Nuprl Lemma : p-fun-exp-injection
∀[A:Type]. ∀[f:A ⟶ (A + Top)]. ∀[n:ℕ]. p-inject(A;A;f^n) supposing p-inject(A;A;f)
Proof
Definitions occuring in Statement :
p-inject: p-inject(A;B;f),
p-fun-exp: f^n,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
p-inject: p-inject(A;B;f),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
p-fun-exp: f^n,
p-id: p-id(),
do-apply: do-apply(f;x),
can-apply: can-apply(f;x),
isl: isl(x),
outl: outl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
equal_wf,
do-apply_wf,
p-fun-exp_wf,
assert_wf,
can-apply_wf,
subtype_rel_dep_function,
top_wf,
subtype_rel_union,
le_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
p-inject_wf,
primrec0_lemma,
true_wf,
primrec-unroll,
eq_int_wf,
bool_wf,
equal-wf-base,
int_subtype_base,
intformeq_wf,
int_formula_prop_eq_lemma,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
p-compose-inject
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,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
cumulativity,
because_Cache,
functionExtensionality,
applyEquality,
unionEquality,
dependent_set_memberEquality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
baseApply,
closedConclusion,
baseClosed,
equalityElimination,
productElimination,
impliesFunctionality
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} (A + Top)]. \mforall{}[n:\mBbbN{}]. p-inject(A;A;f\^{}n) supposing p-inject(A;A;f)
Date html generated:
2018_05_21-PM-06_33_06
Last ObjectModification:
2017_07_26-PM-04_52_01
Theory : general
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