Proof of proveit.physics.quantum.QPE._delta_b_is_zero_or_non_int theorem

import proveit
theory = proveit.Theory() # the theorem's theory
from proveit import n, x, A, defaults
from proveit.logic import Equals, Implies, InSet, Not, NotEquals, Or
from proveit.logic.booleans import unfold_is_bool
from proveit.numbers import zero, one, two, frac, IntervalOO, Less, Neg
from proveit.numbers.number_sets.real_numbers import not_int_if_not_int_in_interval
from proveit.physics.quantum.QPE import _delta_b, _delta_b_in_interval

%proving _delta_b_is_zero_or_non_int

With these allowed/disallowed theorem/theory presumptions (e.g., to avoid circular dependencies), we begin our proof of
_delta_b_is_zero_or_non_int:
(see dependencies)

defaults.assumptions = _delta_b_is_zero_or_non_int.all_conditions()

defaults.assumptions:

_delta_b_in_interval

 ⊢  

_delta_b_in_interval_inst = _delta_b_in_interval.instantiate()

_delta_b_in_interval_inst:  ⊢  

neg_one_less_neg_one_half = Less(Neg(one), Neg(frac(one, two))).prove()

neg_one_less_neg_one_half:  ⊢  

Less(frac(one, two), one).prove()

 ⊢  

Now Prove-It can derive the expanded interval membership for $\delta_{b_r}$:

InSet(_delta_b, IntervalOO(Neg(one), one)).prove()

 ⊢  

Establish that $(\delta = 0) \lor (\lnot[\delta = 0])$ using a version of the Law of the Excluded Middle

unfold_is_bool

 ⊢  

non_zero_assumption = NotEquals(_delta_b, zero)

non_zero_assumption:

zero_assumption = Equals(_delta_b, zero)

zero_assumption:

delta_b_is_zero_or_not = unfold_is_bool.instantiate({A: zero_assumption})

delta_b_is_zero_or_not:  ⊢  

Establish that the assumptions $(\delta = 0)$ and $(\lnot[\delta = 0])$ each imply that $(\delta = 0) \lor (\delta_{b} \notin \mathbb{Z})$

First, assuming that $\delta_b \ne 0$

not_int_if_not_int_in_interval

 ⊢  

not_zero_assumption_gives_not_int = not_int_if_not_int_in_interval.instantiate(
    {n: zero, x: _delta_b},
    assumptions=[*defaults.assumptions, non_zero_assumption])

not_zero_assumption_gives_not_int: ,  ⊢  

Or(zero_assumption, not_zero_assumption_gives_not_int.expr).prove(
        assumptions=[*defaults.assumptions, non_zero_assumption]).as_implication(hypothesis=non_zero_assumption)

 ⊢  

not_zero_implies_not_eq_zero = non_zero_assumption.conclude_as_folded(
        assumptions = [Not(zero_assumption)]).as_implication(hypothesis=Not(zero_assumption))

not_zero_implies_not_eq_zero:  ⊢  

Implies(Not(zero_assumption), Or(zero_assumption, not_zero_assumption_gives_not_int.expr)).prove()

 ⊢  

Second, assuming that $\delta_b = 0$ (the trivial case)

Or(zero_assumption, not_zero_assumption_gives_not_int.expr).prove(
        assumptions=[*defaults.assumptions, zero_assumption]).as_implication(hypothesis=zero_assumption)

 ⊢  

Thus in all cases we have the desired conclusion:

delta_b_is_zero_or_not.derive_via_singular_dilemma(_delta_b_is_zero_or_non_int.instance_expr)

 ⊢  

_delta_b_is_zero_or_non_int may now be readily provable (assuming required theorems are usable).  Simply execute "%qed".

