hcju/isabelleps huggingface dataset:
le_def: fixes p :: "tm" shows "le p = (case p of CP (C c) \<Rightarrow> if 0\<ge>\<^sub>N c then T else F | CP _ \<Rightarrow> Le p | _ \<Rightarrow> Le p)"
coprime_surjTr: fixes A :: "'a set"   and B :: "'a set"   and X :: "'a set"   and Y :: "'a set" assumes "ideal R A"   and "ideal R B"   and "coprime_ideals R A B"   and "X \<in> carrier (R /\<^sub>r A)"   and "Y \<in> carrier (R /\<^sub>r B)" shows "\<exists>r\<in>carrier R. pj R A r = X \<and> pj R B r = Y"
partition_onD4: fixes A :: "'b set"   and P :: "'b set set"   and p :: "'b set"   and q :: "'b set"   and x :: "'b" assumes "partition_on A P"   and "p \<in> P"   and "q \<in> P"   and "x \<in> p"   and "x \<in> q" shows "p = q"
lunit_simps_1: fixes f :: "'d" assumes "D.ide f" shows "D.arr \<l>\<^sub>D[f]"
IH: fixes G :: "('a, 'b) ParityGame_scheme"   and v :: "'a" assumes "card (\<omega>\<^bsub>G\<^esub> ` V\<^bsub>G\<^esub>) < card (\<omega> ` V)"   and "v \<in> V\<^bsub>G\<^esub>"   and "ParityGame G"   and "\<And>v. v \<in> V\<^bsub>G\<^esub> \<Longrightarrow> \<not> \<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v \<rightarrow>\<^bsub>G\<^esub> w)" shows "\<exists>p. v \<in> ParityGame.winning_region G p"
assms_5: shows "l (g (u (\<mu> h))) \<le> h (l (u (\<mu> h)))"
inv_check_left_moving_in_middle_2_on_left_moving_in_middle_B: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and list :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and s' :: "nat" assumes "inv_check_left_moving_in_middle (as, lm) (s, Bk # list, Oc # r) ires" shows "inv_on_left_moving_in_middle_B (as, lm) (s', list, Bk # Oc # r) ires"
hypnat_dvd_all_hypnat_of_nat: shows "\<exists>N>0. \<forall>n\<in>- {0}. hypnat_of_nat n dvd N"
orbit_step: fixes y :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "y \<in> orbit f x"   and "f x \<noteq> y" shows "y \<in> orbit f (f x)"
nonneg: fixes x :: "'a" assumes "x \<in> closure S" shows "0 \<le> f x"
gcd_divides: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z divides x"   and "z divides y"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "z divides somegcd G x y"
sf_size: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln" assumes "\<psi> \<in> sf \<phi>" shows "size \<psi> \<le> size \<phi>"
map_of_update_with_aux: fixes v :: "'d"   and k :: "'c"   and f :: "'d \<Rightarrow> 'd"   and ps :: "('c \<times> 'd) list" shows "map_of (update_with_aux v k f ps) = map_of ps(k \<mapsto> case map_of ps k of None \<Rightarrow> f v | Some v \<Rightarrow> f v)"
INF_lower: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and i :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A" shows "less_eq (Inf (f ` A)) (f i)"
intro: fixes L :: "('a, 'b) gorder_scheme" assumes "\<exists>x. greatest L x (carrier L)" shows "weak_partial_order_top_axioms L"
simps_2: fixes n :: "nat" shows "bernoulli (Suc n) = - 1 / (real n + 2) * (\<Sum>k\<le>n. real (n + 2 choose k) * bernoulli k)"
\<alpha>': shows "\<omega> = \<alpha>' @ \<delta> \<and> i + length (snd r) \<le> length \<alpha>'"
