hcju/isabelleps huggingface dataset:
span_is_subset: fixes S :: "'c set"   and N :: "'c set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule K N V" shows "span S \<subseteq> N"
power2_sum: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.comm_semiring_1 times one plus zero" shows "power.power one times (plus x y) 2 = plus (plus (power.power one times x 2) (power.power one times y 2)) (times (times (numeral.numeral one plus (num.Bit0 num.One)) x) y)"
simple: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and bl :: "'b set" assumes "simple_incidence_system point_set block_collection"   and "bl \<in># block_collection" shows "count block_collection bl = 1"
plus_matrix_def: fixes A :: "'b matrix"   and B :: "'b matrix" shows "A + B = combine_matrix (+) A B"
1: shows "natural_isomorphism (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (\<Phi>'.Y a) (\<Phi>'.Y a') \<Phi>\<Psi>'.map"
zip_add_closed: fixes R :: "'a set"   and as :: "'a list"   and bs :: "'a list" assumes "Ring1 R"   and "set as \<subseteq> R"   and "set bs \<subseteq> R" shows "set (map2 (+) as bs) \<subseteq> R"
normal_simps_5: fixes \<phi>\<^sub>1 :: "'a fm"   and \<phi>\<^sub>2 :: "'a fm" shows "normal (Or \<phi>\<^sub>1 \<phi>\<^sub>2) = (normal \<phi>\<^sub>1 \<and> normal \<phi>\<^sub>2)"
split_face_edge_disj: fixes f :: "face"   and a :: "nat"   and b :: "nat"   and vs :: "nat list"   and f\<^sub>1 :: "face"   and f\<^sub>2 :: "face" assumes "pre_split_face f a b vs"   and "(f\<^sub>1, f\<^sub>2) = split_face f a b vs"   and "3 \<le> |vertices f|"   and "vs = [] \<longrightarrow> (a, b) \<notin> \<E> f \<and> (b, a) \<notin> \<E> f" shows "\<E> f\<^sub>1 \<inter> \<E> f\<^sub>2 = {}"
nonmeets2: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and u :: "'a"   and r :: "'a" assumes "class.interval meets \<I>"   and "\<I> u"   and "\<I> r" shows "\<not> (meets u r \<and> u = r)"
limitin_topspace: fixes X :: "'b topology"   and f :: "'c \<Rightarrow> 'b"   and l :: "'b"   and F :: "'c filter" assumes "limitin X f l F" shows "l \<in> topspace X"
\<pi>k: shows "\<pi> k = x \<or> \<pi> k = y"
stable_one_step_axioms: shows "stable_one_step t1 x incr p B r L t s s0"
insert: fixes Q :: "pi"   and S :: "pi set"   and P :: "pi" assumes "Q \<in> S"   and "(P, S - {Q}) \<in> sumComposeSet" shows "(P \<oplus> Q, S) \<in> sumComposeSet"
induct: fixes f :: "nat \<Rightarrow> 'd"   and g :: "nat \<Rightarrow> 'd" assumes "aeseq_eq f g" obtains "finite {n. f n \<noteq> g n}"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubset> y"   and "y \<sqsubset> z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "x \<sqsubset> z"
induct: fixes P :: "(char list list \<times> int) list \<times> (char list list \<times> int) list \<Rightarrow> bool"   and a0 :: "(char list list \<times> int) list \<times> (char list list \<times> int) list" assumes "\<And>p. P (p, [])"   and "\<And>v va. P ([], v # va)"   and "\<And>xs n p ys m q. \<lbrakk>\<lbrakk>xs = ys; n + m = 0\<rbrakk> \<Longrightarrow> P (p, q); \<lbrakk>xs = ys; n + m \<noteq> 0\<rbrakk> \<Longrightarrow> P (p, q); \<lbrakk>xs \<noteq> ys; (xs, ys) \<in> term_order_rel\<rbrakk> \<Longrightarrow> P (p, (ys, m) # q); \<lbrakk>xs \<noteq> ys; (xs, ys) \<notin> term_order_rel\<rbrakk> \<Longrightarrow> P ((xs, n) # p, q)\<rbrakk> \<Longrightarrow> P ((xs, n) # p, (ys, m) # q)" shows "P a0"
