hcju/isabelleps huggingface dataset:
iso_comp: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and less_eq'' :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less'' :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Q :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "OrderingSetIso less_eq less less_eq' less' P f"   and "OrderingSetIso less_eq' less' less_eq'' less'' Q g"   and "order.greater_eq Q (f ` P)" shows "OrderingSetIso less_eq less less_eq'' less'' P (g \<circ> f)"
bigo_plus_subset: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "O(f + g) \<subseteq> O(f) + O(g)"
in_results'_gpv_restrict_gpvD: fixes x :: "'a"   and \<I>' :: "('b, 'c) \<I>"   and gpv :: "('a, 'b, 'c) gpv" assumes "x \<in> results'_gpv (restrict_gpv \<I>' gpv)" shows "x \<in> results'_gpv gpv"
ring_tOp_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<cdot>\<^sub>r y \<in> carrier R"
ipresIGSubst_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'varSort, 'k, 'l, 'var, 'h, 'm, 'n) model_scheme" shows "ipresIGSubst h termMOD MOD = ipresSubst h MOD"
neutr_eq_iff: fixes a :: "'d"   and b :: "'d" shows "(bot = sup a b) = (a = bot \<and> b = bot)"
rel_spmf_restrict_relpE: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and x :: "'c spmf"   and y :: "'d spmf" assumes "rel_spmf (R \<upharpoonleft> P \<otimes> Q) x y" obtains "rel_spmf R x y" and "pred_spmf P x" and "pred_spmf Q y"
size_Diff1_less: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<in># M" shows "size (M - {#x#}) < size M"
supplement1_Legendre: fixes p :: "nat" assumes "prime p"   and "2 < p" shows "Legendre (- 1) (int p) = (- 1) ^ ((p - 1) div 2)"
reduced_GB_is_GB_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "punit.is_Groebner_basis (punit.reduced_GB F)"
valid_Not: fixes P :: "'state"   and x :: "('b, unit, ('act, 'pred) S_action) formula" shows "P \<Turnstile>\<^sub>S Formula.Not x = (\<not> P \<Turnstile>\<^sub>S x)"
Un_Int_distrib: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "A \<union> B \<inter> C = (A \<union> B) \<inter> (A \<union> C)"
need_second_arg_strict: fixes binop :: "binop"   and b :: "val" assumes "binop \<noteq> CondAnd"   and "binop \<noteq> CondOr" shows "need_second_arg binop b"
weakly_Pareto_optimal_for_men_simple: fixes \<mu> :: "('m \<times> 'w) set" assumes "weakly_Pareto_optimal_for_men \<mu>" shows "simple \<mu>"
simps_2: fixes p :: "fm"   and q :: "fm" shows "d_\<beta> (Or p q) = (\<lambda>k. d_\<beta> p k \<and> d_\<beta> q k)"
pseudonatural_equivalence_axioms: shows "pseudonatural_equivalence (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G F \<Phi>\<^sub>F \<sigma>'.map\<^sub>0 \<sigma>'.map\<^sub>1"
doubleton_eq_iff: fixes a :: "'d"   and b :: "'d"   and c :: "'d"   and d :: "'d" shows "({a, b} = {c, d}) = (a = c \<and> b = d \<or> a = d \<and> b = c)"
simps: fixes a :: "dB" shows "NF a = ((\<exists>ts x. a = dB.Var x \<degree>\<degree> ts \<and> listall NF ts) \<or> (\<exists>t. a = dB.Abs t \<and> NF t))"
rep_eq: fixes x :: "real_alg" shows "real_of x = real_of_3 (rep_real_alg x)"
rel_pmf_reflI: fixes p :: "'c pmf"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> P x x" shows "rel_pmf P p p"
RTranCl_subset2: fixes s :: "'a"   and g :: "'a"   and f :: "'a \<Rightarrow> 'a list"   and h :: "'a \<Rightarrow> 'a list" assumes "s [f]\<rightarrow>* g"   and "\<And>g. s [f]\<rightarrow>* g \<Longrightarrow> set (f g) \<subseteq> set (h g)" shows "s [h]\<rightarrow>* g"
