hcju/isabelleps huggingface dataset:
iwlsFSb_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) Iteration.model_scheme" shows "iwlsFSb MOD \<equiv> igWlsAllDisj MOD \<and> igWlsAbsIsInBar MOD \<and> igConsIPresIGWls MOD \<and> igSubstAllIPresIGWlsAll MOD \<and> igFreshCls MOD \<and> igSubstCls MOD \<and> igAbsRen MOD"
convex_halfspace_lt: fixes a :: "'b"   and b :: "real" shows "convex {x. a \<bullet> x < b}"
assms: shows "i \<in> set (UnsatIndices (preprocess' s start))"
igSwapAllIPresIGWlsAllSTR_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igSwapAllIPresIGWlsAllSTR MOD \<equiv> igSwapIPresIGWlsSTR MOD \<and> igSwapAbsIPresIGWlsAbsSTR MOD"
not_iIN_iMOD_subset: fixes d' :: "nat"   and m :: "nat"   and n' :: "nat"   and r :: "nat" assumes "0 < d'"   and "m \<noteq> Suc 0" shows "\<not> [n'\<dots>,d'] \<subseteq> [ r, mod m ]"
make_affine_infinity: fixes x :: "'b"   and y :: "'b" shows "make_affine (x, y, 0::'b) = Infinity"
\<Delta>\<Phi>_del_min: fixes h :: "'b hp" assumes "hps h \<noteq> []" shows "\<Phi> (del_min (Some h)) - \<Phi> (Some h) \<le> 3 * log 2 (real (size_hps (hps h))) - real (length (hps h)) + 2"
sibling_swapSyms_sibling: fixes t :: "'a tree"   and b :: "'a"   and a :: "'a" assumes "consistent t"   and "sibling t b \<noteq> b"   and "a \<noteq> b" shows "sibling (swapSyms t a (sibling t b)) a = b"
tauPres: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "(P, Q) \<in> Rel" shows "\<tau>.P \<leadsto>[Rel] \<tau>.Q"
rep_eq: fixes x :: "real_alg" shows "sgn_real_alg_rat x = sgn_3 (rep_real_alg x)"
reduced_GB_D2: fixes G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "punit.is_reduced_GB G" shows "punit.is_auto_reduced G"
pointfree_idE: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "f \<circ> g = id" shows "f (g x) = x"
linear_continuous_at: fixes f :: "'d \<Rightarrow> 'e"   and a :: "'d" assumes "bounded_linear f" shows "isCont f a"
sets_PiM_equal_borel: shows "sets (Pi\<^sub>M UNIV (\<lambda>i. borel)) = sets borel"
measure_spmf_def: fixes p :: "'a spmf" shows "measure_spmf p = distr (restrict_space (measure_pmf p) (range Some)) (count_space UNIV) the"
a: shows "(red F)\<^sup>*\<^sup>* (p - q) r"
Invariants_cond_exp_is_integral_fmpt: fixes f :: "(nat \<Rightarrow> real) \<Rightarrow> real" assumes "integrable M' f" shows "AE x in M'. real_cond_exp M' T.Invariants f x = M'.expectation f / M'.prob (space M')"
unfold: fixes s :: "'a"   and h :: "'a \<rightarrow> ('b, 'a) Step" assumes "s \<noteq> \<bottom>" shows "unfold\<cdot>h\<cdot>s = unfold2\<cdot>(unfold\<cdot>h)\<cdot>(h\<cdot>s)"
const: fixes R :: "('a \<Rightarrow> real) set"   and S :: "'a set"   and c :: "real" assumes "function_ring_on R S" shows "(\<lambda>_. c) \<in> R"
range_inter_ts_subseteq: fixes A :: "'a \<Rightarrow> 'b set"   and B :: "'a \<Rightarrow> 'b set" assumes "\<forall>k. A k \<subseteq> B k" shows "\<Rightarrow>\<Inter>A \<subseteq> \<Rightarrow>\<Inter>B"
ev': shows "\<forall>\<^sub>F x in at_right a. ((\<lambda>x. - g x) has_real_derivative - g' x) (at x)"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "nat" assumes "ack x xa = y" obtains n :: "nat" where "x = 0" and "xa = n__" and "y = Suc n__"   | m :: "nat" where "x = Suc ma__" and "xa = 0" and "y = ack ma__ 1"   | m :: "nat" and n :: "nat" where "x = Suc m" and "xa = Suc n" and "y = ack m (ack (Suc m) n)"