iMin_neq_imp_greater: fixes k :: "'a"   and I :: "'a set" assumes "k \<in> I"   and "k \<noteq> iMin I" shows "iMin I < k"
assms_2: shows "\<forall>t\<in>X. \<exists>u\<in>X. (t, u) \<in> R"
unique_session_keys: fixes B :: "agent"   and A :: "agent"   and NA :: "msg"   and K :: "msg"   and NB :: "msg"   and evs :: "event list"   and B' :: "agent"   and A' :: "agent"   and NA' :: "msg"   and NB' :: "msg" assumes "Says Server B \<lbrace>Crypt (shrK A) \<lbrace>NA, Agent A, Agent B, K\<rbrace>, Crypt (shrK B) \<lbrace>NB, Agent A, Agent B, K\<rbrace>\<rbrace> \<in> set evs"   and "Says Server B' \<lbrace>Crypt (shrK A') \<lbrace>NA', Agent A', Agent B', K\<rbrace>, Crypt (shrK B') \<lbrace>NB', Agent A', Agent B', K\<rbrace>\<rbrace> \<in> set evs"   and "evs \<in> otway" shows "A = A' \<and> B = B' \<and> NA = NA' \<and> NB = NB'"
restrict_def: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" shows "restrict f A = (\<lambda>x. if x \<in> A then f x else undefined)"
transp_less: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.preorder less_eq less" shows "transp less"
subst_S: fixes s :: "('f, 'v) Term.term"   and t :: "('f, 'v) Term.term"   and \<sigma> :: "'v \<Rightarrow> ('f, 'v) Term.term" assumes "(s, t) \<in> S" shows "(s \<cdot> \<sigma>, t \<cdot> \<sigma>) \<in> S"
trivial_isomorphism: fixes X :: "'b set set" assumes "ChamberComplex X" shows "ChamberComplexIsomorphism X X id"
cs_not_done: shows "snd (cs final cid) \<noteq> Done"
zero_stalk_def: fixes V :: "'a set" shows "stk1.zero_stalk V \<equiv> stk1.class_of V (zero_str1 V)"
ex_inverse_of_nat_less: fixes x :: "'b" assumes "(0::'b) < x" shows "\<exists>n>0. inverse (of_nat n) < x"
extremum_unique: fixes a :: "'c multiset" shows "(a \<subseteq># {#}) = (a = {#})"
simps: fixes x :: "'a"   and t :: "'a tree"   and dl :: "nat" shows "insert_d x (t, dl) = (case ins (size t + dl) 0 x t of Same \<Rightarrow> t | Bal t' \<Rightarrow> t', dl)"
cqt_basic_2: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>"   and \<psi> :: "'a \<Rightarrow> \<o>" shows "[(\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<equiv> \<psi> \<alpha>) \<^bold>\<equiv> (\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<rightarrow> \<psi> \<alpha>) \<^bold>& (\<^bold>\<forall>\<alpha>. \<psi> \<alpha> \<^bold>\<rightarrow> \<phi> \<alpha>) in v]"
simps_3: fixes x2 :: "'b" shows "extC.Val x2 \<noteq> extC.Bot"
indep_imp_subset_basis: fixes X :: "'a set" assumes "indep_in \<E> X" shows "\<exists>B. \<E>.basis B \<and> X \<subseteq> B"
gcd_is_Max_divisors_nat: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "gcd m n = Max {d. d dvd m \<and> d dvd n}"
trancl_range: fixes r :: "('a \<times> 'a) set" shows "Range (r\<^sup>+) = Range r"
inv_on_f_range: fixes y :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "y \<in> f ` A" shows "inv_on f A y \<in> A"
quotient_eq_iff: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and X :: "'b set"   and Y :: "'b set"   and x :: "'b"   and y :: "'b" assumes "equiv A r"   and "X \<in> A // r"   and "Y \<in> A // r"   and "x \<in> X"   and "y \<in> Y" shows "(X = Y) = ((x, y) \<in> r)"
structCongBisim: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "P \<equiv>\<^sub>s Q" shows "P \<sim> Q"
