hcju/isabelleps huggingface dataset:
l10_6_existence_spec: fixes A :: "'p"   and B :: "'p"   and P' :: "'p" assumes "A \<noteq> B" shows "\<exists>P. P' P ReflectL A B"
bij_betw_linorder_rank: fixes B :: "'b set"   and R :: "('b \<times> 'b) set"   and A :: "'b set" assumes "linorder_on B R"   and "finite A"   and "A \<subseteq> B" shows "bij_betw (linorder_rank R A) A {..<card A}"
lm143: fixes A :: "'b set" assumes "card (Pow A) = 1" shows "A = {}"
alw: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" shows "alw \<phi> \<pi>l = (\<forall>i. \<phi> (map (sdrop i) \<pi>l))"
mps_2: assumes "nmu (dports m1)" shows "php m1"
simple_distinct_map: fixes r :: "Vertex \<Rightarrow> Vertex"   and a :: "Vertex"   and b :: "Vertex"   and c :: "Vertex"   and d :: "Vertex"   and f :: "Vertex" assumes "r \<in> simple_rotations"   and "distinct [a, b, c, d]"   and "\<forall>e\<in>{a, b, c}. r e \<noteq> f" shows "r d = f"
Send: fixes t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list"   and \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list" assumes "send\<langle>t\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>" shows "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet>\<^sub>c (update\<^sub>s\<^sub>t \<S> (send\<langle>t\<rangle>\<^sub>s\<^sub>t # S), \<A> @ [Step (receive\<langle>t\<rangle>\<^sub>s\<^sub>t)])"
Lower_closed: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" shows "Lower L A \<subseteq> carrier L"
del_list_sorted_app: fixes xs :: "'b list"   and x :: "'b"   and ys :: "'b list" assumes "Sorted_Less.sorted (xs @ [x])" shows "del_list x (xs @ ys) = xs @ del_list x ys"
if_C_def: fixes c :: "'c control_state_scheme \<Rightarrow> bool"   and E :: "'c control_state_scheme \<Rightarrow> ('d \<times> 'c control_state_scheme) option"   and F :: "'c control_state_scheme \<Rightarrow> ('d \<times> 'c control_state_scheme) option" shows "_if_SECLEAN c E F = (\<lambda>\<sigma>. if exec_stop \<sigma> then Some (undefined, \<sigma>) else if c \<sigma> then E \<sigma> else F \<sigma>)"
assms_3: shows "map (\<lambda>(uu_, x). hd x) xs = map (\<lambda>(uu_, x). hd x) ys"
incr_sorted: fixes xs :: "'a list" assumes "incr xs" shows "sorted xs"
add_mset_le_left_le: fixes a :: "'b"   and b :: "'b"   and A :: "'b multiset" assumes "a \<le> b" shows "add_mset a A \<le> add_mset b A"
has_integral_mult_right: fixes f :: "'b \<Rightarrow> 'c"   and y :: "'c"   and i :: "'b set"   and c :: "'c" assumes "(f has_integral y) i" shows "((\<lambda>x. c * f x) has_integral c * y) i"
even_power: fixes a :: "'a"   and n :: "nat" shows "even (a ^ n) = (even a \<and> 0 < n)"
compact_valid_path_image: fixes g :: "real \<Rightarrow> 'b" assumes "valid_path g" shows "compact (path_image g)"
set_merge: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list"   and ys :: "'a list" shows "set (Common.merge f xs ys) = set xs \<union> set ys"
domain_def: fixes g :: "('b \<times> real) set" shows "domain g = {x. \<exists>y. (x, y) \<in> g}"
simps: fixes a :: "'d"   and G :: "('d, 'e) monoid_scheme"   and gens :: "'d set" shows "(a \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>) = (a = \<one>\<^bsub>G\<^esub> \<or> (\<exists>x. a = x \<and> x \<in> gens) \<or> (\<exists>x. a = inv\<^bsub>G\<^esub> x \<and> x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>) \<or> (\<exists>x y. a = x \<otimes>\<^bsub>G\<^esub> y \<and> x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub> \<and> y \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>))"
