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G GPDYS.Setting.conditions_of_le_γmax Lemma 3.1 conditions_of_le_γmax GPDYS.Setting.onestep_pos Proposition 4.8 onestep_pos GPDYS.Setting.conditions_of_le_γmax->GPDYS.Setting.onestep_pos GPDYS.Setting.onestep_radial Proposition 4.8 onestep_radial GPDYS.Setting.conditions_of_le_γmax->GPDYS.Setting.onestep_radial GPDYS.Setting.onestep_energy Proposition 4.8 onestep_energy GPDYS.Setting.conditions_of_le_γmax->GPDYS.Setting.onestep_energy GPDYS.Setting.onestep_drop Proposition 4.8 onestep_drop GPDYS.Setting.conditions_of_le_γmax->GPDYS.Setting.onestep_drop GPDYS.Setting.onestep_drop' Proposition 4.8 onestep_drop' GPDYS.Setting.conditions_of_le_γmax->GPDYS.Setting.onestep_drop' GPDYS.Setting.invariant Theorem 3.4 invariant GPDYS.Setting.conditions_of_le_γmax->GPDYS.Setting.invariant GPDYS.Setting.le_γmax_of_conditions Lemma 3.1 le_γmax_of_conditions GPDYS.Setting.lyap_init_le_E0 Lemma 3.1 lyap_init_le_E0 GPDYS.Setting.lyap_init_le_E0->GPDYS.Setting.invariant GPDYS.Setting.kin_le_two_E0 Lemma 3.2 kin_le_two_E0 GPDYS.Setting.h1norm_le Lemma 3.2 h1norm_le GPDYS.Setting.kin_le_two_E0->GPDYS.Setting.h1norm_le GPDYS.Setting.lam_le_two_D Lemma 3.2 lam_le_two_D GPDYS.Setting.kin_le_two_E0->GPDYS.Setting.lam_le_two_D GPDYS.Setting.step_pos Lemma 4.4 step_pos GPDYS.Setting.kin_le_two_E0->GPDYS.Setting.step_pos GPDYS.Setting.one_sub_inner_step_le Lemma 4.4 one_sub_inner_step_le GPDYS.Setting.kin_le_two_E0->GPDYS.Setting.one_sub_inner_step_le GPDYS.Setting.energy_step_le_of_small Lemma 4.7 energy_step_le_of_small GPDYS.Setting.kin_le_two_E0->GPDYS.Setting.energy_step_le_of_small GPDYS.Setting.conditional_descent Lemma 4.5 conditional_descent GPDYS.Setting.kin_le_two_E0->GPDYS.Setting.conditional_descent GPDYS.Setting.gradX_sq_le Lemma 4.6 gradX_sq_le GPDYS.Setting.kin_le_two_E0->GPDYS.Setting.gradX_sq_le GPDYS.Setting.wnorm6_sq_le_P0 Lemma 3.2 wnorm6_sq_le_P0 GPDYS.Setting.h1norm_le->GPDYS.Setting.wnorm6_sq_le_P0 GPDYS.Setting.wnorm6p6_le_P0_cube Lemma 3.2 wnorm6p6_le_P0_cube GPDYS.Setting.wnorm6_sq_le_P0->GPDYS.Setting.wnorm6p6_le_P0_cube GPDYS.Setting.taylorT_le Lemma 4.3 taylorT_le GPDYS.Setting.wnorm6_sq_le_P0->GPDYS.Setting.taylorT_le GPDYS.Setting.abs_inner_nonlin_sub_le Lemma 4.3 abs_inner_nonlin_sub_le GPDYS.Setting.wnorm6_sq_le_P0->GPDYS.Setting.abs_inner_nonlin_sub_le GPDYS.Setting.sq_le_U0 Lemma 3.2 sq_le_U0 GPDYS.Setting.wnorm6p6_le_P0_cube->GPDYS.Setting.sq_le_U0 GPDYS.Setting.nrm2_cube_le Lemma 3.2 nrm2_cube_le GPDYS.Setting.wnorm6p6_le_P0_cube->GPDYS.Setting.nrm2_cube_le GPDYS.Setting.sq_le_U0->GPDYS.Setting.nrm2_cube_le GPDYS.Setting.sq_le_U0->GPDYS.Setting.taylorT_le GPDYS.Setting.sq_le_U0->GPDYS.Setting.abs_inner_nonlin_sub_le GPDYS.Setting.sq_le_U0->GPDYS.Setting.step_pos GPDYS.Setting.wnorm4p4_le_min Lemma 3.2 wnorm4p4_le_min GPDYS.Setting.nrm2_cube_le->GPDYS.Setting.wnorm4p4_le_min GPDYS.Setting.nrm2_nonlin_le Lemma 3.2 nrm2_nonlin_le GPDYS.Setting.nrm2_cube_le->GPDYS.Setting.nrm2_nonlin_le GPDYS.Setting.nrm2_cube_le->GPDYS.Setting.abs_inner_nonlin_sub_le GPDYS.Setting.inner_nonlin_self_le Lemma 3.2 inner_nonlin_self_le