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Theorem Dependencies

proveit.physics.quantum.QPE._fail_sum

see proof

In [1]:
import proveit
# Automation is not needed when querying dependencies:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
from proveit.physics.quantum.QPE import _fail_sum
In [2]:
%dependencies _fail_sum

Unproven conjectures required (directly or indirectly) to prove _fail_sum

proveit.core_expr_types.conditionals.condition_substitution
proveit.core_expr_types.conditionals.dissatisfied_condition_reduction
proveit.core_expr_types.conditionals.satisfied_condition_reduction
proveit.core_expr_types.conditionals.true_condition_elimination
proveit.core_expr_types.expr_arrays.varray_eq_via_elem_eq
proveit.core_expr_types.expr_arrays.varray_neq_via_any_elem_neq
proveit.core_expr_types.operations.operands_substitution_via_tuple
proveit.core_expr_types.tuples.extended_range_from1_len_typical_eq
proveit.core_expr_types.tuples.extended_range_len
proveit.core_expr_types.tuples.general_len
proveit.core_expr_types.tuples.len_of_ranges_with_repeated_indices_from_1
proveit.core_expr_types.tuples.merge
proveit.core_expr_types.tuples.merge_extension
proveit.core_expr_types.tuples.merge_front
proveit.core_expr_types.tuples.partition_back
proveit.core_expr_types.tuples.partition_front
proveit.core_expr_types.tuples.range_fn_transformation
proveit.core_expr_types.tuples.range_from1_len
proveit.core_expr_types.tuples.range_from1_len_is_nat
proveit.core_expr_types.tuples.range_from1_len_typical_eq
proveit.core_expr_types.tuples.range_len
proveit.core_expr_types.tuples.range_len_is_nat
proveit.core_expr_types.tuples.shift_equivalence
proveit.core_expr_types.tuples.tuple_len_0_typical_eq
proveit.core_expr_types.tuples.tuple_neq_via_any_elem_neq
proveit.core_expr_types.tuples.tuple_portion_substitution
proveit.linear_algebra.addition.binary_closure
proveit.linear_algebra.addition.norm_of_sum_of_orthogonal_pair
proveit.linear_algebra.complex_vec_set_is_vec_space
proveit.linear_algebra.inner_products.complex_vec_set_is_hilbert_space
proveit.linear_algebra.inner_products.complex_vec_set_is_inner_prod_space
proveit.linear_algebra.inner_products.inner_prod_scalar_mult_right
proveit.linear_algebra.inner_products.scaled_norm
proveit.linear_algebra.matrices.exponentiation.U_closure
proveit.linear_algebra.matrices.exponentiation.unital2pi_eigen_exp_application
proveit.linear_algebra.matrices.unitaries_are_matrices
proveit.linear_algebra.scalar_multiplication.scalar_mult_closure
proveit.linear_algebra.tensors.norm_preserving_tensor_prod
proveit.linear_algebra.tensors.tensor_prod_disassociation
proveit.linear_algebra.tensors.tensor_prod_is_in_tensor_prod_space
proveit.linear_algebra.tensors.tensor_prod_of_cart_exps_within_cart_exp
proveit.logic.booleans.conjunction.and_if_all
proveit.logic.booleans.conjunction.any_from_and
proveit.logic.booleans.conjunction.conjunction_from_quantification
proveit.logic.booleans.conjunction.disassociate
proveit.logic.booleans.conjunction.redundant_conjunction_general
proveit.logic.booleans.conjunction.some_from_and
proveit.logic.booleans.disjunction.any_if_all
proveit.logic.booleans.disjunction.disassociate
proveit.logic.booleans.disjunction.or_if_any
proveit.logic.booleans.quantification.universality.bundle
proveit.logic.equality.four_chain_transitivity
proveit.logic.sets.enumeration.fold_singleton
proveit.logic.sets.enumeration.unfold
proveit.logic.sets.functions.bijections.bijection_transitivity
proveit.logic.sets.functions.bijections.bijective_by_uniqueness
proveit.logic.sets.functions.bijections.elim_domain_condition
proveit.logic.sets.functions.bijections.membership_unfolding
proveit.logic.sets.functions.injections.subset_injection
proveit.logic.sets.unification.membership_folding
proveit.logic.sets.unification.union_inclusion
proveit.numbers.absolute_value.abs_neg_elim
proveit.numbers.absolute_value.abs_non_neg_elim
proveit.numbers.absolute_value.complex_unit_length
proveit.numbers.absolute_value.weak_upper_bound_asym_interval
proveit.numbers.addition.add_complex_closure
proveit.numbers.addition.add_complex_closure_bin
proveit.numbers.addition.add_int_closure_bin
proveit.numbers.addition.add_nat_closure
proveit.numbers.addition.add_nat_closure_bin
proveit.numbers.addition.add_nat_pos_closure_bin
proveit.numbers.addition.add_nat_pos_from_nonneg
proveit.numbers.addition.add_real_closure
proveit.numbers.addition.add_real_closure_bin
proveit.numbers.addition.association
proveit.numbers.addition.commutation
proveit.numbers.addition.disassociation