groupI: fixes G :: "('c, 'd) monoid_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"   and "\<one>\<^bsub>G\<^esub> \<in> carrier G"   and "\<And>x y z. \<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<exists>y\<in>carrier G. y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>" shows "Group.group G"
ext: fixes g :: "'a"   and f :: "'a" assumes "g \<cdot>\<^sub>A f \<noteq> A.null" shows "A.seq g f"
Mp_dvdm: fixes f :: "int poly"   and g :: "int poly" shows "(Mp f dvdm g) = (f dvdm g)"
onorm_componentwise: fixes f :: "'d \<Rightarrow> 'e" assumes "bounded_linear f" shows "onorm f \<le> (\<Sum>i\<in>Basis. norm (f i))"
split_minD: fixes t :: "('b ivl \<times> 'b) tree"   and x :: "'b ivl"   and t' :: "('b ivl \<times> 'b) tree" assumes "split_min t = (x, t')"   and "t \<noteq> \<langle>\<rangle>" shows "x # Tree2.inorder t' = Tree2.inorder t"
SUP_bot_conv_1: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and B :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Sup (B ` A) = bot) = (\<forall>x\<in>A. B x = bot)"
Neg_mono: fixes A :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A H \<turnstile> B" shows "insert (Neg B) H \<turnstile> Neg A"
cases: fixes x :: "nat" obtains "x = 0"   | v :: "nat" where "x = Suc v"
qAbs_preserves_alpha: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var" assumes "X #= X'"   and "qGood X \<or> qGood X'" shows "qAbs xs x X $= qAbs xs x X'"
assm: shows "\<exists>n. n \<^bold>\<in> i"
is_functor: shows "functor C.CC.comp D.CC.comp FF.map"
alpha_out_var: fixes x :: "'c \<Longrightarrow> 'e" shows "x ;\<^sub>L snd\<^sub>L = $x\<acute>"
dim_le_card: fixes V :: "('b, 'c) vec set"   and W :: "('b, 'c) vec set" assumes "V \<subseteq> vec.span W"   and "finite W" shows "vec.dim V \<le> card W"
mresolution_deriv_def: shows "mresolution_deriv = mresolution_step\<^sup>*\<^sup>*"
disjoint: shows "f ` (S \<inter> U) \<inter> f ` (S \<inter> V) = {}"
int_pow_one: fixes z :: "int" shows "\<one> [^] z = \<one>"
abs_eq: fixes xa :: "'a list"   and x :: "'a \<Rightarrow> 'b" shows "Mapping.tabulate xa x = mapping.Mapping (map_of (map (\<lambda>k. (k, x k)) xa))"
cnj: fixes k :: "nat" shows "cnj (\<chi> k) = \<chi> k"
funpow_dist1_prop: fixes y :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "y \<in> orbit f x" shows "(f ^^ funpow_dist1 f x y) x = y"
eo: shows "prod_list (map ((!) (map_sgas sign)) xa) = 0 \<or> prod_list (map ((!) (map_sgas sign)) xa) = 1"
error_pure: fixes e :: "'m"   and h :: "'l" shows "pure (error e) h"
int_simps_17: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> (#True \<and> P) = P"
c_manifold'_def: fixes charts :: "('c, 'd) chart set"   and k :: "enat" shows "c_manifold' charts k \<equiv> c_manifold charts k"
h: shows "\<forall>q\<in>set qs. \<forall>x\<in>set (characterize_root_list_p p). squash (poly q x) = (if 0 < poly q x then 1 else - 1)"
S4Read: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> Read rmCh mm ires p \<and> $S4 rmhist p \<and> unchanged (e p, c p, r p) \<and> HNext rmhist p \<and> (\<forall>l. $MemInv mm l) \<longrightarrow> S4 rmhist p$ \<and> unchanged (rmhist!p)"
csqrt_eq_0: fixes z :: "complex" shows "(csqrt z = 0) = (z = 0)"
suf_prod_suf': fixes u :: "'b list"   and w :: "'b list"   and z :: "'b list" assumes "u \<le>s w \<cdot> u \<cdot> z" shows "u \<le>s u \<cdot> z"
phis_phi: fixes g :: "'g"   and n :: "'node"   and v :: "'val"   and vs :: "'val list" assumes "phis'_all g (n, v) = Some vs" shows "notriv.phi g v = Some vs"
