Nuprl Lemma : map-square-board-cell
∀[n:ℕ]. ∀[T1,T2:Type]. ∀[f:ℕn ⟶ ℕn ⟶ T1 ⟶ T2]. ∀[b:square-board(n;T1)]. ∀[i,j:ℕn].
(board-cell(map-square-board(i,j,v.f[i;j;v];b);i;j) = f[i;j;board-cell(b;i;j)] ∈ T2)
Proof
Definitions occuring in Statement :
board-cell: board-cell(b;i;j),
map-square-board: map-square-board(i,j,v.f[i; j; v];b),
square-board: square-board(n;T),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2;s3],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
square-board: square-board(n;T),
board-cell: board-cell(b;i;j),
map-square-board: map-square-board(i,j,v.f[i; j; v];b),
nat: ℕ,
top: Top,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
and: P ∧ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
ge: i ≥ j ,
all: ∀x:A. B[x],
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,
prop: ℙ,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_apply: x[s1;s2;s3]
Lemmas referenced :
length_wf,
iff_weakening_equal,
true_wf,
squash_wf,
less_than_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
itermConstant_wf,
intformle_wf,
decidable__le,
select_wf,
lelt_wf,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_properties,
int_seg_properties,
top_wf,
list_wf,
subtype_rel_list,
select-map-index,
nat_wf,
square-board_wf,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesis,
lemma_by_obid,
isectElimination,
natural_numberEquality,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
functionEquality,
universeEquality,
voidElimination,
voidEquality,
applyEquality,
independent_isectElimination,
lambdaEquality,
productElimination,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
cumulativity,
imageElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[T1,T2:Type]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n {}\mrightarrow{} T1 {}\mrightarrow{} T2]. \mforall{}[b:square-board(n;T1)]. \mforall{}[i,j:\mBbbN{}n].
(board-cell(map-square-board(i,j,v.f[i;j;v];b);i;j) = f[i;j;board-cell(b;i;j)])
Date html generated:
2016_05_15-PM-03_14_01
Last ObjectModification:
2016_01_16-AM-10_47_26
Theory : general
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