hcju/isabelleps huggingface dataset:
simps_5: fixes f1 :: "'b"   and f2 :: "final_decision \<Rightarrow> 'b"   and x2 :: "final_decision" shows "(case Decision x2 of Undecided \<Rightarrow> f1 | Decision x \<Rightarrow> f2 x) = f2 x2"
beta_preserves_FV: fixes t :: "sterm"   and t' :: "sterm"   and x :: "char list" assumes "t -> t'"   and "x \<notin> FV t" shows "x \<notin> FV t'"
lower_split_float_interval1: fixes X :: "float interval" shows "lower (fst (split_float_interval X)) = lower X"
permutes_in_image: fixes p :: "'e \<Rightarrow> 'e"   and S :: "'e set"   and x :: "'e" assumes "p permutes S" shows "(p x \<in> S) = (x \<in> S)"
lincomb_obtain_same_length_Rcoeffs: fixes rs :: "'r list"   and ms :: "'m list" assumes "set rs \<subseteq> R"   and "set ms \<subseteq> M" shows "\<exists>ss. set ss \<subseteq> R \<and> length ss = length ms \<and> take (length rs) ss = take (length ms) rs \<and> rs \<bullet>\<cdot> ms = ss \<bullet>\<cdot> ms"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and op :: "op" shows "P (case op of Empty \<Rightarrow> f1 | Incr \<Rightarrow> f2) = ((op = Empty \<longrightarrow> P f1) \<and> (op = Incr \<longrightarrow> P f2))"
assms: shows "v \<in> set (Koszul_syz_sigs_aux bs k)"
GSMP_Ana_key: fixes t :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set"   and K :: "('fun, 'var) Term.term list"   and T :: "('fun, 'var) Term.term list"   and k :: "('fun, 'var) Term.term" assumes "t \<in> GSMP M"   and "Ana t = (K, T)"   and "k \<in> set K" shows "k \<in> GSMP M"
lasocc_x': shows "index (snd (TSdet init h (as @ x # bs @ [x]) (Suc (length as + length bs)))) x = length bs"
simps_28: fixes f1 :: "b_e \<Rightarrow> 'a"   and f2 :: "'a"   and f3 :: "cl \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> e list \<Rightarrow> e list \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> v list \<Rightarrow> e list \<Rightarrow> 'a"   and x41 :: "nat"   and x42 :: "e list"   and x43 :: "e list" shows "(case Label x41 x42 x43 of $x \<Rightarrow> f1 x | Trap \<Rightarrow> f2 | Callcl x \<Rightarrow> f3 x | Label x xa xb \<Rightarrow> f4 x xa xb | Local x xa xb xc \<Rightarrow> f5 x xa xb xc) = f4 x41 x42 x43"
finfun_Bex_except_def: fixes xs :: "'a list"   and A :: "'a \<Rightarrow>f bool"   and P :: "'a \<Rightarrow> bool" shows "finfun_Bex_except xs A P = (\<exists>a. A $ a \<and> a \<notin> set xs \<and> P a)"
ex_kleene_qfp: assumes "omega_complete A (\<sqsubseteq>)" shows "Ex (extreme_bound A (\<sqsubseteq>) {f^n \<bottom> |. n})"
dgrad_p_set_closed_red: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "F \<subseteq> dgrad_p_set d m"   and "p \<in> dgrad_p_set d m"   and "red F p q" shows "q \<in> dgrad_p_set d m"
canonic_int_validates_all_clauses_sel: fixes Sel :: "'at set"   and S :: "'at Literal set set"   and C :: "'at Literal set" assumes "saturated_binary_rule (ordered_sel_resolvent Sel) S"   and "all_fulfill finite S"   and "{} \<notin> S"   and "C \<in> S" shows "canonic_int_ordered (empty_selected_part Sel S) \<Turnstile> C"
pCons_2: shows "degree (pCons a1 p1) = degree (pCons a2 p2)"
simps_2: fixes a :: "ereal" shows "(\<infinity> < a) = False"
