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Theorem Dependencies

proveit.physics.quantum.QPE._alpha_summed

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 _alpha_summed
In [2]:
%dependencies _alpha_summed

Unproven conjectures required (directly or indirectly) to prove _alpha_summed

proveit.core_expr_types.conditionals.satisfied_condition_reduction
proveit.core_expr_types.lambda_maps.general_lambda_substitution
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.general_len
proveit.core_expr_types.tuples.merge_extension
proveit.core_expr_types.tuples.partition_front
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.tuple_len_0_typical_eq
proveit.core_expr_types.tuples.tuple_portion_substitution
proveit.linear_algebra.addition.binary_closure
proveit.linear_algebra.addition.summation_closure
proveit.linear_algebra.addition.vec_sum_index_shift
proveit.linear_algebra.addition.vec_sum_split_after
proveit.linear_algebra.addition.vec_sum_split_first
proveit.linear_algebra.complex_vec_set_is_vec_space
proveit.linear_algebra.inner_products.complex_set_is_hilbert_space
proveit.linear_algebra.inner_products.complex_vec_set_is_hilbert_space
proveit.linear_algebra.matrices.unitaries_are_matrices
proveit.linear_algebra.scalar_multiplication.distribution_over_vec_sum_with_scalar_mult
proveit.linear_algebra.scalar_multiplication.doubly_scaled_as_singly_scaled
proveit.linear_algebra.scalar_multiplication.one_as_scalar_mult_id
proveit.linear_algebra.scalar_multiplication.scalar_mult_closure
proveit.linear_algebra.tensors.factor_scalar_from_tensor_prod
proveit.linear_algebra.tensors.tensor_prod_disassociation
proveit.linear_algebra.tensors.tensor_prod_distribution_over_summation_with_scalar_mult
proveit.linear_algebra.tensors.tensor_prod_factorization_from_add
proveit.linear_algebra.tensors.tensor_prod_is_in_tensor_prod_space
proveit.linear_algebra.tensors.tensor_prod_of_cart_exps_within_cart_exp
proveit.linear_algebra.tensors.tensor_prod_of_vec_spaces_is_vec_space
proveit.logic.booleans.conjunction.any_from_and
proveit.logic.booleans.conjunction.conjunction_from_quantification
proveit.logic.booleans.conjunction.redundant_conjunction_general
proveit.logic.booleans.conjunction.some_from_and
proveit.logic.equality.four_chain_transitivity
proveit.logic.sets.enumeration.in_enumerated_set
proveit.logic.sets.enumeration.true_for_each_then_true_for_all
proveit.logic.sets.enumeration.unfold
proveit.logic.sets.membership.double_negated_membership
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_rational_closure_bin
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.rational_pair_addition
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_triple_12
proveit.numbers.addition.subtraction.add_cancel_triple_13
proveit.numbers.addition.subtraction.add_cancel_triple_23
proveit.numbers.addition.subtraction.add_cancel_triple_32
proveit.numbers.addition.subtraction.negated_add
proveit.numbers.addition.subtraction.nonneg_difference
proveit.numbers.addition.subtraction.nonzero_difference_if_different
proveit.numbers.addition.subtraction.pos_difference
proveit.numbers.addition.subtraction.subtract_from_add
proveit.numbers.addition.term_as_strong_upper_bound
proveit.numbers.addition.weak_bound_via_left_term_bound
proveit.numbers.addition.weak_bound_via_right_term_bound
proveit.numbers.division.distribute_frac_through_sum
proveit.numbers.division.div_as_mult
proveit.numbers.division.div_complex_closure
proveit.numbers.division.div_neq_real
proveit.numbers.division.div_rational_closure
proveit.numbers.division.div_rational_pos_closure
proveit.numbers.division.div_real_closure
proveit.numbers.division.div_real_pos_closure
proveit.numbers.division.frac_cancel_complete
proveit.numbers.division.frac_cancel_left
proveit.numbers.division.frac_one_denom
proveit.numbers.division.mult_frac_cancel_denom_left
proveit.numbers.division.mult_frac_cancel_numer_left
proveit.numbers.division.neg_frac_neg_numerator
proveit.numbers.division.prod_of_fracs
proveit.numbers.division.weak_div_from_denom_bound__all_pos
proveit.numbers.division.weak_div_from_numer_bound__pos_denom
proveit.numbers.exponentiation.complex_power_of_complex_power
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_monotonicity_large_base_less_eq
proveit.numbers.exponentiation.exp_natpos_closure
proveit.numbers.exponentiation.exp_not_eq_zero
proveit.numbers.exponentiation.exp_rational_non_zero__not_zero
proveit.numbers.exponentiation.exp_rational_nonzero_closure
proveit.numbers.exponentiation.exp_zero_eq_one
proveit.numbers.exponentiation.int_exp_of_exp
proveit.numbers.exponentiation.int_exp_of_neg_exp
proveit.numbers.exponentiation.neg_power_as_div
proveit.numbers.exponentiation.product_of_complex_powers
proveit.numbers.exponentiation.product_of_pos_powers
proveit.numbers.exponentiation.product_of_posnat_powers
proveit.numbers.exponentiation.product_of_real_powers
proveit.numbers.exponentiation.real_power_of_product
proveit.numbers.exponentiation.real_power_of_real_power
proveit.numbers.exponentiation.sqrt_complex_closure
proveit.numbers.exponentiation.unit_complex_polar_num_neq_one
proveit.numbers.modular.mod_natpos_in_interval
proveit.numbers.multiplication.association
proveit.numbers.multiplication.commutation
proveit.numbers.multiplication.disassociation
proveit.numbers.multiplication.distribute_through_subtract
proveit.numbers.multiplication.distribute_through_sum
proveit.numbers.multiplication.distribute_through_summation