Node_def: shows "Node = {p. \<exists>f x k. p = (f, x) \<and> f k = Inr 0}"
SucFbd_ASucFbd: fixes r1 :: "('d \<times> 'd) set"   and r2 :: "('e \<times> 'e) set"   and p1 :: "('f \<times> 'f) set" shows "cardSuc (r1 +c r2) <o (p1 +c ctwo) ^c cardSuc (r1 +c r2)"
dvd_productE: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times p (times a b)" obtains x :: "'b" and y :: "'b" where "p = times x y" and "dvd.dvd times x a" and "dvd.dvd times y b"
br_run_conv: fixes \<alpha> :: "('c, 'ls, 'gs) pid_global_config \<Rightarrow> ('c, 'ls, 'gs) global_config"   and I :: "('c, 'ls, 'gs) pid_global_config \<Rightarrow> bool"   and rb :: "nat \<Rightarrow> ('c, 'ls, 'gs) global_config" assumes "br pidgc_\<alpha> (\<lambda>_. True) = br \<alpha> I" shows "sa.is_run rb = (\<exists>ra. rb = \<alpha> \<circ> ra \<and> pid.is_run ra)"
lift_subst: fixes j :: "nat"   and i :: "nat"   and t :: "dB"   and s :: "dB" assumes "j < i + 1" shows "lift (t[s/j]) i = lift t (i + 1)[lift s i/j]"
drop_take: fixes n :: "nat"   and m :: "nat"   and xs :: "'c list" shows "drop n (take m xs) = take (m - n) (drop n xs)"
swap_involutive2: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and y :: "'var"   and x :: "'var" assumes "good X" shows "X #[x \<and> y]_zs #[y \<and> x]_zs = X"
assms_4: shows "\<guillemotleft>f : local.prod a b \<rightarrow> c\<guillemotright>"
min_maxsimpchain_pmaxsimpchain: fixes xs :: "'a set list" assumes "min_maxsimpchain xs" shows "pmaxsimpchain xs"
FunctorMapsTo: fixes F :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) Functor_scheme"   and f :: "'i" assumes "Functor F"   and "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "F ## f maps\<^bsub>CatCod F\<^esub> F @@ dom\<^bsub>CatDom F\<^esub> f to F @@ cod\<^bsub>CatDom F\<^esub> f"
arr_char': fixes f :: "(unit, unit) concrete_category.arr" shows "arr f = (f \<in> MkIde ` {})"
config'_distinct: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and S :: "'a list \<times> 'b"   and qs :: "'a list" shows "distinct (fst (Partial_Cost_Model.config' A S qs)) = distinct (fst S)"
random_bst_empty: shows "random_bst {} = return_pmf \<langle>\<rangle>"
FSwImorph_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'g"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'h"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'g, 'h, 'i) model_scheme" shows "FSwImorph h hA termMOD MOD = termFSwImorph h hA MOD"
changes_R_smods_0_1: fixes q :: "real poly" shows "changes_R_smods 0 q = 0"
hyps_6: fixes b :: "(name \<times> ty) list"   and T :: "ty" assumes "b \<turnstile> e\<^sub>1 : T" shows "b \<turnstile> Lam [x].e : T"
Not_Rex: fixes F :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "\<turnstile> \<not> (\<exists>x. F x) = (\<forall>x. \<not> F x)"
var_rename_wt'_2: fixes \<delta> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and s :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and m :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and n :: "nat" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"   and "s = m \<cdot> \<delta>" shows "s = m \<cdot> var_rename n \<cdot> var_rename_inv n \<circ>\<^sub>s \<delta>"