intros: fixes trsys :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and s :: "'g"   and tl :: "'h"   and s' :: "'g"   and \<tau>move :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool" assumes "trsys s tl s'"   and "\<tau>move s tl s'" shows "\<tau>trsys.silent_move trsys \<tau>move s s'"
that_2: assumes "\<not> discrete X \<and> part X" shows "thesis"
thing2: shows "0 \<le> 9223372039002259455 + sint (SCAST(32 \<rightarrow> 64) w2)"
function_of_def: fixes A :: "'b set"   and enum :: "nat \<Rightarrow> 'b"   and xs :: "'c list" shows "function_of A enum xs = (\<lambda>a. if a \<in> A then xs ! inv_into {0..<length xs} enum a else undefined)"
g_secure_suff_implies_g_secure: fixes P :: "'a process"   and L :: "'a \<Rightarrow> g_level" assumes "g_secure_suff P L" shows "g_secure P L"
separation_t2: fixes "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.t2_space open" shows "(x \<noteq> y) = (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
ACI_nPlus_PLUS: fixes xs1 :: "'b rexp list"   and xs2 :: "'b rexp list" assumes "xs1 \<noteq> []"   and "xs2 \<noteq> []"   and "\<forall>x\<in>set (xs1 @ xs2). \<nexists>r s. x = Plus r s"   and "sorted xs2"   and "distinct xs2" shows "ACI_nPlus (PLUS xs1) (PLUS xs2) = flatten PLUS (set (xs1 @ xs2))"
monad_resumption_optionT: fixes return :: "'a option \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a option \<Rightarrow> 'b) \<Rightarrow> 'b"   and pause :: "'c \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_resumption return bind pause" shows "monad_resumption (return_option return) (bind_option return bind) (pause_option pause)"
Wbis_matchC_M: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" assumes "c \<approx>w d" shows "matchC_M Wbis c d"
can_occur_Recv: fixes i :: "nat"   and p :: "'a"   and q :: "'a"   and s :: "'b"   and s' :: "'b"   and m :: "'c"   and c :: "('a, 'b, 'c) configuration" assumes "can_occur (Recv i p q s s' m) c" shows "states c p = s \<and> channel i = Some (q, p) \<and> (\<exists>xs. msgs c i = Msg m # xs) \<and> recv i p q s s' m"
assms: shows "c = ((Mn n f, xs, ls) # fs, rv)"
stermsl_ctermsl: fixes q :: "('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp" assumes "q \<in> stermsl p" shows "q \<in> ctermsl p"
match_eval_trm: fixes ts :: "'a MFOTL.trm list"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a option"   and xs :: "'a list" assumes "\<forall>t\<in>set ts. \<forall>i\<in>fv_trm t. i < n"   and "Some f = match ts xs" shows "map (MFOTL.eval_trm (tabulate (\<lambda>i. the (f i)) 0 n)) ts = xs"
permute_profile_map_relation: fixes \<sigma> :: "'c \<Rightarrow> 'b"   and R :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool" shows "permute_profile \<sigma> R = (\<lambda>i. map_relation (inv \<sigma>) (R i))"
cinner_mult_left: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and m :: "complex"   and a :: "'b"   and b :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner (of_complex m * a) b = cnj m * cinner a b"
mult_unit_dvd_iff': fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "is_unit a" shows "(a * b dvd c) = (b dvd c)"
maps_to_v: shows "(\<sigma>;;\<tau>) l = Some v"