simps_1: fixes variables_of :: "'c list"   and operators_of :: "'c strips_operator list"   and initial_of :: "'c \<Rightarrow> bool option"   and goal_of :: "'c \<Rightarrow> bool option"   and more :: "'d" shows "\<lparr>strips_problem.variables_of = variables_of, operators_of = operators_of, initial_of = initial_of, goal_of = goal_of, \<dots> = more\<rparr>\<^sub>\<V> = variables_of"
simps: fixes xs :: "('c \<times> 'd) list" shows "get (T xs) = map_of xs"
radj_def: fixes f :: "'d \<Rightarrow> 'c" shows "radj f = (\<lambda>y. \<Squnion> {x. f x \<le> y})"
subformulas\<^sub>\<nu>_list_length: fixes \<phi> :: "'a ltln" shows "length (subformulas\<^sub>\<nu>_list \<phi>) = card (subformulas\<^sub>\<nu> \<phi>)"
imageI: fixes x :: "'k"   and A :: "'k set"   and f :: "'k \<Rightarrow> 'l" assumes "x \<in> A" shows "f x \<in> f ` A"
trancl_induct: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set"   and P :: "'c \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>+"   and "\<And>y. (a, y) \<in> r \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>(a, y) \<in> r\<^sup>+; (y, z) \<in> r; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
IH_1: assumes "i < nr"   and "j < nc"   and "A $$ (i, j) = (0::'a)" shows "pivot_positions_main (mat\<^sub>h A) nr nc i (Suc j) = pivot_positions_main A nr nc i (Suc j)"
length_dominant_term: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and F :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term" shows "length (snd (dominant_term F)) = expansion_level TYPE('a)"
def_equiv_implies_valid: fixes \<Gamma> :: "(name \<times> ty) list"   and t :: "trm"   and s :: "trm"   and T :: "ty" assumes "\<Gamma> \<turnstile> t \<equiv> s : T" shows "valid \<Gamma>"
set_insort_key: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "set (linorder.insort_key less_eq f x xs) = insert x (set xs)"
Pure: fixes A :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and x :: "'h"   and y :: "'i"   and \<I> :: "('j, 'k) \<I>"   and D :: "'l \<Rightarrow> 'm \<Rightarrow> bool" assumes "A x y" shows "eq_\<I>_generat A \<I> D (Pure x) (Pure y)"
distr_eq: shows "distr M S X \<Otimes>\<^sub>M distr M T Y = density (S \<Otimes>\<^sub>M T) (\<lambda>x. ennreal (Px (fst x) * Py (snd x)))"
val_type_expr_sem_rf: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "expr"   and t :: "pdf_type"   and V :: "nat set"   and \<sigma> :: "nat \<Rightarrow> val" assumes "\<Gamma> \<turnstile> e : t"   and "randomfree e"   and "free_vars e \<subseteq> V"   and "\<sigma> \<in> space (state_measure V \<Gamma>)" shows "val_type (expr_sem_rf \<sigma> e) = t"
coprime_pairs_asymptotics: shows "(\<lambda>N. real (card (A N)) - 6 / pi\<^sup>2 * (real N)\<^sup>2) \<in> O(\<lambda>N. real N * ln (real N))"
leadsETo_weaken_R: fixes F :: "'d program"   and A :: "'d set"   and CC :: "'d set set"   and A' :: "'d set"   and B' :: "'d set" assumes "F \<in> A leadsTo[CC] A'"   and "A' \<subseteq> B'" shows "F \<in> A leadsTo[CC] B'"
inject_2: fixes b :: "'c"   and b' :: "'c" shows "(Inr b = Inr b') = (b = b')"
empty_neutral_2: fixes x :: "'b multiset" shows "x + \<Empt> = x"
subst_trop: fixes \<sigma> :: "'c \<Rightarrow> 'd"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'b"   and u :: "('e, 'd) uexpr"   and v :: "('f, 'd) uexpr"   and w :: "('g, 'd) uexpr" shows "\<sigma> \<dagger> trop f u v w = trop f (\<sigma> \<dagger> u) (\<sigma> \<dagger> v) (\<sigma> \<dagger> w)"
hom_in_carrier: fixes h :: "'a \<Rightarrow> 'b"   and G :: "('a, 'c) monoid_scheme"   and H :: "('b, 'd) monoid_scheme"   and x :: "'a" assumes "h \<in> hom G H"   and "x \<in> carrier G" shows "h x \<in> carrier H"
