hcju/isabelleps huggingface dataset:
closing_subst_evalDdb_cong: fixes \<Gamma> :: "nat \<Rightarrow> db"   and \<Gamma>' :: "nat \<Rightarrow> db"   and e :: "db"   and k :: "nat"   and \<rho> :: "nat \<rightarrow> ValD" assumes "\<forall>v. closed (\<Gamma> v) \<and> closed (\<Gamma>' v)"   and "\<forall>v. evalDdb (\<Gamma> v)\<cdot>env_empty_db = evalDdb (\<Gamma>' v)\<cdot>env_empty_db" shows "evalDdb (closing_subst e \<Gamma> k)\<cdot>\<rho> = evalDdb (closing_subst e \<Gamma>' k)\<cdot>\<rho>"
strict_order_iff: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry" shows "(b \<prec> a) = (a = linordered_monoid.max a b \<and> a \<noteq> b)"
assms: fixes w :: "'a" assumes "w \<in> space M" shows "f w = g w"
AE_T_fair: fixes t' :: "'s"   and t :: "'s"   and s :: "'s" assumes "t' \<in> set_pmf (K t)" shows "almost_everywhere (T s) (fair t t')"
wtc_def: fixes cert :: "'a list" shows "wtc cert \<equiv> wtl_cert cert f r \<top> \<bottom> step"
minus_pCons: fixes a :: "'b"   and p :: "'b poly" shows "- pCons a p = pCons (- a) (- p)"
degreen_0: fixes p :: "poly"   and n :: "nat"   and m :: "nat" assumes "isnpolyh p n"   and "m < n" shows "degreen p m = 0"
extf_total: fixes f :: "'s" shows "ext_total (extf f)"
intros_1: fixes s :: "'e"   and A :: "('e, 'f) ioa" assumes "s \<in> ioa.start A" shows "reachable A s"
pphantom_rel_internal: fixes A :: "('c \<times> 'd) set" shows "phantom_rel A = {(Some x, y) |x y. (x, y) \<in> A} \<union> {uu_. \<exists>x y. uu_ = (None, y) \<and> (x, y) \<in> A}"
assms_1: shows "P \<turnstile> \<langle>INIT C ([C],b) \<leftarrow> unit,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v',s\<^sub>1,b\<^sub>1\<rangle>"
l1_implements_a0i_r: shows "implements med01iar a0i l1"
rel_prod_conv: fixes R1 :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and R2 :: "'g \<Rightarrow> 'i \<Rightarrow> bool" shows "rel_prod R1 R2 = (\<lambda>(a, b) (c, d). R1 a c \<and> R2 b d)"
remap_polys_l_def: fixes spec :: "(char list list \<times> int) list" shows "remap_polys_l spec = (\<lambda>\<V> A. SPEC (\<lambda>dom. set_mset (dom_m A) \<subseteq> dom \<and> finite dom) \<bind> (\<lambda>dom. SPEC (\<lambda>_. True) \<bind> (\<lambda>failed. if failed then remap_polys_l_dom_err \<bind> (\<lambda>c. RETURN (error_msg 0 c, \<V>, fmempty)) else FOREACH dom (\<lambda>i (b, \<V>, A'). if i \<in># dom_m A then full_normalize_poly (the (fmlookup A i)) \<bind> (\<lambda>p. weak_equality_l p spec \<bind> (\<lambda>eq. RETURN (\<V> \<union> vars_llist (the (fmlookup A i))) \<bind> (\<lambda>\<V>. RETURN (b \<or> eq, \<V>, fmupd i p A')))) else RETURN (b, \<V>, A')) (False, \<V>, fmempty) \<bind> (\<lambda>(b, \<V>, A). RETURN (if b then CFOUND else CSUCCESS, \<V>, A)))))"
nonneg: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a" assumes "Turn_into_distance f" shows "0 \<le> f x y"
that: fixes a :: "'b" assumes "homotopic_with_canon (\<lambda>x. True) (\<Union> \<F>) T f (\<lambda>x. a)" shows "thesis"
mod_less_eq_dividend: fixes a :: "'b"   and b :: "'b" assumes "(0::'b) \<le> a" shows "a mod b \<le> a"
rel_chain_def: fixes l :: "'a list"   and r :: "('a \<times> 'a) set" shows "rel_chain l r = (\<forall>k<length l - 1. (l ! k, l ! (k + 1)) \<in> r)"
Quotient_part_equivp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "Quotient R Abs Rep T" shows "part_equivp R"
surj_fun_eq: fixes f :: "'b \<Rightarrow> 'a"   and X :: "'b set"   and g1 :: "'a \<Rightarrow> 'c"   and g2 :: "'a \<Rightarrow> 'c" assumes "f ` X = UNIV"   and "\<forall>x\<in>X. (g1 \<cdot> f) x = (g2 \<cdot> f) x" shows "g1 = g2"