subgroupI: fixes H :: "'a set" assumes "H \<subseteq> carrier G"   and "H \<noteq> {}"   and "\<And>a. a \<in> H \<Longrightarrow> \<ominus> a \<in> H"   and "\<And>a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<oplus> b \<in> H" shows "subgroup H (add_monoid G)"
c: shows "Y \<in> obj\<^bsub>NTCatDom \<eta>\<^esub>"
strong_fusion: fixes F :: "i \<Rightarrow> bool" assumes "\<exists>x. F x" shows "\<exists>x. (\<forall>y. F y \<longrightarrow> P y x) \<and> (\<forall>y. P y x \<longrightarrow> (\<exists>z. F z \<and> O y z))"
le_SUP_iff: fixes x :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "(x \<le> Sup (f ` A)) = (\<forall>y<x. \<exists>i\<in>A. y < f i)"
coprime_pairs_asymptotics'': shows "(\<lambda>N. real (card (A N))) \<sim>[sequentially] (\<lambda>N. 6 / pi\<^sup>2 * (real N)\<^sup>2)"
elims: fixes x :: "'b tag_const"   and xa :: "'b tag_const"   and y :: "'b tag_const" assumes "x + xa = y" obtains t\<^sub>1 :: "'b" and t\<^sub>2 :: "'b" where "x = \<tau>\<^sub>c\<^sub>s\<^sub>t t\<^sub>1" and "xa = \<tau>\<^sub>c\<^sub>s\<^sub>t t\<^sub>2" and "y = \<tau>\<^sub>c\<^sub>s\<^sub>t (t\<^sub>1 + t\<^sub>2)"
proj4_def: shows "\<pi>\<^sub>4 \<equiv> \<lambda>(uu_, uu_, flag, uu_, uu_). flag"
subspace_is_vs: fixes W :: "'c set" assumes "subspace K W V" shows "vectorspace K (V.vs W)"
hull_in_free_hull: fixes G :: "'b list set" shows "\<langle>G\<rangle> \<subseteq> \<langle>G\<rangle>\<^sub>F"
real_sqrt_ge_abs2: fixes y :: "real"   and x :: "real" shows "\<bar>y\<bar> \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
\<psi>\<^sub>3_dim: shows "1 < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then \<Sum>k<2 ^ n. (- 1) ^ (f k + i div 2 \<cdot>\<^bsub>n\<^esub>  k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else \<Sum>k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 \<cdot>\<^bsub>n\<^esub>  k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))"
sim_actions_def: shows "sim_actions = llist_all2 (\<lambda>(t, a) (t', a'). t = t' \<and> a \<approx> a')"
adds_antisym: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and lcs :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'a"   and t :: "'a" assumes "class.ulcs_powerprod plus minus zero lcs"   and "plus.adds plus s t"   and "plus.adds plus t s" shows "s = t"
per_def: fixes r :: "'rule" shows "per r \<equiv> \<forall>s r1 sl' s'. s \<in> S \<and> enabled r s \<and> r1 \<in> R - {r} \<and> eff r1 s sl' \<and> s' |\<in>| sl' \<longrightarrow> enabled r s'"
coll_scaleR: fixes x :: "real \<times> real"   and y :: "real \<times> real"   and r :: "real" assumes "coll 0 x y" shows "coll 0 (r *\<^sub>R x) y"
simps_4: fixes F :: "'a formula"   and G :: "'a formula" shows "nnf (\<^bold>\<not> (F \<^bold>\<or> G)) = nnf (\<^bold>\<not> F) \<^bold>\<and> nnf (\<^bold>\<not> G)"
independent_insertI: fixes a :: "('a, 'b) vec"   and S :: "('a, 'b) vec set" assumes "a \<notin> vec.span S"   and "vec.independent S" shows "vec.independent (insert a S)"
CDPlayer_CTRL_Action3_def: shows "CDPlayer_CTRL_Action3 = ({}, CDPlayer_CTRL_Update3)"
worel_SucFbd: fixes r1 :: "('d \<times> 'd) set"   and r2 :: "('e \<times> 'e) set" shows "wo_rel (cardSuc (r1 +c r2))"
