hcju/isabelleps huggingface dataset:
that: fixes n :: "nat" assumes "count M x = Suc n" shows "thesis"
hunion_hinsert_left: fixes x :: "hf"   and A :: "hf"   and B :: "hf" shows "(A \<triangleleft> x) \<squnion> B = A \<squnion> B \<triangleleft> x"
Stable: fixes x :: "nat" assumes "x < n\<^sub>1"   and "token_squats x" shows "\<exists>i. \<forall>m'\<ge>n. rank x m' = Some i"
simps_5: fixes x1 :: "'b" shows "-\<infinity> \<noteq> Fin x1"
inorder_split_maxD: fixes t :: "('b \<times> bal) tree"   and t' :: "('b \<times> bal) tree"   and a :: "'b" assumes "split_max t = (t', a)"   and "t \<noteq> \<langle>\<rangle>"   and "avl t" shows "Tree2.inorder t' @ [a] = Tree2.inorder t"
axioms_2: fixes as\<^sub>0 :: "'b"   and alstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and cs\<^sub>0 :: "'e"   and clstep :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and R :: "'b \<Rightarrow> 'e \<Rightarrow> bool" assumes "simulation as\<^sub>0 alstep cs\<^sub>0 clstep R" shows "df_system cs\<^sub>0 clstep"
assms: fixes x :: "'a"   and k :: "'b set" assumes "(x, k) \<in> s" shows "k \<subseteq> d x"
assms_2: shows "\<Pi>\<^sub>G \<subseteq>\<^sub>m execute_parallel_plan (\<Pi>\<^sub>I) \<pi>"
Id_on_def': fixes A :: "'a \<Rightarrow> bool" shows "Id_on {x. A x} = {(x, y). x = y \<and> A x}"
lift_decompose_literal: fixes L :: "'b literal"   and \<sigma> :: "('b \<times> 'b trm) list"   and t :: "'b trm"   and s :: "'b trm"   and polarity :: "sign"   and \<theta> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list" assumes "decompose_literal (subst_lit L \<sigma>) t s polarity"   and "\<theta> \<doteq> \<sigma> \<lozenge> \<eta>" shows "decompose_literal (subst_lit L \<theta>) (t \<lhd> \<eta>) (s \<lhd> \<eta>) polarity"
carrier_not_empty: shows "carrier (Z 1) \<noteq> {}"
nodes_add_H: shows "nodes (add_edge a w b H) = nodes H"
is_class_sub_Throwable: fixes wf_md :: "'b prog \<Rightarrow> String.literal \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and C :: "String.literal" assumes "wf_prog wf_md P"   and "P \<turnstile> C \<preceq>\<^sup>* Throwable" shows "is_class P C"
assms_4: shows "\<Psi> \<rhd> Q \<sim> P"
rcos_ring_hom_cring: assumes "cring R" shows "ring_hom_cring R (R Quot I) ((+>) I)"
odlist_induct: fixes P :: "'c odlist \<Rightarrow> bool"   and dxs :: "'c odlist" assumes "\<And>dxs. dxs = ODList.empty \<Longrightarrow> P dxs"   and "\<And>dxs x xs. \<lbrakk>dxs = fromList (x # xs); distinct (x # xs); sorted (x # xs); P (fromList xs)\<rbrakk> \<Longrightarrow> P dxs" shows "P dxs"
elims_3: fixes x :: "'b extB"   and xa :: "'b extB" assumes "\<not> x < xa" obtains x :: "'b extB" and y :: "'b extB" where "xa__ = x" and "xa = y" and "\<not> (x \<le> y \<and> \<not> y \<le> x)"
indepH': shows "gs.lin_indpt_list (map of_int_hom.vec_hom fs')"
1: shows "\<nexists>X. simplex (x_mat_eq_vec c A\<^sup>T) = Inr X"
b_e_type_get_local_1: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and i :: "nat" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Get_local i" shows "\<exists>t. ts' = ts @ [t] \<and> local \<C> ! i = t"
ccApprox_nxt_below: fixes t :: "Vars.var ttree"   and x :: "Vars.var" shows "ccApprox (nxt t x) \<sqsubseteq> ccApprox t"
