Nuprl Lemma : equipollent-nat-prod-nsub
∀k:ℕ+. ℕ ~ ℕ × ℕk
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
product: x:A × B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
equipollent: A ~ B,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
nat: ℕ,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f),
pi1: fst(t),
pi2: snd(t),
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
divide_wf,
nat_properties,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
rem_bounds_1,
lelt_wf,
nat_wf,
equal_wf,
int_seg_wf,
biject_wf,
nat_plus_wf,
div_rem_sum,
subtype_rel_sets,
less_than_wf,
nequal_wf,
int_seg_properties,
decidable__le,
intformnot_wf,
intformle_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
squash_wf,
true_wf,
iff_weakening_equal,
add_functionality_wrt_eq,
le_wf,
add_nat_wf,
multiply_nat_wf,
nat_plus_subtype_nat,
int_seg_subtype_nat,
false_wf,
itermAdd_wf,
itermMultiply_wf,
int_term_value_add_lemma,
int_term_value_mul_lemma,
div-cancel3,
rem_invariant,
rem_base_case,
decidable__lt,
mul-commutes,
add-commutes
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
dependent_pairFormation,
lambdaEquality,
independent_pairEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
remainderEquality,
setElimination,
rename,
because_Cache,
natural_numberEquality,
independent_isectElimination,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
applyEquality,
baseClosed,
productElimination,
productEquality,
functionExtensionality,
applyLambdaEquality,
setEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
unionElimination,
imageElimination,
universeEquality,
imageMemberEquality,
multiplyEquality,
addEquality
Latex:
\mforall{}k:\mBbbN{}\msupplus{}. \mBbbN{} \msim{} \mBbbN{} \mtimes{} \mBbbN{}k
Date html generated:
2018_05_21-PM-07_57_26
Last ObjectModification:
2017_07_26-PM-05_34_59
Theory : general
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