distinct_1: fixes x1 :: "'a"   and x2 :: "'b" shows "IObj x1 \<noteq> IMor x2"
seq_decompose: fixes \<sigma> :: "'a \<times> ('b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set"   and \<sigma>1' :: "'b list"   and \<sigma>2' :: "'b list" assumes "\<sigma> \<in> seq ars"   and "labels \<sigma> = \<sigma>1' @ \<sigma>2'" shows "\<exists>\<sigma>1 \<sigma>2. {\<sigma>1, \<sigma>2} \<subseteq> seq ars \<and> \<sigma> = (fst \<sigma>1, snd \<sigma>1 @ snd \<sigma>2) \<and> lst \<sigma>1 = fst \<sigma>2 \<and> lst \<sigma>2 = lst \<sigma> \<and> labels \<sigma>1 = \<sigma>1' \<and> labels \<sigma>2 = \<sigma>2'"
s: shows "s \<notin> \<real>\<^sub>\<le>\<^sub>0"
drop_all_iff: fixes n :: "nat"   and xs :: "'d list" shows "(xs \<up> n = []) = (length xs \<le> n)"
get_writer_def: fixes f :: "'s \<Rightarrow> ('w, 'a, 'm) writerT" shows "local.get_writer f = WriterT (get (\<lambda>s. run_writer (f s)))"
iT_Minus_inext: fixes k :: "nat"   and I :: "nat set"   and n :: "nat" assumes "k \<ominus> I \<noteq> {}"   and "n \<le> k" shows "inext (k - n) (k \<ominus> I) = k - iprev n I"
less_degree_imp: fixes n :: "nat"   and p :: "'b poly" assumes "n < degree p" shows "\<exists>i>n. coeff p i \<noteq> (0::'b)"
qItem_versus_item_simps_3: fixes f :: "('c, 'd, 'varSort, 'var, 'e) qTerm \<Rightarrow> 'f"   and Xa :: "('c, 'd, 'varSort, 'var, 'e) qTerm" assumes "f respectsP (#=)"   and "qGood Xa" shows "Equiv_Relation2.univ f (asTerm Xa) = f Xa"
InvariantWatchCharacterization_def: fixes F :: "Literal list list"   and watch1 :: "nat \<Rightarrow> Literal option"   and watch2 :: "nat \<Rightarrow> Literal option"   and M :: "(Literal \<times> bool) list" shows "InvariantWatchCharacterization F watch1 watch2 M = (\<forall>c w1 w2. 0 \<le> c \<and> c < length F \<and> Some w1 = watch1 c \<and> Some w2 = watch2 c \<longrightarrow> watchCharacterizationCondition w1 w2 M (F ! c) \<and> watchCharacterizationCondition w2 w1 M (F ! c))"
interpolation_poly_int_None: fixes xs_ys :: "(int \<times> int) list"   and alg :: "interpolation_algorithm"   and q :: "int poly" assumes "distinct (map fst xs_ys)"   and "interpolation_poly_int alg xs_ys = None"   and "\<And>x y. (x, y) \<in> set xs_ys \<Longrightarrow> poly q x = y"   and "degree q < length xs_ys" shows "False"
mono: fixes x :: "'c llist llist" shows "mono_llist (\<lambda>lconcat. case x of LNil \<Rightarrow> LNil | LCons xs xss' \<Rightarrow> lappend xs (lconcat xss'))"
rec_pred_def: shows "UF.rec_pred = Cn 1 (Pr 1 z (recf.id 3 1)) [recf.id 1 0, recf.id 1 0]"
natural_transformation_axioms: shows "natural_transformation (\<cdot>) (\<cdot>) ((\<otimes>) \<I>) local.map \<ll>.map"
deg_zero_impl_monom: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "deg R f = 0" shows "f = monom P (coeff P f 0) 0"
"2_9": shows "evalC b (C (replicate i n @ ns)) = evalC b m'"
finrankD: fixes M :: "'v set" assumes "R_finrank M" shows "\<exists>n. \<forall>ms. set ms \<subseteq> M \<and> R_lin_independent ms \<longrightarrow> length ms \<le> n"
cobounded_imp_unbounded: fixes S :: "'a set" assumes "bounded (- S)" shows "\<not> bounded S"
c_sum_le_arg: fixes u :: "nat" shows "c_sum u \<le> u"
peek_st_def2: fixes Pf :: "st \<Rightarrow> 'b assn"   and Y :: "vals"   and s :: "state" shows "peek_st Pf Y s = Pf (snd s) Y s"