iter_widen_inv: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "\<And>x. P x \<Longrightarrow> P (f x)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (x1 \<nabla> x2)"   and "P x"   and "iter_widen f x = Some y" shows "P y"
head_closed: fixes xs :: "'a list" assumes "in_carrier xs" shows "head xs \<in> carrier R"
cdf_nonneg: fixes x :: "real" shows "0 \<le> C x"
sub_args: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm" assumes "s \<in> set (args t)" shows "sub s t"
funcset_carrier: fixes f :: "'a \<Rightarrow> 'b"   and X :: "('a, 'c) partial_object_scheme"   and Y :: "('b, 'd) partial_object_scheme"   and x :: "'a" assumes "f \<in> carrier X \<rightarrow> carrier Y"   and "x \<in> carrier X" shows "f x \<in> carrier Y"
all_finite_subset_image: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and P :: "'b set \<Rightarrow> bool" shows "(\<forall>B. finite B \<and> B \<subseteq> f ` A \<longrightarrow> P B) = (\<forall>B. finite B \<and> B \<subseteq> A \<longrightarrow> P (f ` B))"
dg_is_arrD_2: fixes \<alpha> :: "V"   and \<CC> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "digraph \<alpha> \<CC>"   and "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
attractor_strategy_on_extends: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v0 :: "'a"   and A :: "'a set"   and W :: "'a set"   and A' :: "'a set" assumes "strategy_attracts_via p \<sigma> v0 A W"   and "A \<subseteq> A'" shows "strategy_attracts_via p \<sigma> v0 A' W"
dom_eval: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha> :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota> :: "'d"   and V :: "'b \<Rightarrow> 'd"   and f :: "'b local.term" assumes "evaluation_map C D T \<alpha> \<iota> V"   and "monoidal_language.Arr C f" shows "partial_magma.dom D (evaluation_map.eval D T \<alpha> \<iota> V f) = evaluation_map.eval D T \<alpha> \<iota> V (monoidal_language.Dom C f)"
simps_7: fixes e :: "'a exp"   and F :: "char list"   and D :: "char list" shows "max_vars (e\<bullet>F{D}) = max_vars e"
pairwise_disjoint_boundaries: shows "\<forall>x\<in>boundary ` two_chain. \<forall>y\<in>boundary ` two_chain. x \<noteq> y \<longrightarrow> x \<inter> y = {}"
ordered_model_resolvent_is_resolvent: fixes I :: "'at set" shows "less_restrictive resolvent (ordered_model_resolvent I)"
eq_from_eq_frac: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a" assumes "cq.frac r s = cq.frac r' s'"   and "s \<in> R\<setminus>I"   and "s' \<in> R\<setminus>I"   and "r \<in> R"   and "r' \<in> R" obtains h :: "'a" where "h \<in> R\<setminus>I" and "h \<cdot> (s' \<cdot> r - s \<cdot> r') = \<zero>"
AA: shows "finite {(t, u). s setinterleaves ((t, u), A) \<and> (\<exists>a list. t = a # list \<and> a \<notin> A) \<and> u = []}"
assms_2: assumes "possible t x"   and "possible t' x" shows "(xs \<in> paths (nxt t x)) = (xs \<in> paths (nxt t' x))"
mapping_of_inverse: fixes x :: "'a mpoly" shows "MPoly (mapping_of x) = x"
delete_def: fixes k :: "'b"   and al :: "('b, 'c) assoc_list" shows "Assoc_List.delete k al = Assoc_List (AList.delete_aux k (assoc_list.impl_of al))"
unitary11_gen_real: fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "unitary11_gen M = (\<exists>k. k \<noteq> 0 \<and> congruence M (1, 0, 0, - 1) = cor k *\<^sub>s\<^sub>m (1, 0, 0, - 1))"