equiintegrable_cmul: fixes F :: "('a \<Rightarrow> 'b) set"   and I :: "'a set"   and k :: "real" assumes "F equiintegrable_on I" shows "(\<Union>c\<in>{- k..k}. \<Union>f\<in>F. {\<lambda>x. c *\<^sub>R f x}) equiintegrable_on I"
pochhammer_eq_0_iff: fixes a :: "'a"   and n :: "nat" shows "(pochhammer a n = (0::'a)) = (\<exists>k<n. a = - of_nat k)"
value_mI_genTr3: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set"   and j :: "nat" assumes "0 < n"   and "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "I \<noteq> carrier (O\<^bsub>K P n\<^esub>)"   and "j \<le> n" shows "Zl_mI K P I j \<cdot>\<^sub>r mprod_exp K (K_gamma j) (Kb\<^bsub>K n P\<^esub>) n\<^bsub>K\<^esub>\<^bsup>m_zmax_pdsI K n P I\<^esup> \<in> carrier K"
hypnat_gt_zero_iff2: fixes n :: "nat star" shows "(0 < n) = (\<exists>m. n = m + 1)"
stateful_list_policy_to_list_graph_complies: fixes V :: "'b list"   and E\<^sub>f :: "('b \<times> 'b) list"   and E\<^sub>\<sigma> :: "('b \<times> 'b) list" shows "list_graph_to_graph (stateful_list_policy_to_list_graph \<lparr>hostsL = V, flows_fixL = E\<^sub>f, flows_stateL = E\<^sub>\<sigma>\<rparr>) = stateful_policy_to_network_graph \<lparr>hosts = set V, flows_fix = set E\<^sub>f, flows_state = set E\<^sub>\<sigma>\<rparr>"
INF_empty: fixes f :: "'d \<Rightarrow> 'b \<times> 'c" shows "Inf_prod (f ` {}) = top_prod"
hashcode_list_def: shows "hashcode = foldl (\<lambda>h x. h * 33 + hashcode x) 5381"
times_empty1: fixes B :: "hf" shows "0 * B = 0"
sum_mset_mset_mono: fixes A :: "'a multiset"   and f :: "'a \<Rightarrow> 'b multiset"   and g :: "'a \<Rightarrow> 'b multiset" assumes "\<And>x. x \<in># A \<Longrightarrow> f x \<subseteq># g x" shows "\<Sum>\<^sub># (image_mset f A) \<subseteq># \<Sum>\<^sub># (image_mset g A)"
sim_action_trans: fixes obs :: "('a, 'b) obs_event action"   and obs' :: "('a, 'b) obs_event action"   and obs'' :: "('a, 'b) obs_event action" assumes "obs \<approx> obs'"   and "obs' \<approx> obs''" shows "obs \<approx> obs''"
nprv_negI: fixes \<phi> :: "'fmla"   and F :: "'fmla set" assumes "nprv (insert \<phi> F) fls"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla" shows "nprv F (neg \<phi>)"
cycles_cut: fixes p :: "'state"   and w :: "'action list" assumes "p \<in> nodes"   and "w \<in> cycles p"   and "w \<noteq> []" shows "set w \<inter> cuts \<noteq> {}"
that: fixes r :: "'t \<Rightarrow>\<^sub>0 'b" assumes "r \<in> dgrad_sig_set' (length fs) d"   and "p = rep_list r" shows "thesis"
numeral_neq_neg_one: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and n :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "numeral.numeral one plus n \<noteq> uminus one"
even_bit_succ_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'c"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" shows "bit (plus one a) n = (bit a n \<or> n = 0)"
restrict_map_empty: fixes m :: "'c \<Rightarrow> 'd option"   and S :: "'c set" assumes "dom m \<inter> S = {}" shows "m |` S = Map.empty"