lift_character: fixes \<chi> :: "'a \<Rightarrow> complex"   and z :: "complex" assumes "character (G\<lparr>carrier := {\<one>}\<rparr>) \<chi>"   and "z ^ subgroup_indicator G {\<one>} x = \<chi> (x [^] subgroup_indicator G {\<one>} x)" shows "character (G\<lparr>carrier := adjoin G {\<one>} x\<rparr>) (H.lift_character (\<chi>, z))"
SGs: fixes P :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" assumes "List.member SG P" shows "seq_sem I P = UNIV"
strict_mono_enum: fixes N :: "nat set" assumes "infinite N" shows "strict_mono (enum N)"
finite_derangements: fixes A :: "nat set" assumes "finite A" shows "finite (derangements A)"
FORWEAK_mono_rule: fixes S :: "'c set"   and d :: "'d nres"   and P :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd nres"   and I :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> 'd nres" assumes "S = {} \<Longrightarrow> d \<le> SPEC P"   and "\<And>s. s \<in> S \<Longrightarrow> f s \<le> SPEC (I {s})"   and "\<And>it it' \<sigma>. \<lbrakk>I it \<sigma>; it' \<subseteq> it; it \<subseteq> S\<rbrakk> \<Longrightarrow> I it' \<sigma>"   and "\<And>x it \<sigma>. \<lbrakk>x \<in> S; it \<subseteq> S; I it \<sigma>\<rbrakk> \<Longrightarrow> f x \<le> SPEC (\<lambda>f'. c \<sigma> f' \<le> SPEC (I (insert x it)))"   and "\<And>\<sigma>. I S \<sigma> \<Longrightarrow> P \<sigma>" shows "FORWEAK S d f c \<le> SPEC P"
step_sb: shows "sbh_computation.concurrent_step (ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) (ts\<^sub>s\<^sub>b', m\<^sub>s\<^sub>b', \<S>\<^sub>s\<^sub>b')"
upset_setp_downset_setp_dual: shows "downclosed_set \<circ> (`) \<partial> = upclosed_set"
assms_4: shows "Q\<^sub>2 \<inter> (I\<^sub>1 \<union> I\<^sub>2) \<noteq> {}"
open_Collect_disj: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "open {x. P x}"   and "open {x. Q x}" shows "open {x. P x \<or> Q x}"
Nml_Src: fixes t :: "'c local.term" assumes "Nml t" shows "Nml (Src t)"
h5: shows "tp \<cdot> tr \<cdot> x \<le> tr \<cdot> x \<cdot> tq"
right_unit: fixes a :: "'a set" assumes "a \<in> Partition" shows "a [\<cdot>] Class \<one> = a"
Invariants_vrestr: fixes A :: "'a set" assumes "A \<in> sets Invariants" shows "T --` A = A"
bij_betw_inv_into_LEFT: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and A' :: "'b set"   and B :: "'a set" assumes "bij_betw f A A'"   and "B \<subseteq> A" shows "inv_into A f ` f ` B = B"
fds_subseries_add: fixes P :: "nat \<Rightarrow> bool"   and f :: "'b fds"   and g :: "'b fds" shows "fds_subseries P (f + g) = fds_subseries P f + fds_subseries P g"
subcls: shows "(C', D) \<in> (subcls1 P)\<^sup>+"
fres_galois: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd"   and z :: "'c \<times> 'd" shows "less_eq_prod (times_prod' x y) z = less_eq_prod x (prod.fres z y)"
wlp_and_term_imp_wp: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "local.wlp c Q s \<and> \<pi>: (c, s) \<Rightarrow> t" shows "local.wp c Q s"
M3: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and p :: "'a" assumes "class.interval meets \<I>"   and "\<I> p" shows "\<exists>q r. meets q p \<and> meets p r"
