Nuprl Lemma : at_AFbar
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
∀n:ℕ. ∀s:AF-spread-law(x,y.R[x;y])-consistent-seq(n).
((AFbar() n s) ⇒ (¬{a:T| AF-spread-law(x,y.R[x;y]) n s (inl a)} ))
Proof
Definitions occuring in Statement :
AFbar: AFbar(),
AF-spread-law: AF-spread-law(x,y.R[x; y]),
consistent-seq: R-consistent-seq(n),
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
unit: Unit,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
inl: inl x,
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
AFbar: AFbar(),
AF-spread-law: AF-spread-law(x,y.R[x; y]),
isl: isl(x),
outl: outl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
and: P ∧ Q,
cand: A c∧ B,
true: True,
consistent-seq: R-consistent-seq(n),
int_seg: {i..j-},
nat: ℕ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
isr: isr(x),
bfalse: ff,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_apply: x[s1;s2],
guard: {T},
top: Top
Lemmas referenced :
AFbar_wf,
AF-spread-law_wf,
subtract_wf,
decidable__le,
false_wf,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
decidable__lt,
not-lt-2,
add-mul-special,
zero-mul,
le-add-cancel-alt,
and_wf,
le_wf,
less_than_wf,
unit_wf2,
true_wf,
equal_wf,
set_wf,
consistent-seq_wf,
nat_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
sqequalRule,
setElimination,
rename,
productElimination,
independent_pairFormation,
independent_functionElimination,
natural_numberEquality,
applyEquality,
because_Cache,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
voidElimination,
independent_isectElimination,
addEquality,
minusEquality,
unionEquality,
cumulativity,
productEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
inlEquality,
universeEquality,
functionEquality,
isect_memberEquality,
voidEquality,
intEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}n:\mBbbN{}. \mforall{}s:AF-spread-law(x,y.R[x;y])-consistent-seq(n).
((AFbar() n s) {}\mRightarrow{} (\mneg{}\{a:T| AF-spread-law(x,y.R[x;y]) n s (inl a)\} ))
Date html generated:
2017_04_14-AM-07_27_48
Last ObjectModification:
2017_02_27-PM-02_56_38
Theory : bar-induction
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