filter_compliant_stateful_ACS_correct: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph"   and edgesList :: "('b \<times> 'b) list"   and \<T> :: "'b stateful_policy" assumes "valid_reqs (get_ACS M)"   and "wf_graph G"   and "set edgesList \<subseteq> edges G"   and "all_security_requirements_fulfilled (get_ACS M) G"   and "\<T> = \<lparr>hosts = nodes G, flows_fix = edges G, flows_state = set (filter_compliant_stateful_ACS G M edgesList)\<rparr>" shows "\<forall>F\<in>get_offending_flows (get_ACS M) (stateful_policy_to_network_graph \<T>). F \<subseteq> backflows (filternew_flows_state \<T>)"
aibi: fixes i :: "'a" assumes "i \<in> Basis" shows "a \<bullet> i \<le> b \<bullet> i"
assms: fixes x :: "nat" assumes "x \<in># A" shows "sum_of_2_squares_nat x"
error_accumulation: fixes j :: "nat"   and p :: "nat"   and r :: "real"   and L :: "real"   and B :: "real"   and T :: "real"   and x :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> 'b"   and \<psi> :: "real \<Rightarrow> real \<Rightarrow> 'b \<Rightarrow> 'b"   and x0 :: "'b" assumes "max_stepsize j \<le> root p (\<bar>r\<bar> * L / B / (exp (L * (T - t 0) + 1) - 1))"   and "0 < p"   and "\<And>j. \<lbrakk>t (Suc j) \<le> T; dist (x t j) (grid_function (discrete_evolution \<psi>) x0 t j) \<le> \<bar>r\<bar>\<rbrakk> \<Longrightarrow> dist (\<psi> (stepsize j) (t j) (x t j)) (\<psi> (stepsize j) (t j) (grid_function (discrete_evolution \<psi>) x0 t j)) \<le> L * dist (x t j) (grid_function (discrete_evolution \<psi>) x0 t j)"   and "0 \<le> L"   and "\<And>j. t (Suc j) \<le> T \<Longrightarrow> dist (x t (Suc j)) (discrete_evolution \<psi> (t (Suc j)) (t j) (x t j)) \<le> B * stepsize j ^ (p + 1)"   and "0 \<le> B"   and "dist (x t 0) x0 \<le> B * (exp 1 - 1) * stepsize 0 ^ p / L"   and "t j \<le> T" shows "dist (x t j) (grid_function (discrete_evolution \<psi>) x0 t j) \<le> B / L * (exp (L * (t j - t 0) + 1) - 1) * max_stepsize j ^ p"
MatchF: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi>s :: "('msaux, 'muaux) mformula list"   and mr :: "mregex"   and r :: "Formula.formula Regex.regex"   and mrs :: "mregex list"   and buf :: "event_data option list set list list"   and nts :: "nat list"   and I :: "\<I>"   and aux :: "(nat \<times> event_data option list set list \<times> event_data option list set) list" assumes "case to_mregex r of (mr', \<phi>s') \<Rightarrow> list_all2 (wf_mformula \<sigma> j P V n R) \<phi>s \<phi>s' \<and> mr = mr'"   and "mrs = sorted_list_of_set (LPDs mr)"   and "safe_regex Futu Strict r"   and "wf_mbufn' \<sigma> P V j n R r buf"   and "wf_ts_regex \<sigma> P j r nts"   and "wf_matchF_aux \<sigma> V n R I r aux (Monitor.progress \<sigma> P (formula.MatchF I r) j) 0"   and "Monitor.progress \<sigma> P (formula.MatchF I r) j + length aux = progress_regex \<sigma> P r j" shows "wf_mformula \<sigma> j P V n R (MMatchF I mr mrs \<phi>s buf nts aux) (formula.MatchF I r)"
chain_transp_on_less: fixes f :: "nat \<Rightarrow> 'b"   and A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "nat"   and j :: "nat" assumes "\<forall>i. f i \<in> A \<and> P (f i) (f (Suc i))"   and "transp_on P A"   and "i < j" shows "P (f i) (f j)"
function_one_eval: fixes R :: "('a, 'b) ring_scheme"   and a :: "'c"   and S :: "'c set" assumes "ring_functions R"   and "a \<in> S" shows "\<one>\<^bsub>function_ring S R\<^esub> a = \<one>\<^bsub>R\<^esub>"