A82_Acc: shows "Acc level1 sA82 = {sA93}"
compact_sphere: fixes a :: "'c"   and r :: "real" shows "compact (sphere a r)"
that: fixes y :: "real \<times> real" assumes "c.limit_cycle y"   and "range (c.flow0 y) \<subseteq> trapC" shows "thesis"
atLeastAtMost_subseteq_atLeastLessThan_iff: fixes a :: "'a \<times> ('a, 'b) operation"   and b :: "'a \<times> ('a, 'b) operation"   and c :: "'a \<times> ('a, 'b) operation"   and d :: "'a \<times> ('a, 'b) operation" shows "(hb.atLeastAtMost a b \<subseteq> hb.atLeastLessThan c d) = (weak_hb a b \<longrightarrow> weak_hb c a \<and> hb b d)"
C: fixes Da :: "'a pmf"   and Db :: "'b pmf" shows "set_pmf (Proj1_pmf (Sum_pmf (5 / 10) Da Db)) = set_pmf Da"
disc_25: fixes x51 :: "nat"   and x52 :: "'d"   and x53 :: "'d" shows "isRecvMarker (RecvMarker x51 x52 x53)"
frac_eq: fixes x :: "'a" shows "(frac x = x) = ((0::'a) \<le> x \<and> x < (1::'a))"
completely_multiplicative_fds_deriv: fixes f :: "nat \<Rightarrow> 'a" assumes "completely_multiplicative_function f" shows "fds_deriv (fds f) = - fds (\<lambda>n. f n * mangoldt n) * fds f"
generic_unwinding: fixes P :: "'a process"   and D :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> ('a list \<times> 'a list) set"   and I :: "('b \<times> 'b) set" assumes "ref_union_closed P"   and "view_partition P D R"   and "weakly_future_consistent P I D R"   and "weakly_step_consistent P D R"   and "locally_respects P I D R" shows "secure P I D"
sub: fixes a :: "'b" assumes "a \<in> J" shows "a \<in> B"
plus_numeral: fixes x :: "'a"   and n :: "num" shows "f (x + numeral n) = f x"
s1: shows "c1 - y * A2 \<le> c - y * A"
sings_image: fixes B :: "'b set" shows "sings B = (\<lambda>x. [x]) ` B"
maxsimp_def: fixes X :: "'b set set"   and x :: "'b set" assumes "SimplicialComplex X" shows "SimplicialComplex.maxsimp X x \<equiv> x \<in> X \<and> (\<forall>z\<in>X. order.greater_eq z x \<longrightarrow> z = x)"
mod_eq_dvd_iff_poly: fixes x :: "'b mod_ring poly"   and n :: "'b mod_ring poly"   and y :: "'b mod_ring poly" shows "(x mod n = y mod n) = (n dvd x - y)"
pred_numeral_inc: fixes k :: "num" shows "pred_numeral (Num.inc k) = numeral k"
openI: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>C. \<lbrakk>chain_on P C A; C \<noteq> {}; \<exists>x\<in>A. glb P C x \<and> Q x\<rbrakk> \<Longrightarrow> \<exists>y\<in>C. Q y" shows "open_on P Q A"
par_mono: fixes c\<^sub>1 :: "'a::refinement_lattice"   and d\<^sub>1 :: "'a::refinement_lattice"   and c\<^sub>2 :: "'a::refinement_lattice"   and d\<^sub>2 :: "'a::refinement_lattice" assumes "(c\<^sub>1::'a::refinement_lattice) \<sqsubseteq> (d\<^sub>1::'a::refinement_lattice)"   and "(c\<^sub>2::'a::refinement_lattice) \<sqsubseteq> (d\<^sub>2::'a::refinement_lattice)" shows "(c\<^sub>1::'a::refinement_lattice) \<parallel> (c\<^sub>2::'a::refinement_lattice) \<sqsubseteq> (d\<^sub>1::'a::refinement_lattice) \<parallel> (d\<^sub>2::'a::refinement_lattice)"
BinOp: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>2 :: "char list exp"   and v\<^sub>2 :: "val"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and bop :: "bop"   and v :: "val" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v\<^sub>2,s\<^sub>2\<rangle>"   and "binop (bop, v\<^sub>1, v\<^sub>2) = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>2\<rangle>"
