Nuprl Lemma : finite-double-negation-shift
∀[A:ℙ]. ∀[B:ℕ ⟶ ℙ]. ∀n:ℕ. ((∀i:ℕn. (((B i) ⇒ A) ⇒ A)) ⇒ ((∀i:ℕn. (B i)) ⇒ A) ⇒ A)
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_apply: x[s],
int_seg: {i..j-},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
ge: i ≥ j ,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
less_than: a < b,
label: ...$L... t,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
iff_weakening_equal,
le_wf,
lelt_wf,
decidable__lt,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
decidable__le,
subtract_wf,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermConstant_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
int_seg_properties,
natrec_wf,
nat_wf,
false_wf,
int_seg_subtype_nat,
int_seg_wf,
all_wf,
int_subtype_base,
subtype_base_sq,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
hypothesis,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
functionEquality,
sqequalRule,
lambdaEquality,
applyEquality,
independent_pairFormation,
introduction,
universeEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
dependent_set_memberEquality
Latex:
\mforall{}[A:\mBbbP{}]. \mforall{}[B:\mBbbN{} {}\mrightarrow{} \mBbbP{}]. \mforall{}n:\mBbbN{}. ((\mforall{}i:\mBbbN{}n. (((B i) {}\mRightarrow{} A) {}\mRightarrow{} A)) {}\mRightarrow{} ((\mforall{}i:\mBbbN{}n. (B i)) {}\mRightarrow{} A) {}\mRightarrow{} A)
Date html generated:
2016_05_15-PM-03_20_38
Last ObjectModification:
2016_01_16-AM-10_48_07
Theory : general
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