dg_prod_2_ArrE: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and gf :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>"   and "gf \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Arr\<rparr>" obtains g :: "V" and f :: "V" where "gf = [g, f]\<^sub>\<circ>" and "g \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" and "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
hyps_4: shows "add_mset x' X' = add_mset x X"
suma_left_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p" assumes "A B C D E F SumA G H I" shows "C B A D E F SumA G H I"
multiplicative_dirichlet_prod: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "multiplicative_function f"   and "multiplicative_function g" shows "multiplicative_function (dirichlet_prod f g)"
distinct_3: fixes x11 :: "'d"   and x12 :: "'a \<Rightarrow> 'a set"   and x13 :: "('a, 'b, 'c, 'd) seqp"   and x31 :: "('a, 'b, 'c, 'd) seqp"   and x32 :: "('a, 'b, 'c, 'd) seqp" shows "{x11}\<langle>x12\<rangle> x13 \<noteq> x31 \<oplus> x32"
lab_sink: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat" assumes "Labeling c s t f l" shows "l t = 0"
\<phi>_def: fixes y :: "'d"   and h :: "'c" shows "\<phi> y h = G h \<cdot>\<^sub>D \<eta> y"
vfsequence_vcons_ex: assumes "[a, b]\<^sub>\<circ> \<noteq> []\<^sub>\<circ>" obtains xs' :: "V" and x :: "V" where "[a, b]\<^sub>\<circ> = xs' #\<^sub>\<circ> x" and "vfsequence xs'"
limset_bdd: fixes x :: "real" shows "{lim (\<lambda>k. f (d k) (r n)) |n. x < r n} \<subseteq> {- M..M}"
basis_reduction_swap_2: fixes i :: "nat"   and fs :: "int vec list"   and upw' :: "bool"   and i' :: "nat"   and fs' :: "int vec list" assumes "LLL_invariant False i fs"   and "basis_reduction_swap i fs = (upw', i', fs')"   and "\<alpha> * \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) i\<parallel>\<^sup>2 < \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) (i - 1)\<parallel>\<^sup>2"   and "i < m"   and "i \<noteq> 0" shows "LLL_measure i' fs' < LLL_measure i fs"
hyps_1: fixes q :: "item" assumes "q \<in> Y" shows "item_end q = 0"
split_sel: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "enat \<Rightarrow> 'c"   and enat :: "enat" shows "P (case enat of 0 \<Rightarrow> f1 | eSuc x \<Rightarrow> f2 x) = ((enat = 0 \<longrightarrow> P f1) \<and> (enat = eSuc (epred enat) \<longrightarrow> P (f2 (epred enat))))"
overlap_sort_invar: fixes \<gamma> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "('a, 'c) flow_entry_match \<Rightarrow> 'd"   and ft :: "('a, 'c) flow_entry_match list" shows "check_no_overlap \<gamma> (sort_descending_key k ft) = check_no_overlap \<gamma> ft"
1: shows "equivalence_of_categories (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F G \<eta> \<epsilon>"
LIM_NSLIM_iff: fixes f :: "'a \<Rightarrow> 'b"   and L :: "'b"   and x :: "'a" shows "f \<midarrow>x\<rightarrow> L = f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S L"
arr_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "span_vertical_category C" shows "partial_magma.arr (span_vertical_category.vcomp C) \<mu> = arrow_of_spans C \<mu>"
smallest_ccdi_sets_Int1: fixes a :: "'a set"   and b :: "'a set" assumes "a \<in> M"   and "b \<in> smallest_ccdi_sets \<Omega> M" shows "a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
one_reducible_to_via_refl: fixes A :: "nat set" shows "one_reducible_to_via A A (\<lambda>x. x)"