scale_right_imp_eq: fixes x :: "'b"   and a :: "'a"   and b :: "'a" assumes "x \<noteq> (0::'b)"   and "a *s x = b *s x" shows "a = b"
list_splits_append: fixes xs :: "'b list"   and ys :: "'b list" shows "list_splits (xs @ ys) = map (\<lambda>(xs', x', ys'). (xs', x', ys' @ ys)) (list_splits xs) @ map (\<lambda>(xs', x', ys'). (xs @ xs', x', ys')) (list_splits ys)"
edges_S_rg_edges: shows "edges g \<inter> S (A Q \<pi>) \<times> - S (A Q \<pi>) = edges rg \<inter> S (A Q \<pi>) \<times> - S (A Q \<pi>)"
inner_node_Exit_edge: fixes n :: "'b" assumes "inner_node n" obtains a :: "'a" where "valid_edge a" and "inner_node (sourcenode a)" and "targetnode a = Exit"
eq_fold: fixes A :: "'b set" shows "mset_set A = Finite_Set.fold add_mset {#} A"
nn: shows "0 \<le> \<bar>mantissa x\<bar> * 2 ^ (Suc LENGTH('f) - nat (bitlen \<bar>mantissa x\<bar>)) - 2 ^ LENGTH('f)"
simps_3: fixes y :: "nat" shows "extN.Bot + extN.N y = extN.N y"
CFG_SSA_Transformed_def: fixes \<alpha>e :: "'d \<Rightarrow> ('e \<times> 'f \<times> 'e) set"   and \<alpha>n :: "'d \<Rightarrow> 'e list"   and invar :: "'d \<Rightarrow> bool"   and inEdges' :: "'d \<Rightarrow> 'e \<Rightarrow> ('e \<times> 'f) list"   and Entry :: "'d \<Rightarrow> 'e"   and oldDefs :: "'d \<Rightarrow> 'e \<Rightarrow> 'g set"   and oldUses :: "'d \<Rightarrow> 'e \<Rightarrow> 'g set"   and defs :: "'d \<Rightarrow> 'e \<Rightarrow> 'h set"   and "uses" :: "'d \<Rightarrow> 'e \<Rightarrow> 'h set"   and phis :: "'d \<Rightarrow> 'e \<times> 'h \<Rightarrow> 'h list option"   and var :: "'d \<Rightarrow> 'h \<Rightarrow> 'g" shows "CFG_SSA_Transformed \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs uses phis var \<equiv> CFG_wf \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses \<and> CFG_SSA_wf \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis \<and> CFG_SSA_Transformed_axioms \<alpha>n invar inEdges' oldDefs oldUses defs uses phis var"
A_splay_def: fixes a :: "'b"   and t :: "'b tree" shows "A_splay a t = real (T_splay a t) + \<Phi> (splay a t) - \<Phi> t"
homeomorphic_eq_everything_map: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "homeomorphic_map X Y f = (continuous_map X Y f \<and> open_map X Y f \<and> closed_map X Y f \<and> f ` topspace X = topspace Y \<and> inj_on f (topspace X))"
Tgs_never_says_current_timestamp: fixes evs :: "event list"   and T :: "nat"   and X :: "msg" assumes "CT evs = T"   and "Number T \<in> parts {X}"   and "evs \<in> kerbV" shows "\<forall>A. Says Tgs A X \<notin> set evs"
finite_cone_def: fixes n :: "nat"   and Vs :: "'b vec set" shows "gram_schmidt.finite_cone n Vs = {b. \<exists>c. gram_schmidt.nonneg_lincomb n c (if finite Vs then Vs else {}) b}"
simps_2: fixes x :: "int" shows "take_incr [x] = [x]"
kcdj: shows "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> (GREATEST j. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> ipd (\<pi> j) \<noteq> ipd (\<pi> k))"
simps_6: fixes f1 :: "'c btree \<Rightarrow> 'b"   and f2 :: "'c btree \<Rightarrow> 'c \<Rightarrow> 'c btree \<Rightarrow> 'b"   and x21 :: "'c btree"   and x22 :: "'c"   and x23 :: "'c btree" shows "(case Up\<^sub>i x21 x22 x23 of T\<^sub>i x \<Rightarrow> f1 x | Up\<^sub>i x xa xb \<Rightarrow> f2 x xa xb) = f2 x21 x22 x23"
