Nuprl Lemma : app_permf_comp
∀m,n:ℕ. ∀p,p':ℕm ⟶ ℕm. ∀q,q':ℕn ⟶ ℕn.
((app_permf(m;n;p;q) o app_permf(m;n;p';q')) = app_permf(m;n;p o p';q o q') ∈ (ℕm + n ⟶ ℕm + n))
Proof
Definitions occuring in Statement :
app_permf: app_permf(m;n;p;q),
compose: f o g,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
compose: f o g,
app_permf: app_permf(m;n;p;q),
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
guard: {T},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
int_seg_wf,
nat_wf,
lt_int_wf,
equal-wf-base,
bool_wf,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
lelt_wf,
assert_wf,
less_than_wf,
le_int_wf,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
equal-wf-T-base,
add_nat_wf,
intformless_wf,
int_formula_prop_less_lemma,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
decidable__lt,
decidable__equal_int,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
add-subtract-cancel,
add-member-int_seg2,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
minus-minus,
add-commutes,
zero-add,
le-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation_alt,
cut,
lambdaEquality_alt,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
addEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
inhabitedIsType,
functionIsType,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
independent_isectElimination,
because_Cache,
unionElimination,
equalityElimination,
independent_functionElimination,
productElimination,
equalityIsType1,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
productIsType,
applyLambdaEquality,
imageElimination,
pointwiseFunctionality,
promote_hyp,
minusEquality,
multiplyEquality
Latex:
\mforall{}m,n:\mBbbN{}. \mforall{}p,p':\mBbbN{}m {}\mrightarrow{} \mBbbN{}m. \mforall{}q,q':\mBbbN{}n {}\mrightarrow{} \mBbbN{}n.
((app\_permf(m;n;p;q) o app\_permf(m;n;p';q')) = app\_permf(m;n;p o p';q o q'))
Date html generated:
2019_10_16-PM-00_59_42
Last ObjectModification:
2018_10_08-AM-09_20_29
Theory : perms_1
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