hcju/isabelleps huggingface dataset:
carrier_Fstack: fixes Texp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var ttree"   and Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and S :: "stack_elem list" assumes "TTreeAnalysisCarrier Texp Aexp" shows "carrier (TTreeAnalysis.Fstack Texp as S) \<subseteq> fv S"
hom_bc: shows "homeomorphism (cbox b c) (cbox v w) f2 g2"
simps: fixes db :: "bool discr"   and v\<^sub>1 :: "Value"   and v\<^sub>2 :: "Value" shows "B_project\<cdot>(B\<cdot>db)\<cdot>v\<^sub>1\<cdot>v\<^sub>2 = (if undiscr db then v\<^sub>1 else v\<^sub>2)"
istopology_generate_topology_on: fixes S :: "'c set set" shows "istopology (generate_topology_on S)"
helpers_3: shows "inj_on p (set (filter ((<) z) zs))"
b_def: fixes f :: "'a \<Rightarrow> real \<times> real" shows "b f = (\<Sum>i\<in>Basis. snd (f i) *\<^sub>R i)"
elims_2: fixes x :: "'a ltln" assumes "is_constant x" obtains "x = true\<^sub>n"   | "x = false\<^sub>n"
vsv_vrange_vempty: fixes a :: "V"   and b :: "V" assumes "\<R>\<^sub>\<circ> (ZFC_in_HOL.set {\<langle>a, b\<rangle>}) = []\<^sub>\<circ>" shows "ZFC_in_HOL.set {\<langle>a, b\<rangle>} = []\<^sub>\<circ>"
msb_zero: fixes w :: "32 Word.word" assumes "\<not> msb (- w)"   and "\<not> msb w" shows "uint w = 0"
sqUn_lem2: fixes A :: "'f set option"   and A' :: "'f set option"   and B :: "'f set option" assumes "A \<sqsubseteq> A'" shows "B \<squnion> A \<sqsubseteq> B \<squnion> A'"
PFun_def: fixes A :: "ZF"   and B :: "ZF" shows "PFun A B \<equiv> Sep (Power (CartProd A B)) isFun"
109: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> (y \<squnion> x \<sqinter> bot) = y \<squnion> x"
vec_index_vCons: fixes a :: "'b"   and v :: "'b vec"   and n :: "nat" shows "vCons a v $ n = (if n = 0 then a else v $ (n - 1))"
wordinterval_CIDR_split_distinct: fixes i :: "'b wordinterval" shows "distinct (wordinterval_CIDR_split_prefixmatch i)"
in_E': fixes x :: "'v"   and y :: "'v" shows "((x, y) \<in> E') = (x \<in> X \<and> (x, y) \<in> \<^bold>E)"
int_shiftr_numeral_1: fixes w' :: "num" shows "1 >> numeral w' = 0"
lit_fun_simps_4: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c" shows "\<guillemotleft>f x\<guillemotright> = uop f \<guillemotleft>x\<guillemotright>"
\<beta>_boundedness_diag_lt: fixes y :: "'a"   and m :: "int"   and x :: "'a" assumes "- int (k y) \<le> m"   and "m \<le> int (k x)"   and "x \<in> X"   and "y \<in> X" shows "\<exists>U\<subseteq>\<R>. \<Union> U = {u \<in> V. u x - u y < real_of_int m}"
group_comm_groupI: assumes "\<And>x y. \<lbrakk>x \<in> carrier (add_monoid G); y \<in> carrier (add_monoid G)\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>add_monoid G\<^esub> y = y \<otimes>\<^bsub>add_monoid G\<^esub> x" shows "comm_group (add_monoid G)"
assms_2: shows "dim_col A = prod_list (nths ds (- rmodes))"
distr_pair_PiM_eq_PiM: fixes I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure"   and i' :: "'b" assumes "\<And>i. i \<in> I \<Longrightarrow> prob_space (M i)"   and "prob_space (M i')" shows "distr (M i' \<Otimes>\<^sub>M Pi\<^sub>M I M) (Pi\<^sub>M (insert i' I) M) (\<lambda>(x, X). X(i' := x)) = Pi\<^sub>M (insert i' I) M"