isGlb_def: fixes S :: "'b set"   and po :: "'b potype" shows "isGlb S po \<equiv> \<lambda>G. G \<in> pset po \<and> (\<forall>y\<in>S. (G, y) \<in> order po) \<and> (\<forall>z\<in>pset po. (\<forall>y\<in>S. (z, y) \<in> order po) \<longrightarrow> (z, G) \<in> order po)"
im_sheaf_morphisms_def: fixes U :: "'a set set"   and V :: "'a set set" shows "im_sheaf_morphisms U V \<equiv> sheaf_spec_morphisms (identity Spec \<^sup>\<inverse> Spec U) (identity Spec \<^sup>\<inverse> Spec V)"
56: fixes x :: "'a"   and y :: "'a" shows "(- x \<squnion> y) \<sqinter> x = y \<sqinter> x"
mem_Sigma_iff: fixes a :: "'b"   and b :: "'c"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" shows "((a, b) \<in> Sigma A B) = (a \<in> A \<and> b \<in> B a)"
construct_partition_on: fixes B :: "'a set"   and C :: "'a set" assumes "finite B"   and "finite C"   and "B \<inter> C = {}" shows "construct_partition_on B C = {P. partition_on (B \<union> C) P}"
hom_mult: fixes h :: "'b \<Rightarrow> 'c"   and G :: "('b, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme"   and x :: "'b"   and y :: "'b" assumes "h \<in> hom G H"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"
leg0_uniquely_isomorphic_2: shows "\<exists>!\<theta>. \<guillemotleft>\<theta> : s\<^sub>0 \<star> chine \<Rightarrow> r\<^sub>0\<guillemotright>"
traces_def: fixes P :: "'a process" shows "traces P \<equiv> Domain (failures P)"
Abs_filter_inject: fixes x :: "'b set"   and y :: "'b set" assumes "x \<in> filters"   and "y \<in> filters" shows "(Filters.filter.Abs_filter x = Filters.filter.Abs_filter y) = (x = y)"
vec_def: fixes x :: "'a" shows "vec x = (\<chi>i. x)"
real_vector_rpr_axioms: shows "real_vector_rpr carrier relation"
has_integral_linear: fixes f :: "'c \<Rightarrow> 'd"   and y :: "'d"   and S :: "'c set"   and h :: "'d \<Rightarrow> 'e" assumes "(f has_integral y) S"   and "bounded_linear h" shows "(h \<circ> f has_integral h y) S"
sublist_append: fixes l :: "nat"   and m :: "nat"   and r :: "nat"   and xs :: "'b list" assumes "l \<le> m"   and "m \<le> r"   and "r \<le> length xs" shows "sublist l m xs @ sublist m r xs = sublist l r xs"
not_less_Nil: fixes x :: "'b list" shows "\<not> list_less x []"
assms: shows "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
set_mr_cos_def: fixes M :: "('e, 'f, 'g) Module_scheme"   and H :: "'e set" shows "set_mr_cos M H = {X. \<exists>a\<in>carrier M. X = a \<uplus>\<^bsub>M\<^esub> H}"
tendsto_le: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and F :: "'d filter"   and f :: "'d \<Rightarrow> 'b"   and x :: "'b"   and g :: "'d \<Rightarrow> 'b"   and y :: "'b" assumes "class.linorder_topology less_eq less open"   and "F \<noteq> bot"   and "filterlim f (topological_space.nhds open x) F"   and "filterlim g (topological_space.nhds open y) F"   and "\<forall>\<^sub>F x in F. less_eq (g x) (f x)" shows "less_eq y x"