GPDYS.Setting.wnorm4p4_le_min->GPDYS.Setting.inner_nonlin_self_le GPDYS.Setting.nrm2_nonlin_le->GPDYS.Setting.one_sub_inner_step_le GPDYS.Setting.inner_nonlin_self_le->GPDYS.Setting.lam_le_two_D GPDYS.Setting.inner_nonlin_self_le->GPDYS.Setting.step_pos GPDYS.Setting.abs_R_le Lemma 4.4 abs_R_le GPDYS.Setting.lam_le_two_D->GPDYS.Setting.abs_R_le GPDYS.Setting.GX_step Lemma 4.1 GX_step GPDYS.Setting.GX_step_sub Lemma 4.1 GX_step_sub GPDYS.Setting.GX_step->GPDYS.Setting.GX_step_sub GPDYS.Setting.GX_delta Lemma 4.1 GX_delta GPDYS.Setting.GX_step->GPDYS.Setting.GX_delta GPDYS.Setting.residual_identity Lemma 4.2 residual_identity GPDYS.Setting.GX_step->GPDYS.Setting.residual_identity GPDYS.Setting.GX_step->GPDYS.Setting.step_pos GPDYS.Setting.radial_identity Lemma 4.1 radial_identity GPDYS.Setting.GX_step_sub->GPDYS.Setting.radial_identity GPDYS.Setting.descent_equality_case Lemma 4.5 descent_equality_case GPDYS.Setting.GX_step_sub->GPDYS.Setting.descent_equality_case GPDYS.Setting.GX_step_sub->GPDYS.Setting.gradX_sq_le GPDYS.Setting.energy_barrier Lemma 4.7 energy_barrier GPDYS.Setting.GX_step_sub->GPDYS.Setting.energy_barrier GPDYS.Setting.energy_identity Lemma 4.2 energy_identity GPDYS.Setting.GX_delta->GPDYS.Setting.energy_identity GPDYS.Setting.radial_identity->GPDYS.Setting.energy_step_le_of_small GPDYS.Setting.radial_identity->GPDYS.Setting.conditional_descent GPDYS.Setting.radial_identity->GPDYS.Setting.gradX_sq_le GPDYS.Setting.nrmGsq_step_sub_le Lemma 4.6 nrmGsq_step_sub_le GPDYS.Setting.radial_identity->GPDYS.Setting.nrmGsq_step_sub_le GPDYS.Setting.energy_identity->GPDYS.Setting.energy_step_le_of_small GPDYS.Setting.energy_identity->GPDYS.Setting.conditional_descent GPDYS.Setting.residual_identity->GPDYS.Setting.conditional_descent GPDYS.Setting.taylorT_le->GPDYS.Setting.conditional_descent GPDYS.Setting.abs_inner_nonlin_sub_le->GPDYS.Setting.conditional_descent GPDYS.Setting.nrm2_step_le_one Lemma 4.4 nrm2_step_le_one GPDYS.Setting.step_pos->GPDYS.Setting.nrm2_step_le_one GPDYS.Setting.nrm2sq_delta_le_two_mul Lemma 4.4 nrm2sq_delta_le_two_mul GPDYS.Setting.step_pos->GPDYS.Setting.nrm2sq_delta_le_two_mul GPDYS.Setting.step_pos->GPDYS.Setting.energy_step_le_of_small GPDYS.Setting.step_pos->GPDYS.Setting.conditional_descent GPDYS.Setting.step_pos->GPDYS.Setting.gradX_sq_le GPDYS.Setting.step_pos->GPDYS.Setting.nrmGsq_step_sub_le GPDYS.Setting.φcont_continuousOn Lemma 4.7 φcont_continuousOn GPDYS.Setting.step_pos->GPDYS.Setting.φcont_continuousOn GPDYS.Setting.step_pos->GPDYS.Setting.energy_barrier GPDYS.Setting.step_pos->GPDYS.Setting.onestep_pos GPDYS.Setting.nrm2_step_le_one->GPDYS.Setting.nrm2sq_delta_le_two_mul GPDYS.Setting.nrm2_step_le_one->GPDYS.Setting.energy_step_le_of_small GPDYS.Setting.nrm2_step_le_one->GPDYS.Setting.conditional_descent GPDYS.Setting.nrm2_step_le_one->GPDYS.Setting.gradX_sq_le GPDYS.Setting.nrm2_step_le_one->GPDYS.Setting.nrmGsq_step_sub_le GPDYS.Setting.nrm2_step_le_one->GPDYS.Setting.onestep_radial GPDYS.Setting.one_sub_inner_step_nonneg Lemma 4.4 one_sub_inner_step_nonneg GPDYS.Setting.nrm2_step_ge Lemma 4.4 nrm2_step_ge GPDYS.Setting.one_sub_inner_step_le->GPDYS.Setting.nrm2_step_ge