gc_invar_gc: fixes Ns :: "('f \<times> 'l) set llist"   and i :: "enat" assumes "chain (\<leadsto>GC) Ns"   and "active_subset (lhd Ns) = {}"   and "i < llength Ns" shows "gc_invar Ns i"
ucast_less_ucast: fixes x :: "'b word"   and y :: "'b word" assumes "LENGTH('b) \<le> LENGTH('c)" shows "(UCAST('b \<rightarrow> 'c) x < UCAST('b \<rightarrow> 'c) y) = (x < y)"
fmpt_power: fixes n :: "nat" shows "fmpt M (T ^^ n)"
laplace_expansion_row: fixes A :: "'b mat"   and n :: "nat"   and i :: "nat" assumes "A \<in> carrier_mat n n"   and "i < n" shows "det A = (\<Sum>j<n. A $$ (i, j) * cofactor A i j)"
0: shows "\<forall>a\<in>set as. hd_coeff a \<noteq> 0"
path_conditions_disjoint_points_iff: fixes x :: "'a"   and y :: "'a" shows "(x ; 1 \<cdot> (x\<^sup>T ; 1 + y ; 1) \<cdot> y\<^sup>T ; 1 = 0 \<and> start_points x \<cdot> end_points y = 0) = (x ; 1 \<cdot> y\<^sup>T ; 1 = 0)"
elims: fixes x :: "'b\<^sub>\<bottom>"   and y :: "'b option" assumes "up2option x = y" obtains "x = Ibottom" and "y = None"   | a :: "'b" where "x = Iup a" and "y = Some a"
comp_intro: fixes P :: "'c \<times> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'d"   and Q :: "'d \<times> 'e \<Rightarrow> bool"   and c :: "'e" assumes "P (a, b)"   and "Q (b, c)" shows "(P ;; Q) (a, c)"
insert_before_def: shows "insert_before \<equiv> l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_insert_before get_parent get_child_nodes set_child_nodes get_ancestors adopt_node set_disconnected_nodes get_disconnected_nodes get_owner_document"
reachable_eq_imp: fixes r :: "'d" shows "eq_imp (\<lambda>r' s. (mut_roots s, mut_ghost_honorary_root s, \<Union> (ran ` obj_fields ` set_option (sys_heap s r')), filter (\<lambda>s. is_mw_Mutate s \<or> is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s))) (reachable r)"
LFO_refine: fixes li :: "'a list"   and l :: "'b set"   and A :: "('a \<times> 'b) set"   and R :: "('d \<times> 'c) set"   and fi :: "'a \<Rightarrow> 'd \<Rightarrow> 'd nres"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c nres"   and s0i :: "'d"   and s0 :: "'c" assumes "(li, l) \<in> \<langle>A\<rangle>list_set_rel"   and "\<And>x xi s si. \<lbrakk>(xi, x) \<in> A; (si, s) \<in> R\<rbrakk> \<Longrightarrow> fi xi si \<le> \<Down> R (f x s)"   and "(s0i, s0) \<in> R" shows "nfoldli li (\<lambda>_. True) fi s0i \<le> \<Down> R (FOREACH l f s0)"
rep_eq: fixes x :: "'b scene" shows "Rep_scene (- x) = (\<lambda>xa y. Rep_scene x y xa)"
scopeFresh: fixes x :: "name"   and P :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>P \<sim> P"
assms_1: shows "\<Gamma> \<turnstile> s1 \<approx>\<^sub>L s2"
pred_mono: fixes P :: "'b \<Rightarrow> bool"   and Pa :: "'b \<Rightarrow> bool" assumes "P \<le> Pa" shows "pred_stream P \<le> pred_stream Pa"
viette2'_monic: fixes b :: "real"   and c :: "real"   and \<xi> :: "real" assumes "b\<^sup>2 - 4 * c = 0"   and "\<xi>\<^sup>2 + b * \<xi> + c = 0" shows "\<xi> * \<xi> = c"
sub_non_empty_blocks: fixes \<U> :: "'c set"   and \<A> :: "'c set multiset"   and \<V> :: "'c set"   and \<B> :: "'c set multiset"   and A :: "'c set" assumes "sub_design \<U> \<A> \<V> \<B>"   and "A \<in># \<A>" shows "A \<noteq> {}"
Dirichlet_L_0_even: assumes "\<chi> (n - 1) = 1" shows "Dirichlet_L n \<chi> 0 = 0"
