Nuprl Lemma : funinv-ap-equals
∀[n:ℕ]. ∀[f:ℕn →⟶ ℕn]. ∀[a,b:ℕn]. (inv(f) b) = a ∈ ℤ supposing (f a) = b ∈ ℤ
Proof
Definitions occuring in Statement :
injection: A →⟶ B,
funinv: inv(f),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
ge: i ≥ j ,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
lelt: i ≤ j < k,
int_seg: {i..j-},
nat: ℕ,
injection: A →⟶ B,
guard: {T},
prop: ℙ,
subtype_rel: A ⊆r B,
cand: A c∧ B,
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
nat_wf,
lelt_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
squash_wf,
sq_stable__equal,
equal_wf,
sq_stable__and,
int_seg_wf,
injection_wf,
int_subtype_base,
equal-wf-base-T,
funinv_wf3,
funinv-property
Rules used in proof :
voidEquality,
voidElimination,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
independent_isectElimination,
unionElimination,
axiomEquality,
dependent_functionElimination,
lambdaFormation,
independent_functionElimination,
isect_memberEquality,
imageElimination,
baseClosed,
imageMemberEquality,
productElimination,
natural_numberEquality,
rename,
setElimination,
lambdaEquality,
applyLambdaEquality,
sqequalRule,
applyEquality,
intEquality,
productEquality,
because_Cache,
independent_pairFormation,
equalitySymmetry,
equalityTransitivity,
dependent_set_memberEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n \mrightarrow{}{}\mrightarrow{} \mBbbN{}n]. \mforall{}[a,b:\mBbbN{}n]. (inv(f) b) = a supposing (f a) = b
Date html generated:
2018_05_21-PM-08_16_32
Last ObjectModification:
2017_12_15-PM-00_54_32
Theory : general
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