Nuprl Lemma : fpf-sub-join-right2
∀[A:Type]. ∀[B,C:A ─→ Type]. ∀[eq:EqDecider(A)]. ∀[f:a:A fp-> B[a]]. ∀[g:a:A fp-> C[a]].
g ⊆ f ⊕ g supposing ∀x:A. (((↑x ∈ dom(f)) ∧ (↑x ∈ dom(g))) ⇒ ((B[x] ⊆r C[x]) c∧ (f(x) = g(x) ∈ C[x])))
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf-sub: f ⊆ g,
fpf-ap: f(x),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
assert: ↑b,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ─→ B[x],
universe: Type,
equal: s = t ∈ T
Lemmas :
assert_witness,
fpf-dom_wf,
fpf-join_wf,
top_wf,
assert_wf,
all_wf,
subtype-fpf2,
subtype_top,
subtype_rel_wf,
fpf-ap_wf,
fpf_wf,
deq_wf,
fpf-join-dom2,
bool_wf,
equal-wf-T-base,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot
\mforall{}[A:Type]. \mforall{}[B,C:A {}\mrightarrow{} Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[f:a:A fp-> B[a]]. \mforall{}[g:a:A fp-> C[a]].
g \msubseteq{} f \moplus{} g supposing \mforall{}x:A. (((\muparrow{}x \mmember{} dom(f)) \mwedge{} (\muparrow{}x \mmember{} dom(g))) {}\mRightarrow{} ((B[x] \msubseteq{}r C[x]) c\mwedge{} (f(x) = g(x))))
Date html generated:
2015_07_17-AM-09_20_35
Last ObjectModification:
2015_01_28-AM-07_49_29
Home
Index