less_bot_Bot_is_Value: fixes x :: "'b bot" assumes "bot.Bot < x" shows "\<exists>z. x = bot.Value z"
simps_2: fixes A :: "'a mat"   and k :: "nat" shows "A ^\<^sub>m Suc k = A ^\<^sub>m k * A"
alg_dvd_sum: fixes A :: "'a set"   and y :: "'b"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> y alg_dvd f x" shows "y alg_dvd sum f A"
intros_1: fixes s :: "char list \<Rightarrow> int option" shows "(SKIP, s) \<Rightarrow>\<^sub>A Suc 0 \<Down> s"
strict_linear_order_on_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "strict_linear_order_on A r \<equiv> trans r \<and> irrefl r \<and> total_on A r"
ext_inject: fixes owns :: "room \<Rightarrow> guest option"   and currk :: "room \<Rightarrow> key"   and issued :: "key set"   and cards :: "guest \<Rightarrow> (key \<times> key) set"   and roomk :: "room \<Rightarrow> key"   and isin :: "room \<Rightarrow> guest set"   and safe :: "room \<Rightarrow> bool"   and more :: "'a"   and owns' :: "room \<Rightarrow> guest option"   and currk' :: "room \<Rightarrow> key"   and issued' :: "key set"   and cards' :: "guest \<Rightarrow> (key \<times> key) set"   and roomk' :: "room \<Rightarrow> key"   and isin' :: "room \<Rightarrow> guest set"   and safe' :: "room \<Rightarrow> bool"   and more' :: "'a" shows "(\<lparr>owns = owns, currk = currk, issued = issued, cards = cards, roomk = roomk, isin = isin, safe = safe, \<dots> = more\<rparr> = \<lparr>owns = owns', currk = currk', issued = issued', cards = cards', roomk = roomk', isin = isin', safe = safe', \<dots> = more'\<rparr>) = (owns = owns' \<and> currk = currk' \<and> issued = issued' \<and> cards = cards' \<and> roomk = roomk' \<and> isin = isin' \<and> safe = safe' \<and> more = more')"
linear: fixes y :: "'a"   and x :: "'a" shows "y \<le> x \<or> x \<le> y"
splitFace_holds_edges_sym: fixes g' :: "graph"   and v :: "nat"   and a :: "nat"   and f' :: "face"   and ws :: "nat list" assumes "pre_splitFace g' v a f' ws"   and "minGraphProps g'" shows "edges_sym (snd (snd (splitFace g' v a f' ws)))"
simps_2: fixes M :: "'d"   and xvec :: "name list"   and N :: "'d" shows "bn (M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle>) = xvec"
reach_in: fixes u :: "'a"   and v :: "'a" assumes "u \<in> S"   and "v \<in> S" shows "u \<rightarrow>\<^sup>* v"
matrix_equation: fixes p :: "real poly"   and signs :: "rat list list"   and qs :: "real poly list"   and subsets :: "nat list list" assumes "p \<noteq> 0"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set signs"   and "\<forall>q. List.member qs q \<longrightarrow> coprime p q"   and "all_list_constr subsets (length qs)" shows "satisfy_equation p qs subsets signs"
lex_linorder: fixes D :: "'a set"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res" assumes "linorder_on D cmp" shows "linorder_on (lists D) (cmp_lex cmp)"
continuous_on_inner: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. f x \<bullet> g x)"
ev_blocks_def: shows "local.ev_blocks \<equiv> local.ev_blocks_part n"
bottom_pow: fixes A :: "'b set" shows "order.bottom (Pow A) = {}"
alpha_Tree_permute_rep_commute: fixes p :: "perm"   and t\<^sub>\<alpha> :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha>" shows "p \<bullet> Formula.rep_Tree\<^sub>\<alpha> t\<^sub>\<alpha> =\<^sub>\<alpha> Formula.rep_Tree\<^sub>\<alpha> (p \<bullet> t\<^sub>\<alpha>)"
unique_sat_slice_mem: fixes \<phi> :: "'c MFOTL.formula"   and S :: "'c list set"   and n :: "nat"   and \<sigma> :: "(char list \<times> 'c list) trace"   and v :: "'c list"   and i :: "nat" assumes "safe_formula \<phi>"   and "gen_unique \<phi>"   and "S \<noteq> {}"   and "mergeable_envs n S"   and "MFOTL.nfv \<phi> \<le> n"   and "MFOTL.sat (map_\<Gamma> (\<lambda>D. D \<inter> relevant_events \<phi> S) \<sigma>) v i \<phi>" shows "\<exists>v'\<in>S. \<forall>x\<in>fv \<phi>. v' ! x = v ! x"
