Nuprl Lemma : term-opr_functionality
∀[opr:Type]. ∀[t,t':term(opr)].
(term-opr(t) = term-opr(t') ∈ opr) supposing (alpha-eq-terms(opr;t;t') and (¬↑isvarterm(t)))
Proof
Definitions occuring in Statement :
alpha-eq-terms: alpha-eq-terms(opr;a;b),
term-opr: term-opr(t),
isvarterm: isvarterm(t),
term: term(opr),
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
or: P ∨ Q,
prop: ℙ,
not: ¬A,
implies: P ⇒ Q,
false: False,
exists: ∃x:A. B[x],
and: P ∧ Q,
varterm: varterm(v),
isvarterm: isvarterm(t),
isl: isl(x),
squash: ↓T,
true: True,
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
alpha-eq-terms: alpha-eq-terms(opr;a;b),
alpha-aux: alpha-aux(opr;vs;ws;a;b),
mkterm: mkterm(opr;bts),
subtype_rel: A ⊆r B,
nat: ℕ,
bound-term: bound-term(opr),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
term-opr: term-opr(t),
outr: outr(x),
pi1: fst(t),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
so_lambda: λ2x.t[x],
so_apply: x[s],
nil: [],
it: ⋅,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
sq_type: SQType(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
less_than: a < b
Lemmas referenced :
term-cases,
alpha-eq-terms_wf,
istype-assert,
isvarterm_wf,
istype-void,
assert_functionality_wrt_uiff,
btrue_wf,
squash_wf,
true_wf,
term_wf,
istype-universe,
mkterm_wf,
subtype_rel_list,
bound-term_wf,
less_than_wf,
bound-term-size_wf,
term-size_wf,
list_wf,
varname_wf,
istype-less_than,
iff_weakening_uiff,
assert_wf,
equal_wf,
term-opr_wf,
not_wf,
subtype_rel_self,
iff_weakening_equal,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
set_wf,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__equal_int,
subtract_wf,
itermSubtract_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
le_wf,
equal-wf-base,
length_wf_nat,
alpha-aux_wf,
rev-append_wf,
nil_wf,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
unionElimination,
hypothesis,
universeIsType,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsType,
because_Cache,
independent_functionElimination,
productElimination,
independent_isectElimination,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
voidElimination,
hyp_replacement,
lambdaFormation_alt,
setEquality,
setElimination,
rename,
productEquality,
setIsType,
intWeakElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
functionIsTypeImplies,
equalityIstype,
promote_hyp,
hypothesis_subsumption,
dependent_set_memberEquality_alt,
applyLambdaEquality,
baseApply,
closedConclusion,
intEquality,
sqequalBase,
productIsType,
spreadEquality
Latex:
\mforall{}[opr:Type]. \mforall{}[t,t':term(opr)].
(term-opr(t) = term-opr(t')) supposing (alpha-eq-terms(opr;t;t') and (\mneg{}\muparrow{}isvarterm(t)))
Date html generated:
2020_05_19-PM-09_55_48
Last ObjectModification:
2020_05_13-PM-03_29_42
Theory : terms
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