Pred: fixes P :: "'state"   and \<phi> :: "unit" assumes "P \<turnstile>\<^sub>S \<phi>" shows "S_transform.S_transition P \<langle>S_action.Pred \<phi>,P\<rangle>"
assms: shows "\<forall>X\<in>XX. well_founded X (\<sqsubset>) \<and> (\<forall>x\<in>X. \<forall>y\<in>\<Union> XX. y \<sqsubset> x \<longrightarrow> y \<in> X)"
streamsE: fixes s :: "'d stream"   and A :: "'d set" assumes "s \<in> streams A" obtains "shd s \<in> A" and "stl s \<in> streams A"
exec_all_def1: fixes P :: "jvm_prog"   and \<sigma> :: "jvm_state"   and \<sigma>' :: "jvm_state" shows "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>' = ((\<sigma>, \<sigma>') \<in> (exec_1 P)\<^sup>*)"
closure_box: fixes a :: "'b"   and b :: "'b" assumes "box a b \<noteq> {}" shows "closure (box a b) = cbox a b"
std_apart_apart': fixes x :: "char list"   and l :: "fterm literal"   and y :: "char list" assumes "x \<in> vars\<^sub>l (l \<cdot>\<^sub>l (\<lambda>x. \<epsilon> (y @ x)))" shows "\<exists>x'. x = y @ x'"
Fourier_sum_0_integrals: fixes f :: "real \<Rightarrow> real"   and n :: "nat" assumes "f absolutely_integrable_on {- pi..pi}" shows "(\<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k 0) = (integral\<^sup>L (lebesgue_on {- pi..pi}) f / 2 + (\<Sum>k = Suc 0..n div 2. LINT x|lebesgue_on {- pi..pi}. cos (real k * x) * f x)) / pi"
simps_5: fixes \<Sigma> :: "nat \<Rightarrow> 'a set"   and project :: "'a \<Rightarrow> 'a"   and lookup :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "nat"   and r :: "'b rexp"   and s :: "'b rexp" assumes "project \<Sigma> project" shows "project.lang \<Sigma> project lookup n (Plus r s) = project.lang \<Sigma> project lookup n r \<union> project.lang \<Sigma> project lookup n s"
assms: shows "ct_list_eq (to_ct_list as) (xs @ ys)"
LCons: fixes xs :: "'a llist" assumes "lfinite xs"   and "\<not> lnull xs"   and "P (ltl xs)" shows "P xs"
adjunction_data_in_bicategory_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" shows "adjunction_data_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon> \<equiv> bicategory V H \<a> \<i> src trg \<and> adjunction_data_in_bicategory_axioms V H src f g \<eta> \<epsilon>"
g: fixes x :: "'a" assumes "x \<in> S" shows "g differentiable at x"
set_locals_def2: fixes l :: "lname \<Rightarrow> val option"   and g :: "oref \<Rightarrow> obj option"   and l' :: "lname \<Rightarrow> val option" shows "set_locals l (st g l') = st g l"
monic_prod: fixes as :: "'b set"   and f :: "'b \<Rightarrow> 'c poly" assumes "\<And>a. a \<in> as \<Longrightarrow> monic (f a)" shows "monic (prod f as)"
regular_in_p_image_iff: fixes x :: "'a" shows "regular x = in_p_image x"
closed_segment_subset_existence_ivl: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> Y.existence_ivl0 x0" shows "{0--t} \<subseteq> Y.existence_ivl0 x0"
countableE_bij: fixes S :: "'b set" assumes "countable S" obtains f :: "nat \<Rightarrow> 'b" and C :: "nat set" where "bij_betw f C S"
trie_MapOps_def: shows "trie_MapOps \<equiv> \<lparr>MapOps.empty = empty_trie, lookup = lookup_trie, update = trie_update\<rparr>"