less_eq_list_induct: fixes x1 :: "'f list"   and x2 :: "'f list"   and Pa :: "'f list \<Rightarrow> 'f list \<Rightarrow> bool" assumes "x1 \<preceq> x2"   and "\<And>ys. Pa [] ys"   and "\<And>xs ys y. \<lbrakk>xs \<preceq> ys; Pa xs ys\<rbrakk> \<Longrightarrow> Pa xs (y # ys)"   and "\<And>x y xs ys. \<lbrakk>x = y; xs \<preceq> ys; Pa xs ys\<rbrakk> \<Longrightarrow> Pa (x # xs) (y # ys)" shows "Pa x1 x2"
futures_def: fixes P :: "'c process"   and xs :: "'c list" shows "futures P xs \<equiv> {(ys, Y). (xs @ ys, Y) \<in> failures P}"
monom_zero: shows "monom 0 (0::'a) = 0"
map_comp: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'b"   and v :: "'c tree" shows "map_tree g (map_tree f v) = map_tree (g \<circ> f) v"
e2stack_0_iff_Nil: fixes e :: "nat"   and ss :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" assumes "e = encode_config (ss, rv)" shows "(e2stack e = 0) = (ss = [])"
mult_coprime_aux: fixes a :: "nat"   and b :: "nat" assumes "1 < a"   and "1 < b"   and "coprime a b" shows "f (a * b) = f a * f b"
h: shows "((\<exists>l\<in>EqUni ` set eq. evalUni (elimVarUni_atom L' l) x) \<or> (\<exists>l\<in>LessUni ` set les. evalUni (elimVarUni_atom L' l) x) \<or> (\<exists>l\<in>LeqUni ` set leq. evalUni (elimVarUni_atom L' l) x) \<or> (\<exists>l\<in>NeqUni ` set neq. evalUni (elimVarUni_atom L' l) x)) = ((\<exists>l\<in>set eq. evalUni (elimVarUni_atom L' (EqUni l)) x) \<or> (\<exists>l\<in>set les. evalUni (elimVarUni_atom L' (LessUni l)) x) \<or> (\<exists>l\<in>set leq. evalUni (elimVarUni_atom L' (LeqUni l)) x) \<or> (\<exists>l\<in>set neq. evalUni (elimVarUni_atom L' (NeqUni l)) x))"
eq1: shows "lookup (proj_poly (component_of_term u) p * (monomial c (pp_of_term v) when component_of_term v = component_of_term u)) (pp_of_term u) = (lookup (proj_poly (component_of_term u) p * monomial c (pp_of_term v)) (pp_of_term u) when component_of_term v = component_of_term u)"
Ints_nonzero_abs_less1: fixes x :: "'b" assumes "x \<in> \<int>"   and "\<bar>x\<bar> < (1::'b)" shows "x = (0::'b)"
simps_2: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm" shows "args (App s t) = args s @ [t]"
Bsub: shows "B \<subseteq> {x. a \<bullet> x = 0}"
desargues_config_not_col_12: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "desargues_config A B C A' B' C' M N P R" shows "\<not> col C B B'"
Graph6: fixes M :: "node list"   and E :: "(nat \<times> nat) list"   and R :: "nat"   and T :: "nat" assumes "Proper_Edges (M, E)"   and "R < length E"   and "T < length M" shows "Proper_Edges (M, E[R := (fst (E ! R), T)])"
state_measure_nonempty: fixes V :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type" shows "space (state_measure V \<Gamma>) \<noteq> {}"
WTrtCallNT: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and es :: "expr list"   and Ts :: "ty list"   and Copt :: "char list option"   and M :: "char list"   and T :: "ty" assumes "P,E,h \<turnstile> e : NT"   and "P,E,h \<turnstile> es [:] Ts" shows "P,E,h \<turnstile> Call e Copt M es : T"
functor_axioms: shows "functor (\<cdot>) (\<cdot>) C.map"
in_dgrad_p_set_iff: fixes p :: "'a \<Rightarrow>\<^sub>0 'd"   and d :: "'a \<Rightarrow> nat"   and m :: "nat" shows "(p \<in> punit.dgrad_p_set d m) = (\<forall>v\<in>keys p. d (local.punit.pp_of_term v) \<le> m)"
set_spmf_cond_spmf_fst: fixes p :: "('e \<times> 'd) spmf"   and a :: "'e" shows "set_spmf (cond_spmf_fst p a) = snd ` (set_spmf p \<inter> {a} \<times> UNIV)"
fundamental_theorem_of_linear_inequalities_3: fixes A :: "'a vec set" assumes "A \<subseteq> carrier_vec n"   and "finite A" shows "cone A = {b \<in> carrier_vec n. \<nexists>c. c \<in> carrier_vec n \<and> (\<forall>a\<^sub>i\<in>A. (0::'a) \<le> c \<bullet> a\<^sub>i) \<and> c \<bullet> b < (0::'a)}"