proveit.numbers.multiplication.elim_one_any
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_int_closure_bin
proveit.numbers.multiplication.mult_nat_closure_bin
proveit.numbers.multiplication.mult_nat_pos_closure_bin
proveit.numbers.multiplication.mult_neg_any
proveit.numbers.multiplication.mult_neg_left
proveit.numbers.multiplication.mult_neg_right
proveit.numbers.multiplication.mult_not_eq_zero
proveit.numbers.multiplication.mult_rational_closure_bin
proveit.numbers.multiplication.mult_rational_pos_closure_bin
proveit.numbers.multiplication.mult_real_closure_bin
proveit.numbers.multiplication.mult_zero_any
proveit.numbers.multiplication.mult_zero_right
proveit.numbers.multiplication.rightward_commutation
proveit.numbers.multiplication.strong_bound_via_right_factor_bound
proveit.numbers.multiplication.weak_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_weak_bound
proveit.numbers.negation.negated_zero
proveit.numbers.negation.pos_times_neg
proveit.numbers.negation.real_closure
proveit.numbers.negation.real_neg_closure
proveit.numbers.number_sets.complex_numbers.exp_neg2pi_i_x
proveit.numbers.number_sets.complex_numbers.i_is_complex
proveit.numbers.number_sets.complex_numbers.int_within_complex
proveit.numbers.number_sets.complex_numbers.nonzero_complex_is_complex_nonzero
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.in_interval
proveit.numbers.number_sets.integers.int_interval_within_int
proveit.numbers.number_sets.integers.interval_lower_bound
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_within_int
proveit.numbers.number_sets.integers.nonneg_int_is_natural
proveit.numbers.number_sets.integers.nonpos_int_is_int_nonpos
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.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.nonzero_int_within_rational_nonzero
proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
proveit.numbers.number_sets.rational_numbers.rational_nonzero_within_rational
proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational_nonzero
proveit.numbers.number_sets.real_numbers.all_in_interval_co__is__real
proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
proveit.numbers.number_sets.real_numbers.all_in_interval_oo__is__real
proveit.numbers.number_sets.real_numbers.e_is_real_pos
proveit.numbers.number_sets.real_numbers.in_IntervalCO
proveit.numbers.number_sets.real_numbers.in_IntervalOC
proveit.numbers.number_sets.real_numbers.in_IntervalOO
proveit.numbers.number_sets.real_numbers.interval_cc_lower_bound
proveit.numbers.number_sets.real_numbers.interval_co_lower_bound
proveit.numbers.number_sets.real_numbers.interval_co_upper_bound
proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
proveit.numbers.number_sets.real_numbers.interval_oo_lower_bound
proveit.numbers.number_sets.real_numbers.interval_oo_upper_bound
proveit.numbers.number_sets.real_numbers.nat_pos_within_real
proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
proveit.numbers.number_sets.real_numbers.not_int_if_not_int_in_interval
proveit.numbers.number_sets.real_numbers.pi_is_real_pos
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_pos_within_real
proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
proveit.numbers.number_sets.real_numbers.relax_IntervalCO
proveit.numbers.number_sets.real_numbers.rescale_interval_co_membership
proveit.numbers.number_sets.real_numbers.zero_is_real
proveit.numbers.numerals.decimals.add_1_1
proveit.numbers.numerals.decimals.add_2_2
proveit.numbers.numerals.decimals.add_4_4
proveit.numbers.numerals.decimals.less_0_1
proveit.numbers.numerals.decimals.less_1_2
proveit.numbers.numerals.decimals.mult_2_2
proveit.numbers.numerals.decimals.mult_4_4
proveit.numbers.numerals.decimals.mult_8_2
proveit.numbers.numerals.decimals.nat3
proveit.numbers.numerals.decimals.nat4
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.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.ordering.less_add_right
proveit.numbers.ordering.less_eq_add_right_strong
proveit.numbers.ordering.less_is_not_eq_nat
proveit.numbers.ordering.relax_less
proveit.numbers.rounding.floor_increasing_less_eq
proveit.numbers.rounding.floor_of_integer
proveit.numbers.rounding.floor_of_real_below_int
proveit.numbers.rounding.real_minus_floor_upper_bound
proveit.numbers.rounding.round_in_terms_of_floor
proveit.numbers.summation.gen_finite_geom_sum
proveit.numbers.summation.summation_complex_closure
proveit.physics.quantum.QFT.invFT_is_unitary
proveit.physics.quantum.QFT.invFT_on_matrix_elem
proveit.physics.quantum.algebra.ket_one_in_qubit_space
proveit.physics.quantum.algebra.ket_zero_in_qubit_space
proveit.physics.quantum.algebra.num_bra_is_lin_map
proveit.physics.quantum.algebra.num_ket_in_register_space
proveit.physics.quantum.algebra.prepend_num_ket_with_one_ket
proveit.physics.quantum.algebra.prepend_num_ket_with_zero_ket
proveit.physics.quantum.algebra.qmult_disassociation
proveit.physics.quantum.algebra.qmult_distribution_over_summation
proveit.physics.quantum.algebra.qmult_matrix_is_linmap
proveit.physics.quantum.algebra.qmult_op_complex_closure
proveit.physics.quantum.algebra.qmult_op_is_linmap
proveit.physics.quantum.algebra.qmult_op_ket_in_QmultCodomain
proveit.physics.quantum.algebra.qmult_op_ket_is_ket
proveit.physics.quantum.algebra.qmult_op_op_is_op
proveit.physics.quantum.algebra.qmult_pulling_scalar_out_front
proveit.physics.quantum.algebra.scalar_mult_absorption
proveit.physics.quantum.algebra.scalar_mult_factorization
proveit.physics.quantum.algebra.single_qubit_num_ket