leg_simps_1: shows "C.arr assoc\<mu>\<nu>\<pi>.dom.leg0"
equal_pair_count_0: fixes ps :: "('b \<times> 'b) list" assumes "fst (hd ps) \<noteq> snd (hd ps)" shows "equal_pair_count ps = 0"
admits_triangular_mat_1x1: fixes A :: "'a mat" assumes "A \<in> carrier_mat 1 1" shows "admits_triangular_reduction A"
uf_graph_relD: fixes a :: "('a \<times> 'a) set"   and b :: "'edge set"   and c :: "'edge set" assumes "((a, b), c) \<in> uf_graph_rel" shows "b = c \<and> uf_graph_invar (a, b)"
in_disc_1: shows "of_complex z1 \<in> unit_disc"
affine_hull_Int: fixes S :: "'a set"   and T :: "'a set" assumes "\<not> affine_dependent (S \<union> T)" shows "affine hull (S \<inter> T) = affine hull S \<inter> affine hull T"
polyn_add1: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and d :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "pol_coeff S c"   and "pol_coeff S d" shows "polyn_expr R X (fst c) c \<plusminus> polyn_expr R X (fst d) d = polyn_expr R X (max (fst c) (fst d)) (add_cf S c d)"
pmf_nonneg: fixes p :: "'b pmf"   and x :: "'b" shows "0 \<le> pmf p x"
path_in_\<alpha>n: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and g :: "'a"   and ns :: "'b list"   and n :: "'b" assumes "graph_path \<alpha>e \<alpha>n invar inEdges'"   and "graph_path_base.path \<alpha>n invar inEdges' g ns"   and "n \<in> set ns" shows "n \<in> set (\<alpha>n g)"
wcode_on_right_moving_1_BkE: fixes ires :: "cell list"   and rs :: "nat"   and b :: "cell list"   and ba :: "cell list"   and aa :: "cell list"   and list :: "cell list" assumes "wcode_on_right_moving_1 ires rs (b, Bk # ba)"   and "Bk # b = aa \<and> list = b" shows "wcode_on_right_moving_1 ires rs (aa, ba)"
Rep_hyp2_inject: fixes x :: "hyp2"   and y :: "hyp2" shows "(Rep_hyp2 x = Rep_hyp2 y) = (x = y)"
symp_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "symp r = (\<forall>x y. r x y \<longrightarrow> r y x)"
valid_weak_action_modality_strong: fixes X :: "'a"   and P :: "'state"   and \<alpha> :: "'act"   and x :: "('idx, 'pred, 'act) formula" assumes "finite (supp X)" shows "P \<Turnstile> \<langle>\<langle>\<alpha>\<rangle>\<rangle>x = (\<exists>\<alpha>'' x' P'. Act \<alpha> x = Act \<alpha>'' x' \<and> P \<Rightarrow>\<langle>\<alpha>''\<rangle> P' \<and> P' \<Turnstile> x' \<and> bn \<alpha>'' \<sharp>* X)"
order_trans_rules_24: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
set_iterator_I: fixes l0 :: "'d list"   and S :: "'d set"   and iti :: "('e \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e" assumes "distinct l0"   and "S = set l0"   and "iti = foldli l0" shows "set_iterator iti S"
list_of_oalist_cases: fixes y :: "('a \<times> 'b) list \<times> 'a key_order" assumes "y \<in> Collect ko.oalist_inv" obtains x :: "('a, 'b) oalist" where "y = list_of_oalist x"
eval_expands_to: fixes f :: "real \<Rightarrow> real"   and F :: "'a"   and basis :: "(real \<Rightarrow> real) list" assumes "(f expands_to F) basis" shows "\<forall>\<^sub>F x in at_top. eval F x = f x"
aodv_ex_labelE: fixes p :: "(state, msg, pseqp, pseqp label) seqp"   and P :: "pseqp label \<Rightarrow> (state, msg, pseqp, pseqp label) seqp \<Rightarrow> bool" assumes "\<forall>l\<in>labels \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V p. P l p" obtains "\<exists>p l. P l p"