Nfunc_LeastTr2_3: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set"   and i :: "nat" assumes "\<forall>l\<le>n. G \<guillemotright> f l "   and "\<forall>l\<le>n. G \<guillemotright> g l "   and "i \<le> n"   and "f i \<noteq> g i" shows "f (LEAST k. f k \<noteq> g k) \<noteq> g (LEAST k. f k \<noteq> g k)"
assms: shows "\<exists>p r t. j = make_full_observation drinks (Some 1) r p t"
cnf_foldr_and: fixes fs :: "'b formula list" shows "cnf (foldr (\<^bold>\<and>) fs (\<^bold>\<not> \<bottom>)) = \<Union> (cnf ` set fs)"
simps: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat" shows "semi_mojmir_def.state_rank \<Sigma> \<delta> q\<^sub>0 w q n = (if semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n \<noteq> {} \<and> \<not> semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 q then Some (card (semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w q n)) else None)"
homotopic_with_compose_continuous_left: fixes p :: "('b \<Rightarrow> 'd) \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and X :: "'b set"   and Y :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and Z :: "'d set" assumes "homotopic_with_canon (\<lambda>f. p (h \<circ> f)) X Y f g"   and "continuous_on Y h"   and "h ` Y \<subseteq> Z" shows "homotopic_with_canon p X Z (h \<circ> f) (h \<circ> g)"
next_events_def: fixes P :: "'c process"   and xs :: "'c list" shows "next_events P xs \<equiv> {x. xs @ [x] \<in> traces P}"
comp_subst_id_subst: fixes \<sigma> :: "'s" shows "\<sigma> \<odot> id_subst = \<sigma>"
intros: fixes p :: "'c \<times> 'd" shows "sndsp p (snd p)"
circ_tang_vec_def: fixes \<mu> :: "complex"   and E :: "complex"   and p :: "bool" shows "circ_tang_vec \<mu> E p = sgn_bool p * \<i> * (E - \<mu>)"
SET_CR3: fixes evs3 :: "event list"   and C :: "agent"   and k :: "nat"   and NC2 :: "nat"   and KC1 :: "nat"   and SKi :: "nat"   and X :: "agent"   and NC1 :: "nat"   and i :: "nat"   and EKi :: "nat" assumes "evs3 \<in> set_cr"   and "C = Cardholder k"   and "Nonce NC2 \<notin> used evs3"   and "Key KC1 \<notin> used evs3"   and "KC1 \<in> symKeys"   and "Gets C \<lbrace>sign (invKey SKi) \<lbrace>Agent X, Nonce NC1\<rbrace>, cert (CA i) EKi onlyEnc (priSK RCA), cert (CA i) SKi onlySig (priSK RCA)\<rbrace> \<in> set evs3"   and "Says C (CA i) \<lbrace>Agent C, Nonce NC1\<rbrace> \<in> set evs3" shows "Says C (CA i) (EXHcrypt KC1 EKi \<lbrace>Agent C, Nonce NC2\<rbrace> (Pan (pan C))) # Notes C \<lbrace>Key KC1, Agent (CA i)\<rbrace> # evs3 \<in> set_cr"
iT_Minus_Max: fixes I :: "nat set"   and k :: "nat" assumes "I \<noteq> {}"   and "iMin I \<le> k" shows "Max (k \<ominus> I) = k - iMin I"
MetaOperationCallTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and \<tau> :: "'a OCL_Types.type"   and op :: "metaop"   and \<sigma> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E MetaOperationCall \<tau> op : \<sigma>" obtains "mataop_type \<tau> op \<sigma>"
hyps_2: fixes b :: "bool" assumes "iconf (shp s) e"   and "P,shp s \<turnstile>\<^sub>b (e,b) \<surd>" shows "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>throw a,(h', l', sh'),False\<rangle>"
