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Theorem Dependencies

proveit.numbers.exponentiation.sqrt2_is_not_rational

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.numbers.exponentiation import sqrt2_is_not_rational
In [2]:
%dependencies sqrt2_is_not_rational

Unproven conjectures required (directly or indirectly) to prove sqrt2_is_not_rational

proveit.logic.equality.four_chain_transitivity
proveit.numbers.absolute_value.abs_rational_nonzero_closure
proveit.numbers.divisibility.GCD_one_def
proveit.numbers.divisibility.common_exponent_introduction
proveit.numbers.divisibility.common_factor_elimination
proveit.numbers.divisibility.even__if__power_is_even
proveit.numbers.divisibility.left_factor_divisibility
proveit.numbers.division.div_rational_nonzero_closure
proveit.numbers.division.div_real_closure
proveit.numbers.division.frac_one_denom
proveit.numbers.division.mult_frac_cancel_denom_left
proveit.numbers.exponentiation.exp_eq_real
proveit.numbers.exponentiation.exp_natpos_closure
proveit.numbers.exponentiation.exp_rational_non_zero__not_zero
proveit.numbers.exponentiation.nth_power_of_nth_root
proveit.numbers.exponentiation.posnat_power_of_product
proveit.numbers.exponentiation.square_abs_rational_simp
proveit.numbers.exponentiation.square_expansion
proveit.numbers.multiplication.elim_one_left
proveit.numbers.multiplication.elim_one_right
proveit.numbers.multiplication.mult_real_closure_bin
proveit.numbers.multiplication.right_mult_eq_real
proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
proveit.numbers.number_sets.complex_numbers.real_within_complex
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.natural_numbers.nat_pos_within_nat
proveit.numbers.number_sets.natural_numbers.natural_lower_bound
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.nonzero_int_within_rational_nonzero
proveit.numbers.number_sets.rational_numbers.nonzero_rational_is_rational_nonzero
proveit.numbers.number_sets.rational_numbers.rational_membership_is_bool
proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
proveit.numbers.number_sets.rational_numbers.reduced_nat_pos_ratio
proveit.numbers.number_sets.real_numbers.nat_pos_within_real
proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
proveit.numbers.number_sets.real_numbers.rational_within_real
proveit.numbers.numerals.decimals.add_1_1
proveit.numbers.numerals.decimals.less_0_1
proveit.numbers.numerals.decimals.less_1_2
proveit.numbers.numerals.decimals.posnat1
proveit.numbers.numerals.decimals.posnat2
proveit.numbers.ordering.less_is_not_eq_nat

Axioms required (directly or indirectly) to prove sqrt2_is_not_rational

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_t_f
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.quantification.existence.exists_def
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.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

Conservative definitions used (but not logically required) to prove sqrt2_is_not_rational

proveit.logic.sets.membership.not_in_set_def
proveit.numbers.numerals.decimals.tuple_len_2_typical_eq

No theorems/conjectures depend upon sqrt2_is_not_rational