abs_summable_on_reindex: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c"   and A :: "'a set" assumes "(\<lambda>x. f (g x)) abs_summable_on A" shows "f abs_summable_on g ` A"
Scope: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and x :: "name" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "x \<sharp> \<Psi>"   and "x \<sharp> \<alpha>" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>P \<longmapsto> \<alpha> \<prec> \<lparr>\<nu>x\<rparr>P'"
integrable_const: fixes a :: "'d" shows "integrable M (\<lambda>x. a)"
i_Exec_Stream_expand_shrink_map_nth: fixes k :: "nat"   and f :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e"   and n :: "nat" assumes "0 < k" shows "((f \<circ> i_Exec_Comp_Stream trans_fun (input \<odot> k) c) \<div> k) n = last_message (map f (f_Exec_Comp_Stream trans_fun (input n # NoMsg\<^bsup>k - Suc 0\<^esup>) (f_Exec_Comp trans_fun (input \<Down> n \<odot> k) c)))"
stream_all2_weaken: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c stream"   and ys :: "'d stream"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "stream_all2 P xs ys"   and "\<And>x y. P x y \<Longrightarrow> Q x y" shows "stream_all2 Q xs ys"
AKcryptSK_def: fixes authK :: "nat"   and servK :: "nat"   and evs :: "event list" shows "AKcryptSK authK servK evs \<equiv> \<exists>A B Ts. Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace>\<rbrace>) \<in> set evs"
1: shows "mat (row_length A1 * row_length B1) (length A2 * length B2) (A1 \<circ> A2 \<otimes> B1 \<circ> B2)"
intros_4: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "Enc X Y \<in> analz H"   and "Y \<in> synth (analz H)" shows "X \<in> analz H"
ImpL_in_SNa: fixes M :: "trm"   and N :: "trm"   and a :: "coname"   and y :: "name"   and z :: "name" assumes "SNa M"   and "SNa N" shows "SNa (ImpL <a>.M y.N z)"
s0_\<gamma>e_s'': shows "s0\<^bsub>SES\<^esub> (\<gamma> @ [e])\<Longrightarrow>\<^bsub>SES\<^esub> s''"
linear_factors: shows "char_poly A = (\<Prod>a\<leftarrow>as. [:- a, 1::'a:])"
subst_compose_def: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'd) Term.term"   and \<tau> :: "'d \<Rightarrow> ('b, 'c) Term.term" shows "\<sigma> \<circ>\<^sub>s \<tau> = (\<lambda>x. \<sigma> x \<cdot> \<tau>)"
mult_cancel_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_no_zero_divisors_cancel plus zero times" shows "(times c a = times c b) = (c = zero \<or> a = b)"
N_def: fixes fs_init :: "int vec list" shows "LLL.N fs_init = Missing_Lemmas.max_list (map (nat \<circ> sq_norm) fs_init)"
set_disconnected_nodes_get_ancestors: fixes ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" shows "\<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_ancestors_locs. r h h')"
is_serial_sol_then_valid_plan_encoded: fixes \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat" assumes "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<phi> prob_with_noop abs_prob  t" shows "valid_plan (decode_abs_plan (rem_noops (map (strips_op_to_sasp (prob_with_noop abs_prob)) (concat (\<Phi>\<inverse> \<phi> prob_with_noop abs_prob  \<A> t)))))"
m_st_height: fixes X :: "char list set"   and S :: "ivl Abs_State.st" assumes "finite X"   and "set (Abs_State.dom S) \<subseteq> X" shows "m_st m_ivl S \<le> 2 * card X"
UP_root_div_def: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c"   and a :: "'c" assumes "UP_cring R" shows "UP_cring.UP_root_div R f a = Cring_Poly.compose R (poly_shift (UP_cring.taylor R a f)) (X_poly_minus R a)"
