hcju/isabelleps huggingface dataset:
conj_inclusion_2: shows "\<phi> \<notin> K \<inter> (K \<^bold>* .\<not> (\<phi> .\<and>. \<psi>))"
isolated_singularity_at_transform: fixes g :: "complex \<Rightarrow> complex"   and z :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "isolated_singularity_at g z"   and "\<forall>\<^sub>F w in at z. g w = f w" shows "isolated_singularity_at f z"
equality: fixes r :: "('a, 'b) web_scheme"   and r' :: "('a, 'b) web_scheme" assumes "edge r = edge r'"   and "weight r = weight r'"   and "A r = A r'"   and "B r = B r'"   and "web.more r = web.more r'" shows "r = r'"
frestrictionD_1: fixes a :: "V"   and b :: "V"   and r :: "V"   and A :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<restriction>\<^sub>\<bullet> A" shows "a \<in>\<^sub>\<circ> A"
assms: fixes w :: "'a set"   and x :: "'a" assumes "openin (top_of_set S) w"   and "x \<in> w" shows "\<exists>u v. openin (top_of_set S) u \<and> P v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w"
span_def: fixes S :: "'d set" shows "cspan S = csubspace hull S"
assms_4: shows "f \<in> space\<^sub>N (\<LL> p M)"
Am_ub3: fixes t :: "'b tree" assumes "bst t" shows "S34.Am t \<le> log (root 3 4) (real (size1 t)) + 1"
prems_2: shows "wf\<^sub>t\<^sub>r\<^sub>m (Fun f T)"
policy_range_comp_def: fixes f :: "'f \<Rightarrow> 'e"   and p :: "'d \<Rightarrow> 'f decision option" shows "f o\<^sub>f p = (\<lambda>x. case p x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow y\<rfloor> \<Rightarrow> \<lfloor>allow (f y)\<rfloor> | \<lfloor>deny y\<rfloor> \<Rightarrow> \<lfloor>deny (f y)\<rfloor>)"
reduced_word_problem_eq_hd_step: fixes w :: "'w"   and a :: "'w"   and ss :: "'w list"   and ts :: "'w list" assumes "\<And>y ss ts. \<lbrakk>order.greater (S_length w) (S_length y); y \<noteq> (0::'w); S_reduced_for y ss; S_reduced_for y ts\<rbrakk> \<Longrightarrow> \<exists>xss. flip_altsublist_chain (ss # xss @ [ts])"   and "S_reduced_for w (a # ss)"   and "S_reduced_for w (a # ts)" shows "\<exists>xss. flip_altsublist_chain ((a # ss) # xss @ [a # ts])"
lnth_LCons': fixes x :: "'c"   and xs :: "'c llist"   and n :: "nat" shows "lnth (LCons x xs) n = (if n = 0 then x else lnth xs (n - 1))"
intro: fixes a :: "'c"   and G :: "('d, 'c) pre_digraph" assumes "a \<in> arcs G" shows "bidel_arc_axioms G a"
simps_2: fixes l :: "'b"   and ls :: "'b list"   and f :: "'b \<Rightarrow> 'a" shows "ListSum (l # ls) f = f l + ListSum ls f"
ThrowNull\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>throw e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NullPointer,s\<^sub>1\<rangle>"
wellformed_item_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and x :: "item" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.wellformed_item \<RR> Doc x = (item_rule x \<in> \<RR> \<and> item_origin x \<le> item_end x \<and> item_end x \<le> length Doc \<and> item_dot x \<le> length (item_rhs x))"
idx_spec: fixes xs :: "'a list" shows "decr xs = (\<forall>i<length xs - 1. xs ! Suc i \<le> xs ! i)"
strategy_attracts_viaE: fixes A :: "'a set"   and W :: "'a set" assumes "strategy_attracts_via p \<sigma> v0 A W" shows "visits_via P A W"
univ_part: shows "UNIV = {j::nat. (i::nat) = j} \<union> {j::nat. i < j} \<union> {j::nat. j < i}"
span_insert: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and S :: "'b set" assumes "module scale" shows "module.span scale (insert a S) = {x. \<exists>k. x - scale k a \<in> module.span scale S}"
