Nuprl Lemma : mul-list-positive
∀L:ℕ+ List. 0 < Π(L)
Proof
Definitions occuring in Statement :
mul-list: Π(ns) ,
list: T List,
nat_plus: ℕ+,
less_than: a < b,
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
guard: {T},
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
cons: [a / b],
le: A ≤ B,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
mul-list: Π(ns) ,
reduce: reduce(f;k;as),
list_ind: list_ind
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
member-less_than,
intformeq_wf,
int_formula_prop_eq_lemma,
nat_plus_wf,
list-cases,
mul_list_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
istype-nat,
colength_wf_list,
istype-void,
istype-le,
list_wf,
mul-list_wf,
subtype_rel_list,
subtract-1-ge-0,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
mul_bounds_1b,
nat_plus_properties,
decidable__lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
functionIsTypeImplies,
inhabitedIsType,
unionElimination,
imageMemberEquality,
baseClosed,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityIstype,
dependent_set_memberEquality_alt,
applyEquality,
intEquality,
because_Cache,
instantiate,
imageElimination,
baseApply,
closedConclusion,
sqequalBase,
cumulativity,
multiplyEquality
Latex:
\mforall{}L:\mBbbN{}\msupplus{} List. 0 < \mPi{}(L)
Date html generated:
2020_05_20-AM-08_07_15
Last ObjectModification:
2019_12_13-AM-10_06_16
Theory : general
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