Nuprl Lemma : coW-equiv_weakening
∀[A:𝕌']. ∀B:A ⟶ Type. ∀w,w':coW(A;a.B[a]). coW-equiv(a.B[a];w;w') supposing w = w' ∈ coW(A;a.B[a])
Proof
Definitions occuring in Statement :
coW-equiv: coW-equiv(a.B[a];w;w'),
coW: coW(A;a.B[a]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
coW-equiv: coW-equiv(a.B[a];w;w'),
coW-game: coW-game(a.B[a];w;w'),
sg-legal2: Legal2(x;y),
sg-legal1: Legal1(x;y),
sg-pos: Pos(g),
pi1: fst(t),
pi2: snd(t),
exists: ∃x:A. B[x],
so_apply: x[s1;s2],
sg-init: InitialPos(g),
or: P ∨ Q,
simple-game: SimpleGame,
nat: ℕ,
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
sq_stable: SqStable(P),
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
coW-equiv_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
implies-sg-win2,
coW-game_wf,
equal_wf,
copath_wf,
exists_wf,
or_wf,
copath-length_wf,
copathAgree_wf,
sg-init_wf,
copath-nil_wf,
all_wf,
coW_wf,
ifthenelse_wf,
lt_int_wf,
nat_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
set_wf,
equal-wf-base,
and_wf,
less_than_irreflexivity,
sq_stable__copathAgree,
not-lt-2,
le_antisymmetry_iff,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
le-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
axiomEquality,
hypothesis,
thin,
rename,
applyEquality,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
sqequalRule,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination,
dependent_pairFormation,
cumulativity,
spreadEquality,
productEquality,
functionEquality,
setEquality,
functionExtensionality,
independent_pairEquality,
setElimination,
intEquality,
addEquality,
dependent_pairEquality,
dependent_set_memberEquality,
independent_pairFormation,
unionElimination,
equalityElimination,
promote_hyp,
voidElimination,
inrFormation,
applyLambdaEquality,
hyp_replacement,
inlFormation,
isect_memberEquality,
voidEquality,
minusEquality
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}B:A {}\mrightarrow{} Type. \mforall{}w,w':coW(A;a.B[a]). coW-equiv(a.B[a];w;w') supposing w = w'
Date html generated:
2019_06_20-PM-00_57_57
Last ObjectModification:
2019_01_02-PM-01_34_32
Theory : co-recursion-2
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