Pareto_strictI: fixes x :: "'alt"   and y :: "'alt"   and i :: "'agent" assumes "\<And>i. i \<in> agents \<Longrightarrow> R i x y"   and "i \<in> agents"   and "x \<prec>[R i] y" shows "x \<prec>[Pareto R] y"
Times_in_interior_subtopology: fixes x :: "'b"   and y :: "'c"   and U :: "('b \<times> 'c) set"   and S :: "'b set"   and T :: "'c set" assumes "(x, y) \<in> U"   and "openin (top_of_set (S \<times> T)) U" obtains V :: "'b set" and W :: "'c set" where "openin (top_of_set S) V" and "x \<in> V" and "openin (top_of_set T) W" and "y \<in> W" and "V \<times> W \<subseteq> U"
eq_implies: fixes x :: "'b"   and y :: "'b"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "x = y" shows "equivpartp r y x"
h: shows "\<forall>f\<in>F. Sup_pres f"
commutative_monoid_axioms: shows "commutative_monoid R (+) \<zero>"
lead_coeff_minus: fixes p :: "'b poly" shows "lead_coeff (- p) = - lead_coeff p"
Snd: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<langle>X, Y\<rangle> \<in> parts H" shows "Y \<in> parts H"
CatchMiss: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>1 :: "('b, 'a, 'c) com"   and s :: "'b"   and n :: "nat"   and t :: "('b, 'c) xstate"   and c\<^sub>2 :: "('b, 'a, 'c) com" assumes "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> =n\<Rightarrow> t"   and "\<not> isAbr t" shows "\<Gamma>\<turnstile> \<langle>Catch c\<^sub>1 c\<^sub>2,Normal s\<rangle> =n\<Rightarrow> t"
rel_pmf_OO_trans: fixes R :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and p :: "'h pmf"   and q :: "'i pmf"   and S :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and r :: "'j pmf" assumes "rel_pmf R p q"   and "rel_pmf S q r" shows "rel_pmf (R OO S) p r"
old_inheritable: shows "G \<turnstile>Method old inheritable_in pid C "
that: fixes f :: "nat \<Rightarrow> nat" assumes "range f = A"   and "\<forall>i. f i < f (Suc i)" shows "thesis"
cInf_insert_If: 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 X :: "'c set"   and a :: "'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "preorder.bdd_below less_eq X" shows "Inf (insert a X) = (if X = {} then a else inf a (Inf X))"
abs_summable_on_Sigma_project2: fixes f :: "'b \<times> 'c \<Rightarrow> 'd"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'c set"   and x :: "'b" assumes "f abs_summable_on Sigma A B"   and "x \<in> A" shows "(\<lambda>y. f (x, y)) abs_summable_on B x"
simps_2: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and S :: "'a list" shows "map_hd f (a # S) = f a # S"
shift_bounds_cl_Suc_ivl: fixes g :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "sum g {Suc m..Suc n} = (\<Sum>i = m..n. g (Suc i))"
suffixE: fixes xs :: "'c list"   and ys :: "'c list" assumes "suffix xs ys" obtains zs :: "'c list" where "ys = zs @ xs"
new_character_data_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_character_data_ptr :: "(_) character_data_ptr"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'ah" assumes "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr"   and "ptr \<noteq> cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^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 new_character_data_ptr" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
deterministic_RP_saturated: shows "saturated_upto grounded_R"
bisimw: shows "wset s1' t = None \<or> x1' \<approx>w x2'"