cil_set_induct_fix\<alpha>: fixes w :: "'b list"   and w1 :: "'b list"   and \<alpha> :: "'b \<Rightarrow> 'c set \<times> 'c set"   and w2 :: "'b list"   and P :: "'b list \<Rightarrow> ('b \<Rightarrow> 'c set \<times> 'c set) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "w \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2"   and "P [] \<alpha> [] []"   and "\<And>e w' w1' w2. \<lbrakk>w' \<in> w1' \<otimes>\<^bsub>\<alpha>\<^esub> w2; fst (\<alpha> e) \<inter> mon_pl (map \<alpha> w2) = {}; P w' \<alpha> w1' w2\<rbrakk> \<Longrightarrow> P (e # w') \<alpha> (e # w1') w2"   and "\<And>e w' w2' w1. \<lbrakk>w' \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2'; fst (\<alpha> e) \<inter> mon_pl (map \<alpha> w1) = {}; P w' \<alpha> w1 w2'\<rbrakk> \<Longrightarrow> P (e # w') \<alpha> w1 (e # w2')" shows "P w \<alpha> w1 w2"
exec_gpv_resource_of_oracle: fixes gpv :: "('h, 'i, 'j) gpv"   and "oracle" :: "'k \<Rightarrow> 'i \<Rightarrow> ('j \<times> 'k) spmf"   and s :: "'k" shows "exec_gpv run_resource gpv (RES oracle s) = map_spmf (map_prod id (RES oracle)) (exec_gpv oracle gpv s)"
square_free_m_cong: fixes f :: "int poly"   and g :: "int poly" assumes "pn.square_free_m f"   and "pn.eq_m f g" shows "pn.square_free_m g"
qk_uk_norm: fixes k :: "'c"   and A :: "(('a, 'c) vec, 'b) vec" shows "(1 / norm (column k (Gram_Schmidt_matrix A))) *\<^sub>R column k (Gram_Schmidt_matrix A) = column k (fst (QR_decomposition A))"
weak_lower_semilattice_axioms: shows "weak_lower_semilattice (fpl \<P> \<H>)"
graph_max_final_ex: fixes n :: "nat" shows "\<exists>f\<in>set (finals (graph n)). |vertices f| = n"
pos: fixes eff :: "'a \<Rightarrow> 'b \<Rightarrow> 'b fset \<Rightarrow> bool"   and rules :: "'a stream"   and S :: "'b set"   and rs :: "'a stream"   and r :: "'a" assumes "RuleSystem eff rules S"   and "i.fair rules rs"   and "r \<in> i.R rules" shows "shd (sdrop (RuleSystem.pos rs r) rs) = r"
GammaNegExists: fixes A :: "('a, 'b) form"   and t :: "'a FOL_Fitting.term"   and G :: "('a, 'b) form list" assumes "\<stileturn> Neg (A[t/0]) # G" shows "\<stileturn> Neg (Exists A) # G"
castOp_def: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" shows "castOp s X \<equiv> SOME delta_inp_binp. wlsInp (fst3 delta_inp_binp) (snd3 delta_inp_binp) \<and> wlsBinp (fst3 delta_inp_binp) (trd3 delta_inp_binp) \<and> s = stOf (fst3 delta_inp_binp) \<and> X = Op (fst3 delta_inp_binp) (snd3 delta_inp_binp) (trd3 delta_inp_binp)"
2: fixes l :: "'lbl"   and l' :: "'lbl" assumes "l \<notin> set (remdups (map (the_LabelN \<circ> fst) (filter (Not \<circ> is_LabelS) A)))" shows "GSMP (trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj l A) \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t (proj_unl l A)) \<subseteq> GSMP (trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj l' A) \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t (proj_unl l' A))"
div_2_mult_2_bds_2: fixes d :: "nat"   and n :: "nat" assumes "0 < d" shows "\<lfloor>real n / real d\<rfloor> - 2 * \<lfloor>real (n div 2) / real d\<rfloor> \<le> 1"
Sigma_cong: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d \<Rightarrow> 'e set"   and D :: "'d \<Rightarrow> 'e set" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sigma A C = Sigma B D"
lcp_first_mismatch: fixes a :: "'b"   and b :: "'b"   and w :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "a \<noteq> b" shows "w \<cdot> [a] \<cdot> u \<and>\<^sub>p w \<cdot> [b] \<cdot> v = w"
