Nuprl Lemma : assert-eq_mFO
∀[x,y:mFOL()]. uiff(↑eq_mFO(x;y);x = y ∈ mFOL())
Proof
Definitions occuring in Statement :
eq_mFO: eq_mFO(x;y),
mFOL: mFOL(),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
all: ∀x:A. B[x],
so_apply: x[s],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
eq_mFO: eq_mFO(x;y),
mFO-equal: mFO-equal(x),
mFOatomic: name(vars),
mFOL_ind: mFOL_ind,
mFO-dest-atomic: mFO-dest-atomic,
mFOatomic?: mFOatomic?(v),
pi1: fst(t),
mFOatomic-name: mFOatomic-name(v),
pi2: snd(t),
mFOatomic-vars: mFOatomic-vars(v),
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
btrue: tt,
mFOconnect: mFOconnect(knd;left;right),
bfalse: ff,
assert: ↑b,
uiff: uiff(P;Q),
and: P ∧ Q,
false: False,
mFOconnect?: mFOconnect?(v),
not: ¬A,
mFOquant: mFOquant(isall;var;body),
mFOquant?: mFOquant?(v),
guard: {T},
mFO-dest-connective: mFO-dest-connective,
mFOconnect-knd: mFOconnect-knd(v),
mFOconnect-left: mFOconnect-left(v),
mFOconnect-right: mFOconnect-right(v),
band: p ∧b q,
mFO-dest-quantifier: mFO-dest-quantifier,
mFOquant-isall: mFOquant-isall(v),
mFOquant-var: mFOquant-var(v),
mFOquant-body: mFOquant-body(v),
or: P ∨ Q,
sq_type: SQType(T),
deq: EqDecider(T),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
true: True,
rev_uimplies: rev_uimplies(P;Q),
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
exists: ∃x:A. B[x],
bnot: ¬bb,
squash: ↓T,
nequal: a ≠ b ∈ T
Lemmas referenced :
mFOL-induction,
mFOL_wf,
uiff_wf,
assert_wf,
eq_mFO_wf,
equal_wf,
mFOatomic_wf,
equal-wf-base-T,
list_subtype_base,
int_subtype_base,
atom_subtype_base,
istype-void,
btrue_wf,
mFOconnect?_wf,
bfalse_wf,
btrue_neq_bfalse,
mFOconnect_wf,
istype-assert,
mFOquant?_wf,
mFOquant_wf,
istype-int,
bool_wf,
list_wf,
istype-atom,
assert_witness,
eq_atom_wf,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqtt_to_assert,
band_wf,
assert_of_eq_atom,
list-deq_wf1,
int-deq_wf,
iff_transitivity,
equal-wf-base,
iff_weakening_uiff,
assert_of_band,
mFOatomic-name_wf,
mFOatomic?_wf,
deq_property,
list-deq_wf,
mFOatomic-vars_wf,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_atom,
mFOconnect-left_wf,
mFOconnect-right_wf,
assert_elim,
mFOconnect-knd_wf,
eq_bool_wf,
assert_of_eq_bool,
mFOquant-var_wf,
mFOquant-body_wf,
eq_int_wf,
assert_of_eq_int,
mFOquant-isall_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
sqequalRule,
lambdaEquality_alt,
functionEquality,
hypothesis,
hypothesisEquality,
universeIsType,
independent_functionElimination,
lambdaFormation_alt,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
independent_isectElimination,
because_Cache,
inhabitedIsType,
productElimination,
independent_pairFormation,
voidElimination,
equalitySymmetry,
dependent_set_memberEquality_alt,
equalityTransitivity,
productIsType,
equalityIstype,
applyLambdaEquality,
setElimination,
rename,
isectIsType,
dependent_functionElimination,
functionIsType,
independent_pairEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
unionElimination,
instantiate,
cumulativity,
productEquality,
atomEquality,
promote_hyp,
hyp_replacement,
natural_numberEquality,
sqequalBase,
equalityElimination,
dependent_pairFormation_alt,
imageElimination,
imageMemberEquality
Latex:
\mforall{}[x,y:mFOL()]. uiff(\muparrow{}eq\_mFO(x;y);x = y)
Date html generated:
2020_05_20-AM-09_08_18
Last ObjectModification:
2020_01_25-AM-10_46_49
Theory : minimal-first-order-logic
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