is_rcoset_replist_Cons: fixes H :: "'g set"   and g :: "'g"   and gs :: "'g list" shows "is_rcoset_replist H (g # gs) = (g \<in> G \<and> set gs \<subseteq> G \<and> rcoset_rel H `` {g} \<notin> set (map (\<lambda>x. rcoset_rel H `` {x}) gs) \<and> distinct (map (\<lambda>x. rcoset_rel H `` {x}) gs) \<and> G = rcoset_rel H `` {g} \<union> (\<Union>x\<in>set gs. rcoset_rel H `` {x}))"
analz_spies_pubK: fixes b :: "keymode"   and A :: "agent"   and evs :: "event list" shows "Key (publicKey b A) \<in> analz (knows Spy evs)"
simps: fixes e1 :: "'b"   and a1 :: "'c"   and r1 :: "nat"   and ts1 :: "('b, 'c) SkewBinomialTree list"   and e2 :: "'b"   and a2 :: "'c"   and r2 :: "nat"   and ts2 :: "('b, 'c) SkewBinomialTree list" shows "link (Node e1 a1 r1 ts1) (Node e2 a2 r2 ts2) = (if a1 \<le> a2 then Node e1 a1 (Suc r1) (Node e2 a2 r2 ts2 # ts1) else Node e2 a2 (Suc r2) (Node e1 a1 r1 ts1 # ts2))"
wf_progressE: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x" assumes "s \<in> wf_state"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "\<not> final x" obtains taa :: "('l, 't, 'x, 'm, 'w, 'o) thread_action" and x' :: "'x" and m' :: "'m" where "t \<turnstile> \<langle>x, shr s\<rangle> -taa\<rightarrow> \<langle>x', m'\<rangle>"
cod_simps_3: shows "H\<nu>H\<pi>\<rho>.cod.leg1 = H\<nu>H\<pi>\<rho>.leg1"
mbhd_invar_def: fixes x :: "'b"   and l :: "'b list" shows "mbhd_invar x l \<equiv> distinct l \<and> x \<notin> set (tl l)"
continuous_on_closed_vimage: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "closed s" shows "continuous_on s f = (\<forall>B. closed B \<longrightarrow> closed (f -` B \<inter> s))"
distinct6_rot_CW: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and D :: "Points"   and E :: "Points"   and F :: "Points" assumes "distinct6 A B C D E F" shows "distinct6 F A B C D E"
negligible_iff_measure0: fixes S :: "'c set" assumes "S \<in> lmeasurable" shows "negligible S = (Sigma_Algebra.measure lebesgue S = 0)"
leg0_commutes: shows "cod.leg0 \<cdot> chine = leg0"
d: fixes p :: "'a"   and q :: "'a" shows "((p, q) \<in> d) = (\<exists>k l u v. k \<parallel> l \<and> l \<parallel> p \<and> p \<parallel> u \<and> u \<parallel> v \<and> k \<parallel> q \<and> q \<parallel> v)"
news_psubst: fixes c :: "'a"   and z :: "('a, 'b) form list"   and d :: "'c"   and f :: "'a \<Rightarrow> 'c" assumes "news c z"   and "d \<notin> f ` \<Union> (params ` set z)" shows "news d (map (psubst (f(c := d))) z)"
in_notinI: fixes j :: "'b"   and i :: "'b" assumes "(j, i) \<notin> s \<or> j = i"   and "i \<in> Field s"   and "j \<in> Field s" shows "(i, j) \<in> s"
\<sigma>i: shows "\<sigma> (f, Suc (length bef + length aft)) ! i = length bef"
intro: assumes "\<pi> \<sim>[at_top] (\<lambda>x. x / ln x)" shows "prime_number_theorem"
All_cong: fixes H :: "fm set"   and A :: "fm"   and A' :: "fm"   and i :: "name" assumes "H \<turnstile> A IFF A'"   and "\<forall>C\<in>H. atom i \<sharp> C" shows "H \<turnstile> SyntaxN.All i A IFF SyntaxN.All i A'"
subset_segment_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "(closed_segment a b \<subseteq> closed_segment c d) = (a \<in> closed_segment c d \<and> b \<in> closed_segment c d)"
simps_2: fixes f :: "POPLmark.type \<Rightarrow> POPLmark.type"   and T :: "POPLmark.type" shows "mapB f (TVarB T) = TVarB (f T)"
norm_scaleC: fixes a :: "complex" shows "norm (a *\<^sub>C x) = cmod a * norm x"
