Nuprl Lemma : div_mono1
∀[i,k:ℕ]. (i ÷ k < i) supposing (1 < k and 0 < i)
Proof
Definitions occuring in Statement :
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
divide: n ÷ m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
nat_plus: ℕ+,
guard: {T},
int_nzero: ℤ-o,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
less_than: a < b,
squash: ↓T
Lemmas referenced :
divide_wf,
mul_preserves_le,
int_term_value_add_lemma,
int_term_value_mul_lemma,
itermAdd_wf,
itermMultiply_wf,
multiply-is-int-iff,
add-is-int-iff,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
le-add-cancel,
zero-add,
add-associates,
add-commutes,
add-swap,
add_functionality_wrt_le,
less-iff-le,
not-lt-2,
false_wf,
rem_bounds_1,
nequal_wf,
div_rem_sum,
int_formula_prop_not_lemma,
intformnot_wf,
le_weakening2,
less_than_transitivity2,
div_base_case,
nat_wf,
equal_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
member-less_than,
less_than_wf,
decidable__lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
unionElimination,
isectElimination,
natural_numberEquality,
sqequalRule,
isect_memberEquality,
divideEquality,
because_Cache,
lambdaFormation,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
productElimination,
independent_functionElimination,
addEquality,
applyEquality,
multiplyEquality,
imageElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}[i,k:\mBbbN{}]. (i \mdiv{} k < i) supposing (1 < k and 0 < i)
Date html generated:
2016_05_15-PM-04_46_39
Last ObjectModification:
2016_01_16-AM-11_24_56
Theory : general
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