Nuprl Lemma : binary_map_ind_wf
∀[T,Key,A:Type]. ∀[R:A ⟶ binary_map(T;Key) ⟶ ℙ]. ∀[v:binary_map(T;Key)]. ∀[E:{x:A| R[x;bm_E()]} ].
∀[T:key:Key
⟶ value:T
⟶ cnt:ℤ
⟶ left:binary_map(T;Key)
⟶ right:binary_map(T;Key)
⟶ {x:A| R[x;left]}
⟶ {x:A| R[x;right]}
⟶ {x:A| R[x;bm_T(key;value;cnt;left;right)]} ].
(binary_map_ind(v;E;key,value,cnt,left,right,rec1,rec2.T[key;value;cnt;left;right;rec1;rec2]) ∈ {x:A| R[x;v]} )
Proof
Definitions occuring in Statement :
binary_map_ind: binary_map_ind(v;E;key,value,cnt,left,right,rec1,rec2.T[key;value;cnt;left;right;rec1;rec2]),
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,
so_apply: x[a;b;c;d;e;f;g],
so_apply: x[s1;s2],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
binary_map_ind: binary_map_ind(v;E;key,value,cnt,left,right,rec1,rec2.T[key;value;cnt;left;right;rec1;rec2]),
so_apply: x[a;b;c;d;e;f;g],
so_apply: x[s1;s2],
binary_map-definition,
binary_map-induction,
uniform-comp-nat-induction,
binary_map-ext,
eq_atom: x =a y,
bool_cases_sqequal,
eqff_to_assert,
any: any x,
btrue: tt,
bfalse: ff,
it: ⋅,
top: Top,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
prop: ℙ,
guard: {T},
or: P ∨ Q,
squash: ↓T,
subtype_rel: A ⊆r B
Latex:
\mforall{}[T,Key,A:Type]. \mforall{}[R:A {}\mrightarrow{} binary\_map(T;Key) {}\mrightarrow{} \mBbbP{}]. \mforall{}[v:binary\_map(T;Key)]. \mforall{}[E:\{x:A| R[x;bm\_E()]\} ].
\mforall{}[T:key:Key
{}\mrightarrow{} value:T
{}\mrightarrow{} cnt:\mBbbZ{}
{}\mrightarrow{} left:binary\_map(T;Key)
{}\mrightarrow{} right:binary\_map(T;Key)
{}\mrightarrow{} \{x:A| R[x;left]\}
{}\mrightarrow{} \{x:A| R[x;right]\}
{}\mrightarrow{} \{x:A| R[x;bm\_T(key;value;cnt;left;right)]\} ].
(binary\_map\_ind(v;E;key,value,cnt,left,right,rec1,rec2.T[key;value;cnt;left;right;rec1;rec2])
\mmember{} \{x:A| R[x;v]\} )
Date html generated:
2016_05_17-PM-01_37_46
Last ObjectModification:
2016_01_17-AM-11_21_16
Theory : binary-map
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