simps_5: fixes i :: "'lbl strand_label"   and t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term"   and A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and D :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" shows "tr'\<^sub>p\<^sub>c ((i, insert\<langle>t,s\<rangle>) # A) D = tr'\<^sub>p\<^sub>c A (List.insert (i, t, s) D)"
simps: fixes a :: "nat"   and b :: "nat" shows "balanced1_arith a b = (10 * (b + 1) \<le> 25 * (a + 1))"
elims_2: fixes G :: "('a, 'b) graph"   and x :: "'a"   and xa :: "('a \<times> 'b \<times> 'a) list"   and xb :: "'a" assumes "valid_unMultigraph G"   and "valid_unMultigraph.is_trail G x xa xb" obtains v :: "'a" and v' :: "'a" where "x = va__" and "xa = []" and "xb = v'__" and "va__ = v'__ \<and> v'__ \<in> nodes G"   | v :: "'a" and v1 :: "'a" and w :: "'b" and v2 :: "'a" and ps :: "('a \<times> 'b \<times> 'a) list" and v' :: "'a" where "x = v" and "xa = (v1, w, v2) # ps" and "xb = v'" and "v = v1 \<and> (v1, w, v2) \<in> edges G \<and> (v1, w, v2) \<notin> set ps \<and> (v2, w, v1) \<notin> set ps \<and> valid_unMultigraph.is_trail G v2 ps v'"
AE_iff_emeasure_eq_1: fixes P :: "'a cfg stream \<Rightarrow> bool"   and s :: "'a cfg" assumes "MDP.MC.T.random_variable s (count_space UNIV) P" shows "almost_everywhere (MDP.MC.T s) P = (emeasure (MDP.MC.T s) {x \<in> space (MDP.MC.T s). P x} = 1)"
similar_mat_witD2_3: fixes A :: "'c mat"   and n :: "nat"   and m :: "nat"   and B :: "'c mat"   and P :: "'c mat"   and Q :: "'c mat" assumes "A \<in> carrier_mat n m"   and "similar_mat_wit A B P Q" shows "A = P * B * Q"
Inf_from_q_def: fixes q :: "'q" shows "Inf_from_q q = inference_system.Inf_from (Inf_q q)"
igSwapAbsIPresIGWlsAbs_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSwapAbsIPresIGWlsAbs MOD \<equiv> \<forall>zs z1 z2 us s A. isInBar (us, s) \<and> igWlsAbs MOD (us, s) A \<longrightarrow> igWlsAbs MOD (us, s) (igSwapAbs MOD zs z1 z2 A)"
nn_integral_spmf: fixes A :: "'c set"   and p :: "'c spmf" shows "(\<Sum>\<^sup>+ x\<in>A. ennreal (spmf p x)) = emeasure (measure_spmf p) A"
ai_def: shows "ai = f2 / (f2 - f2'')"
map_simp_ide: fixes a :: "'a" assumes "Go\<tau>.A.ide a" shows "Go\<tau>.map a = B (G a) (\<tau> a)"
lt_max_keys: fixes v :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'd" assumes "v \<in> keys p" shows "v \<preceq>\<^sub>t lt p"
poly_0_coeff_0: fixes p :: "'b poly" shows "poly p (0::'b) = coeff p 0"
nipdj: shows "ipd (\<pi> j) \<noteq> \<pi> m"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "'a" assumes "\<mu> x xa = y" obtains i :: "nat" and j :: "nat" where "x = i" and "xa = j" and "y = (if j < i then fs ! i \<bullet> gso j / \<parallel>gso j\<parallel>\<^sup>2 else if i = j then 1::'a else (0::'a))"
sigma_sets_subset: fixes a :: "'b set set"   and M :: "'b measure" assumes "a \<subseteq> sets M" shows "sigma_sets (space M) a \<subseteq> sets M"
raa_cor_1: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> in v]"   and "[\<^bold>\<not>\<psi> in v] \<Longrightarrow> [\<^bold>\<not>\<phi> in v]"   and "[\<phi> in v]" shows "[\<psi> in v]"
chain_iterates_above: fixes a :: "'a" assumes "a \<in> Field leq" shows "iterates_above a \<in> Chains leq"
bdd_above_mono: fixes B :: "'b multiset set"   and A :: "'b multiset set" assumes "subset_mset.bdd_above B"   and "A \<subseteq> B" shows "subset_mset.bdd_above A"
flowsto_trans: fixes A :: "('e \<times> 'f \<Rightarrow>\<^sub>L 'e) set"   and S :: "real set"   and B :: "('e \<times> 'f \<Rightarrow>\<^sub>L 'e) set"   and C :: "('e \<times> 'f \<Rightarrow>\<^sub>L 'e) set"   and T :: "real set"   and D :: "('e \<times> 'f \<Rightarrow>\<^sub>L 'e) set"   and E :: "('e \<times> 'f \<Rightarrow>\<^sub>L 'e) set" assumes "flowsto A S B C"   and "flowsto C T D E" shows "flowsto A {s + t |s t. s \<in> S \<and> t \<in> T} (B \<union> D \<union> C) E"
