Theorem Dependencies¶

proveit.logic.sets.inclusion.superset_membership_from_proper_subset ¶

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.logic.sets.inclusion import superset_membership_from_proper_subset

In [2]:

%dependencies superset_membership_from_proper_subset

Axioms required (directly or indirectly) to prove superset_membership_from_proper_subset

proveit.logic.booleans.eq_true_elim

proveit.logic.booleans.eq_true_intro

proveit.logic.equality.equals_symmetry

proveit.logic.equality.equals_transitivity

proveit.logic.equality.substitution

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

proveit.logic.sets.inclusion.subset_eq_def

proveit.logic.sets.inclusion.proper_subset_def

Theorems/conjectures that depend directly on superset_membership_from_proper_subset

proveit.core_expr_types.tuples.tuple_eq_via_elem_eq

proveit.logic.booleans.disjunction.closure

proveit.logic.booleans.quantification.universality.forall_with_conditions__is_bool

proveit.logic.equality.multi_substitution

proveit.numbers.exponentiation.sqrt2_is_not_rational

proveit.numbers.number_sets.integers.zero_is_int

proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos

proveit.numbers.rounding.ceil_of_real_is_real

proveit.numbers.rounding.floor_of_real_is_real

proveit.numbers.rounding.real_minus_floor_interval

proveit.numbers.rounding.round_of_real_is_real

proveit.numbers.summation.finite_geom_sum

proveit.numbers.summation.sum_first_n_int

proveit.physics.quantum.QPE._Psi_ket_is_normalized_vec

proveit.physics.quantum.QPE._Psi_output

proveit.physics.quantum.QPE._alpha_ideal_case

proveit.physics.quantum.QPE._alpha_m_evaluation

proveit.physics.quantum.QPE._alpha_m_mod_as_geometric_sum

proveit.physics.quantum.QPE._alpha_m_mod_evaluation

proveit.physics.quantum.QPE._alpha_sqrd_upper_bound

proveit.physics.quantum.QPE._alpha_summed

proveit.physics.quantum.QPE._best_floor_is_in_m_domain

proveit.physics.quantum.QPE._best_floor_is_int

proveit.physics.quantum.QPE._best_guarantee

proveit.physics.quantum.QPE._best_guarantee_delta_nonzero

proveit.physics.quantum.QPE._best_round_is_int

proveit.physics.quantum.QPE._delta_b_floor_diff_in_interval

proveit.physics.quantum.QPE._delta_b_in_interval

proveit.physics.quantum.QPE._delta_b_is_real

proveit.physics.quantum.QPE._delta_b_is_zero_or_non_int

proveit.physics.quantum.QPE._delta_b_not_eq_scaledNonzeroInt

proveit.physics.quantum.QPE._e_value_ge_two

proveit.physics.quantum.QPE._e_value_in_e_domain

proveit.physics.quantum.QPE._fail_sum

proveit.physics.quantum.QPE._fail_sum_prob_conds_equiv_lemma

proveit.physics.quantum.QPE._failure_upper_bound

proveit.physics.quantum.QPE._failure_upper_bound_lemma

proveit.physics.quantum.QPE._modabs_in_full_domain_simp

proveit.physics.quantum.QPE._non_int_delta_b_diff

proveit.physics.quantum.QPE._outcome_prob

proveit.physics.quantum.QPE._pfail_in_real

proveit.physics.quantum.QPE._phase_from_best_with_delta_b

proveit.physics.quantum.QPE._phase_is_real

proveit.physics.quantum.QPE._precision_guarantee

proveit.physics.quantum.QPE._precision_guarantee_lemma_01

proveit.physics.quantum.QPE._precision_guarantee_lemma_02

proveit.physics.quantum.QPE._psi_t_formula

proveit.physics.quantum.QPE._psi_t_ket_is_normalized_vec

proveit.physics.quantum.QPE._psi_t_output

proveit.physics.quantum.QPE._sample_space_bijection

proveit.physics.quantum.QPE._scaled_abs_delta_b_floor_diff_interval

proveit.physics.quantum.QPE._scaled_delta_b_floor_in_interval

proveit.physics.quantum.QPE._scaled_delta_b_is_zero_or_non_int

proveit.physics.quantum.QPE._scaled_delta_b_round_in_interval

proveit.physics.quantum.QPE._success_complements_failure

proveit.physics.quantum.QPE._two_pow_t_is_nat_pos

proveit.physics.quantum.QPE._two_pow_t_minus_one_is_nat_pos

proveit.physics.quantum.QPE.qpe_exact