hyps_2: fixes P' :: "('a, 'b, 'c) psi"   and b :: "'d" assumes "\<tau> \<prec> C = \<tau> \<prec> P'" shows "Prop b \<Psi> (P \<parallel> !P) P'"
P2_real_model'_def: fixes M :: "'grp \<times> 'grp"   and \<sigma> :: "bool"   and z :: "'e"   and \<A> :: "(bool \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> ((('grp \<times> 'grp \<times> 'grp) \<times> nat) \<times> 'f) spmf) \<times> ('grp \<times> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> 'd spmf)" shows "P2_real_model' M \<sigma> z \<A> = (let (x0, x1) = M; (\<A>1, \<A>2, \<A>3) = \<A> in \<A>1 \<sigma> z \<bind> (\<lambda>((h0, h1, a, b0, b1), s). assert_spmf (h0 \<in> carrier \<G> \<and> h1 \<in> carrier \<G> \<and> a \<in> carrier \<G> \<and> b0 \<in> carrier \<G> \<and> b1 \<in> carrier \<G>) \<bind> (\<lambda>_. \<A>2 (h0, h1, a, b0, b1) s \<bind> (\<lambda>(((in1, in2, in3), r), s'). let (h, a, b) = (h0 \<otimes> inv h1, a, b0 \<otimes> inv b1) in funct_DH_ZK (h, a, b) ((in1, in2, in3), r) \<bind> (\<lambda>(out_zk_funct, uu_). assert_spmf out_zk_funct \<bind> (\<lambda>_. P2_real_model_end M ((h0, h1, a, b0, b1), s') \<A>3))))))"
perm_append_swap: fixes xs :: "'d list"   and ys :: "'d list" shows "xs @ ys <~~> ys @ xs"
dual_assertion_conjunctive: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "dual x \<in> conjunctive.conjunctive inf inf times"
WTrtAAssNT: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and a :: "'addr expr"   and i :: "'addr expr"   and e :: "'addr expr"   and T' :: "ty" assumes "P,E,h \<turnstile> a : NT"   and "P,E,h \<turnstile> i : Integer"   and "P,E,h \<turnstile> e : T'" shows "P,E,h \<turnstile> a\<lfloor>i\<rceil> := e : Void"
union: fixes M :: "'a set set" assumes "\<forall>S\<in>M. S \<subseteq> V \<and> P S" shows "P (\<Union> M)"
ref_def: fixes P :: "('c, 'd) action"   and Q :: "('c, 'd) action" shows "P \<sqsubseteq> Q \<equiv> \<forall>A. (action.relation_of Q \<longrightarrow> action.relation_of P) A"
fps_deriv_def: fixes f :: "'a fps" shows "fps_deriv f = Abs_fps (\<lambda>n. of_nat (n + 1) * fps_nth f (n + 1))"
Q_impl_struct: fixes Q :: "value \<Rightarrow> value \<Rightarrow> bool"   and t\<^sub>1 :: "value"   and t\<^sub>2 :: "value" assumes "value_struct_rel Q"   and "Q t\<^sub>1 t\<^sub>2" shows "veq_structure t\<^sub>1 t\<^sub>2"
cDiff_disjoint: fixes A :: "'f cset"   and B :: "'f cset" shows "cInt A (cDiff B A) = cempty"
mat_ge_gt_trans: fixes sd :: "nat"   and n :: "nat"   and A :: "'a mat"   and B :: "'a mat"   and C :: "'a mat" assumes "sd \<le> n"   and "A \<ge>\<^sub>m B"   and "mat_gt (\<succ>) sd B C"   and "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n" shows "mat_gt (\<succ>) sd A C"
cr_commute_def: shows "cr_commute \<equiv> \<lambda>x y. x = apply_commute y"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "int \<Rightarrow> 'a"   and f2 :: "char list \<Rightarrow> 'a"   and f3 :: "'a"   and f4 :: "char list \<Rightarrow> 'a"   and f5 :: "int \<Rightarrow> 'a"   and f6 :: "int \<Rightarrow> 'a"   and f7 :: "int \<Rightarrow> 'a"   and instr :: "instr" shows "P (case instr of LOADI x \<Rightarrow> f1 x | LOAD x \<Rightarrow> f2 x | ADD \<Rightarrow> f3 | STORE x \<Rightarrow> f4 x | JMP x \<Rightarrow> f5 x | JMPLESS x \<Rightarrow> f6 x | JMPGE x \<Rightarrow> f7 x) = ((\<forall>x1. instr = LOADI x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. instr = LOAD x2 \<longrightarrow> P (f2 x2)) \<and> (instr = ADD \<longrightarrow> P f3) \<and> (\<forall>x4. instr = STORE x4 \<longrightarrow> P (f4 x4)) \<and> (\<forall>x5. instr = JMP x5 \<longrightarrow> P (f5 x5)) \<and> (\<forall>x6. instr = JMPLESS x6 \<longrightarrow> P (f6 x6)) \<and> (\<forall>x7. instr = JMPGE x7 \<longrightarrow> P (f7 x7)))"