cont2cont_Pair: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "cont f"   and "cont g" shows "cont (\<lambda>x. (f x, g x))"
winding_number_homotopic_loops_null_eq: fixes p :: "real \<Rightarrow> complex"   and \<zeta> :: "complex" assumes "path p"   and "\<zeta> \<notin> path_image p" shows "(winding_number p \<zeta> = 0) = (\<exists>a. homotopic_loops (- {\<zeta>}) p (\<lambda>t. a))"
cbox_Pair_eq: fixes a :: "'b"   and c :: "'c"   and b :: "'b"   and d :: "'c" shows "cbox (a, c) (b, d) = cbox a b \<times> cbox c d"
limk: shows "\<forall>i<d. \<forall>j<d. (\<lambda>n. g (n + k) $$ (i, j)) \<longlonglongrightarrow> A $$ (i, j)"
cardinalityPreservation: fixes XX :: "'b set set" assumes "\<forall>X\<in>XX. finite X"   and "is_non_overlapping XX" shows "card (\<Union> XX) = sum card XX"
"1_2": shows "subseq as' (rem_condless_act s [] (rem_effectless_act as))"
parallel_reduction_fnE: fixes x :: "'a"   and T :: "PreSimplyTyped.type"   and A :: "'a trm"   and X :: "'a trm" assumes "Fn x T A >> X" shows "X = Fn x T A \<or> (\<exists>A'. A >> A' \<and> X = Fn x T A')"
min_inf: fixes x :: "'a"   and y :: "'a" shows "min x y = x \<sqinter> y"
ascP_def: fixes f :: "'d list \<Rightarrow> 'd list" shows "ascP f = (\<forall>xs. f xs = f [] @ xs)"
V_def: shows "V = arith_prog_fb (int_of_fbint y) d"
WODBackwardSliceCorrect: fixes prog :: "cmd" shows "BackwardSlice sourcenode targetnode kind (valid_edge prog) (_Entry_) (Defs prog) (Uses prog) id (While_CFG_wf.wod_backward_slice prog)"
sinvar_all_edges_normal_form_sr_def: fixes P :: "'a \<Rightarrow> 'v \<Rightarrow> 'a \<Rightarrow> 'v \<Rightarrow> bool" shows "sinvar_all_edges_normal_form_sr P \<equiv> \<forall>G nP. sinvar G nP = (\<forall>(s, r)\<in>edges G. P (nP s) s (nP r) r)"
P: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "(f \<otimes>\<^bsub>P\<^esub> X_poly R [^]\<^bsub>P\<^esub> n) (n + k) = f k"
qstar_def: fixes x :: "'a \<times> 'b" shows "sd_star x = Sup_prod (range (sd_power x))"
norm_def: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and k :: "nat \<Rightarrow> 'd"   and n :: "nat" shows "DBM_Normalization.norm M k n \<equiv> \<lambda>i j. let ub = if 0 < i then k i else (0::'d); lb = if 0 < j then - k j else (0::'d) in if i \<le> n \<and> j \<le> n then norm_lower (norm_upper (M i j) ub) lb else M i j"
not_ordLeq_ordLess: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "r \<le>o r'" shows "(r', r) \<notin> ordLess"
simps: fixes a :: "'a" shows "the_Inl (Inl a) = a"
csucc_def: fixes \<kappa> :: "V" shows "csucc \<kappa> \<equiv> LEAST \<kappa>''. Ord \<kappa>'' \<and> Card \<kappa>'' \<and> \<kappa> < \<kappa>''"
pfa_is_empty_def: shows "pfa_is_empty \<equiv> arl_is_empty"
cexpr_typing_opE: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and oper :: "pdf_operator"   and e :: "cexpr"   and t :: "pdf_type" assumes "\<Gamma> \<turnstile>\<^sub>c oper $$\<^sub>c e : t" obtains t :: "pdf_type" where "\<Gamma> \<turnstile>\<^sub>c e : t" and "op_type oper t = Some ta__"
\<theta>_dom: shows "subst_domain \<theta> = insert x (subst_domain \<sigma>)"