GPDYS.Setting.one_sub_inner_step_le->GPDYS.Setting.nrm2sq_delta_le_two_mul GPDYS.Setting.nrm2sq_delta_le Lemma 4.4 nrm2sq_delta_le GPDYS.Setting.one_sub_inner_step_le->GPDYS.Setting.nrm2sq_delta_le GPDYS.Setting.inner_step_le_nrm2 Lemma 4.4 inner_step_le_nrm2 GPDYS.Setting.inner_step_le_nrm2->GPDYS.Setting.nrm2_step_ge GPDYS.Setting.nrm2_step_ge->GPDYS.Setting.energy_step_le_of_small GPDYS.Setting.nrm2_step_ge->GPDYS.Setting.conditional_descent GPDYS.Setting.nrm2_step_ge->GPDYS.Setting.onestep_radial GPDYS.Setting.nrm2sq_delta_le_two_mul->GPDYS.Setting.nrm2sq_delta_le GPDYS.Setting.nrm2sq_delta_le->GPDYS.Setting.energy_step_le_of_small GPDYS.Setting.nrm2sq_delta_le->GPDYS.Setting.conditional_descent GPDYS.Setting.energy_step_le_of_small->GPDYS.Setting.energy_barrier GPDYS.Setting.conditional_descent->GPDYS.Setting.energy_barrier GPDYS.Setting.conditional_descent->GPDYS.Setting.onestep_drop GPDYS.Setting.conditional_descent->GPDYS.Setting.onestep_drop' GPDYS.Setting.descent_equality_case->GPDYS.Setting.energy_barrier GPDYS.Setting.gradX_sq_le->GPDYS.Setting.onestep_drop GPDYS.Setting.nrmGsq_step_sub_le->GPDYS.Setting.onestep_drop' GPDYS.Setting.zcont_hyp Lemma 4.7 zcont_hyp GPDYS.Setting.zcont_hyp->GPDYS.Setting.φcont_continuousOn GPDYS.Setting.zcont_hyp->GPDYS.Setting.energy_barrier GPDYS.Setting.φcont_continuousOn->GPDYS.Setting.energy_barrier GPDYS.Setting.energy_barrier->GPDYS.Setting.onestep_energy GPDYS.Setting.onestep_pos->GPDYS.Setting.invariant GPDYS.Setting.onestep_radial->GPDYS.Setting.invariant GPDYS.Setting.onestep_energy->GPDYS.Setting.onestep_drop GPDYS.Setting.onestep_energy->GPDYS.Setting.onestep_drop' GPDYS.Setting.onestep_drop->GPDYS.Setting.invariant GPDYS.Setting.drop Theorem 3.4 drop GPDYS.Setting.onestep_drop->GPDYS.Setting.drop GPDYS.Setting.picone Remark 2.3 picone GPDYS.Setting.energy_ge_of_critical Remark 2.3 energy_ge_of_critical GPDYS.Setting.picone->GPDYS.Setting.energy_ge_of_critical GPDYS.Setting.hessForm_ge Theorem 5.1 hessForm_ge GPDYS.Setting.picone->GPDYS.Setting.hessForm_ge GPDYS.Setting.critical_pos Remark 2.3 critical_pos GPDYS.Setting.eq_of_critical_nonneg Remark 2.3 eq_of_critical_nonneg GPDYS.Setting.critical_pos->GPDYS.Setting.eq_of_critical_nonneg GPDYS.Setting.main_theorem Theorem 3.4 main_theorem GPDYS.Setting.critical_pos->GPDYS.Setting.main_theorem GPDYS.Setting.energy_le_of_critical Remark 2.3 energy_le_of_critical GPDYS.Setting.energy_ge_of_critical->GPDYS.Setting.energy_le_of_critical GPDYS.Setting.eq_of_nonneg_le_energy Remark 2.3 eq_of_nonneg_le_energy GPDYS.Setting.energy_ge_of_critical->GPDYS.Setting.eq_of_nonneg_le_energy GPDYS.Setting.energy_le_of_critical->GPDYS.Setting.eq_of_critical_nonneg GPDYS.Setting.energy_le_of_critical->GPDYS.Setting.main_theorem GPDYS.Setting.rate_step Theorem 5.1 rate_step GPDYS.Setting.energy_le_of_critical->GPDYS.Setting.rate_step GPDYS.Setting.eq_or_eq_neg_of_energy_le Remark 2.3 eq_or_eq_neg_of_energy_le GPDYS.Setting.energy_le_of_critical->GPDYS.Setting.eq_or_eq_neg_of_energy_le GPDYS.Setting.eq_of_nonneg_le_energy->GPDYS.Setting.eq_of_critical_nonneg GPDYS.Setting.eq_of_nonneg_le_energy->GPDYS.Setting.main_theorem