Nuprl Lemma : FOL-sequent-evidence-trivial
Uniform evidence for a trivially true sequent where the conclusion is
a given member of the hypotheses.⋅
∀hyps:mFOL() List. ∀i:ℕ||hyps||. FOL-sequent-evidence{i:l}(<hyps, hyps[i]>)
Proof
Definitions occuring in Statement :
FOL-sequent-evidence: FOL-sequent-evidence{i:l}(s),
mFOL: mFOL(),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
all: ∀x:A. B[x],
pair: <a, b>,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
FOL-sequent-evidence: FOL-sequent-evidence{i:l}(s),
FO-uniform-evidence: FO-uniform-evidence(vs;fmla),
FOL-sequent-abstract: FOL-sequent-abstract(s),
FOSatWith+: Dom,S,a +|= fmla,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
nat: ℕ,
cand: A c∧ B,
FOL-hyps-meaning: FOL-hyps-meaning(Dom;S;a;hyps),
le: A ≤ B,
mFOL-sequent: mFOL-sequent()
Lemmas referenced :
FOL-abstract_wf,
mFOL-sequent-freevars-subset-1,
subtype_rel_FOAssignment,
mFOL-freevars_wf,
FOSatWith+_wf,
lelt_wf,
top_wf,
subtype_rel_list,
select-map,
length-map,
le_wf,
select-tuple_wf,
list_wf,
int_seg_wf,
FOStruct+_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
length_wf,
int_seg_properties,
mFOL_wf,
mFOL-sequent-freevars_wf,
FOAssignment_wf,
select_wf,
FOL-hyps-meaning_wf,
tuple-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalRule,
introduction,
lambdaEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
independent_isectElimination,
hypothesis,
hypothesisEquality,
independent_pairEquality,
setElimination,
rename,
natural_numberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
applyEquality,
cumulativity,
universeEquality,
dependent_set_memberEquality,
productEquality
Latex:
\mforall{}hyps:mFOL() List. \mforall{}i:\mBbbN{}||hyps||. FOL-sequent-evidence\{i:l\}(<hyps, hyps[i]>)
Date html generated:
2016_07_08-PM-05_21_17
Last ObjectModification:
2016_01_16-PM-04_32_23
Theory : minimal-first-order-logic
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