setD: fixes s1 :: "'s1"   and s2 :: "'s2"   and x :: "'a"   and y :: "'b"   and s1' :: "'s1"   and y' :: "'b"   and s2' :: "'s2" assumes "X s1 s2"   and "x \<in> outs_\<I> \<I>"   and "((y, s1'), y', s2') \<in> set_spmf (joint_oracle s1 s2 x)" shows "(y, s1') \<in> set_spmf (oracle1 s1 x) \<and> (y', s2') \<in> set_spmf (oracle2 s2 x)"
simps_1: fixes A :: "('a list \<Rightarrow> 'b pmf) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> ((nat \<times> nat list) \<times> 'b) pmf)"   and s :: "('a list \<times> 'b) pmf" shows "Partial_Cost_Model.config'_rand A s [] = s"
linear_poly_sum: fixes p :: "linear_poly"   and v :: "nat \<Rightarrow> 'b" shows "p \<lbrace> v \<rbrace> = (\<Sum>x\<in>vars p. coeff p x *R v x)"
finiteDomainImpliesFinite: fixes f :: "('c \<times> 'd) set" assumes "runiq f" shows "finite (Domain f) = finite f"
prems_2: fixes x :: "nat" assumes "x < card r" shows "x \<in> r"
listrel1p_def: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'g list" shows "listrel1p r xs ys = ((xs, ys) \<in> listrel1 {(x, y). r x y})"
A\<^sub>i: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "SkolemizedAxiomsSetMcL" shows "x \<cdot> (y \<cdot> z) \<cong> (x \<cdot> y) \<cdot> z"
no_outstanding_write_to_read_only_memory_nth_update: fixes i :: "nat"   and sb :: "'a memref list"   and p :: "'a"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "no_write_to_read_only_memory \<S>\<^sub>s\<^sub>b sb" shows "no_outstanding_write_to_read_only_memory \<S>\<^sub>s\<^sub>b (ts\<^sub>s\<^sub>b[i := (p, is, xs, sb, \<D>, \<O>, \<R>)])"
IHAPP_3: shows "{#[entry fg p]. p \<in># P#} \<subseteq># c'"
is_lub_thelub_ex: fixes S :: "'b set"   and x :: "'b" assumes "\<exists>u. S <<| u"   and "S <| x" shows "lub S \<sqsubseteq> x"
Mid_perm: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A B C" shows "Tarski_neutral_dimensionless.Midpoint Bet Cong A B C \<and> Tarski_neutral_dimensionless.Midpoint Bet Cong A C B"
crel_vs1_execute_None: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e"   and b :: "'f Heap"   and heap :: "heap" assumes "crel_vs1 R a b"   and "execute b heap = None"   and "P heap" shows "False"
section_map_def: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "section_map X Y f \<equiv> \<exists>g. retraction_maps Y X g f"
concat_eq_append_conv: fixes xss :: "'b list list"   and ys :: "'b list"   and zs :: "'b list" shows "(concat xss = ys \<cdot> zs) = (if xss = \<epsilon> then ys = \<epsilon> \<and> zs = \<epsilon> else \<exists>xss1 xs xs' xss2. xss = xss1 \<cdot> (xs \<cdot> xs') # xss2 \<and> ys = concat xss1 \<cdot> xs \<and> zs = xs' \<cdot> concat xss2)"
ok_ok: fixes e :: "'b environment" assumes "ok e" shows "\<exists>x. e = Env x"
map_fun_def: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'd" shows "(f ---> g) h = g \<circ> h \<circ> f"
scc_root_disc_le: fixes v :: "'a"   and scc :: "'a set"   and x :: "'a" assumes "scc_root s v scc"   and "x \<in> scc"   and "x \<in> dom (discovered s)" shows "\<delta> s v \<le> \<delta> s x"
unfold: fixes x :: "rat list" assumes "x \<in> set (take_indices B (find_nonzeros_from_input_vec C))" shows "x \<in> A"
