Nuprl Lemma : fpf-sub-val2
∀[A,A':Type].
∀[B:A ⟶ Type]
∀eq:EqDecider(A'). ∀f,g:a:A fp-> B[a]. ∀x:A'.
∀[P,Q:a:A ⟶ B[a] ⟶ ℙ].
((∀x:A. ∀z:B[x]. (P[x;z] ⇒ Q[x;z])) ⇒ z != f(x) ==> P[x;z] ⇒ z != g(x) ==> Q[x;z] supposing g ⊆ f)
supposing strong-subtype(A;A')
Proof
Definitions occuring in Statement :
fpf-sub: f ⊆ g,
fpf-val: z != f(x) ==> P[a; z],
fpf: a:A fp-> B[a],
deq: EqDecider(T),
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s1;s2],
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
pi1: fst(t),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
fpf-val: z != f(x) ==> P[a; z],
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
list_ind: list_ind,
reduce: reduce(f;k;as),
deq-member: x ∈b L,
fpf-sub: f ⊆ g,
fpf-ap: f(x),
pi2: snd(t)
Lemmas referenced :
strong-subtype_wf,
deq_wf,
fpf_wf,
all_wf,
fpf-sub_wf,
strong-subtype-deq-subtype,
fpf-sub_witness,
strong-subtype_witness,
fpf_ap_pair_lemma,
subtype_rel_list,
assert-deq-member,
strong-subtype-l_member-type,
deq-member_wf,
assert_wf,
l_member_wf
Rules used in proof :
universeEquality,
functionEquality,
independent_isectElimination,
cumulativity,
functionExtensionality,
applyEquality,
lambdaEquality,
sqequalRule,
because_Cache,
lambdaFormation,
rename,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
productElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
equalityTransitivity,
setEquality,
dependent_set_memberEquality
Latex:
\mforall{}[A,A':Type].
\mforall{}[B:A {}\mrightarrow{} Type]
\mforall{}eq:EqDecider(A'). \mforall{}f,g:a:A fp-> B[a]. \mforall{}x:A'.
\mforall{}[P,Q:a:A {}\mrightarrow{} B[a] {}\mrightarrow{} \mBbbP{}].
((\mforall{}x:A. \mforall{}z:B[x]. (P[x;z] {}\mRightarrow{} Q[x;z]))
{}\mRightarrow{} z != f(x) ==> P[x;z] {}\mRightarrow{} z != g(x) ==> Q[x;z] supposing g \msubseteq{} f)
supposing strong-subtype(A;A')
Date html generated:
2020_05_20-AM-09_02_53
Last ObjectModification:
2020_01_07-PM-00_54_54
Theory : finite!partial!functions
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