elims: fixes x :: "ternaryvalue"   and y :: "ternaryvalue" assumes "eval_ternary_Not x = y" obtains "x = TernaryTrue" and "y = TernaryFalse"   | "x = TernaryFalse" and "y = TernaryTrue"   | "x = TernaryUnknown" and "y = TernaryUnknown"
map_of_eq_dom: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'b) list" assumes "map_of xs = map_of ys" shows "fst ` set xs = fst ` set ys"
sigma_algebra_Pow: fixes sp :: "'a set" shows "sigma_algebra sp (Pow sp)"
dickson_leI: fixes d :: "'a \<Rightarrow> nat"   and s :: "'a"   and m :: "nat"   and t :: "'a" assumes "d s \<le> m"   and "d t \<le> m"   and "s \<preceq> t" shows "dickson_le d m s t"
nprovContain: shows "provRules \<subseteq> nprovRules"
power_mono_even: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "nat"   and a :: "'a"   and b :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "even n"   and "less_eq (abs a) (abs b)" shows "less_eq (power.power one times a n) (power.power one times b n)"
aaaa: shows "count_list bs z + count_list (take (Suc i) cs) z \<le> 1"
rPar1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>Q) P \<alpha> P' A\<^sub>P \<Psi>\<^sub>P"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<alpha>"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* Q"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* \<Psi>\<^sub>P"   and "bn \<alpha> \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C"   and "bn \<alpha> \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) \<alpha> (P' \<parallel> Q) (A\<^sub>P @ A\<^sub>Q) (\<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q)"
Oops2_not_AKcryptSK: fixes evs :: "event list"   and A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and SK :: "nat" assumes "evs \<in> kerbIV_gets"   and "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs" shows "\<not> AKcryptSK servK SK evs"
strictly_maximal_literal_def: fixes atom_ordering :: "('a \<times> 'a) set"   and S :: "'a Literal set"   and A :: "'a Literal" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.strictly_maximal_literal atom_ordering S A \<equiv> A \<in> S \<and> (\<forall>B. B \<in> S \<and> A \<noteq> B \<longrightarrow> propositional_atoms.literal_ordering atom_ordering B A)"
wf_ciD1: fixes ci :: "'a instr \<Rightarrow> (nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog \<Rightarrow> 'b \<Rightarrow> 'a val list \<Rightarrow> 'a val list \<Rightarrow> String.literal \<Rightarrow> String.literal \<Rightarrow> nat \<Rightarrow> 'a frame list \<Rightarrow> bool" assumes "wf_ci ci" shows "ci_stk_offer ci"
mc: fixes Cn :: "'b set \<Rightarrow> 'b set"   and A :: "'b set"   and \<gamma> :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set set"   and \<phi> :: "'b" assumes "Tarskian_logic Cn" shows "Tarskian_logic.meet_contraction A \<gamma> \<phi> \<equiv> \<Inter> (\<gamma> A \<phi>)"
limu: shows "((\<lambda>n. real_of_ereal (u n)) \<longlongrightarrow> l) F"
tm_def: shows "consume (Type tn) [TAny, TSome T_i32] = Type tm"