snapshot_state_must_have_been_reached: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and p :: "'d"   and u :: "'e"   and i :: "nat" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "ps final p = Some u"   and "\<not> ps (distributed_system.s channel trans send recv init t i) p \<noteq> None"   and "ps (distributed_system.s channel trans send recv init t (i + 1)) p \<noteq> None"   and "i < length t" shows "states (distributed_system.s channel trans send recv init t i) p = u"
assms_4: shows "automaton.trans A = oseqp_sos \<Gamma> i"
psp_ensures: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> A ensures A'"   and "F \<in> B co B'" shows "F \<in> A \<inter> B' ensures A' \<inter> B \<union> (B' - B)"
intro: fixes S :: "'c set"   and G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c" assumes "S \<subseteq> V\<^bsub>G\<^esub>"   and "v0 \<notin> S"   and "v1 \<notin> S"   and "\<And>xs. v0 \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> v1 \<Longrightarrow> set xs \<inter> S \<noteq> {}" shows "Separation_axioms G v0 v1 S"
33: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<squnion> (x \<sqinter> z \<squnion> x \<sqinter> (y \<squnion> z)) = x \<sqinter> (y \<squnion> z)"
mult_less_cancel2: fixes m :: "nat"   and k :: "nat"   and n :: "nat" shows "(m *n k < n *n k) = (0 < k \<and> m < n)"
preserves_reflects_arr: fixes f :: "'a" shows "Go\<tau>.B.arr (\<tau> f) = Go\<tau>.A.arr f"
le_lt_convs_4: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b" assumes "comparator comp" shows "(comp x y = Lt) = lt_of_comp comp x y"
basis_expansion_unique: fixes f :: "'e \<Rightarrow> 'd"   and x :: "('d, 'e) vec" shows "((\<Sum>i\<in>UNIV. f i *s axis i (1::'d)) = x) = (\<forall>i. f i = x $ i)"
fmmap_fmupd: fixes f :: "'f \<Rightarrow> 'e"   and x :: "'d"   and y :: "'f"   and m :: "('d, 'f) fmap" shows "fmmap f (fmupd x y m) = fmupd x (f y) (fmmap f m)"
map_indets_def: fixes f :: "'b \<Rightarrow> 'd" shows "map_indets f = poly_subst (\<lambda>x. monomial (1::'c) (monomial 1 (f x)))"
MP_same: fixes H :: "fm set"   and A :: "fm"   and B :: "fm" assumes "H \<turnstile> A IMP B"   and "H \<turnstile> A" shows "H \<turnstile> B"
inv: fixes a :: "'a"   and H :: "'a set" assumes "a \<in> H" shows "comm.additive.inverse a \<in> comm.additive.generate H"
thm_neg_eqE: fixes v :: "i"   and x :: "\<nu>"   and y :: "\<nu>" shows "[x\<^sup>P \<^bold>\<noteq>\<^sub>E y\<^sup>P \<^bold>\<equiv> \<^bold>\<not>x\<^sup>P \<^bold>=\<^sub>E y\<^sup>P in v]"
arc_join_eq_alt: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "pathfinish g1 = pathstart g2" shows "arc (g1 +++ g2) = (arc g1 \<and> arc g2 \<and> path_image g1 \<inter> path_image g2 = {pathstart g2})"
invar: shows "\<forall>i. enat i < llength Ns \<longrightarrow> gc_invar Ns (enat i)"
expands_to_hd'': fixes f :: "real \<Rightarrow> real"   and C :: "'a"   and e :: "real"   and xs :: "('a \<times> real) msllist"   and g :: "real \<Rightarrow> real"   and b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "(f expands_to MS (MSLCons (C, e) xs) g) (b # basis)" shows "(eval C expands_to C) basis"
Pow_Int_eq: fixes A :: "'c set"   and B :: "'c set" shows "Pow (A \<inter> B) = Pow A \<inter> Pow B"