trms\<^sub>s\<^sub>t_update\<^sub>s\<^sub>t_eq_snd: fixes t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list"   and \<S> :: "('fun, 'var) strand_step list set"   and \<S>' :: "('fun, 'var) strand_step list set"   and \<A>' :: "('fun, 'var) extstrand_step list"   and \<A> :: "('fun, 'var) extstrand_step list" assumes "send\<langle>t\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>"   and "\<S>' = update\<^sub>s\<^sub>t \<S> (send\<langle>t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<A>' = \<A> @ [Step (receive\<langle>t\<rangle>\<^sub>s\<^sub>t)]" shows "\<Union> (trms\<^sub>s\<^sub>t ` \<S>) \<union> \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set \<A>) = \<Union> (trms\<^sub>s\<^sub>t ` \<S>') \<union> \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set \<A>')"
tan_mono_lt_eq: fixes x :: "real"   and y :: "real" assumes "- (pi / 2) < x"   and "x < pi / 2"   and "- (pi / 2) < y"   and "y < pi / 2" shows "(tan x < tan y) = (x < y)"
red_step: fixes prog :: "cmd"   and l :: "nat"   and c :: "cmd"   and s :: "char list \<Rightarrow> val option"   and c' :: "cmd"   and s' :: "char list \<Rightarrow> val option" assumes "labels prog l c"   and "\<langle>c,s\<rangle> \<rightarrow> \<langle>c',s'\<rangle>" shows "\<exists>l'. prog \<turnstile> \<langle>c,s,l\<rangle> \<leadsto> \<langle>c',s',l'\<rangle> \<and> labels prog l' c'"
incomparable_minimal_antichain: fixes A :: "'a set" shows "incomparable (minimal_antichain A)"
S1: assumes "u = 0" shows "c_tl u = sgn1 (c_len u - 1) * (c_pair (c_len u - 2) (c_snd (c_snd (u - 1))) + 1)"
ax: fixes R :: "('a, 'b) sequent list \<times> ('a, 'b) sequent \<Rightarrow> bool"   and r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and R' :: "('a, 'b) sequent list \<Rightarrow> ('a, 'b) sequent \<Rightarrow> bool"   and M :: "'b"   and N :: "'b"   and seq :: "('a, 'b) sequent" assumes "R r"   and "case r of (x, xa) \<Rightarrow> Axp x xa" shows "extp R R' M N (extendRule seq r)"
cases: fixes x :: "'a list" obtains "xb__ = []"   | x :: "'a" and ys :: "'a list" where "xb__ = x # ys"
OF_lm_noa_none: fixes \<gamma> :: "'b set \<Rightarrow> 'c \<Rightarrow> bool"   and ft :: "('b, 'a) flow_entry_match list"   and p :: "'c" assumes "OF_match_linear \<gamma> ft p = NoAction" shows "\<forall>e\<in>set ft. \<not> \<gamma> (ofe_fields e) p"
bound_typ_inst_composed_subst: fixes S :: "nat \<Rightarrow> typ"   and R :: "nat \<Rightarrow> typ"   and sch :: "type_scheme" shows "bound_typ_inst ($ S \<circ> R) ($ S sch) = $ S (bound_typ_inst R sch)"
lossless_def: fixes \<A> :: "('grp \<Rightarrow> (('grp \<times> 'grp) \<times> 'b) spmf) \<times> ('grp \<times> 'grp \<Rightarrow> 'b \<Rightarrow> bool spmf)" shows "local.ind_cpa.lossless \<A> = ((\<forall>pk. lossless_spmf (fst \<A> pk)) \<and> (\<forall>cipher \<sigma>. lossless_spmf (snd \<A> cipher \<sigma>)))"
crel_vs_rel_state: fixes R0 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R1 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and x :: "'f \<Rightarrow> 'h"   and y :: "'g \<Rightarrow> 'i Heap" assumes "(R0 ===> state_dp_consistency.crel_vs R1 OO local.rel_state (=)) x y" shows "state_dp_consistency.rel_fun_lifted R0 R1 x (state_of \<circ> y)"
landau_trans_34: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<in> O[F](g)"   and "g \<in> o[F](h)" shows "f \<in> o[F](h)"
prv_dsj_cases: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi> \<in> fmla"   and "prv (dsj \<phi>1 \<phi>2)"   and "prv (imp \<phi>1 \<chi>)"   and "prv (imp \<phi>2 \<chi>)" shows "prv \<chi>"