proveit.numbers.addition.elim_zero_any
proveit.numbers.addition.elim_zero_left
proveit.numbers.addition.elim_zero_right
proveit.numbers.addition.leftward_commutation
proveit.numbers.addition.strong_bound_via_left_term_bound
proveit.numbers.addition.subtraction.add_cancel_basic
proveit.numbers.addition.subtraction.add_cancel_general
proveit.numbers.addition.subtraction.add_cancel_general_rev
proveit.numbers.addition.subtraction.add_cancel_reverse
proveit.numbers.addition.subtraction.add_cancel_triple_12
proveit.numbers.addition.subtraction.add_cancel_triple_13
proveit.numbers.addition.subtraction.add_cancel_triple_21
proveit.numbers.addition.subtraction.add_cancel_triple_23
proveit.numbers.addition.subtraction.add_cancel_triple_31
proveit.numbers.addition.subtraction.add_cancel_triple_32
proveit.numbers.addition.subtraction.nonneg_difference
proveit.numbers.addition.subtraction.pos_difference
proveit.numbers.addition.subtraction.subtract_from_add
proveit.numbers.addition.subtraction.subtract_from_add_reversed
proveit.numbers.addition.weak_bound_via_left_term_bound
proveit.numbers.division.div_as_mult
proveit.numbers.division.div_complex_closure
proveit.numbers.division.div_rational_nonzero_closure
proveit.numbers.division.div_real_pos_closure
proveit.numbers.division.frac_one_denom
proveit.numbers.division.mult_frac_cancel_denom_left
proveit.numbers.exponentiation.complex_x_to_first_power_is_x
proveit.numbers.exponentiation.exp_complex_closure
proveit.numbers.exponentiation.exp_int_closure
proveit.numbers.exponentiation.exp_nat_pos_rev
proveit.numbers.exponentiation.exp_natpos_closure
proveit.numbers.exponentiation.exp_rational_non_zero__not_zero
proveit.numbers.exponentiation.exp_real_closure_nat_power
proveit.numbers.exponentiation.exponentiated_one
proveit.numbers.exponentiation.neg_power_as_div
proveit.numbers.exponentiation.product_of_posnat_powers
proveit.numbers.exponentiation.product_of_real_powers
proveit.numbers.exponentiation.sqrt_complex_closure
proveit.numbers.exponentiation.sqrt_real_pos_closure
proveit.numbers.modular.interval_left_shift_bijection
proveit.numbers.modular.mod_abs_x_reduce_to_abs_x
proveit.numbers.modular.mod_natpos_in_interval
proveit.numbers.modular.redundant_mod_elimination_in_sum_in_modabs
proveit.numbers.multiplication.association
proveit.numbers.multiplication.commutation
proveit.numbers.multiplication.disassociation
proveit.numbers.multiplication.distribute_through_sum
proveit.numbers.multiplication.elim_one_left
proveit.numbers.multiplication.elim_one_right
proveit.numbers.multiplication.leftward_commutation
proveit.numbers.multiplication.mult_complex_closure_bin
proveit.numbers.multiplication.mult_nat_closure_bin
proveit.numbers.multiplication.mult_real_closure_bin
proveit.numbers.multiplication.mult_zero_right
proveit.numbers.multiplication.strong_bound_via_right_factor_bound
proveit.numbers.negation.complex_closure
proveit.numbers.negation.distribute_neg_through_binary_sum
proveit.numbers.negation.double_negation
proveit.numbers.negation.int_closure
proveit.numbers.negation.int_neg_closure
proveit.numbers.negation.mult_neg_one_left
proveit.numbers.negation.nat_closure
proveit.numbers.negation.nat_pos_closure
proveit.numbers.negation.negated_zero
proveit.numbers.negation.real_closure
proveit.numbers.number_sets.complex_numbers.i_is_complex
proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
proveit.numbers.number_sets.complex_numbers.real_within_complex
proveit.numbers.number_sets.complex_numbers.zero_is_complex
proveit.numbers.number_sets.integers.disjoint_intervals
proveit.numbers.number_sets.integers.in_interval
proveit.numbers.number_sets.integers.int_interval_within_int
proveit.numbers.number_sets.integers.int_not_in_interval
proveit.numbers.number_sets.integers.interval_lower_bound
proveit.numbers.number_sets.integers.interval_subset_eq
proveit.numbers.number_sets.integers.interval_upper_bound
proveit.numbers.number_sets.integers.nat_pos_within_int
proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
proveit.numbers.number_sets.integers.nat_within_int
proveit.numbers.number_sets.integers.neg_int_is_int_neg
proveit.numbers.number_sets.integers.neg_int_within_int
proveit.numbers.number_sets.integers.negative_if_in_neg_int
proveit.numbers.number_sets.integers.nonneg_int_is_natural
proveit.numbers.number_sets.integers.nonpos_int_is_int_nonpos
proveit.numbers.number_sets.integers.pos_int_is_natural_pos
proveit.numbers.number_sets.integers.zero_set_within_int
proveit.numbers.number_sets.natural_numbers.fold_forall_natural_pos
proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
proveit.numbers.number_sets.natural_numbers.natural_lower_bound
proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
proveit.numbers.number_sets.natural_numbers.zero_set_within_nat
proveit.numbers.number_sets.rational_numbers.int_within_rational
proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
proveit.numbers.number_sets.rational_numbers.neg_int_within_rational_neg
proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
proveit.numbers.number_sets.real_numbers.all_in_interval_co__is__real
proveit.numbers.number_sets.real_numbers.e_is_real_pos
proveit.numbers.number_sets.real_numbers.nat_pos_within_real
proveit.numbers.number_sets.real_numbers.nonneg_if_in_real_nonneg
proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
proveit.numbers.number_sets.real_numbers.pi_is_real_pos
proveit.numbers.number_sets.real_numbers.rational_neg_within_real_neg
proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
proveit.numbers.number_sets.real_numbers.rational_within_real
proveit.numbers.number_sets.real_numbers.real_neg_within_real
proveit.numbers.number_sets.real_numbers.real_pos_within_real
proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonneg
proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
proveit.numbers.number_sets.real_numbers.zero_is_real
proveit.numbers.numerals.decimals.add_1_1
proveit.numbers.numerals.decimals.add_2_1
proveit.numbers.numerals.decimals.add_3_1
proveit.numbers.numerals.decimals.less_0_1
proveit.numbers.numerals.decimals.nat3
proveit.numbers.numerals.decimals.nat4
proveit.numbers.numerals.decimals.nat5
proveit.numbers.numerals.decimals.nat6
proveit.numbers.numerals.decimals.nat8
proveit.numbers.numerals.decimals.posnat1
proveit.numbers.numerals.decimals.posnat2
proveit.numbers.numerals.decimals.posnat3
proveit.numbers.numerals.decimals.posnat4
proveit.numbers.numerals.decimals.posnat5
proveit.numbers.numerals.decimals.posnat6
proveit.numbers.numerals.decimals.posnat8
proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
proveit.numbers.numerals.decimals.tuple_len_4_typical_eq
proveit.numbers.numerals.decimals.tuple_len_5_typical_eq
proveit.numbers.numerals.decimals.tuple_len_6_typical_eq
proveit.numbers.numerals.decimals.tuple_len_8_typical_eq
proveit.numbers.ordering.less_add_right
proveit.numbers.ordering.less_add_right_weak_int
proveit.numbers.ordering.less_eq_add_right
proveit.numbers.ordering.less_eq_add_right_strong
proveit.numbers.ordering.less_is_not_eq_nat
proveit.numbers.ordering.less_or_greater_eq
proveit.numbers.ordering.not_less_eq_from_less
proveit.numbers.ordering.not_less_from_less_eq
proveit.numbers.ordering.relax_equal_to_less_eq
proveit.numbers.ordering.relax_less
proveit.physics.quantum.QFT.invFT_is_unitary
proveit.physics.quantum.algebra.ket_one_in_qubit_space
proveit.physics.quantum.algebra.ket_one_norm
proveit.physics.quantum.algebra.ket_zero_and_one_have_zero_inner_prod
proveit.physics.quantum.algebra.ket_zero_in_qubit_space
proveit.physics.quantum.algebra.ket_zero_norm
proveit.physics.quantum.algebra.normalization_preservation
proveit.physics.quantum.algebra.num_ket_in_register_space
proveit.physics.quantum.algebra.num_ket_neq
proveit.physics.quantum.algebra.qmult_matrix_is_linmap
proveit.physics.quantum.algebra.qmult_op_ket_is_ket
proveit.physics.quantum.algebra.state_space_preservation
proveit.physics.quantum.circuits.born_rule_on_register
proveit.physics.quantum.circuits.circuit_equiv_temporal_sub
proveit.physics.quantum.circuits.concat_onto_ideal_expt
proveit.physics.quantum.circuits.equiv_reversal
proveit.physics.quantum.circuits.equiv_transitivity
proveit.physics.quantum.circuits.input_consolidation
proveit.physics.quantum.circuits.output_consolidation
proveit.physics.quantum.circuits.phase_kickbacks_on_register
proveit.physics.quantum.circuits.prob_eq_via_equiv
proveit.physics.quantum.circuits.qcircuit_eq
proveit.physics.quantum.circuits.qcircuit_neq
proveit.physics.quantum.circuits.qcircuit_output_part_neq
proveit.physics.quantum.circuits.register_meas_bijection
proveit.physics.quantum.circuits.register_meas_sample_space
proveit.physics.quantum.circuits.rhs_prob_via_equiv
proveit.physics.quantum.circuits.trivial_expansion_below
proveit.physics.quantum.circuits.unary_multi_output_reduction
proveit.physics.quantum.circuits.unitary_gate_operation
proveit.statistics.prob_of_all_as_sum
proveit.statistics.prob_of_all_events_transformation
proveit.statistics.prob_of_disjoint_events_is_prob_sum
proveit.statistics.sample_space_via_bijections