a_Hcarr: fixes h :: "'a" assumes "h \<in> a_kernel R S h" shows "h \<in> carrier R"
unfold_congs_2: fixes r :: "('d, 'e, 'f) automaton_scheme"   and r' :: "('d, 'e, 'f) automaton_scheme"   and v' :: "('d \<times> 'e \<times> 'd) set"   and f :: "('d \<times> 'e \<times> 'd) set \<Rightarrow> ('d \<times> 'e \<times> 'd) set"   and f' :: "('d \<times> 'e \<times> 'd) set \<Rightarrow> ('d \<times> 'e \<times> 'd) set" assumes "r = r'"   and "automaton.trans r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "trans_update f r = trans_update f' r'"
typeof_lit_typeof: fixes v :: "val"   and T :: "ty"   and h :: "heap" assumes "typeof v = \<lfloor>T\<rfloor>" shows "typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>"
vdunion_vdoubleton: fixes a :: "V"   and b :: "V"   and A :: "V \<Rightarrow> V" shows "VSigma (set {a, b}) A = set {a} \<times>\<^sub>\<circ> A a \<union>\<^sub>\<circ> set {b} \<times>\<^sub>\<circ> A b"
carrier_eq: shows "alts = {x. Pareto (prefs_from_table xs) x x}"
GFP_lemma3: fixes f :: "'a \<Rightarrow> 'a" assumes "Mono f"   and "f \<in> carrier L \<rightarrow> carrier L" shows "f (GFP f) \<sqsubseteq> GFP f"
coeff_Q': fixes k :: "nat" shows "poly.coeff Q' k = (if n < k then 0 else (- 1) ^ (n - k) * e' (n - k))"
prj_simps_8: shows "C.cod f_src_src.Prj\<^sub>0\<^sub>1 = src.apex"
prj_chine_assoc_2: shows "\<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>1\<^sub>0 \<cdot> \<mu>_\<nu>\<pi>_\<rho>.chine_assoc = \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0\<^sub>1"
simps_5: fixes t1 :: "PrimScheme"   and t2 :: "PrimScheme" shows "sch_to_pr (Comp_op t1 t2) = (\<lambda>x. sch_to_pr t1 (sch_to_pr t2 x))"
prodrg_tOp: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) Ring_scheme" assumes "\<forall>k\<in>I. Ring (A k)" shows "(\<cdot>\<^sub>r\<^bsub>r\<Pi>\<^bsub>I\<^esub> A\<^esub>) = prod_tOp I A"
Confidentiality_Tgs: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"   and "Key authK \<notin> analz (knows Spy evs)"   and "\<not> expiredSK Ts evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbIV_gets" shows "Key servK \<notin> analz (knows Spy evs)"
finite_dom_lookup: fixes t :: "('a, 'b) RBT.rbt" shows "finite (dom (RBT.lookup t))"
rbt_less_simps_2: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a"   and c :: "color"   and lt :: "('a, 'b) rbt"   and kt :: "'a"   and v :: "'b"   and rt :: "('a, 'b) rbt" shows "ord.rbt_less less k (Branch c lt kt v rt) = (less kt k \<and> ord.rbt_less less k lt \<and> ord.rbt_less less k rt)"
star_denest_var: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "times_prod' (prod.qstar x) (prod.qstar (times_prod' y (prod.qstar x))) = prod.qstar (sup_prod x y)"
mult_1_left: fixes a :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "Var \<circ>\<^sub>s a = a"
diff_delta_arctan_upper_11_def: shows "diff_delta_arctan_upper_11 \<equiv> \<lambda>x. 1 / (x\<^sup>2 * (1 + x\<^sup>2))"
simps_2: fixes E :: "'b uprod set"   and u :: "'b"   and x :: "'b uprod"   and xs :: "'b uprod list"   and v :: "'b" shows "epath E u (x # xs) v = ((\<exists>w. u \<noteq> w \<and> Upair u w = x \<and> epath E w xs v) \<and> x \<in> E)"