GPDYS.Setting.eq_of_nonneg_le_energy->GPDYS.Setting.eq_or_eq_neg_of_energy_le GPDYS.Setting.eq_of_critical_nonneg->GPDYS.Setting.main_theorem GPDYS.Setting.invariant->GPDYS.Setting.drop GPDYS.Setting.accumulation_point_critical Theorem 3.4 accumulation_point_critical GPDYS.Setting.invariant->GPDYS.Setting.accumulation_point_critical GPDYS.Setting.invariant->GPDYS.Setting.main_theorem GPDYS.Setting.invariant->GPDYS.Setting.rate_step GPDYS.Setting.rate_theorem Theorem 5.1 rate_theorem GPDYS.Setting.invariant->GPDYS.Setting.rate_theorem GPDYS.Setting.sum_drop_le Theorem 3.4 sum_drop_le GPDYS.Setting.drop->GPDYS.Setting.sum_drop_le GPDYS.Setting.drop->GPDYS.Setting.rate_step GPDYS.Setting.g_tendsto_zero Theorem 3.4 g_tendsto_zero GPDYS.Setting.sum_drop_le->GPDYS.Setting.g_tendsto_zero GPDYS.Setting.g_tendsto_zero->GPDYS.Setting.accumulation_point_critical GPDYS.Setting.accumulation_point_critical->GPDYS.Setting.main_theorem GPDYS.Setting.exists_groundState Corollary exists_groundState GPDYS.Setting.main_theorem->GPDYS.Setting.exists_groundState GPDYS.Setting.rate_theorem' Theorem 5.1 rate_theorem' GPDYS.Setting.main_theorem->GPDYS.Setting.rate_theorem' GPDYS.Setting.exists_groundState_sign Remark 2.3 exists_groundState_sign GPDYS.Setting.exists_groundState->GPDYS.Setting.exists_groundState_sign GPDYS.Setting.nu_ge Theorem 5.1 nu_ge GPDYS.Setting.hessForm_ge->GPDYS.Setting.nu_ge GPDYS.Setting.local_quadratic_growth Theorem 5.1 local_quadratic_growth GPDYS.Setting.hessForm_ge->GPDYS.Setting.local_quadratic_growth GPDYS.Setting.local_PL Theorem 5.1 local_PL GPDYS.Setting.hessForm_ge->GPDYS.Setting.local_PL GPDYS.Setting.nu_ge->GPDYS.Setting.local_quadratic_growth GPDYS.Setting.nu_ge->GPDYS.Setting.local_PL GPDYS.Setting.nu_ge->GPDYS.Setting.rate_step GPDYS.Setting.nu_ge->GPDYS.Setting.rate_theorem' GPDYS.Setting.energy_sub_eq Theorem 5.1 energy_sub_eq GPDYS.Setting.energy_sub_eq->GPDYS.Setting.local_quadratic_growth GPDYS.Setting.energy_sub_eq->GPDYS.Setting.local_PL GPDYS.Setting.inner_g_sub_eq Theorem 5.1 inner_g_sub_eq GPDYS.Setting.inner_g_sub_eq->GPDYS.Setting.local_PL GPDYS.Setting.local_quadratic_growth->GPDYS.Setting.rate_theorem GPDYS.Setting.local_PL->GPDYS.Setting.rate_step GPDYS.Setting.rate_step->GPDYS.Setting.rate_theorem GPDYS.Setting.rate_theorem->GPDYS.Setting.rate_theorem' GPDYS.Setting.rate_theorem_μmax Theorem 5.1 rate_theorem_μmax GPDYS.Setting.rate_theorem'->GPDYS.Setting.rate_theorem_μmax GPDYS.Setting.eq_or_eq_neg_of_energy_le->GPDYS.Setting.exists_groundState_sign

Dependency graph of the formalization

Each box is a declaration whose docstring refers to a result in the paper. Arrows point from a lemma to the results whose proof uses it, so the two main theorems are at the top. Click a box to see the paper reference, the statement in Lean (proof hidden), and its dependencies among the displayed results. Edges follow compiled constant references through helpers; the elaborated types include every inherited hypothesis.

Section 5 (Theorems 3.4, 5.1)   Lemma 4.7 / Prop. 4.8   ground state (Remark 2.3)   other lemmas

Full catalog of all statements: STATEMENTS.html.