boundOutputFreshSimps_2: fixes yvec :: "name list"   and xvec :: "name list"   and B :: "('d, 'e, 'f) boundOutput" assumes "yvec \<sharp>* xvec" shows "yvec \<sharp>* (\<lparr>\<nu>*xvec\<rparr>B) = yvec \<sharp>* B"
3: shows "smap f (v ## gtrace r v) = s @- sconst k"
assms_2: shows "r \<notin> UNIV\<^sup>\<star>"
1: fixes c :: "('test, 'atom, 'choice) cmd" assumes "c \<in> set (take n2 cl)" shows "0 < brn c"
polyfun_matricize: fixes T :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a tensor"   and ds :: "nat list"   and N :: "nat set"   and I :: "nat set"   and nr :: "nat"   and nc :: "nat"   and i :: "nat"   and j :: "nat" assumes "\<And>x. dims (T x) = ds"   and "\<And>is. is \<lhd> ds \<Longrightarrow> polyfun N (\<lambda>x. Tensor.lookup (T x) is)"   and "\<And>x. dim_row (matricize I (T x)) = nr"   and "\<And>x. dim_col (matricize I (T x)) = nc"   and "i < nr"   and "j < nc" shows "polyfun N (\<lambda>x. matricize I (T x) $$ (i, j))"
simps_2: fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "take n (x # xs) = (case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # take m xs)"
diff_neutralizes: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and R :: "'c set"   and r1 :: "'c"   and r2 :: "'c" assumes "Group.group G"   and "subgroup H G"   and "R \<in> rcosets\<^bsub>G\<^esub> H"   and "r1 \<in> R"   and "r2 \<in> R" shows "r1 \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> r2 \<in> H"
augment_with_path_def: fixes p :: "(nat \<times> nat) list" shows "augment_with_path p \<equiv> augment (augmentingFlow p)"
measurable_def: fixes A :: "'a measure"   and B :: "'b measure" shows "A \<rightarrow>\<^sub>M B = {f \<in> space A \<rightarrow> space B. \<forall>y\<in>sets B. f -` y \<inter> space A \<in> sets A}"
fresh_6: fixes a :: "name"   and x1 :: "name"   and x2 :: "('d, 'e, 'f) psi" shows "a \<sharp> \<lparr>\<nu>x1\<rparr>x2 = a \<sharp> [x1].x2"
nnfis_minus_nnfis_integral_1: fixes a :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and b :: "real"   and g :: "'b \<Rightarrow> real" assumes "a \<in> nnfis f M"   and "b \<in> nnfis g M"   and "measure_space M" shows "integrable (\<lambda>t. f t - g t) M"
comm_semiring_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "comm_semiring_hom hom \<equiv> semiring_hom hom"
SpecStuck: fixes s :: "'d"   and r :: "('d \<times> 'd) set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option" assumes "\<forall>t. (s, t) \<notin> r" shows "\<Gamma>\<turnstile> \<langle>Spec r,Normal s\<rangle> \<Rightarrow> Stuck"
countable_datatype: fixes Rep :: "'b \<Rightarrow> ('c, unit) Old_Datatype.node set"   and Abs :: "('c, unit) Old_Datatype.node set \<Rightarrow> 'b"   and rep_set :: "('c, unit) Old_Datatype.node set \<Rightarrow> bool" assumes "type_definition Rep Abs (Collect rep_set)"   and "\<And>x. rep_set x \<Longrightarrow> Countable.finite_item x" shows "OFCLASS('b, countable_class)"
elemE: fixes S :: "('d, 'e) eq_object_scheme"   and a :: "'d"   and A :: "'d set" assumes "a .\<in>\<^bsub>S\<^esub> A" obtains a' :: "'d" where "a' \<in> A" and "a .=\<^bsub>S\<^esub> a'"
alt_seq_Zp_cauchy: fixes p :: "int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "padic_integers p"   and "padic_integers.is_Zp_cauchy p s" shows "padic_integers.is_Zp_cauchy p (alt_seq s)"
hq_h_rw: shows "h ^ CARD('a) - h = (\<Prod>c\<in>UNIV. h - [:c:])"