%qed

proveit.physics.quantum.QPE._delta_b_is_zero_or_non_int has been proven.

	step type	requirements	statement
0	generalization	1	⊢
1	instantiation	2, 9, 3, 4, 5, 6	⊢
	: , : , :
2	theorem		⊢
	proveit.logic.booleans.disjunction.singular_constructive_dilemma
3	instantiation	7, 9	⊢
	:
4	instantiation	8, 9	⊢
	:
5	deduction	10	⊢
6	instantiation	11, 12, 13	⊢
	: , :
7	conjecture		⊢
	proveit.logic.booleans.negation.closure
8	conjecture		⊢
	proveit.logic.booleans.unfold_is_bool
9	instantiation	14	⊢
	: , :
10	instantiation	15, 33, 16	⊢
	: , :
11	theorem		⊢
	proveit.logic.equality.rhs_via_equality
12	deduction	17	⊢
13	instantiation	18, 19	⊢
	: , : , :
14	axiom		⊢
	proveit.logic.equality.equality_in_bool
15	theorem		⊢
	proveit.logic.booleans.disjunction.or_if_only_left
16	instantiation	20, 21	⊢
	: , :
17	instantiation	22, 23, 24	,  ⊢
	: , :
18	axiom		⊢
	proveit.logic.equality.substitution
19	instantiation	25	⊢
	: , :
20	conjecture		⊢
	proveit.logic.sets.membership.double_negated_membership
21	instantiation	148, 149, 26	⊢
	: , : , :
22	theorem		⊢
	proveit.logic.booleans.disjunction.or_if_only_right
23	instantiation	27, 31	⊢
	: , :
24	instantiation	28, 29, 30, 31	,  ⊢
	: , :
25	axiom		⊢
	proveit.logic.equality.not_equals_def
26	instantiation	32, 75, 33	⊢
	: , : , :
27	theorem		⊢
	proveit.logic.equality.unfold_not_equals
28	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.not_int_if_not_int_in_interval
29	conjecture		⊢
	proveit.numbers.number_sets.integers.zero_is_int
30	instantiation	34, 35, 36, 90, 37	⊢
	: , : , :
31	assumption		⊢
32	theorem		⊢
	proveit.logic.equality.sub_left_side_into
33	assumption		⊢
34	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.in_IntervalOO
35	instantiation	53, 39, 38	⊢
	: , :
36	instantiation	53, 39, 107	⊢
	: , :
37	instantiation	40, 41, 42	⊢
	: , :
38	instantiation	148, 144, 43	⊢
	: , : , :
39	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
40	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
41	instantiation	44, 98, 45, 46, 47*, 48*	⊢
	: , : , :
42	instantiation	49, 50, 51	⊢
	: , : , :
43	instantiation	148, 146, 52	⊢
	: , : , :
44	conjecture		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
45	instantiation	53, 90, 109	⊢
	: , :
46	instantiation	54, 98, 90, 109, 55, 56	⊢
	: , : , :
47	instantiation	64, 57, 58, 59	⊢
	: , : , : , :
48	instantiation	112, 60, 61	⊢
	: , : , :
49	theorem		⊢
	proveit.logic.equality.sub_right_side_into
50	instantiation	62, 90, 109, 63, 70	⊢
	: , : , :
51	instantiation	64, 72, 65, 66	⊢
	: , : , : , :
52	instantiation	67, 126	⊢
	:
53	conjecture		⊢
	proveit.numbers.addition.add_real_closure_bin
54	conjecture		⊢
	proveit.numbers.ordering.less_add_right
55	instantiation	68, 98, 109, 99	⊢
	: , : , :
56	instantiation	69, 70	⊢
	: , :
57	instantiation	71, 80, 96, 72	⊢
	: , : , :
58	instantiation	91	⊢
	:
59	instantiation	84, 73	⊢
	: , :
60	instantiation	74, 75, 150, 134, 76, 77, 81, 80, 78	⊢
	: , : , : , : , : , :
61	instantiation	79, 80, 81, 82	⊢
	: , : , :
62	conjecture		⊢
	proveit.numbers.ordering.less_eq_add_right_strong
63	instantiation	83, 98, 109, 99	⊢
	: , : , :
64	conjecture		⊢
	proveit.logic.equality.four_chain_transitivity
65	instantiation	91	⊢
	:
66	instantiation	84, 85	⊢
	: , :
67	conjecture		⊢
	proveit.numbers.negation.int_closure
68	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
69	conjecture		⊢
	proveit.numbers.ordering.relax_less
70	instantiation	86, 128	⊢
	:
71	conjecture		⊢
	proveit.numbers.addition.subtraction.negated_add
72	instantiation	87, 126, 147, 88*	⊢
	: , : , : , :
73	instantiation	92, 89	⊢
	:
74	conjecture		⊢
	proveit.numbers.addition.disassociation
75	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
76	conjecture		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
77	instantiation	129	⊢