MWLsSteps_det_cases_6: fixes \<iota> :: "nat"   and V :: "('exp, 'id) MWLsCom list"   and m :: "'id \<Rightarrow> 'val"   and \<alpha> :: "('exp, 'id) MWLsCom list"   and p :: "('exp, 'id) MWLsCom option"   and m' :: "'id \<Rightarrow> 'val" assumes "SRabbr (spawn\<^bsub>\<iota>\<^esub> V) m \<alpha> p m'" obtains "\<alpha> = V" and "p = None" and "m' = m"
subseteq_guards_Catch: fixes c :: "('d, 'e, 'f) com"   and c1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com" assumes "c \<subseteq>\<^sub>g Catch c1 c2" shows "\<exists>c1' c2'. c = Catch c1' c2' \<and> (c1' \<subseteq>\<^sub>g c1) \<and> (c2' \<subseteq>\<^sub>g c2)"
Interp_imp_INTERP: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and C :: "'b literal multiset"   and D :: "'b literal multiset"   and N :: "'b literal multiset set" assumes "ground_resolution_with_selection S"   and "C \<le> D"   and "ground_resolution_with_selection.Interp S N D \<Turnstile> C" shows "ground_resolution_with_selection.INTERP S N \<Turnstile> C"
dom_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "CCC.dom f = (if arr (fst f) \<and> CC.arr (snd f) then (local.dom (fst f), CC.dom (snd f)) else (null, CC.null))"
H: fixes xs' :: "(char list list \<times> 'a) list"   and x :: "char list list"   and aaa :: "char list list"   and ba :: "'a" assumes "\<forall>x\<in>set xs'. sorted_wrt var_order (fst x)"   and "sorted_wrt (rel2p (Id \<union> var_order_rel)) x"   and "(aaa, ba) \<in> set xs'" shows "sorted_wrt (rel2p (Id \<union> var_order_rel)) aaa"
collapse: fixes tree :: "'a tree" assumes "tree \<noteq> \<langle>\<rangle>" shows "\<langle>un_Node1 tree, value tree, un_Node3 tree\<rangle> = tree"
assms: fixes a :: "'a"   and b :: "'b"   and c :: "'c"   and d :: "'d" assumes "ab = (a, b)"   and "cd = (c, d)" shows "f a b c d \<le> SPEC \<Phi>"
similar_mat_witD_6: fixes n :: "nat"   and A :: "'c mat"   and B :: "'c mat"   and P :: "'c mat"   and Q :: "'c mat" assumes "n = dim_row A"   and "similar_mat_wit A B P Q" shows "P \<in> carrier_mat n n"
a: shows "Ide a \<and> Arr a \<and> Dom a = a \<and> Cod a = a \<and> ide \<lbrace>a\<rbrace> \<and> ide \<lbrace>\<^bold>\<lfloor>a\<^bold>\<rfloor>\<rbrace> \<and> \<guillemotleft>\<lbrace>a\<^bold>\<down>\<rbrace> : \<lbrace>a\<rbrace> \<rightarrow> \<lbrace>\<^bold>\<lfloor>a\<^bold>\<rfloor>\<rbrace>\<guillemotright>"
pivot_funD_3: fixes A :: "'b mat"   and nr :: "nat"   and f :: "nat \<Rightarrow> nat"   and nc :: "nat"   and i :: "nat" assumes "dim_row A = nr"   and "pivot_fun A f nc"   and "i < nr"   and "Suc i < nr" shows "f i < f (Suc i) \<or> f (Suc i) = nc"
SUP_commute: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and B :: "'c set"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup ((\<lambda>i. Sup (f i ` B)) ` A) = Sup ((\<lambda>j. Sup ((\<lambda>i. f i j) ` A)) ` B)"
transfer_bind: fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and Q :: "'h \<Rightarrow> 'i \<Rightarrow> bool" shows "(local.rel_state R ===> (R ===> local.rel_state Q) ===> local.rel_state Q) (\<bind>) (\<bind>)"
root_int_floor_pos: fixes x :: "int"   and p :: "nat" assumes "0 \<le> x" shows "root_int_floor_pos p x = \<lfloor>root p (real_of_int x)\<rfloor>"
assms: shows "h \<turnstile> ok attach_shadow_root element_ptr shadow_root_mode"