Nuprl Lemma : int-prod_wf_absval_1
∀[n:ℕ]. ∀[f:ℕn ⟶ {s:ℤ| |s| = 1 ∈ ℤ} ]. (Π(f[x] | x < n) ∈ {s:ℤ| |s| = 1 ∈ ℤ} )
Proof
Definitions occuring in Statement :
int-prod: Π(f[x] | x < k),
absval: |i|,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
int_seg: {i..j-},
sq_type: SQType(T),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
true: True,
squash: ↓T,
so_apply: x[s],
all: ∀x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
prop: ℙ,
nat: ℕ,
subtype_rel: A ⊆r B,
int-prod: Π(f[x] | x < k),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermConstant_wf,
itermMultiply_wf,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
decidable__equal_int,
subtype_base_sq,
iff_weakening_equal,
subtype_rel_self,
absval_mul,
istype-universe,
true_wf,
squash_wf,
istype-nat,
int_seg_wf,
istype-int,
equal-wf-base,
int_subtype_base,
istype-le,
istype-void,
absval_pos,
absval_wf,
equal_wf,
primrec_wf
Rules used in proof :
Error :dependent_pairFormation_alt,
approximateComputation,
unionElimination,
applyLambdaEquality,
cumulativity,
promote_hyp,
productElimination,
independent_isectElimination,
imageMemberEquality,
universeEquality,
instantiate,
imageElimination,
Error :isectIsTypeImplies,
Error :isect_memberEquality_alt,
Error :universeIsType,
Error :functionIsType,
axiomEquality,
Error :setIsType,
independent_functionElimination,
dependent_functionElimination,
because_Cache,
multiplyEquality,
sqequalBase,
baseClosed,
baseApply,
Error :equalityIstype,
voidElimination,
Error :lambdaFormation_alt,
independent_pairFormation,
Error :dependent_set_memberEquality_alt,
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
Error :inhabitedIsType,
rename,
setElimination,
Error :lambdaEquality_alt,
applyEquality,
hypothesis,
hypothesisEquality,
intEquality,
setEquality,
closedConclusion,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \{s:\mBbbZ{}| |s| = 1\} ]. (\mPi{}(f[x] | x < n) \mmember{} \{s:\mBbbZ{}| |s| = 1\} )
Date html generated:
2019_06_20-PM-01_18_30
Last ObjectModification:
2019_06_19-AM-10_34_48
Theory : int_2
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