char_one: shows "principal_char (Characters (Characters G)) \<one>\<^bsub>Characters (Characters G)\<^esub> = 1"
perp_col4: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and Q :: "'a"   and R :: "'a"   and S :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "P \<noteq> Q"   and "R \<noteq> S"   and "Tarski_neutral_dimensionless.Col Bet A B P"   and "Tarski_neutral_dimensionless.Col Bet A B Q"   and "Tarski_neutral_dimensionless.Col Bet C D R"   and "Tarski_neutral_dimensionless.Col Bet C D S"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B C D" shows "Tarski_neutral_dimensionless.Perp Bet Cong P Q R S"
lenlex_irreflexive: fixes r :: "('b \<times> 'b) set"   and xs :: "'b list" assumes "\<And>x. (x, x) \<notin> r" shows "(xs, xs) \<notin> lenlex r"
C_eq_All_untilSorted_withSimpsQ: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and l :: "(int \<times> int \<times> protocol) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "Cp (list2FWpolicy (qsort (removeShadowRules2 (remdups (rm_MT_rules Cp (insertDeny (removeShadowRules1 (policy2list p)))))) l)) = Cp p"
disconnected_nodes_eq2: fixes document_ptr :: "(_) document_ptr" assumes "document_ptr \<noteq> owner_document" shows "|h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r = |h' \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r"
translate_plus: fixes a :: "real \<times> real"   and x :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" shows "ccw' (a + x) (b + x) (c + x) = ccw' a b c"
strongdB_aux: fixes d :: "'d"   and R :: "(('exp, 'id) MWLfCom list \<times> ('exp, 'id) MWLfCom list) set"   and i :: "nat"   and V :: "('exp, 'id) MWLfCom list"   and V' :: "('exp, 'id) MWLfCom list"   and m1 :: "'id \<Rightarrow> 'val"   and W :: "('exp, 'id) MWLfCom list"   and m2 :: "'id \<Rightarrow> 'val"   and m1' :: "'id \<Rightarrow> 'val" assumes "Strong_d_Bisimulation d R"   and "i < length V"   and "(V, V') \<in> R"   and "SRabbr (V ! i) m1 W m2"   and "m1 =\<^bsub>d\<^esub> m1'" shows "\<exists>W' m2'. SRabbr (V' ! i) m1' W' m2' \<and> (W, W') \<in> R \<and> m2 =\<^bsub>d\<^esub> m2'"
poly_add_replicate_zero'_2: fixes p :: "'a list"   and n :: "nat" assumes "set p \<subseteq> carrier R" shows "poly_add (replicate n \<zero>) p = local.normalize p"
chain_subdiv_chain_refl: fixes C :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "boundary_chain C" shows "chain_subdiv_chain C C"
asymptotics_aux_2: fixes x :: "real"   and i :: "nat"   and s :: "real" assumes "x\<^sub>1 \<le> x"   and "i < k"   and "s \<equiv> if 0 \<le> p then 1 else - 1" shows "(bs ! i * x + (hs ! i) x) powr p \<le> (bs ! i * x + s * hb * x * ln x powr - (1 + e)) powr p"
distinct_45: fixes pi :: "pi"   and name' :: "name"   and pi' :: "pi" shows "\<tau>.pi \<noteq> <\<nu>name'>pi'"
integral: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c" assumes "domain R"   and "a \<otimes>\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub>"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "a = \<zero>\<^bsub>R\<^esub> \<or> b = \<zero>\<^bsub>R\<^esub>"
nectxt_imp_supt_ctxt: fixes C :: "('d, 'e) ctxt"   and t :: "('d, 'e) Term.term" assumes "C \<noteq> \<box>" shows "C\<langle>t\<rangle> \<rhd> t"
arc_in_arc_or_complement: fixes x :: "'a"   and y :: "'a" assumes "arc x"   and "arc y"   and "\<not> x \<le> y" shows "x \<le> - y"