fixp_lowerbound: fixes f :: "'a \<Rightarrow> 'a"   and z :: "'a" assumes "monotone (\<lambda>x y. y \<le> x) (\<lambda>x y. y \<le> x) f"   and "z \<le> f z" shows "z \<le> fixp f"
roots_bound: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c" assumes "UP_domain R"   and "f \<in> carrier (UP R)"   and "f \<noteq> \<zero>\<^bsub>UP R\<^esub>"   and "finite (carrier R)" shows "finite {a \<in> carrier R. eval R R id a f = \<zero>\<^bsub>R\<^esub>} \<and> card {a \<in> carrier R. eval R R id a f = \<zero>\<^bsub>R\<^esub>} \<le> deg R f"
append_cols_nth: fixes A :: "'a mat"   and n :: "nat"   and a :: "nat"   and B :: "'a mat"   and b :: "nat"   and i :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat n a"   and "B \<in> carrier_mat n b"   and "i < n"   and "j < a + b" shows "(A @\<^sub>c B) $$ (i, j) = (if j < dim_col A then A $$ (i, j) else B $$ (i, j - a))"
lookup_pair_eq_0: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and k :: "'a" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs" shows "(comparator.lookup_pair comp xs k = (0::'b)) = (k \<notin> fst ` set xs)"
uminus_l_inv_vec: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "- v + v = 0\<^sub>v n"
suf_nemp: fixes v :: "'b list"   and u :: "'b list" assumes "v \<noteq> \<epsilon>" shows "u \<cdot> v \<noteq> \<epsilon>"
simps_2: fixes n :: "nat" shows "enumchain (Suc n) = (LEAST m. enumchain n < m \<and> P (enumchain n) m)"
Cauchy_theorem_convex_simple: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and g :: "real \<Rightarrow> complex" assumes "f holomorphic_on S"   and "convex S"   and "valid_path g"   and "path_image g \<subseteq> S"   and "pathfinish g = pathstart g" shows "(f has_contour_integral 0) g"
strict_arr_char: fixes f :: "'c local.term set" shows "arr f = (\<F>C.arr f \<and> Diag (\<F>C.DOM f) \<and> Diag (\<F>C.COD f))"
lub_upper: fixes A :: "'f llist set"   and x :: "'f llist" assumes "Complete_Partial_Order.chain lprefix A"   and "x \<in> A" shows "lprefix x (lSup A)"
0: shows "rawpsubstT (tms r1 r2) (zip (map Var us) (map snd txs)) = tms (rawpsubstT r1 (zip (map Var us) (map snd txs))) (rawpsubstT r2 (zip (map Var us) (map snd txs)))"
PO_l2_inv1: shows "reach l2 \<subseteq> l2_inv1"
ccTTree_cong_below: fixes S :: "Vars.var set"   and G :: "CoCalls"   and G' :: "CoCalls" assumes "G G|` S \<sqsubseteq> G' G|` S" shows "ccTTree S G \<sqsubseteq> ccTTree S G'"
f_closed: fixes x :: "'b"   and y :: "'a" assumes "x \<in> B"   and "y \<in> D" shows "x \<cdot> y \<in> D"
dg_Par_Dom_app: fixes \<alpha> :: "V"   and x :: "V" assumes "arr_Par \<alpha> x" shows "dg_Par \<alpha>\<lparr>Dom\<rparr>\<lparr>x\<rparr> = x\<lparr>ArrDom\<rparr>"
of_complex_inner_1': fixes x :: "complex" shows "\<langle>1::'b, of_complex x\<rangle> = x"
index_component_mult: fixes i :: "nat"   and v :: "'a vec"   and w :: "'a vec" assumes "i < dim_vec v"   and "i < dim_vec w" shows "component_mult v w $ i = v $ i * w $ i"
terminal_path_is_path: fixes v :: "'d"   and v' :: "'d"   and pth :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list" assumes "terminal_path v v' pth" shows "path v v' pth"
cring_simprules_24: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> (y \<otimes>\<^bsub>R\<^esub> z) = y \<otimes>\<^bsub>R\<^esub> (x \<otimes>\<^bsub>R\<^esub> z)"