Axioms required (directly or indirectly) to prove _fail_sum

proveit.core_expr_types.conditionals.condition_replacement
proveit.core_expr_types.conditionals.true_case_reduction
proveit.core_expr_types.lambda_maps.lambda_substitution
proveit.core_expr_types.operations.operands_substitution
proveit.core_expr_types.tuples.empty_range_def
proveit.core_expr_types.tuples.range_extension_def
proveit.core_expr_types.tuples.tuple_eq_def
proveit.core_expr_types.tuples.tuple_len_incr
proveit.linear_algebra.scalar_multiplication.scalar_mult_extends_number_mult
proveit.logic.booleans.bools_def
proveit.logic.booleans.conjunction.and_t_t
proveit.logic.booleans.disjunction.empty_disjunction
proveit.logic.booleans.disjunction.left_in_bool
proveit.logic.booleans.disjunction.multi_disjunction_def
proveit.logic.booleans.disjunction.or_f_f
proveit.logic.booleans.disjunction.or_f_t
proveit.logic.booleans.disjunction.or_t_f
proveit.logic.booleans.disjunction.or_t_t
proveit.logic.booleans.disjunction.right_in_bool
proveit.logic.booleans.eq_true_elim
proveit.logic.booleans.eq_true_intro
proveit.logic.booleans.false_not_true
proveit.logic.booleans.implication.affirmation_via_contradiction
proveit.logic.booleans.implication.denial_via_contradiction
proveit.logic.booleans.implication.iff_def
proveit.logic.booleans.negation.negation_elim
proveit.logic.booleans.negation.not_f
proveit.logic.booleans.negation.not_t
proveit.logic.booleans.negation.operand_is_bool
proveit.logic.booleans.true_axiom
proveit.logic.equality.equality_in_bool
proveit.logic.equality.equals_reflexivity
proveit.logic.equality.equals_symmetry
proveit.logic.equality.equals_transitivity
proveit.logic.equality.not_equals_def
proveit.logic.equality.substitution
proveit.logic.sets.enumeration.enum_set_def
proveit.logic.sets.functions.images.set_image_def
proveit.logic.sets.inclusion.proper_subset_def
proveit.logic.sets.inclusion.subset_eq_def
proveit.logic.sets.membership.not_in_set_def
proveit.numbers.number_sets.natural_numbers.successor_in_nats
proveit.numbers.number_sets.natural_numbers.zero_in_nats
proveit.numbers.numerals.decimals.one_def
proveit.numbers.ordering.less_eq_def
proveit.numbers.ordering.max_def_bin
proveit.numbers.ordering.transitivity_less_less
proveit.numbers.rounding.floor_is_an_int
proveit.physics.quantum.QPE.QPE1_def
proveit.physics.quantum.QPE.QPE_def
proveit.physics.quantum.QPE._eigen_uu
proveit.physics.quantum.QPE._normalized_ket_u
proveit.physics.quantum.QPE._phase_in_interval
proveit.physics.quantum.QPE._s_in_nat_pos
proveit.physics.quantum.QPE._t_in_natural_pos
proveit.physics.quantum.QPE._u_ket_register
proveit.physics.quantum.QPE._unitary_U

Conservative definitions used (but not logically required) to prove _fail_sum

proveit.physics.quantum.QPE._psi_t_def
proveit.physics.quantum.QPE._Psi_def
proveit.physics.quantum.QPE._alpha_m_def
proveit.physics.quantum.QPE._best_floor_def
proveit.physics.quantum.QPE._fail_def
proveit.physics.quantum.QPE._mod_add_def
proveit.physics.quantum.QPE._sample_space_def

Theorems/conjectures that depend directly on _fail_sum

proveit.physics.quantum.QPE._failure_upper_bound_lemma