is_poincare_line_conjugate_circline: fixes H :: "circline" assumes "is_poincare_line H" shows "is_poincare_line (conjugate_circline H)"
flip_compare_lists: fixes as :: "real list"   and bs :: "real list" assumes "length as = length bs" shows "flip_cmp_result (compare_lists as bs) = compare_lists bs as"
rel_with_target_impl_reflC_transC_TRel_is_strong_reduction_simulation: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "strong_reduction_simulation Rel (STCal Source Target)"   and "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel"   and "\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>*" shows "strong_reduction_simulation (TRel\<^sup>*) Target"
fi: shows "(fi, i) \<in> set (fact (d, j))"
IH_2: fixes i :: "nat" assumes "MFOTL.future_reach \<phi>2 \<noteq> \<infinity>" shows "\<exists>j. i \<le> Monitor.progress \<sigma> \<phi>2 j"
simps_6: fixes x :: "unit + 'c \<times> 'c list_R1" shows "set_list_R1 (list_R1 x) = (\<Union>uu\<in>setr x. fsts uu \<union> \<Union> (set_list_R1 ` snds uu))"
LIMSEQ_unique: fixes "open" :: "'a set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.t2_space open"   and "filterlim X (topological_space.nhds open a) sequentially"   and "filterlim X (topological_space.nhds open b) sequentially" shows "a = b"
analz_insert_Decrypt: fixes K :: "nat"   and H :: "msg set"   and X :: "msg" assumes "Key (invKey K) \<in> analz H" shows "analz (insert (Crypt K X) H) = insert (Crypt K X) (analz (insert X H))"
s: shows "e \<in> insert (routing_oiface a, fst (range_prefix_match (routing_match a) s)) (set (routing_port_ranges as (snd (range_prefix_match (routing_match a) s))))"
order_refl: fixes x :: "'x \<Rightarrow>\<^sub>0 nat" shows "lex_pm x x"
mon_cI: fixes s :: "'f list"   and c :: "'f list multiset"   and m :: "'g"   and fg :: "('f, 'h, 'i, 'g, 'j) flowgraph_rec_scheme" assumes "s \<in># c"   and "m \<in> mon_s fg s" shows "m \<in> mon_c fg c"
fin\<F>: fixes X :: "nat set" assumes "X \<in> \<F>" shows "finite X"
vconn_implies_hpar_1: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<Rightarrow> g\<guillemotright>" shows "src \<mu> = src f"
eucl_rel_poly_unique_mod: fixes x :: "'b poly"   and y :: "'b poly"   and q1 :: "'b poly"   and r1 :: "'b poly"   and q2 :: "'b poly"   and r2 :: "'b poly" assumes "eucl_rel_poly x y (q1, r1)"   and "eucl_rel_poly x y (q2, r2)" shows "r1 = r2"
assms_2: fixes js :: "nat list" assumes "js \<in> set jss" shows "js \<noteq> []"
case_cong: fixes exception :: "'a exception"   and exception' :: "'a exception"   and f1 :: "'b"   and g1 :: "'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "exception = exception'"   and "exception' = Fault \<Longrightarrow> f1 = g1"   and "\<And>x2. exception' = OK x2 \<Longrightarrow> f2 x2 = g2 x2" shows "(case exception of Fault \<Rightarrow> f1 | OK x2 \<Rightarrow> f2 x2) = (case exception' of Fault \<Rightarrow> g1 | OK x2 \<Rightarrow> g2 x2)"
iFROM_iprev': fixes t :: "nat"   and n :: "nat" assumes "t \<in> [n\<dots>]" shows "iprev (Suc t) [n\<dots>] = t"
dom_comp_subset: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'd option" shows "g ` dom (f \<circ> g) \<subseteq> dom f"
delete_None_set_of_conv: fixes x :: "'b"   and t :: "'b tree" shows "(delete x t = None) = (x \<notin> set_of t)"
quorum_assm: fixes quorum_of :: "'b \<Rightarrow> 'b set \<Rightarrow> bool"   and p :: "'b"   and Q :: "'b set"   and p' :: "'b" assumes "personal_quorums quorum_of"   and "quorum_of p Q"   and "p' \<in> Q" shows "quorum_of p' Q"
neg_0_equal_iff_equal: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.group_add minus plus zero uminus" shows "(zero = uminus a) = (zero = a)"
