Nuprl Lemma : fpf-sub-set
∀[A:Type]. ∀[P:A ⟶ ℙ]. ∀[B:A ⟶ Type]. ∀[eq:EqDecider(A)]. ∀[f,g:a:{a:A| P[a]} fp-> B[a]]. f ⊆ g supposing f ⊆ g
Proof
Definitions occuring in Statement :
fpf-sub: f ⊆ g,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
prop: ℙ,
fpf-sub: f ⊆ g,
all: ∀x:A. B[x],
fpf: a:A fp-> B[a],
fpf-dom: x ∈ dom(f),
pi1: fst(t),
iff: P ⇐⇒ Q,
and: P ∧ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
nat: ℕ,
ge: i ≥ j ,
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
squash: ↓T
Lemmas referenced :
fpf-sub_witness,
subtype-fpf-general,
fpf-sub_wf,
strong-subtype-deq-subtype,
strong-subtype-set2,
fpf_wf,
deq_wf,
assert-deq-member,
subtype_rel_list,
select_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
set_wf,
and_wf,
equal_wf,
assert_wf,
deq-member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
cumulativity,
instantiate,
hypothesis,
independent_functionElimination,
setEquality,
because_Cache,
setElimination,
rename,
independent_isectElimination,
universeEquality,
functionEquality,
lambdaFormation,
dependent_functionElimination,
productElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageMemberEquality,
baseClosed,
imageElimination,
equalitySymmetry,
dependent_set_memberEquality,
equalityTransitivity,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[A:Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbP{}]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[f,g:a:\{a:A| P[a]\} fp-> B[a]].
f \msubseteq{} g supposing f \msubseteq{} g
Date html generated:
2018_05_21-PM-09_18_45
Last ObjectModification:
2018_02_09-AM-10_17_16
Theory : finite!partial!functions
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