rep_eq: fixes x :: "nat"   and xa :: "int"   and xb :: "int" shows "real_of_float (rapprox_rat x xa xb) = truncate_up x (real_of_int xa / real_of_int xb)"
elims: fixes x :: "mode"   and y :: "'a ltln \<Rightarrow> 'a ltln" assumes "simplify x = y" obtains "x = Nop" and "y = id"   | "x = Fast" and "y = rewrite_iter_fast"   | "x = Slow" and "y = rewrite_iter_slow"
ind_hyp: fixes x :: "'a" shows "sorted_by cmp (x # xs) = ((\<forall>a\<in>set xs. cmp x a) \<and> sorted_by cmp xs)"
simps_9: fixes x :: "pdf_type" shows "op_type pdf_operator.Not x = (case x of BOOL \<Rightarrow> Some BOOL | PRODUCT pdf_type1 x \<Rightarrow> Map.empty x | _ \<Rightarrow> None)"
currentLevelPrefixToLevel: fixes level :: "nat"   and M :: "('b \<times> bool) list" shows "currentLevel (prefixToLevel level M) \<le> level"
run_asset_v_msg: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "let (s', S, A, B, C, D) = run_asset_v n s; SK = (Some S, {A, B}, {C, D}) in {(Owner n, Crypt (SesK SK) (Pwd n)), (Asset n, Crypt (SesK SK) (Num 0))} \<subseteq> s'"
withdraw_claim_length_stable: fixes ts :: "traffic"   and d :: "cars"   and ts' :: "traffic"   and v :: "view"   and c :: "cars" shows "ts \<^bold>\<midarrow>wdc( d ) \<^bold>\<rightarrow> ts' \<longrightarrow> len v ts c = len v ts' c"
diff_numeral_special_8: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and m :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "minus (uminus (numeral.numeral one plus m)) (uminus one) = neg_numeral.sub minus plus one num.One m"
leg1_simps_3: shows "src t\<^sub>1 = src t\<^sub>0"
Extend_satisfies0: fixes k :: "'k"   and i :: "nat"   and a :: "'a"   and \<AA> :: "'i"   and P :: "'v" assumes "\<not> find0 k i a"   and "LEQ k i (#\<^sub>V \<AA>)"   and "lformula0 a \<or> len P \<le> Length \<AA>" shows "(Extend k i \<AA> P \<Turnstile>\<^sub>0 a) = (\<AA> \<Turnstile>\<^sub>0 decr0 k i a)"
less_action: fixes P :: "('a, 'b) action"   and Q :: "('a, 'b) action" shows "P < Q \<equiv> P \<sqsubseteq> Q \<and> Q \<notsqsubseteq> P"
Ri_mono: fixes N :: "'a literal multiset set"   and N' :: "'a literal multiset set" assumes "N \<subseteq> N'" shows "sr_ext_Ri N \<subseteq> sr_ext_Ri N'"
get_child_nodes_delete_shadow_root: fixes ptr' :: "(_) object_ptr"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "ptr' \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr"   and "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'"   and "r \<in> get_child_nodes_locs ptr'" shows "r h h'"
last_message_replicate_NoMsg: fixes n :: "nat" shows "last_message NoMsg\<^bsup>n\<^esup> = NoMsg"