ord_works: fixes a :: "nat"   and n :: "nat" shows "[a ^ Pocklington.ord n a = 1] (mod n) \<and> (\<forall>m. 0 < m \<and> m < Pocklington.ord n a \<longrightarrow> [a ^ m \<noteq> 1] (mod n))"
sumAssoc: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" shows "(P \<oplus> Q) \<oplus> R \<sim>\<^sup>s P \<oplus> Q \<oplus> R"
fds_deriv_shift: fixes c :: "'b"   and f :: "'b fds" shows "fds_deriv (fds_shift c f) = fds_shift c (fds_deriv f)"
intros_3: fixes s :: "dB"   and s' :: "dB"   and t :: "dB"   and t' :: "dB" assumes "s => s'"   and "t => t'" shows "s \<degree> t => s' \<degree> t'"
hequiv_proj: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and i :: "'b"   and a :: "'b"   and b :: "'b" assumes "Hintikka A H"   and "Nom a at i in' H"   and "a \<in> A"   and "Nom b at i in' H"   and "b \<in> A" shows "proj (hequiv_rel A H) a = proj (hequiv_rel A H) b"
norm_power2: fixes q :: "octo" shows "(norm q)\<^sup>2 = Ree (Octonions.cnj q * q)"
hisPred_def: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid"   and n' :: "nat" shows "hisPred t n nid n' \<equiv> GREATEST n''. \<exists>nid'. (n'', nid') \<in> his t n nid \<and> n'' < n'"
clinearI: fixes f :: "'b \<Rightarrow> 'c" assumes "\<And>b1 b2. f (b1 + b2) = f b1 + f b2"   and "\<And>r b. f (r *\<^sub>C b) = r *\<^sub>C f b" shows "clinear f"
FP_1: shows "part {..<brn (fst cf2)} (F ` P)"
sinc_neg: fixes x :: "real" shows "sinc (- x) = sinc x"
basis_finmap_eq: assumes "local.basis_proj \<noteq> {}" shows "local.basis_finmap = range (\<lambda>f. \<Pi>' i\<in>domain f. from_nat_into local.basis_proj ((f)\<^sub>F i))"
h2: shows "(\<Sum>i = 0..<n + 1. f i / B ^ i) * B ^ n = (\<Sum>i = 0..<n + 1. f i * B ^ (n - i))"
step_def: shows "step = (\<lambda>(n, d). (d, n + d - 2 * (n mod d)))"
pred_inject_2: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c" shows "pred_option P (Some a) = P a"
partition_filter2: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "snd (partition P xs) = filter (Not \<circ> P) xs"
P: fixes J :: "'i set"   and H :: "'i set" assumes "J \<subseteq> H"   and "finite H"   and "H \<subseteq> I" shows "P J = distr (P H) (Pi\<^sub>M J M) (\<lambda>f. restrict f J)"
WTrt_shext_mono: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty"   and sh' :: "sheap" assumes "P,E,h,sh \<turnstile> e : T"   and "sh \<unlhd>\<^sub>s sh'"   and "\<not> sub_RI e" shows "P,E,h,sh' \<turnstile> e : T"
word_of_int_uint: fixes w :: "'a Word.word" shows "word_of_int (uint w) = w"
simps_2: fixes a :: "'c"   and t1 :: "'c bt"   and t2 :: "'c bt" shows "postorder (Br a t1 t2) = postorder t1 @ postorder t2 @ [a]"
pi_edges_it: fixes S :: "'G" shows "proper_it (edges_it S) (edges_it S)"
disj_left_commute: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "b OR a OR c = a OR b OR c"
span_insert_cong: fixes A :: "'a set"   and B :: "'a set"   and p :: "'a" assumes "ideal A = ideal B" shows "ideal (insert p A) = ideal (insert p B)"
equiv2: shows "w2 \<equiv>\<^sub>E r'\<^sub>2"
distinct_rtrancl_list_impl: fixes L :: "('b \<times> 'b) list"   and ls :: "'b list" assumes "distinct L"   and "distinct ls" shows "distinct (rtrancl_list_impl L ls)"
HEndPhase0_Inv2a_dblock_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase0 s s' p"   and "Inv1 s"   and "Inv2a_inner s q"   and "Inv2c_inner s p" shows "Inv2a_innermost s' q (dblock s' q)"
region_set'_def: fixes R :: "('d \<Rightarrow> 'e) set"   and r :: "'d list"   and c :: "'e" shows "region_set' R r c = {[r\<rightarrow>c]v |v. v \<in> R}"