fls_shift_def: fixes n :: "int"   and f :: "'b fls" shows "fls_shift n f \<equiv> Abs_fls (\<lambda>k. f $$ (k + n))"
indep_subset: fixes X :: "'a set"   and Y :: "'a set" assumes "indep X"   and "Y \<subseteq> X" shows "indep Y"
WT_converter_id: fixes \<I> :: "('y, 'z) \<I>" shows "\<I>, \<I> \<turnstile>\<^sub>C 1\<^sub>C \<surd>"
assms_1: fixes a :: "'a" assumes "a \<in> A" shows "b dvd a"
f': shows "f' = (Rep_a ---> Rep_b ---> Abs_c) f"
span_in_category_axioms: shows "span_in_category (\<cdot>) (Cod (\<mu> \<star> \<nu> \<star> \<pi>))"
c0_2: shows "ennreal (1 - d2 / 2) < emeasure M {x \<in> space M. upper_asymptotic_density {n. \<exists>l\<in>{1..n}. u n x - u (n - l) x \<le> - c0 * real l} < d2 / 2}"
vector_space_def: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd" shows "vector_space scale \<equiv> ((\<forall>a x y. scale a (x + y) = scale a x + scale a y) \<and> (\<forall>a b x. scale (a + b) x = scale a x + scale b x)) \<and> (\<forall>a b x. scale a (scale b x) = scale (a * b) x) \<and> (\<forall>x. scale (1::'c) x = x)"
maximal_independent_subset_extend: fixes S :: "'a set"   and V :: "'a set" assumes "S \<subseteq> V"   and "cindependent S" obtains B :: "'a set" where "S \<subseteq> B" and "B \<subseteq> V" and "cindependent B" and "V \<subseteq> cspan B"
Collect_const_case_prod: fixes P :: "bool" shows "{(a, b). P} = (if P then UNIV else {})"
failure_refine_def: fixes P :: "'b process"   and Q :: "'b process" shows "P \<sqsubseteq>\<^sub>F Q \<equiv> F Q \<subseteq> F P"
aj_le_ci: fixes j :: "nat"   and i :: "nat"   and m :: "nat"   and n :: "nat" assumes "odd j"   and "\<bar>real i / 2 ^ m - real j / 2 ^ n\<bar> < 1 / 2 ^ n" shows "a (real j / 2 ^ n) \<le> c (real i / 2 ^ m)"
init_qp_notempty: shows "init qp \<noteq> {}"
af_subsequence_M: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set"   and n :: "nat" shows "af (\<phi> M\<^sub>n \<psi>) (prefix (Suc n) w) = af \<psi> (prefix (Suc n) w) and\<^sub>n af \<phi> (prefix (Suc n) w) or\<^sub>n af (\<phi> M\<^sub>n \<psi>) (w [1 \<rightarrow> Suc n])"
equation_2: fixes xa :: "'a" shows "eq_i_o xa = Predicate.bind (Predicate.single xa) Predicate.single"
equal_vec_def: fixes x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" shows "equal_class.equal x y = (\<forall>i. x $ i = y $ i)"
gauss_cnj_i: shows "gauss_cnj \<i>\<^sub>\<int> = - \<i>\<^sub>\<int>"
antisym: fixes x :: "'a\<^sub>m"   and y :: "'a\<^sub>m" assumes "bo x y"   and "bo y x"   and "x \<in> UNIV" shows "x = y"
worec_fixpoint: fixes r :: "('d \<times> 'd) set"   and H :: "('d \<Rightarrow> 'e) \<Rightarrow> 'd \<Rightarrow> 'e" assumes "wo_rel r"   and "wo_rel.adm_wo r H" shows "wo_rel.worec r H = H (wo_rel.worec r H)"
division_of_closed: fixes s :: "'a set set"   and i :: "'a set" assumes "s division_of i" shows "closed i"
wf_dom_ss0_w0: shows "wf_dom ss0 w0"
Gamma_one_half_complex: shows "Gamma (1 / 2) = complex_of_real (sqrt pi)"
finite_emeasure_space: fixes M :: "'b measure" assumes "finite_measure M" shows "emeasure M (space M) \<noteq> top"
sat_sub_by_sat: fixes c\<^sub>2 :: "('a, 'b) conf"   and c\<^sub>1 :: "('a, 'b) conf" assumes "Conf.sat c\<^sub>2"   and "c\<^sub>2 \<sqsubseteq> c\<^sub>1" shows "Conf.sat c\<^sub>1"
all_total_imp_totalize: fixes F :: "'c program" assumes "all_total F" shows "totalize F = F"