IH: shows "erfc x = exp (- x\<^sup>2) / sqrt pi * (\<Sum>i<n. (- 1) ^ i * fact (2 * i) / (4 ^ i * fact i) / x ^ (2 * i + 1)) + erf_remainder n x"
at_neg': fixes v :: "view"   and c :: "cars"   and \<phi> :: "traffic \<Rightarrow> view \<Rightarrow> bool"   and ts :: "traffic" shows "(\<not> (\<forall>v'. v = c > v' \<longrightarrow> \<phi> ts v')) = (\<forall>v'. v = c > v' \<longrightarrow> \<not> \<phi> ts v')"
not_strict_SD_iff: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "p \<in> lotteries_on carrier"   and "q \<in> lotteries_on carrier" shows "(\<not> p \<prec>[SD le] q) = (\<exists>u. is_vnm_utility u \<and> measure_pmf.expectation q u \<le> measure_pmf.expectation p u)"
eq_sym_conv: fixes a :: "'c"   and b :: "'c" shows "(a = b) = (b = a)"
cong_lt_per2__lt: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C"   and "Tarski_neutral_dimensionless.Per Bet Cong A' B' C'"   and "Cong A B A' B'"   and "Tarski_neutral_dimensionless.Lt Bet Cong B C B' C'" shows "Tarski_neutral_dimensionless.Lt Bet Cong A C A' C'"
tabulation_in_maps_axioms: shows "tabulation_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg r r.tab (tab\<^sub>0 r) (tab\<^sub>1 r)"
neg_always_eventually: fixes p :: "'a" shows "\<box> p = - \<diamond> (- p)"
empty_replicate: fixes n :: "nat"   and x :: "'c" shows "([] = replicate n x) = (n = 0)"
awp_iff_constrains: fixes A :: "'b set"   and F :: "'b program"   and B :: "'b set" shows "(A \<subseteq> awp F B) = (F \<in> A co B)"
map_sel_1: fixes a :: "('f, 'g, 'h) Resumption.resumption"   and f1 :: "'f \<Rightarrow> 'i"   and f2 :: "'g \<Rightarrow> 'j" assumes "Resumption.resumption.is_Done a" shows "Resumption.resumption.result (Resumption.resumption.map_resumption f1 f2 a) = map_option f1 (Resumption.resumption.result a)"
intro: fixes G :: "('c, 'd) graph" assumes "forest G"   and "connected_graph G" shows "tree G"
nfv_simps_11: fixes n :: "nat" shows "Formula.nfv_regex (regex.Skip n) = 0"
intro: fixes G :: "('a, 'b) monoid_scheme" assumes "Group.comm_monoid G"   and "Group.group G" shows "comm_group G"
vertex_3_char_def: shows "vertex_3 \<equiv> CHR ''C''"
simps: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float list"   and f :: "floatarith" shows "tmf_ivl_cs prec ord I a f = those (map (tmf_c prec (map interval_of a) f) [0..<ord] @ [tmf_c prec I f ord])"
sl_def: shows "sl = (\<lambda>(A, r, f). (err A, le r, lift2 f))"
compl_sup_top: fixes x :: "'a" shows "- x \<squnion> x = top"
eigenvector_char_matrix: fixes A :: "'c mat"   and n :: "nat"   and v :: "'c vec"   and e :: "'c" assumes "A \<in> carrier_mat n n" shows "eigenvector A v e = (v \<in> carrier_vec n \<and> v \<noteq> 0\<^sub>v n \<and> char_matrix A e *\<^sub>v v = 0\<^sub>v n)"
P_ltl_step_1: fixes P :: "nat option \<times> value option list \<times> nat \<Rightarrow>f value option \<Rightarrow> bool"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list" assumes "P (None, [], r)"   and "l = STR ''coin'' \<longrightarrow> P (Some 1, [value_plus (r $ 2) (Some (hd i))], r(2 $:= value_plus (r $ 2) (Some (i ! 0))))"   and "value_gt (Some (Num 100)) (r $ 2) = trilean.true \<longrightarrow> P (Some 1, [], r)"   and "\<not>? value_gt (Some (Num 100)) (r $ 2) = trilean.true \<longrightarrow> P (Some 2, [r $ 1], r)" shows "P (ltl_step drinks (Some 1) r (l, i))"