transp_le: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.preorder less_eq less" shows "transp less_eq"
checkVarValue_bounded: fixes val :: "integer" shows "checkVarValue type val \<in> {min_var_value..max_var_value}"
dependent_mono: fixes B :: "'b set"   and A :: "'b set" assumes "dependent B"   and "B \<subseteq> A" shows "dependent A"
is_OK_returns_heap_I: fixes h :: "'ap"   and f :: "('ap, 'aq, 'ar) prog"   and h' :: "'ap" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'" shows "h \<turnstile> ok f"
exprTypingLow: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and e :: "expr"   and x :: "char list" assumes "\<Gamma> \<turnstile> e : Low"   and "x \<in> fv e" shows "\<Gamma> x = Some Low"
mat_empty_row_length: fixes nr :: "nat"   and nc :: "nat"   and M :: "'a list list" assumes "mat nr nc M"   and "M = []" shows "mat 0 nc M"
strict_implies_order: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry" assumes "b \<prec> a" shows "b \<preceq> a"
cone_carrier: fixes A :: "'a vec set" assumes "A \<subseteq> carrier_vec n" shows "cone A \<subseteq> carrier_vec n"
terminal_TNil: fixes x1 :: "'a" shows "terminal (TNil x1) = x1"
dom_id: fixes a :: "((int \<times> int) set set, int) Combinators"   and p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "noDenyAll (a # p)"   and "separated (a # p)"   and "p \<noteq> []"   and "x \<notin> dom (C (list2FWpolicy p))"   and "x \<in> dom (C a)" shows "x \<notin> dom (C (list2FWpolicy (insertDenies p)))"
simps: fixes cms :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list"   and mem :: "'Var\<^sub>C \<Rightarrow> 'Val" shows "conc.sound_mode_use (cms, mem) = (list_all (\<lambda>cm. conc.locally_sound_mode_use (cm, mem)) cms \<and> conc.globally_sound_mode_use (cms, mem))"
Bonk_Schramm_extension_unfolded_wo_props_4: shows "\<forall>a b. (b, middle a b) \<in> Bonk_Schramm_extension_unfolded_wo"
set_Rep_Abs: shows "A = range (Rep \<circ> Abs)"
unbounded_line_def: fixes a :: "'b"   and b :: "'b" shows "unbounded_line a b = {x. \<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b}"
has_field_def: fixes P :: "'a prog"   and C :: "String.literal"   and F :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and D :: "String.literal" shows "P \<turnstile> C has F:T (fm) in D \<equiv> \<exists>FDTs. P \<turnstile> C has_fields FDTs \<and> map_of FDTs (F, D) = \<lfloor>(T, fm)\<rfloor>"
vsv_ex1: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (xs #\<^sub>\<circ> x)" shows "\<exists>!b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> xs #\<^sub>\<circ> x"
K'_def: fixes K :: "('c \<times> ('d \<times> real) list) list"   and S :: "'d list"   and s :: "'c" assumes "Closed_Kernel_From K S" shows "Closed_Kernel_From.K' K S s \<equiv> case map_of (map (\<lambda>(s, \<mu>). (s, pmf_of_alist \<mu>)) K) s of None \<Rightarrow> return_pmf (hd S) | Some s \<Rightarrow> s"
extract_Nil_code: fixes P :: "'c \<Rightarrow> bool" shows "List.extract P [] = None"
tail_algebra_0: fixes M :: "'b measure" assumes "shift_operator_ergodic M" shows "shift_operator_ergodic.tail_algebra M 0 = Pi\<^sub>M UNIV (\<lambda>_. M)"
gt_tpoly_imp_ge: fixes q :: "('v pvar, hmultiset) tpoly"   and p :: "('v pvar, hmultiset) tpoly" assumes "q >\<^sub>p p" shows "q \<ge>\<^sub>p p"
maintain_D_invar: fixes u :: "'v" assumes "u \<notin> S"   and "D u < \<infinity>"   and "\<forall>v. v \<notin> S \<longrightarrow> D u \<le> D v" shows "D_invar (relax_outgoing u D) (Set.insert u S)"