focus_sum: fixes X :: "'b set"   and f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and I :: "'d set" shows "focus X (sum f I) = (\<Sum>i\<in>I. focus X (f i))"
consistent_imp_rank_ge: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" assumes "consistent A b" shows "(if \<exists>a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'b) then mod_type_class.to_nat (GREATEST a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'b)) + 1 else 0) \<le> rank A"
simulation1: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and s1 :: "'g"   and s2 :: "'i"   and tl1 :: "'h"   and s1' :: "'g" assumes "delay_bisimulation_obs trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2"   and "bisim s1 s2"   and "trsys1 s1 tl1 s1'"   and "\<not> \<tau>move1 s1 tl1 s1'" shows "\<exists>s2' s2'' tl2. (\<tau>trsys.silent_move trsys2 \<tau>move2)\<^sup>*\<^sup>* s2 s2' \<and> trsys2 s2' tl2 s2'' \<and> \<not> \<tau>move2 s2' tl2 s2'' \<and> bisim s1' s2'' \<and> tlsim tl1 tl2"
hbt_delete_1: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "hbt t" shows "hbt (delete x t)"
add_left_cancel: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.cancel_semigroup_add plus" shows "(plus a b = plus a c) = (b = c)"
qmpt_factorE_2: fixes proj :: "'a \<Rightarrow> 'c"   and M2 :: "'c measure"   and T2 :: "'c \<Rightarrow> 'c" assumes "qmpt_factor proj M2 T2" shows "AE x in M. proj (T x) = T2 (proj x)"
density_RN_deriv: fixes N :: "'b measure" assumes "absolutely_continuous (distr M M' T) N"   and "sets N = sets (distr M M' T)" shows "density (distr M M' T) (RN_deriv (distr M M' T) N) = N"
absorb_iff2: fixes b :: "'a"   and a :: "'a" shows "(b \<sqsubseteq> a) = (a \<sqinter> b = b)"
total_on_refl_on_consistent_into: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "total_onP A r"   and "refl_onP A r"   and "order_consistent r s"   and "x \<in> A"   and "y \<in> A"   and "s x y" shows "r x y"
reducecoeffh: fixes t :: "Ferrack.num"   and g :: "int"   and bs :: "real list" assumes "dvdnumcoeff t g"   and "0 < g" shows "real_of_int g * Inum bs (reducecoeffh t g) = Inum bs t"
measure_completion: fixes S :: "'a set"   and M :: "'a measure" assumes "S \<in> sets M" shows "Sigma_Algebra.measure (completion M) S = Sigma_Algebra.measure M S"
trivialpair_in_USdB: fixes d :: "'d" shows "[] \<approx>\<^bsub>d\<^esub> []"
assms: shows "\<forall>a b. r (a, b\<lparr>ok := True\<rparr>) \<longrightarrow> r (a, b\<lparr>ok := False\<rparr>)"
RP_model: fixes St :: "'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set"   and St' :: "'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set"   and I :: "'a set" assumes "St \<leadsto> St'" shows "(I \<Turnstile>s grounding_of_clss (N_of_state St' \<union> P_of_state St' \<union> Q_of_state St')) = (I \<Turnstile>s grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St))"
simps_2: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and xs :: "'c list" shows "remdups_gen f (x # xs) = x # remdups_gen f (filter (\<lambda>y. f x \<noteq> f y) xs)"
assms_5: shows "has_greatest_postfixpoint (h \<circ> g)"
Inr_def: shows "Inr = Abs_sum \<circ> Inr_Rep"
ntsubst_preserves_free: fixes \<theta> :: "('d + 'e, 'f) trm"   and \<sigma> :: "'e \<Rightarrow> ('d, 'f) trm" assumes "dfree \<theta>"   and "\<And>i. dfree (\<sigma> i)" shows "dfree (TsubstFO \<theta> \<sigma>)"
group_homomorphism_\<eta>_cong: fixes \<eta>' :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> R \<Longrightarrow> \<eta>' x = \<eta> x"   and "\<eta>' \<in> extensional R" shows "group_homomorphism \<eta>' R (+) \<zero> R' (+') \<zero>'"