map_G_rel_cong: fixes L1 :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and L2 :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and Co1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co2 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Contra1 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Contra2 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and xf :: "('j, 'l, 'n, 'p, 'r, 't, 'v) G"   and yf :: "('k, 'm, 'o, 'q, 's, 'u, 'v) G"   and L1' :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and x :: "'j \<Rightarrow> 'w"   and y :: "'k \<Rightarrow> 'x"   and L2' :: "'y \<Rightarrow> 'z \<Rightarrow> bool"   and xa :: "'l \<Rightarrow> 'y"   and ya :: "'m \<Rightarrow> 'z"   and Co1' :: "'aa \<Rightarrow> 'ab \<Rightarrow> bool"   and xb :: "'n \<Rightarrow> 'aa"   and yb :: "'o \<Rightarrow> 'ab"   and Co2' :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and xc :: "'p \<Rightarrow> 'ac"   and yc :: "'q \<Rightarrow> 'ad"   and Contra1' :: "'ae \<Rightarrow> 'af \<Rightarrow> bool"   and xd :: "'ae \<Rightarrow> 'r"   and yd :: "'af \<Rightarrow> 's"   and Contra2' :: "'ag \<Rightarrow> 'ah \<Rightarrow> bool"   and xe :: "'ag \<Rightarrow> 't"   and ye :: "'ah \<Rightarrow> 'u" assumes "rel_G L1 L2 Co1 Co2 Contra1 Contra2 xf yf"   and "\<And>xa ya. L1 xa ya \<Longrightarrow> L1' (x xa) (y ya)"   and "\<And>x y. L2 x y \<Longrightarrow> L2' (xa x) (ya y)"   and "\<And>x y. Co1 x y \<Longrightarrow> Co1' (xb x) (yb y)"   and "\<And>x y. Co2 x y \<Longrightarrow> Co2' (xc x) (yc y)"   and "\<And>x y. Contra1' x y \<Longrightarrow> Contra1 (xd x) (yd y)"   and "\<And>x y. Contra2' x y \<Longrightarrow> Contra2 (xe x) (ye y)" shows "rel_G L1' L2' Co1' Co2' Contra1' Contra2' (map_G x xa xb xc xd xe xf) (map_G y ya yb yc yd ye yf)"
cod_char: fixes f :: "'c \<times> 'c" shows "C'.VV.cod f = (if C'.VV.arr f then C'.VxV.cod f else C'.VxV.null)"
Var_BetaD: fixes x :: "name"   and t :: "Lambda_Z.term" assumes "Var x \<rightarrow>\<^sub>\<beta> t" shows "False"
sscanl_lfp_fusion': fixes g :: "'b \<rightarrow> 'a"   and f :: "'b \<rightarrow> 'c \<rightarrow> 'b"   and f' :: "'a \<rightarrow> 'c \<rightarrow> 'a"   and z :: "'b"   and xs :: "[:'c:]" assumes "g\<cdot>\<bottom> = \<bottom>"   and "\<And>acc x. x \<noteq> \<bottom> \<Longrightarrow> g\<cdot>(f\<cdot>acc\<cdot>x) = f'\<cdot>(g\<cdot>acc)\<cdot>x" shows "smap\<cdot>g\<cdot>(sscanl\<cdot>f\<cdot>z\<cdot>xs) = sscanl\<cdot>f'\<cdot>(g\<cdot>z)\<cdot>xs"
Sup2: fixes A :: "'a Assertion set"   and z :: "'a Assertion" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<le> z" shows "\<Squnion>A \<le> z"
nodes_alt_def: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set" shows "gnodes A w V = \<Union> (greachable A w V ` {v. v \<in> ginitial A \<inter> V})"
intros_3: fixes \<phi> :: "'a ltln" assumes "\<phi> = \<phi>" shows "\<phi> \<turnstile>\<^sub>s \<phi>"
is_unifier_def: fixes subst_atm :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and \<sigma> :: "'b"   and AA :: "'a set" shows "substitution_ops.is_unifier subst_atm \<sigma> AA = (card (substitution_ops.subst_atms subst_atm AA \<sigma>) \<le> 1)"
coeff_iff_polynomial_cond: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p1 :: "'c list"   and p2 :: "'c list" assumes "ring R"   and "polynomial\<^bsub>R\<^esub> K p1"   and "polynomial\<^bsub>R\<^esub> K p2" shows "(p1 = p2) = (ring.coeff R p1 = ring.coeff R p2)"
f3_def: shows "f3 \<equiv> \<lambda>A. THE x. \<exists>a b c. A = {a, b, c} \<and> a < b \<and> b < c \<and> x = f {(a, b), (a, c)}"