Nuprl Lemma : Sudoku-induction
∀P:SudokuPuzzle() ⟶ ℙ
((∀p:SudokuPuzzle(). ((∀q:SudokuPuzzle(). P[q] supposing Sudoku-size(q) < Sudoku-size(p)) ⇒ P[p]))
⇒ (∀p:SudokuPuzzle(). P[p]))
Proof
Definitions occuring in Statement :
Sudoku-size: Sudoku-size(P),
SudokuPuzzle: SudokuPuzzle(),
less_than: a < b,
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T
Lemmas referenced :
SudokuPuzzle_wf,
istype-less_than,
Sudoku-size_wf,
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,
istype-le,
subtract-1-ge-0,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
rename,
universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalRule,
functionIsType,
because_Cache,
isectIsType,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
universeEquality,
setElimination,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
independent_pairFormation,
voidElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
functionExtensionality,
isect_memberEquality_alt,
applyLambdaEquality,
unionElimination,
imageElimination,
productElimination
Latex:
\mforall{}P:SudokuPuzzle() {}\mrightarrow{} \mBbbP{}
((\mforall{}p:SudokuPuzzle()
((\mforall{}q:SudokuPuzzle(). P[q] supposing Sudoku-size(q) < Sudoku-size(p)) {}\mRightarrow{} P[p]))
{}\mRightarrow{} (\mforall{}p:SudokuPuzzle(). P[p]))
Date html generated:
2020_05_20-AM-08_04_43
Last ObjectModification:
2020_02_25-PM-02_28_06
Theory : general
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