Nuprl Lemma : finite-double-negation-shift2
∀[A:ℙ]. ∀n:ℕ. ∀[B:ℕn ⟶ ℙ]. ((∀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},
subtype_rel: A ⊆r B,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
true: True,
less_than: a < b,
squash: ↓T
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_wf,
subtype_rel_self,
istype-nat,
natrec_wf,
nat_wf,
subtype_rel_function,
int_seg_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
intformle_wf,
itermConstant_wf,
intformeq_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
subtract_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__lt,
istype-le,
istype-less_than,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
setElimination,
rename,
because_Cache,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
independent_functionElimination,
sqequalRule,
functionIsType,
universeIsType,
hypothesisEquality,
applyEquality,
universeEquality,
inhabitedIsType,
isectIsType,
productElimination,
lambdaEquality_alt,
isectEquality,
functionEquality,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
dependent_set_memberEquality_alt,
productIsType,
addEquality,
minusEquality,
multiplyEquality,
imageElimination
Latex:
\mforall{}[A:\mBbbP{}]. \mforall{}n:\mBbbN{}. \mforall{}[B:\mBbbN{}n {}\mrightarrow{} \mBbbP{}]. ((\mforall{}i:\mBbbN{}n. (((B i) {}\mRightarrow{} A) {}\mRightarrow{} A)) {}\mRightarrow{} ((\mforall{}i:\mBbbN{}n. (B i)) {}\mRightarrow{} A) {}\mRightarrow{} A)
Date html generated:
2020_05_20-AM-08_05_00
Last ObjectModification:
2019_10_31-AM-10_48_54
Theory : general
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