tso_store_refs_simps_5: fixes m :: "'b"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state"   and m' :: "'b"   and r :: "'a"   and obj :: "('c, 'd, 'a) object" shows "mut_m.tso_store_refs m (s(mutator m' := s (mutator m')\<lparr>roots := insert r (roots (s (mutator m')))\<rparr>, sys := s sys\<lparr>heap := sys_heap s(r \<mapsto> obj)\<rparr>)) = mut_m.tso_store_refs m s"
abs_eq: shows "(\<sqinter>) = (\<lambda>x y. Abs_atoms (Rep_atoms x \<inter> Rep_atoms y))"
not_empty_pairE: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'d prog"   and A :: "('e \<times> 'f) set" assumes "heap addr2thread_id thread_id2addr allocate typeof_addr heap_write P"   and "A \<noteq> {}" obtains a :: "'e" and b :: "'f" where "(a, b) \<in> A"
repr_independenceD: fixes H :: "'a set"   and y :: "'a"   and x :: "'a" assumes "subgroup H G"   and "y \<in> carrier G"   and "H #> x = H #> y" shows "y \<in> H #> x"
filternew_flows_state_def: fixes \<T> :: "'a stateful_policy" shows "filternew_flows_state \<T> \<equiv> {(s, r). (s, r) \<in> flows_state \<T> \<and> (r, s) \<notin> flows_fix \<T>}"
method_def: fixes P :: "'b prog"   and C :: "char list"   and M :: "char list" shows "method P C M \<equiv> THE (D, b, Ts, T, m). P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D"
Lp_bounded_bounded_support: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure"   and C :: "real"   and p :: "ennreal" assumes "f \<in> borel_measurable M"   and "AE x in M. \<bar>f x\<bar> \<le> C"   and "emeasure M {x \<in> space M. f x \<noteq> 0} \<noteq> \<infinity>" shows "f \<in> space\<^sub>N (\<LL> p M)"
de_morgan4': fixes p :: "'a"   and q :: "'a" assumes "test p"   and "test q" shows "!(!p \<cdot> !q) = p + q"
induct: fixes P :: "('d, 'c) ipurge_rec \<Rightarrow> bool"   and a0 :: "('d, 'c) ipurge_rec" assumes "\<And>I D u x xs ys. \<lbrakk>D x \<in> sources I D u (x # xs) \<Longrightarrow> P \<lparr>Pol = I, Map = D, Dom = u, In = xs, Out = ys @ [x]\<rparr>; D x \<notin> sources I D u (x # xs) \<Longrightarrow> P \<lparr>Pol = I, Map = D, Dom = u, In = ipurge_tr I D (D x) xs, Out = ys\<rparr>\<rbrakk> \<Longrightarrow> P \<lparr>Pol = I, Map = D, Dom = u, In = x # xs, Out = ys\<rparr>"   and "\<And>v va vb vd ve. P \<lparr>Pol = v, Map = va, Dom = vb, In = [], Out = vd, \<dots> = ve\<rparr>" shows "P a0"
reflection: shows "rel_weakly_reflects_barbs (indRelTEQ TRel) (STCalWB SWB TWB)"
get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) shadow_root_ptr"   and getter :: "('c, 'a, 'b, 'k option) RShadowRoot_scheme \<Rightarrow> 'm" assumes "h \<turnstile> ok get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr getter" shows "ptr |\<in>| shadow_root_ptr_kinds h"
lir_fps_correct: fixes f :: "nat \<Rightarrow> 'a"   and g :: "('a \<times> nat \<times> 'a) list"   and cs :: "'a list"   and fs :: "'a list" assumes "linear_inhomogenous_recurrence f (eval_polyexp g) cs fs" shows "map_option fps_of_ratfps (lir_fps cs fs g) = Some (Abs_fps f)"
fun_belowI: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>x. f x \<sqsubseteq> g x" shows "f \<sqsubseteq> g"
par_distinct: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Par C D" shows "A \<noteq> B \<and> C \<noteq> D"
subS: shows "{z. norm z \<le> m \<and> (\<exists>t\<in>P. norm (z - t) \<le> e)} \<subseteq> S"