satPB_Un: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and \<Phi>1 :: "(('fsym, 'tp) efsym, 'psym) lit list set"   and \<Phi>2 :: "(('fsym, 'tp) efsym, 'psym) lit list set" shows "TE.satPB \<xi> (\<Phi>1 \<union> \<Phi>2) = (TE.satPB \<xi> \<Phi>1 \<and> TE.satPB \<xi> \<Phi>2)"
extension_lemma_submanifoldE: fixes f :: "'a \<Rightarrow> 'c"   and B :: "'a set" assumes "diff_fun k (charts_submanifold S) f"   and "closed B"   and "B \<subseteq> sub.carrier" obtains f' :: "'a \<Rightarrow> 'c" where "diff_fun k charts f'" and "\<And>x. x \<in> B \<Longrightarrow> f' x = f x" and "csupport_on carrier f' \<inter> carrier \<subseteq> sub.carrier"
A: shows "sinks_aux I D (U \<union> V) xs = sinks_aux I D U xs \<union> sinks_aux I D V (ipurge_tr_aux I D U xs)"
Int_insert_left: fixes a :: "'d"   and B :: "'d set"   and C :: "'d set" shows "insert a B \<inter> C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
add_edge_was_path: fixes a :: "'v"   and b :: "'w"   and c :: "'v"   and x :: "'v"   and p :: "('v \<times> 'w \<times> 'v) list"   and y :: "'v" assumes "is_path_undir (add_edge a b c G) x p y"   and "(a, b, c) \<notin> set p"   and "(c, b, a) \<notin> set p"   and "a \<in> V"   and "c \<in> V" shows "is_path_undir G x p y"
11: shows "Bex (sset (smap f (gtrace (sdrop n (fromN 1 ||| r)) (gtarget (stake n (fromN 1 ||| r)) (0, p))))) even"
semiring_simprules_6: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "semiring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"
is_Groebner_basis_singleton: fixes f :: "'t \<Rightarrow>\<^sub>0 'c" shows "is_Groebner_basis {f}"
WtFT_def: fixes cl :: "('g, 'h, 'i) cmd list" shows "WtFT cl \<equiv> \<Sum>n\<in>theFT cl. 1 / real (length cl)"
surjective: fixes r :: "'a point_scheme" shows "r = \<lparr>cx = cx r, cy = cy r, \<dots> = point.more r\<rparr>"
tm_poly_def: fixes taylor_model :: "taylor_model" shows "tm_poly taylor_model = (case taylor_model of TaylorModel x1 x2 \<Rightarrow> x1)"
mult_imp_less_multiset_ext\<^sub>D\<^sub>M: fixes N :: "'b multiset"   and A :: "'b set"   and M :: "'b multiset"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "set_mset N \<subseteq> A"   and "set_mset M \<subseteq> A"   and "\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. R x y \<longrightarrow> R y z \<longrightarrow> R x z"   and "(M, N) \<in> mult {(x, y). x \<in> A \<and> y \<in> A \<and> R x y}" shows "less_multiset_ext\<^sub>D\<^sub>M R M N"
disc_1: fixes x1 :: "'g" shows "isl (Inl x1)"
try_spmf_lossless: fixes p :: "'h spmf"   and q :: "'h spmf" assumes "lossless_spmf p" shows "TRY p ELSE q = p"
subnodes_ordered: fixes n :: "ref"   and t :: "dag"   and p :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and var :: "ref \<Rightarrow> nat" assumes "n \<in> set_of t"   and "Dag p low high t"   and "ordered t var" shows "var n \<le> var p"
