Nuprl Lemma : fpf-val-single1
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[x:A]. ∀[v,P:Top]. (z != x : v(x) ==> P[a;z] ~ True ⇒ P[x;v])
Proof
Definitions occuring in Statement :
fpf-single: x : v,
fpf-val: z != f(x) ==> P[a; z],
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s1;s2],
implies: P ⇒ Q,
true: True,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
fpf-single: x : v,
fpf-val: z != f(x) ==> P[a; z],
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
fpf-dom: x ∈ dom(f),
pi1: fst(t),
deq: EqDecider(T),
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uall: ∀[x:A]. B[x],
eqof: eqof(d),
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bor: p ∨bq,
ifthenelse: if b then t else f fi ,
assert: ↑b,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
prop: ℙ,
rev_implies: P ⇐ Q,
false: False
Lemmas referenced :
fpf_ap_pair_lemma,
deq_member_cons_lemma,
deq_member_nil_lemma,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
equal_wf,
eqtt_to_assert,
safe-assert-deq,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
top_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
applyEquality,
setElimination,
rename,
hypothesisEquality,
lambdaFormation,
unionElimination,
equalityElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
baseClosed,
cumulativity,
independent_functionElimination,
because_Cache,
productElimination,
independent_isectElimination,
independent_pairFormation,
addLevel,
impliesFunctionality,
levelHypothesis,
universeEquality,
isect_memberFormation,
sqequalAxiom
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[x:A]. \mforall{}[v,P:Top]. (z != x : v(x) ==> P[a;z] \msim{} True {}\mRightarrow{} P[x;v])
Date html generated:
2018_05_21-PM-09_25_19
Last ObjectModification:
2018_02_09-AM-10_21_08
Theory : finite!partial!functions
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