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[Merged by Bors] - feat(data/real/sqrt): sqrt x < y ↔ x < y^2
#13546
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YaelDillies
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Prove `real.sqrt_lt_iff` and generalize `real.lt_sqrt`.
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sqrt x < y ↔ x < y^2sqrt x < y ↔ x < y^2
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* origin/master: (394 commits) feat(data/set/[basic|prod]): make `×ˢ` bind more strongly, and define `mem.out` (#13422) feat(order/basic): Simple shortcut lemmas (#13421) chore(number_theory/dioph): Cleanup (#13403) feat(analysis/normed_space/exponential): ring homomorphisms are preserved by the exponential (#13402) feat(algebraic_geometry/projective_spectrum): degree zero part of a localized ring (#13398) feat(set_theory/cardinal): A set of cardinals is small iff it's bounded (#13373) feat(data/polynomial/{derivative, iterated_deriv}): reduce assumptions (#13368) feat(algebra/monoid_algebra/grading): Use the new graded_algebra API (#13360) feat(algebra/group/to_additive): let @[to_additive] mimic alias’s docstrings (#13330) feat(set_theory/cofinality): Basic fundamental sequences (#13326) feat(special_functions/pow): continuity of real to complex power (#13244) feat(group_theory/torsion): extension closedness, and torsion scalars in modules (#13172) feat(category_theory/path_category): canonical quotient of a path category (#13159) refactor(number_theory/padics/padic_norm): Switch nat and rat definitions (#12454) feat(analysis/normed): more lemmas about the sup norm on pi types and matrices (#13536) fix(category_theory/monoidal): improve hygiene in coherence tactic (#13507) feat(src/number_theory/cyclotomic/discriminant): add discr_prime_pow_ne_two (#13465) chore(algebra/group/type_tags): missing simp lemmas (#13553) feat(measure_theory): allow measurability to prove ae_strongly_measurable (#13427) refactor(algebra/hom/group): generalize a few lemmas to `monoid_hom_class` (#13447) chore(data/list/cycle): Add basic `simp` lemmas + minor golfing (#13533) feat(algebra/hom/non_unital_alg): introduce notation for non-unital algebra homomorphisms (#13470) chore(algebra/group/defs): Declare `field_simps` attribute earlier (#13543) feat(analysis/normed/normed_field): add `one_le_(nn)norm_one` for nontrivial normed rings (#13556) refactor(analysis/calculus/cont_diff): reorder the file (#13468) move(set_theory/*): Organize in folders (#13530) chore(number_theory/zsqrtd/basic): simplify le_total proof (#13555) feat(group_theory/perm/basic): Iterating a permutation is the same as taking a power (#13554) feat(data/real/sqrt): `sqrt x < y ↔ x < y^2` (#13546) feat(algebra/hom/group and *): introduce `mul_hom M N` notation `M →ₙ* N` (#13526) feat(group_theory/schreier): Schreier's lemma in terms of `group.fg` and `group.rank` (#13361) feat(linear_algebra/trace): dual_tensor_hom is an equivalence + basis-free characterization of the trace (#10372) feat(order/filter/basic): allow functions between different types in lemmas about [co]map by a constant function (#13542) feat(data/finset/basic): simp `to_finset_eq_empty` (#13531) feat(topology/algebra/algebra): ℚ-scalar multiplication is continuous (#13458) chore(model_theory/encoding): Improve the encoding of terms (#13532) feat(topology/separation): Finite sets in T2 spaces (#12845) feat(analysis/inner_product_space/gram_schmidt_ortho): Gram-Schmidt Orthogonalization and Orthonormalization (#12857) chore(algebra/big_operators/fin): golf finset.prod_range (#13535) chore(analysis/normed_space/star): make an argument explicit (#13523) feat(*): `op_op_op_comm` lemmas (#13528) chore(data/real/nnreal): add commuted version of `nnreal.mul_finset_sup` (#13512) chore(*/matrix): order `m` and `n` alphabetically (#13510) feat(analysis/calculus/specific_functions): trivial extra lemmas (#13516) feat(analysis): lemmas about nnnorm and nndist (#13498) feat(data/int/basic): add lemma `int.abs_le_one_iff` (#13513) feat(category_theory/limits): add characteristic predicate for zero objects (#13511) feat(order/filter/n_ary): Add lemma equating map₂ to map on the product (#13490) fix(analysis/locally_convex/balanced_hull_core): minimize import (#13450) feat(order/cover): define `wcovby` (#13424) ...
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Prove
real.sqrt_lt_iffand generalizereal.lt_sqrt.