pr0_def: fixes a :: "'a"   and b :: "'a" shows "\<p>\<^sub>0[a, b] \<equiv> if ide a \<and> ide b then \<r>[a] \<cdot> (a \<otimes> \<t>[b]) else null"
bounded_Pair_image: fixes f :: "'c \<Rightarrow> 'b"   and s :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "bounded (f ` s)"   and "bounded (g ` s)" shows "bounded ((\<lambda>x. (f x, g x)) ` s)"
set_takeWhileD: fixes x :: "'a"   and P :: "'a \<Rightarrow> bool"   and xs :: "'a list" assumes "x \<in> set (takeWhile P xs)" shows "x \<in> set xs \<and> P x"
ChamberComplexApartmentSystemTriangle_axioms_def: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set" shows "ChamberComplexApartmentSystemTriangle_axioms X \<A> A B B' C D E z \<equiv> ((A \<in> \<A> \<and> B \<in> \<A> \<and> B' \<in> \<A>) \<and> (SimplicialComplex.maxsimp X C \<and> SimplicialComplex.maxsimp X D) \<and> SimplicialComplex.maxsimp X E \<and> z \<lhd> C) \<and> ((z \<lhd> D \<and> z \<lhd> E) \<and> C \<in> A \<inter> B \<and> D \<in> A \<inter> B') \<and> (E \<in> B \<inter> B' \<and> D \<noteq> C) \<and> E \<noteq> D \<and> C \<noteq> E"
eq: shows "\<forall>\<^sub>F x in at_top. q * (p / q * (ln (f x) / ln (g x)) - 1) * ln (g x) = p * ln (f x) - q * ln (g x)"
relA: shows "HMA_M (from_hma\<^sub>m A) A"
add_ci: fixes l :: "'a list"   and P :: "pol"   and Q :: "pol" assumes "in_carrier l" shows "Ipol l (P \<langle>+\<rangle> Q) = Ipol l P \<oplus> Ipol l Q"
extreme_bound_singleton: fixes x :: "'a" assumes "x \<in> A" shows "extreme_bound A (\<lambda>x y. sympartp (\<sqsubseteq>) y x) {x} x"
hyps_for_collect: fixes n :: "('form, 'rule) graph_node"   and p :: "('form, 'var) antecedent" shows "fset (hyps_for n p) = {h. hyps n h = Some p}"
space_measure_of: fixes A :: "'a set set"   and \<Omega> :: "'a set"   and \<mu> :: "'a set \<Rightarrow> ennreal" assumes "A \<subseteq> Pow \<Omega>" shows "space (measure_of \<Omega> A \<mu>) = \<Omega>"
VDM_Complete: fixes c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes " \<Turnstile> c : A " shows "{} \<rhd> c : A"
p: shows "monom 1 n - 1 \<noteq> 0"
l_type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def: fixes type_wf :: "(_) heap \<Rightarrow> bool" shows "l_type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t type_wf \<equiv> \<forall>h. type_wf h \<longrightarrow> DocumentClass.type_wf h"
primeD2: fixes p :: "nat"   and m :: "nat" assumes "prime p"   and "m dvd p" shows "m = 1 \<or> m = p"
vmc_path_lset_induction: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and Q :: "'c llist \<Rightarrow> bool"   and S :: "'c set" assumes "vmc_path G P v0 p \<sigma>"   and "Q P"   and "v0 \<in> S"   and "\<And>P v0. \<lbrakk>vmc_path_no_deadend G P v0 p \<sigma>; v0 \<in> S; Q P\<rbrakk> \<Longrightarrow> Q (ltl P) \<and> vmc_path_no_deadend.w0 P \<in> S" shows "lset P \<subseteq> S"
append1_eq_conv: fixes xs :: "'c list"   and x :: "'c"   and ys :: "'c list"   and y :: "'c" shows "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
finite_atms_of: fixes C :: "'b literal multiset" shows "finite (atms_of C)"
ord_spmfI: fixes pq :: "('a \<times> 'a) spmf"   and p :: "'a spmf"   and q :: "'a spmf" assumes "\<And>x y. (x, y) \<in> set_spmf pq \<Longrightarrow> _struct  x y"   and "map_spmf fst pq = p"   and "map_spmf snd pq = q" shows "p \<sqsubseteq> q"
star2: fixes x :: "('a \<times> 'a) set" shows "(Id \<union> x)\<^sup>* = x\<^sup>*"
formulaEntailsFormulaThatEntailsClause: fixes formula1 :: "Literal list list"   and formula2 :: "Literal list list"   and clause :: "Literal list" assumes "formulaEntailsFormula formula1 formula2"   and "formulaEntailsClause formula2 clause" shows "formulaEntailsClause formula1 clause"
