Nuprl Lemma : injection_le
∀[k,m:ℕ]. k ≤ m supposing ∃f:ℕk ⟶ ℕm. Inj(ℕk;ℕm;f)
Proof
Definitions occuring in Statement :
inject: Inj(A;B;f),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
exists: ∃x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
nat_plus: ℕ+,
less_than: a < b,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
inject: Inj(A;B;f),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
less_than'_wf,
exists_wf,
int_seg_wf,
inject_wf,
nat_wf,
sq_stable__le,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
not-le-2,
le-add-cancel-alt,
lelt_wf,
int_seg_properties,
le_wf,
int_subtype_base,
equal_wf,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
minus-zero,
omega-shadow,
mul-distributes,
mul-associates,
mul-commutes,
eq_int_wf,
decidable__lt,
not-lt-2,
le-add-cancel2,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
set_subtype_base,
int_seg_subtype,
le_antisymmetry_iff,
le_weakening,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
all_wf,
not-equal-2,
not-equal-implies-less,
subtype_rel_self,
assert_wf,
bnot_wf,
not_wf,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
ifthenelse_wf,
equal_functionality_wrt_subtype_rel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
functionExtensionality,
applyEquality,
because_Cache,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
independent_pairFormation,
addEquality,
voidEquality,
intEquality,
minusEquality,
dependent_set_memberEquality,
applyLambdaEquality,
dependent_pairFormation,
sqequalIntensionalEquality,
promote_hyp,
multiplyEquality,
equalityElimination,
instantiate,
cumulativity,
impliesFunctionality
Latex:
\mforall{}[k,m:\mBbbN{}]. k \mleq{} m supposing \mexists{}f:\mBbbN{}k {}\mrightarrow{} \mBbbN{}m. Inj(\mBbbN{}k;\mBbbN{}m;f)
Date html generated:
2017_04_14-AM-07_33_32
Last ObjectModification:
2017_02_27-PM-03_11_31
Theory : fun_1
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