Nuprl Lemma : mFOL-proveable-formula-evidence-ext2
Main Lemma. See mFOL-proveable-evidence also.⋅
∀[fmla:mFOL()]. (mFOL-proveable-formula(fmla) ⇒ mFOL-evidence(fmla))
Proof
Definitions occuring in Statement :
mFOL-proveable-formula: mFOL-proveable-formula(fmla),
mFOL-evidence: mFOL-evidence(fmla),
mFOL: mFOL(),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
member: t ∈ T,
eq_int: (i =z j),
btrue: tt,
it: ⋅,
bfalse: ff,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
select-tuple: x.n,
pi1: fst(t),
subtract: n - m,
pi2: snd(t),
lt_int: i <z j,
le_int: i ≤z j,
genrec-ap: genrec-ap,
spreadn: spread3,
ifunion: ifunion(b; t),
mFOL-proveable-formula-evidence-ext,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
top: Top,
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
or: P ∨ Q,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
mFOL-proveable-formula-evidence-ext,
lifting-strict-decide,
istype-void,
has-value_wf_base,
istype-base,
is-exception_wf,
istype-universe,
lifting-strict-spread,
strict4-spread,
lifting-strict-atom_eq,
strict4-decide,
lifting-strict-int_eq,
lifting-strict-less,
value-type-has-value,
int-value-type,
lifting-strict-isaxiom
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination,
independent_pairFormation,
lambdaFormation_alt,
callbyvalueApply,
universeIsType,
baseApply,
closedConclusion,
hypothesisEquality,
applyExceptionCases,
inrFormation_alt,
imageMemberEquality,
imageElimination,
inlFormation_alt,
because_Cache,
inhabitedIsType,
sqequalSqle,
divergentSqle,
callbyvalueDecide,
unionElimination,
sqleReflexivity,
equalityIsType1,
dependent_functionElimination,
independent_functionElimination,
decideExceptionCases,
axiomSqleEquality,
exceptionSqequal,
callbyvalueLess,
productElimination,
intEquality,
lessExceptionCases,
callbyvalueSpread,
spreadExceptionCases
Latex:
\mforall{}[fmla:mFOL()]. (mFOL-proveable-formula(fmla) {}\mRightarrow{} mFOL-evidence(fmla))
Date html generated:
2019_10_16-AM-11_43_40
Last ObjectModification:
2018_10_12-AM-11_13_33
Theory : minimal-first-order-logic
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