hyp2_axiom7: shows "\<forall>a b c p q. B\<^sub>K a p c \<and> B\<^sub>K b q c \<longrightarrow> (\<exists>x. B\<^sub>K p x b \<and> B\<^sub>K q x a)"
agree_supset: fixes B :: "('d + 'd) set"   and A :: "('d + 'd) set"   and \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and \<nu>' :: "(real, 'd) vec \<times> (real, 'd) vec" assumes "B \<subseteq> A"   and "Vagree \<nu> \<nu>' A" shows "Vagree \<nu> \<nu>' B"
ideD_3: fixes a :: "'j" assumes "J.ide a" shows "J.cod a = a"
rtranclE: fixes a :: "'d"   and b :: "'d"   and r :: "('d \<times> 'd) set" assumes "(a, b) \<in> r\<^sup>*" obtains "a = b"   | y :: "'d" where "(a, y) \<in> r\<^sup>*" and "(y, b) \<in> r"
left_diff_distrib': fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "times (minus b c) a = minus (times b a) (times c a)"
ordering_oLimit: fixes f :: "nat \<Rightarrow> ordinal" shows "OrdinalVeblen.ordering A (oLimit f) = oLimit (\<lambda>n. OrdinalVeblen.ordering A (f n))"
tuple_in_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : a \<rightarrow> c\<guillemotright>" shows "\<guillemotleft>\<langle>f, g\<rangle> : a \<rightarrow> b \<otimes> c\<guillemotright>"
b2_def: shows "b2 = ball x e \<inter> {y. (y - a) \<bullet> rot (a - b) < 0}"
le_less_trans: fixes x :: "'c list"   and y :: "'c list"   and z :: "'c list" assumes "suffix x y"   and "strict_suffix y z" shows "strict_suffix x z"
eval_var_no_jumpE: fixes Env :: "env"   and s0 :: "state"   and var :: "Term.var"   and w :: "val"   and upd :: "val \<Rightarrow> state \<Rightarrow> state"   and s1 :: "state"   and j :: "jump"   and T :: "ty"   and s :: "state"   and j' :: "jump"   and val :: "val" assumes "prg Env\<turnstile>s0 \<midarrow>var=\<succ>(w, upd)\<rightarrow> s1"   and "fst s0 \<noteq> Some (Jump j)"   and "Env,empty_dt\<Turnstile>var\<Colon>=T"   and "wf_prog (prg Env)" obtains "fst s1 \<noteq> Some (Jump j)" and "\<lbrakk>normal s1; fst s \<noteq> Some (Jump j')\<rbrakk> \<Longrightarrow> fst (upd val s) \<noteq> Some (Jump j')"
sub_RI_blocks_body: fixes vs :: "val list"   and pns :: "char list list"   and Ts :: "ty list"   and body :: "char list exp" assumes "length vs = length pns"   and "length Ts = length pns" shows "sub_RI (blocks (pns, Ts, vs, body)) = sub_RI body"
simps_1: fixes v :: "'a"   and u :: "'b"   and us :: "('a \<times> 'b) list"   and as :: "'a \<Rightarrow> 'b" shows "update_assignment ((v, u) # us) as = (update_assignment us as)(v := u)"
subspace_def: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'c set" assumes "module scale" shows "module.subspace scale S = ((0::'c) \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. scale c x \<in> S))"
maxim_equality: fixes r :: "('d \<times> 'd) set"   and A :: "'d set"   and x :: "'d" assumes "wo_rel r"   and "wo_rel.isMaxim r A x" shows "wo_rel.maxim r A = x"
ipurge_ref_def: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and xs :: "'c list"   and X :: "'c set" shows "ipurge_ref I D u xs X \<equiv> {x \<in> X. (u, D x) \<notin> I \<and> (\<forall>v\<in>sinks I D u xs. (v, D x) \<notin> I)}"
load_tmps_fresh: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "load_tmps is \<inter> dom \<theta> = {}"
finfun_rec_upd: fixes a :: "'b"   and f :: "'b \<Rightarrow>f 'a"   and a' :: "'b"   and b' :: "'a" shows "finfun_rec (\<lambda>b. b) (\<lambda>a'. If (a = a')) (f(a' $:= b')) = (if a = a' then b' else finfun_rec (\<lambda>b. b) (\<lambda>a'. If (a = a')) f)"
map_Suc_upt: fixes m :: "nat"   and n :: "nat" shows "map Suc [m..<n] = [Suc m..<Suc n]"