	: , :
78	instantiation	148, 138, 98	⊢
	: , : , :
79	conjecture		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_23
80	instantiation	148, 138, 109	⊢
	: , : , :
81	instantiation	148, 138, 90	⊢
	: , : , :
82	instantiation	91	⊢
	:
83	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
84	theorem		⊢
	proveit.logic.equality.equals_reversal
85	instantiation	92, 96	⊢
	:
86	conjecture		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
87	conjecture		⊢
	proveit.numbers.addition.rational_pair_addition
88	instantiation	112, 93, 94	⊢
	: , : , :
89	instantiation	95, 96	⊢
	:
90	instantiation	97, 98, 109, 99	⊢
	: , : , :
91	axiom		⊢
	proveit.logic.equality.equals_reflexivity
92	conjecture		⊢
	proveit.numbers.addition.elim_zero_left
93	instantiation	120, 150, 100, 101, 102, 103	⊢
	: , : , : , :
94	instantiation	104, 105, 106	⊢
	:
95	conjecture		⊢
	proveit.numbers.negation.complex_closure
96	instantiation	148, 138, 107	⊢
	: , : , :
97	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
98	instantiation	108, 109	⊢
	:
99	instantiation	110, 111	⊢
	:
100	instantiation	129	⊢
	: , :
101	instantiation	129	⊢
	: , :
102	instantiation	112, 113, 114	⊢
	: , : , :
103	conjecture		⊢
	proveit.numbers.numerals.decimals.mult_2_2
104	conjecture		⊢
	proveit.numbers.division.frac_cancel_complete
105	instantiation	148, 138, 115	⊢
	: , : , :
106	instantiation	116, 117	⊢
	:
107	instantiation	148, 144, 118	⊢
	: , : , :
108	conjecture		⊢
	proveit.numbers.negation.real_closure
109	instantiation	148, 144, 119	⊢
	: , : , :
110	conjecture		⊢
	proveit.physics.quantum.QPE._delta_b_in_interval
111	assumption		⊢
112	axiom		⊢
	proveit.logic.equality.equals_transitivity
113	instantiation	120, 150, 121, 122, 123, 124	⊢
	: , : , : , :
114	conjecture		⊢
	proveit.numbers.numerals.decimals.add_2_2
115	instantiation	148, 144, 125	⊢
	: , : , :
116	conjecture		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
117	conjecture		⊢
	proveit.numbers.numerals.decimals.posnat4
118	instantiation	148, 146, 126	⊢
	: , : , :
119	instantiation	148, 127, 128	⊢
	: , : , :
120	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
121	instantiation	129	⊢
	: , :
122	instantiation	129	⊢
	: , :
123	instantiation	130, 132	⊢
	:
124	instantiation	131, 132	⊢
	:
125	instantiation	148, 146, 133	⊢
	: , : , :
126	instantiation	148, 149, 134	⊢
	: , : , :
127	conjecture		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
128	instantiation	135, 136, 137	⊢
	: , :
129	conjecture		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
130	conjecture		⊢
	proveit.numbers.multiplication.elim_one_left
131	conjecture		⊢
	proveit.numbers.multiplication.elim_one_right
132	instantiation	148, 138, 139	⊢
	: , : , :
133	instantiation	148, 149, 140	⊢
	: , : , :
134	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
135	conjecture		⊢
	proveit.numbers.division.div_rational_pos_closure
136	instantiation	148, 142, 141	⊢
	: , : , :
137	instantiation	148, 142, 143	⊢
	: , : , :
138	conjecture		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
139	instantiation	148, 144, 145	⊢
	: , : , :
140	conjecture		⊢
	proveit.numbers.numerals.decimals.nat4
141	conjecture		⊢
	proveit.numbers.numerals.decimals.posnat1
142	conjecture		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
143	conjecture		⊢
	proveit.numbers.numerals.decimals.posnat2
144	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
145	instantiation	148, 146, 147	⊢
	: , : , :
146	conjecture		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
147	instantiation	148, 149, 150	⊢
	: , : , :
148	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
149	conjecture		⊢
	proveit.numbers.number_sets.integers.nat_within_int
150	conjecture		⊢
	proveit.numbers.numerals.decimals.nat2
*equality replacement requirements