cases: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and a1 :: "'c rule list"   and a2 :: "state"   and a3 :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>a1, a2\<rangle> \<Rightarrow>\<^sub>s a3" obtains t :: "state" where "a1 = []" and "a2 = ta__" and "a3 = ta__"   | m :: "'c match_expr" and rs :: "'c rule list" where "a1 = Rule m__ Accept # rs__" and "a2 = Undecided" and "a3 = Decision FinalAllow" and "matches \<gamma> m__ p"   | m :: "'c match_expr" and rs :: "'c rule list" where "a1 = Rule ma__ Drop # rsa__" and "a2 = Undecided" and "a3 = Decision FinalDeny" and "matches \<gamma> ma__ p"   | m :: "'c match_expr" and rs :: "'c rule list" where "a1 = Rule mb__ Reject # rsb__" and "a2 = Undecided" and "a3 = Decision FinalDeny" and "matches \<gamma> mb__ p"   | m :: "'c match_expr" and rs :: "'c rule list" and t :: "state" where "a1 = Rule mc__ Log # rsc__" and "a2 = Undecided" and "a3 = tb__" and "matches \<gamma> mc__ p" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rsc__, Undecided\<rangle> \<Rightarrow>\<^sub>s tb__"   | m :: "'c match_expr" and rs :: "'c rule list" and t :: "state" where "a1 = Rule md__ Empty # rsd__" and "a2 = Undecided" and "a3 = tc__" and "matches \<gamma> md__ p" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rsd__, Undecided\<rangle> \<Rightarrow>\<^sub>s tc__"   | m :: "'c match_expr" and rs :: "'c rule list" and t :: "state" and a :: "action" where "a1 = Rule me__ a # rse__" and "a2 = Undecided" and "a3 = td__" and "\<not> matches \<gamma> me__ p" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rse__, Undecided\<rangle> \<Rightarrow>\<^sub>s td__"   | m :: "'c match_expr" and chain :: "char list" and rs\<^sub>1 :: "'c rule list" and m' :: "'c match_expr" and rs\<^sub>2 :: "'c rule list" and rrs :: "'c rule list" and t :: "state" where "a1 = Rule mf__ (Call chain__) # rrs__" and "a2 = Undecided" and "a3 = te__" and "matches \<gamma> mf__ p" and "\<Gamma> chain__ = Some (rs\<^sub>1 @ Rule m' Return # rs\<^sub>2)" and "matches \<gamma> m' p" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow>\<^sub>s Undecided" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rrs__, Undecided\<rangle> \<Rightarrow>\<^sub>s te__"   | m :: "'c match_expr" and chain :: "char list" and rs :: "'c rule list" and X :: "final_decision" and rrs :: "'c rule list" where "a1 = Rule mg__ (Call chaina__) # rrsa__" and "a2 = Undecided" and "a3 = Decision X" and "matches \<gamma> mg__ p" and "\<Gamma> chaina__ = Some rsf__" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rsf__, Undecided\<rangle> \<Rightarrow>\<^sub>s Decision X"   | m :: "'c match_expr" and chain :: "char list" and rs :: "'c rule list" and rrs :: "'c rule list" and t :: "state" where "a1 = Rule m (Call chain) # rrs" and "a2 = Undecided" and "a3 = t" and "matches \<gamma> m p" and "\<Gamma> chain = Some rs" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>s Undecided" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rrs, Undecided\<rangle> \<Rightarrow>\<^sub>s t"
Int_absorb1: fixes B :: "'d set"   and A :: "'d set" assumes "B \<subseteq> A" shows "A \<inter> B = B"
sel_rews_3: fixes app :: "'e\<cdot>'f error\<cdot>'g" shows "runErrorT\<cdot>(ErrorT\<cdot>app) = app"
crel_vs_bind_eq: fixes v :: "'a"   and s :: "['a]"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and sf :: "'a \<Rightarrow> ['c]" assumes "crel_vs (=) v s"   and "crel_vs R (f v) (sf v)" shows "crel_vs R (f v) (s \<bind> sf)"
rel_frontier_def: fixes S :: "'a set" shows "rel_frontier S = closure S - rel_interior S"