hyp: assumes "i < j"   and "\<not> is_zero_row (mod_type_class.from_nat j) A"   and "j < nrows A" shows "(LEAST n. A $ mod_type_class.from_nat i $ n \<noteq> (0::'a)) < (LEAST n. A $ mod_type_class.from_nat j $ n \<noteq> (0::'a))"
indeces_periodN: fixes w :: "'b list"   and n :: "nat" assumes "w \<noteq> \<epsilon>"   and "n \<noteq> 0"   and "\<And>i. i + n < \<^bold>|w\<^bold>| \<Longrightarrow> w ! i = w ! (i + n)" shows "periodN w n"
vmc_path_ldropn: fixes n :: "nat" assumes "enat n < llength P'" shows "vmc_path G (ldropn n P') (P' $ n) p \<sigma>'"
gbinomial_Vandermonde: fixes a :: "'a"   and b :: "'a"   and n :: "nat" shows "(\<Sum>k = 0..n. (a gchoose k) * (b gchoose (n - k))) = a + b gchoose n"
rotate4: fixes A4 :: "fm"   and A1 :: "fm"   and A2 :: "fm"   and A3 :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A4 (insert A1 (insert A2 (insert A3 H))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 H))) \<turnstile> B"
rank_sup: fixes a :: "hf"   and b :: "hf" shows "rank (a \<squnion> b) = rank a \<squnion> rank b"
rupture_one_not_zero: fixes K :: "'a list set"   and p :: "'a list list" assumes "subfield K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "0 < degree p" shows "\<one>\<^bsub>Rupt\<^bsub>K [X]\<^esub> K p\<^esub> \<noteq> \<zero>\<^bsub>Rupt\<^bsub>K [X]\<^esub> K p\<^esub>"
induct: fixes x1 :: "'c"   and x2 :: "('c, 'd) bintree list"   and P :: "'c \<Rightarrow> ('c, 'd) bintree list \<Rightarrow> bool" assumes "is_heap_list x1 x2"   and "\<And>h. P h []"   and "\<And>h ts t. \<lbrakk>is_heap_list h ts; P h ts; is_heap_list (priority t) (children t); P (priority t) (children t); h \<le> priority t\<rbrakk> \<Longrightarrow> P h (t # ts)" shows "P x1 x2"
right_neutral: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and a :: "'b" assumes "Groups.monoid f z" shows "f a z = a"
all_subset_image: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and P :: "'d set \<Rightarrow> bool" shows "(\<forall>B\<subseteq>f ` A. P B) = (\<forall>B\<subseteq>A. P (f ` B))"
ESem_considers_fv': fixes Fn :: "('a \<rightarrow> 'a) \<rightarrow> 'a"   and Fn_project :: "'a \<rightarrow> 'a \<rightarrow> 'a"   and B :: "bool discr \<rightarrow> 'a"   and B_project :: "'a \<rightarrow> 'a \<rightarrow> 'a \<rightarrow> 'a"   and tick :: "'a \<rightarrow> 'a"   and e :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> 'a" shows "semantic_domain.ESem Fn Fn_project B B_project tick e\<cdot>\<rho> = semantic_domain.ESem Fn Fn_project B B_project tick e\<cdot>(\<rho> f|` fv e)"
hyps: fixes \<sigma>' :: "'state"   and coll :: "'coll"   and \<sigma> :: "'state" assumes "(\<sigma>', coll) \<in> csmall_nstep P \<sigma> n"   and "R P coll"   and "Q P \<sigma>" shows "R' P \<sigma> \<sigma>' coll"
lift_rcond_def: fixes b :: "('b, 'c) uexpr" shows "\<lceil>b\<rceil>\<^sub>\<leftarrow> = \<lceil>b\<rceil>\<^sub><"
assms_1: shows "functor (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>[\<^sub>A\<^sub>2,\<^sub>B\<^sub>]) F"
suntil_reps: fixes y :: "('s \<times> ('c \<Rightarrow> real)) stream" assumes "\<forall>s\<in>sset (smap abss y). s \<in> \<S>"   and "(holds \<phi>' suntil holds \<psi>') (s' ## smap abss y)" shows "(holds \<phi> suntil holds \<psi>) (s ## y)"
gPB_def: shows "gPB = gC ` \<Phi> \<union> Fax \<union> Wax"
that: fixes d :: "real" assumes "0 < d"   and "content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) < e" shows "thesis"
l1_refines_trans_s0: shows "{R01s} TS.trans s0, TS.trans l1 {> R01s}"