Nuprl Lemma : fappend_wf
∀[n,m:ℕ]. ∀[f:ℕn ⟶ ℕm]. ∀[x:ℕm]. (f[n:=x] ∈ ℕn + 1 ⟶ ℕm)
Proof
Definitions occuring in Statement :
fappend: f[n:=x],
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
lelt: i ≤ j < k,
and: P ∧ Q,
guard: {T},
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
le: A ≤ B,
less_than: a < b,
fappend: f[n:=x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
eq_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
equal_wf,
bnot_wf,
not_wf,
int_seg_wf,
int_seg_properties,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
lelt_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
equalityTransitivity,
equalitySymmetry,
baseClosed,
intEquality,
applyEquality,
functionExtensionality,
natural_numberEquality,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
isect_memberFormation,
lambdaFormation,
equalityElimination,
independent_functionElimination,
impliesFunctionality,
addEquality,
axiomEquality,
functionEquality
Latex:
\mforall{}[n,m:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}m]. \mforall{}[x:\mBbbN{}m]. (f[n:=x] \mmember{} \mBbbN{}n + 1 {}\mrightarrow{} \mBbbN{}m)
Date html generated:
2017_04_17-AM-09_50_45
Last ObjectModification:
2017_02_27-PM-05_46_00
Theory : num_thy_1
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