oprod_minus_Id2: fixes r' :: "('d \<times> 'd) set"   and r :: "('e \<times> 'e) set" assumes "r' \<subseteq> Id" shows "r *o r' - Id \<subseteq> {((x1, y), x2, y). (x1, x2) \<in> r - Id \<and> y \<in> Field r'}"
flip_twice: fixes i :: "nat"   and b :: "bool list" shows "flip i (flip i b) = b"
l3_inv2I: fixes x :: "l3_state" assumes "Keys_bad (ik x) (l3_state.bad x)" shows "x \<in> l3_inv2"
l1_iagreement_Init_def: shows "l1_iagreement_Init \<equiv> {s. \<forall>A B N. signalsInit s (Commit A B N) \<le> signalsInit s (Running A B N)}"
keys_to_list_def: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "ordered_term.keys_to_list ord_term p = ordered_term.pps_to_list ord_term (keys p)"
subset_insertI: fixes B :: "'b set"   and a :: "'b" shows "B \<subseteq> a \<triangleright> B"
Trivial_Abinds_below_fv: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "ABinds \<Gamma>\<cdot>ae \<sqsubseteq> (\<lambda>x. up\<cdot>0) f|` fv \<Gamma>"
0: shows "\<forall>(k, \<gamma>)\<in>boundary diamond_cube. valid_path \<gamma>"
unique_connecting_path_set_2: fixes v :: "'c"   and w :: "'c" assumes "v \<in> V\<^bsub>T\<^esub>"   and "w \<in> V\<^bsub>T\<^esub>" shows "w \<in> set (v \<leadsto>\<^bsub>T\<^esub> w)"
map_poly_degree_less: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b poly" assumes "f (lead_coeff p) = (0::'a)"   and "degree p \<noteq> 0" shows "degree (map_poly f p) < degree p"
RNext_enabled: fixes mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and p :: "PrIds"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals" assumes "\<forall>l. basevars (mm!l, rtrner ch!p, rs!p)" shows "\<turnstile> rs!p = #NotAResult \<and> Calling ch p \<and> (\<forall>l. MemInv mm l) \<longrightarrow> Enabled (<RNext ch mm rs p>_(rtrner ch!p, rs!p))"
E_inf_def: fixes s :: "'s"   and f :: "'s stream \<Rightarrow> ennreal" shows "E_inf s f = (\<Sqinter>cfg\<in>cfg_on s. integral\<^sup>N (T cfg) f)"
elims_2: fixes x :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.rank_invar x" obtains "x = []"   | t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" where "x = [t__]"   | t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" and t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" where "x = t # t' # bq" and "SkewBinomialHeapStruc.rank t < SkewBinomialHeapStruc.rank t' \<and> SkewBinomialHeapStruc.rank_invar (t' # bq)"
mono_extend_set_offending_flows: fixes V :: "'a set"   and E :: "('a \<times> 'a) set"   and E' :: "('a \<times> 'a) set"   and F' :: "('a \<times> 'a) set"   and nP :: "'a \<Rightarrow> 'b" assumes "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "E' \<subseteq> E"   and "F' \<in> set_offending_flows \<lparr>nodes = V, edges = E'\<rparr> nP" shows "\<exists>F\<in>set_offending_flows \<lparr>nodes = V, edges = E\<rparr> nP. F' \<subseteq> F"
is_lasso_prpl_conv: fixes G :: "('e, 'f) b_graph_rec_scheme"   and prpl :: "'e list \<times> 'e list" assumes "b_graph G" shows "b_graph.is_lasso_prpl G prpl = (snd prpl \<noteq> [] \<and> b_graph.is_lasso G (lasso_of_prpl prpl))"
makespan_def': fixes T :: "nat \<Rightarrow> nat" shows "makespan T = Max (T ` {1..m})"
UNIV_eq: shows "UNIV = Abs_bit1 ` {0..<int CARD('a bit1)}"