fact_ge_zero: fixes n :: "nat" shows "(0::'b) \<le> fact n"
permutes_superset: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and T :: "'b set" assumes "p permutes S"   and "\<forall>x\<in>S - T. p x = x" shows "p permutes T"
h9: shows "MPoly_Type.degree (sum f {0..n}) x = n"
eq_iff: fixes x :: "'b mod_ring"   and y :: "'b mod_ring" shows "(to_int_mod_ring x = to_int_mod_ring y) = (x = y)"
bres_canc1: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "less_eq_prod (times_prod' x (prod.bres x y)) y"
WHILEIT_nofail_imp_rwof_nofail: fixes m0 :: "'a nres"   and I :: "'a \<Rightarrow> bool"   and cond :: "'a \<Rightarrow> bool"   and step :: "'a \<Rightarrow> 'a nres"   and s :: "'a" assumes "nofail (m0 \<bind> WHILE\<^sub>T\<^bsup>I\<^esup> cond step)"   and "rwof m0 cond step s"   and "cond s" shows "nofail (step s)"
prems_2: shows "\<not> ps (S t (Suc i)) p \<noteq> None"
equivalence_functor_F: shows "equivalence_functor hom.comp hom'.comp (\<lambda>\<mu>'. e \<star> \<mu>' \<star> src \<mu>)"
simps: fixes a1 :: "'a list"   and a2 :: "'b sequent" shows "((a1, a2) \<in> Ax) = ((\<exists>i. a1 = [] \<and> a2 = ( \<LM> At i  \<RM> \<Rightarrow>* \<LM> At i  \<RM>)) \<or> a1 = [] \<and> a2 = ( \<LM> ff  \<RM> \<Rightarrow>* \<Empt>))"
unit_factor_mult_unit_left: fixes a :: "'a"   and b :: "'a" assumes "is_unit a" shows "unit_factor (a * b) = a * unit_factor b"
"case": fixes f :: "'d match_expr \<Rightarrow> action \<Rightarrow> 'c"   and x1 :: "'d match_expr"   and x2 :: "action" shows "(case Rule x1 x2 of Rule x xa \<Rightarrow> f x xa) = f x1 x2"
zero_isomorphic_to_FG_zero: assumes "V = 0" shows "isomorphic (*) 0"
prems: shows "\<forall>x\<in>insert a A. \<forall>y\<in>insert a A. f x y = (if y = x then g x else (0::'b))"
FunctorMapsTo: fixes f :: "'c" assumes "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"
Seq: fixes P\<^sub>1 :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>1 :: "com"   and P\<^sub>2 :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>2 :: "com"   and P\<^sub>3 :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<turnstile>\<^sub>t {P\<^sub>1} c\<^sub>1 {P\<^sub>2}"   and "\<turnstile>\<^sub>t {P\<^sub>2} c\<^sub>2 {P\<^sub>3}" shows "\<turnstile>\<^sub>t {P\<^sub>1} c\<^sub>1;; c\<^sub>2 {P\<^sub>3}"
CHAOS_subset_FD: fixes A :: "'d set"   and B :: "'d set" assumes "A \<subseteq> B" shows "CHAOS B \<sqsubseteq>\<^sub>F\<^sub>D CHAOS A"
mono_Det_sym: fixes P :: "'b process"   and Q :: "'b process"   and S :: "'b process" assumes "P \<sqsubseteq> Q" shows "(S [+] P) \<sqsubseteq> (S [+] Q)"
DBM_val_bounded_len_3: fixes v :: "'c \<Rightarrow> nat"   and u :: "'c \<Rightarrow> 'd"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat"   and c :: "'c"   and d :: "'c"   and cs :: "'c list" assumes "u \<turnstile>\<^bsub>v,n\<^esub> m"   and "v c \<le> n"   and "v d \<le> n"   and "\<forall>c\<in>set cs. v c \<le> n" shows "dbm_entry_val u (Some c) (Some d) (len m (v c) (v d) (map v cs))"
cinner_extensionality: fixes \<psi> :: "'a"   and \<phi> :: "'a" assumes "\<And>\<gamma>. \<gamma> \<bullet>\<^sub>C \<psi> = \<gamma> \<bullet>\<^sub>C \<phi>" shows "\<psi> = \<phi>"
abs_eq: fixes xa :: "nat \<Rightarrow>\<^sub>0 nat"   and x :: "'a" shows "MPoly_Type.monom xa x = MPoly (monomial x xa)"
set_4: fixes x2 :: "'d" shows "vars_hd (Sym x2) = {}"