Axioms required (directly or indirectly) to prove _alpha_summed

proveit.core_expr_types.conditionals.conditional_substitution
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.linear_algebra.addition.scalar_sum_extends_number_sum
proveit.linear_algebra.addition.vec_sum_single
proveit.linear_algebra.scalar_multiplication.scalar_mult_extends_number_mult
proveit.linear_algebra.tensors.unary_tensor_prod_def
proveit.logic.booleans.bools_def
proveit.logic.booleans.conjunction.and_t_t
proveit.logic.booleans.disjunction.left_in_bool
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.negation.negation_elim
proveit.logic.booleans.negation.not_f
proveit.logic.booleans.negation.not_t
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.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.transitivity_less_less
proveit.numbers.rounding.floor_is_an_int
proveit.numbers.rounding.floor_of_x_less_eq_x
proveit.numbers.rounding.round_is_an_int
proveit.physics.quantum.QPE._phase_in_interval
proveit.physics.quantum.QPE._t_in_natural_pos
proveit.physics.quantum.algebra.multi_qmult_def

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

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._best_round_def
proveit.physics.quantum.QPE._delta_b_def
proveit.physics.quantum.QPE._mod_add_def

Theorems/conjectures that depend directly on _alpha_summed

proveit.physics.quantum.QPE._alpha_sqrd_upper_bound