g': shows "internal_equivalence f' g' \<phi>' \<psi>'"
sturm_squarefree'_adjacent_root_propagate_left: fixes p :: "real poly"   and i :: "nat"   and x :: "real" assumes "p \<noteq> 0"   and "i < length (sturm_squarefree' p) - 1"   and "poly (sturm_squarefree' p ! i) x = 0"   and "poly (sturm_squarefree' p ! (i + 1)) x = 0" shows "\<forall>j\<le>i + 1. poly (sturm_squarefree' p ! j) x = 0"
rk_increment'_def: fixes error :: "int"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "nat"   and A :: "nat \<Rightarrow> nat \<Rightarrow> real"   and b :: "nat \<Rightarrow> real"   and c :: "nat \<Rightarrow> real"   and h :: "real"   and t :: "real"   and x :: "'b" shows "rk_increment' error f s A b c h t x = eucl_down error (\<Sum>j = 1..s. b j *\<^sub>R rk_eval A c f t h x j)"
not_pred_eq: fixes P :: "unit pred" shows "not_pred P = (if eval P () then \<bottom> else single ())"
sum_union_disjoint': fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite A"   and "finite B"   and "A \<inter> B = {}"   and "A \<union> B = C" shows "sum g C = sum g A + sum g B"
Healthy_Idempotent: fixes H :: "'d upred \<Rightarrow> 'd upred"   and P :: "'d upred" assumes "Idempotent H" shows "H P is H"
wtl: shows "wtl ins c 0 s\<^sub>0 \<noteq> \<top>"
maxim_Un: fixes r :: "('d \<times> 'd) set"   and A :: "'d set"   and B :: "'d set" assumes "wo_rel r"   and "finite A"   and "A \<noteq> {}"   and "A \<subseteq> Field r"   and "finite B"   and "B \<noteq> {}"   and "B \<subseteq> Field r" shows "wo_rel.maxim r (A \<union> B) = max2 r (wo_rel.maxim r A) (wo_rel.maxim r B)"
subst_var_eq_def: fixes v :: "nat"   and lp' :: "linear_poly"   and eq :: "nat \<times> linear_poly" shows "subst_var_eq v lp' eq \<equiv> (lhs eq, subst_var v lp' (rhs eq))"
find_index_less_size_conv: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "(find_index P xs < length xs) = (\<exists>x\<in>set xs. P x)"
cases: fixes x :: "checker_type" obtains ts :: "t list" where "x = Type ts__"   | ts :: "ct list" where "x = TopType ts"   | "x = Bot"
module_with_axioms_def: fixes plus\<^sub>M :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and scale :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" shows "module_with_axioms plus\<^sub>M scale \<equiv> ((\<forall>a x y. scale a (plus\<^sub>M x y) = plus\<^sub>M (scale a x) (scale a y)) \<and> (\<forall>a b x. scale (a + b) x = plus\<^sub>M (scale a x) (scale b x))) \<and> (\<forall>a b x. scale a (scale b x) = scale (a * b) x) \<and> (\<forall>x. scale (1::'d) x = x)"
cap_non_negative: shows "\<forall>u v. (0::'capacity) \<le> c (u, v)"
lines_def: fixes P :: "Points"   and Q :: "Points" shows "lines P Q \<equiv> {l. incid P l \<and> incid Q l}"
from_nat_not_eq: fixes a :: "nat"   and b :: "'a" assumes "a \<noteq> mod_type_class.to_nat b"   and "a < CARD('a)" shows "mod_type_class.from_nat a \<noteq> b"
generalise_output_def: fixes t :: "transition"   and regX :: "nat"   and outputX :: "nat" shows "generalise_output t regX outputX = \<lparr>Label = Label t, Arity = Arity t, Guards = Guards t, Outputs = (Outputs t)[outputX := V (R regX)], Updates = Updates t\<rparr>"
assms_4: shows "a \<bullet> w \<noteq> 0"
add: fixes b1 :: "'b"   and b2 :: "'b" shows "f (b1 + b2) = f b1 + f b2"
