ⟶ B[x],
inl: inl x,
int: ℤ,
sqle: s ≤ t,
axiom: Ax
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
pi2: snd(t),
subtype_rel: A ⊆r B,
prop: ℙ,
pi1: fst(t),
and: P ∧ Q,
sq_stable: SqStable(P),
squash: ↓T,
it: ⋅,
btrue: tt,
mono: mono(T),
is-above: is-above(T;a;z),
exists: ∃x:A. B[x],
cand: A c∧ B,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
true: True
Lemmas referenced :
sq_stable__req,
uimplies_subtype,
real_wf,
assert-rat-term-eq,
rat_term_to_real_wf,
istype-sqle,
rat-term-eq_wf,
bool_subtype_base,
istype-int,
rat_term_wf,
bool-mono,
btrue_wf,
bool_wf,
subtype_base_sq,
req_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
inhabitedIsType,
hypothesis,
lambdaFormation_alt,
productElimination,
sqequalRule,
hypothesisEquality,
applyEquality,
independent_isectElimination,
hyp_replacement,
equalitySymmetry,
dependent_set_memberEquality_alt,
independent_pairFormation,
equalityTransitivity,
productIsType,
equalityIstype,
applyLambdaEquality,
setElimination,
rename,
lambdaEquality_alt,
universeIsType,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
functionIsType,
dependent_pairFormation_alt,
sqequalBase,
instantiate,
cumulativity,
natural_numberEquality
Latex:
\mforall{}[r1,r2:rat\_term()]. \mforall{}[f:\mBbbZ{} {}\mrightarrow{} \mBbbR{}].
((snd(rat\_term\_to\_real(f;r1))) = (snd(rat\_term\_to\_real(f;r2)))) supposing
((fst(rat\_term\_to\_real(f;r1))) and
(fst(rat\_term\_to\_real(f;r2))) and
(inl Ax \mleq{} rat-term-eq(r1;r2)))
Date html generated:
2019_10_29-AM-09_54_19
Last ObjectModification:
2019_04_01-PM-07_02_46
Theory : reals
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