Nuprl Lemma : free-word-inv-append2
∀[X:Type]. ∀[x:free-word(X)]. (x + free-word-inv(x) = 0 ∈ free-word(X))
Proof
Definitions occuring in Statement :
free-word-inv: free-word-inv(w),
free-0: 0,
free-append: w + w',
free-word: free-word(X),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
free-word: free-word(X),
all: ∀x:A. B[x],
prop: ℙ,
implies: P ⇒ Q,
cand: A c∧ B,
free-word-inv: free-word-inv(w),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
free-append: w + w',
guard: {T},
free-0: 0,
nil: [],
it: ⋅,
word-equiv: word-equiv(X;w1;w2),
exists: ∃x:A. B[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
quotient: x,y:A//B[x; y],
squash: ↓T,
true: True,
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
so_lambda: λ2x.t[x],
so_apply: x[s],
transitive-reflexive-closure: R^*,
append: as @ bs,
so_lambda: so_lambda3,
top: Top,
so_apply: x[s1;s2;s3],
or: P ∨ Q,
infix_ap: x f y,
word-rel: word-rel(X;w1;w2),
inverse-letters: a = -b,
sq_type: SQType(T),
false: False
Lemmas referenced :
word-equiv-equiv,
list_wf,
word-equiv_wf,
map_wf,
equal_wf,
reverse_wf,
quotient-member-eq,
append_wf,
nil_wf,
transitive-reflexive-closure_wf,
word-rel_wf,
subtype_rel_self,
equal-wf-base,
squash_wf,
true_wf,
free-word_wf,
last_induction,
list_ind_nil_lemma,
reverse_nil_lemma,
map_nil_lemma,
transitive-closure_wf,
transitive-reflexive-closure_transitivity,
cons_wf,
transitive-reflexive-closure-base-case,
reverse_append_sq,
subtype_rel_list,
top_wf,
map_append_sq,
reverse-cons,
map_cons_lemma,
list_ind_cons_lemma,
inverse-letters_wf,
length_wf,
length-append,
exists_wf,
subtype_base_sq,
int_subtype_base,
or_wf,
append_assoc_sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
unionEquality,
hypothesis,
promote_hyp,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
because_Cache,
independent_pairFormation,
sqequalRule,
lambdaEquality,
unionElimination,
inrEquality,
inlEquality,
dependent_functionElimination,
independent_functionElimination,
independent_isectElimination,
dependent_pairFormation,
productEquality,
applyEquality,
instantiate,
universeEquality,
pointwiseFunctionality,
pertypeElimination,
productElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
isect_memberEquality,
axiomEquality,
voidElimination,
voidEquality,
inlFormation,
applyLambdaEquality,
cumulativity,
intEquality,
inrFormation
Latex:
\mforall{}[X:Type]. \mforall{}[x:free-word(X)]. (x + free-word-inv(x) = 0)
Date html generated:
2020_05_20-AM-08_22_41
Last ObjectModification:
2018_08_21-PM-02_02_24
Theory : free!groups
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