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],
decidable: Dec(P),
or: P ∨ Q,
nat: ℕ,
int_nzero: ℤ-o,
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,
prop: ℙ,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
uiff: uiff(P;Q),
squash: ↓T,
less_than: a < b
Lemmas referenced :
decidable__lt,
istype-less_than,
member-less_than,
divide_wfa,
nat_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
int_subtype_base,
nequal_wf,
istype-nat,
div_base_case,
intformnot_wf,
int_formula_prop_not_lemma,
div_rem_sum,
rem_bounds_1,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
mul_preserves_le,
divide_wf,
satisfiable-full-omega-tt,
equal_wf,
add-is-int-iff,
multiply-is-int-iff,
itermMultiply_wf,
itermAdd_wf,
int_term_value_mul_lemma,
int_term_value_add_lemma,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
hypothesis,
unionElimination,
isectElimination,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
Error :isect_memberEquality_alt,
Error :dependent_set_memberEquality_alt,
Error :lambdaFormation_alt,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
Error :universeIsType,
Error :equalityIstype,
applyEquality,
baseClosed,
sqequalBase,
equalitySymmetry,
intEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
equalityTransitivity,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
computeAll,
voidEquality,
isect_memberEquality,
lambdaEquality,
dependent_pairFormation,
imageElimination,
lambdaFormation,
divideEquality,
multiplyEquality,
lemma_by_obid,
productElimination
Latex:
\mforall{}[i,k:\mBbbN{}]. (i \mdiv{} k < i) supposing (1 < k and 0 < i)
Date html generated:
2019_06_20-PM-02_31_15
Last ObjectModification:
2019_03_06-AM-10_53_24
Theory : num_thy_1
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