const: fixes c :: "'n \<Rightarrow> real" shows "P (\<lambda>x. \<chi>i. c i * x $ i)"
set_val_is_l_set_val: shows "l_set_val ShadowRootClass.type_wf set_val set_val_locs"
Diag_TensorE_4: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Diag (t \<^bold>\<otimes> u)" shows "Diag u"
matrix_vector_mul_lid: fixes x :: "('d, 'e) vec" shows "mat (1::'d) *v x = x"
simps: fixes fs :: "int vec iarray"   and m :: "nat"   and dmus :: "int iarray iarray"   and i :: "nat" shows "dmu_array fs m dmus i = (if i = m then dmus else let dmus' = dmu_array_row fs dmus i in dmu_array fs m dmus' (Suc i))"
basic_trans_rules_25: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b"   and "b = c" shows "a \<le> c"
zmax_def: fixes x :: "int"   and y :: "int" shows "zmax x y = (if x \<le> y then y else x)"
128: fixes x :: "'a"   and y :: "'a" shows "- - x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
cod_simps_3: shows "fgh.\<nu>\<pi>.cod.leg1 = fgh.\<nu>\<pi>.leg1"
2: shows "pairwise (\<lambda>u u'. disjnt (elts (lift (x * u) x)) (elts (lift (x * u') x))) (elts y)"
fin_cut_same_def: fixes x :: "'b"   and xs :: "'b list" shows "fin_cut_same x xs = take (LEAST n. drop n xs = replicate (length xs - n) x) xs"
ord_iso_def: shows "ord_iso = bij \<sqinter> mono \<sqinter> (mono \<circ> the_inv)"
clear: shows "set (take_indices (cols (M_mat signs subsets)) (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))) \<subseteq> set (cols (M_mat signs subsets))"
concurrent_store_store_independent: fixes i :: "'a"   and e1 :: "'b"   and mo :: "'a"   and r :: "'a"   and e2 :: "'b"   and mo2 :: "'a"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat" assumes "\<not> hb (i, Store e1 mo i) (r, Store e2 mo2 r)"   and "\<not> hb (r, Store e2 mo2 r) (i, Store e1 mo i)"   and "xs prefix of j"   and "(i, Store e1 mo i) \<in> set (node_deliver_messages xs)"   and "(r, Store e2 mo2 r) \<in> set (node_deliver_messages xs)" shows "i \<noteq> mo2 \<and> r \<noteq> mo"
strand_sem_split_right_1: fixes M :: "('fun, 'var) Term.term set"   and S :: "('fun, 'var) strand_step list"   and S' :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<lbrakk>M; S @ S'\<rbrakk>\<^sub>c \<theta>" shows "\<lbrakk>M \<union> (ik\<^sub>s\<^sub>t S \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>); S'\<rbrakk>\<^sub>c \<theta>"
validn_def: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and n :: "nat"   and c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" shows "(G \<Turnstile>\<^sub>n c : A) = ( \<Turnstile>\<^sub>n G   \<longrightarrow>  \<Turnstile>\<^sub>n c : A )"
ctr_simps_relation_top_empty_eq: shows "(\<lambda>x. x \<in> UNIV) = top"
gt_wt: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wt t >\<^sub>p wt s" shows "t >\<^sub>t s"
disc_2: fixes x2 :: "'e" shows "\<not> isl (Inr x2)"
set_n_lists: fixes n :: "nat"   and xs :: "'c list" shows "set (List.n_lists n xs) = {ys. length ys = n \<and> set ys \<subseteq> set xs}"
pnormal_def: fixes p :: "'a list" shows "pnormal p = (pnormalize p = p \<and> p \<noteq> [])"
rep_eq: fixes x :: "real" shows "enn2ereal (ennreal x) = 0 \<squnion> ereal x"
finite_Sup_prod: fixes A :: "('a \<times> 'b) set" shows "Sup A = Finite_Set.fold sup bot A"
HInfinite_minus_iff: fixes x :: "'a star" shows "(- x \<in> HInfinite) = (x \<in> HInfinite)"