rsr3: fixes x :: "('a \<times> 'a) set" shows "(rel_ad (rel_ad x))\<^sup>= = Id"
Inter_lower: fixes B :: "'e set"   and A :: "'e set set" assumes "B \<in> A" shows "\<Inter> A \<subseteq> B"
has_horizontal_homs: shows "horizontal_homs (\<bullet>) src trg"
A2: shows "\<forall>(P, Q)\<in>RelA. CondA (P, Q)"
ab: shows "(a, b) \<in> (set TI)\<^sup>+"
is_aligned_mult_triv2: fixes x :: "'b word"   and n :: "nat" shows "is_aligned (x * 2 ^ n) n"
cmp_imp_equal: fixes x :: "'a"   and y :: "'a" assumes "x \<in> D2"   and "y \<in> D2"   and "cmp2 x y = EQUAL" shows "x = y"
c_manifolds_order_le: fixes l :: "enat" assumes "l \<le> l" shows "c_manifolds l charts1 charts1"
mod_diff_mult_self1: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k * m \<le> n" shows "(n - k * m) mod m = n mod m"
a: fixes n :: "'node" shows "phiDefNodes_aux g v (filter (\<lambda>n. (n, v) \<notin> dom (Mapping.lookup phis)) (\<alpha>n g)) n \<subseteq> set (\<alpha>n g)"
assms_2: shows "g \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"
Par_Wbis_discrR1: fixes c :: "('test, 'atom) com"   and d1 :: "('test, 'atom) com"   and d2 :: "('test, 'atom) com" assumes "c \<approx>w d1"   and "discr d2" shows "c \<approx>w d1 | d2"
non_volatile_owned_or_read_only_append: fixes pending_write :: "bool"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and xs :: "'b memref list"   and ys :: "'b memref list" shows "non_volatile_owned_or_read_only pending_write \<S> \<O> (xs @ ys) = (non_volatile_owned_or_read_only pending_write \<S> \<O> xs \<and> non_volatile_owned_or_read_only (pending_write \<or> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b xs \<noteq> {}) (share xs \<S>) (acquired True xs \<O>) ys)"
eq_imp_rtranclp: fixes x :: "'a"   and y :: "'a"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "x = y" shows "r\<^sup>*\<^sup>* x y"
assms: shows "\<forall>\<^sub>F x in F. f x \<noteq> (0::'b)"
gbinomial_binomial: fixes n :: "nat"   and k :: "nat" shows "n gchoose k = n choose k"
eventually_all_iff: fixes P :: "nat \<Rightarrow> 'b \<Rightarrow> bool" shows "(\<forall>\<^sub>F x in \<U>. \<forall>y. P x y) = (\<forall>Y. \<forall>\<^sub>F x in \<U>. P x (Y x))"
assms: shows "\<eta>: ValF\<cdot>f \<mapsto> unitK\<cdot>(ValKF\<cdot>f')"
bigomega_iff_bigo: fixes g :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" shows "(g \<in> \<Omega>[F](f)) = (f \<in> O[F](g))"
app_subst_id_type_scheme: shows "$ id_subst = (\<lambda>sch. sch)"
inject_16: fixes x17 :: "nat"   and y17 :: "nat" shows "(Var x17 = Var y17) = (x17 = y17)"
simps: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and X :: "'c set"   and gpv :: "('a, 'b, 'c) gpv" shows "pred_gpv' P Q X gpv = ((\<forall>x. x \<in> results_gpv (\<I>_uniform UNIV X) gpv \<longrightarrow> P x) \<and> (\<forall>x. x \<in> outs_gpv (\<I>_uniform UNIV X) gpv \<longrightarrow> Q x))"
execn_strip: shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"
polynomialI: fixes p :: "'a list list"   and K :: "'a list set" assumes "set p \<subseteq> K"   and "lead_coeff p \<noteq> \<zero>\<^bsub>K [X]\<^esub>" shows "polynomial\<^bsub>K [X]\<^esub> K p"
get_child_nodes_def: shows "get_child_nodes \<equiv> l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_child_nodes"
length_nearest_neighbors: fixes n :: "nat"   and p :: "(real, 'b) vec"   and kdt :: "'b kdt" shows "length (nearest_neighbors n p kdt) = min n (KD_Tree.size_kdt kdt)"
spmf_mk_lossless: fixes p :: "'c spmf"   and x :: "'c" shows "spmf (mk_lossless p) x = spmf p x / weight_spmf p"