Nuprl Lemma : binary_map-induction
∀[T,Key:Type]. ∀[P:binary_map(T;Key) ⟶ ℙ].
(P[bm_E()]
⇒ (∀key:Key. ∀value:T. ∀cnt:ℤ. ∀left,right:binary_map(T;Key).
(P[left] ⇒ P[right] ⇒ P[bm_T(key;value;cnt;left;right)]))
⇒ {∀v:binary_map(T;Key). P[v]})
Proof
Definitions occuring in Statement :
bm_T: bm_T(key;value;cnt;left;right),
bm_E: bm_E(),
binary_map: binary_map(T;Key),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
so_lambda: λ2x.t[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
ext-eq: A ≡ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
sq_type: SQType(T),
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
bm_E: bm_E(),
binary_map_size: binary_map_size(p),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
bm_T: bm_T(key;value;cnt;left;right),
spreadn: let a,b,c,d,e = u in v[a; b; c; d; e],
cand: A c∧ B,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T
Latex:
\mforall{}[T,Key:Type]. \mforall{}[P:binary\_map(T;Key) {}\mrightarrow{} \mBbbP{}].
(P[bm\_E()]
{}\mRightarrow{} (\mforall{}key:Key. \mforall{}value:T. \mforall{}cnt:\mBbbZ{}. \mforall{}left,right:binary\_map(T;Key).
(P[left] {}\mRightarrow{} P[right] {}\mRightarrow{} P[bm\_T(key;value;cnt;left;right)]))
{}\mRightarrow{} \{\mforall{}v:binary\_map(T;Key). P[v]\})
Date html generated:
2016_05_17-PM-01_37_40
Last ObjectModification:
2016_01_17-AM-11_21_08
Theory : binary-map
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