hcju/isabelleps huggingface dataset:
order_le_less_subst2: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd"   and c :: "'d" assumes "a \<le> b"   and "f b < c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "f a < c"
smc_dg_smc_10: fixes \<aa> :: "V" shows "smc_dg (smc_10 \<aa>) = dg_10 \<aa>"
max_array_size_def: shows "max_array_size = (65535::'a)"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "P 0"   and "P (Suc 0)"   and "\<And>h. \<lbrakk>P (h + 1); P h\<rbrakk> \<Longrightarrow> P (Suc (Suc h))" shows "P a0"
card_le_inj: fixes A :: "'d set"   and B :: "'e set" assumes "finite A"   and "finite B"   and "card A \<le> card B" shows "\<exists>f. f ` A \<subseteq> B \<and> inj_on f A"
insert_compr: fixes a :: "'a"   and B :: "'a set" shows "insert a B = {x. x = a \<or> x \<in> B}"
matches_list_And_concat: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m1 :: "'b match_expr"   and a :: "action"   and p :: "'c"   and m1' :: "'b match_expr list"   and m2 :: "'b match_expr"   and m2' :: "'b match_expr list" assumes "matches \<gamma> m1 a p = match_list \<gamma> m1' a p"   and "matches \<gamma> m2 a p = match_list \<gamma> m2' a p" shows "matches \<gamma> (MatchAnd m1 m2) a p = match_list \<gamma> (concat (map (\<lambda>x. map (MatchAnd x) m2') m1')) a p"
generate_valid_topology_SOME_nodes: fixes M :: "'b SecurityInvariant_configured list"   and V :: "'b set"   and E :: "('b \<times> 'b) set" shows "nodes (generate_valid_topology_SOME M \<lparr>nodes = V, edges = E\<rparr>) = V"
fold_congs_1: fixes r :: "('c, 'd) partial_object_scheme"   and r' :: "('c, 'd) partial_object_scheme"   and v' :: "'c set"   and f :: "'c set \<Rightarrow> 'c set"   and f' :: "'c set \<Rightarrow> 'c set" assumes "r = r'"   and "carrier r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "carrier_update f r = carrier_update f' r'"
lproject_to_lselect: fixes A :: "'b set"   and w :: "'b llist" shows "lproject A w = lselect (liset A w) w"
Lp_space: fixes p :: "real"   and M :: "'b measure" assumes "0 < p" shows "space\<^sub>N (\<LL> (ennreal p) M) = {f \<in> borel_measurable M. integrable M (\<lambda>x. \<bar>f x\<bar> powr p)}"
finite_mult_not_eq_zero_rightI: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "finite {a. f a \<noteq> (0::'b)}" shows "finite {a. f a * g a \<noteq> (0::'b)}"
is_projection_on_is_cadjoint: fixes \<pi> :: "'b \<Rightarrow> 'b"   and M :: "'b set" assumes "is_projection_on \<pi> M"   and "closed_csubspace M" shows "is_cadjoint \<pi> \<pi>"
nat_of_inverse: fixes x :: "'b sat" shows "Abs_sat (nat_of x) = x"
ll_on_open_congI: fixes Y :: "'a set"   and S :: "real set"   and g :: "real \<Rightarrow> 'a \<Rightarrow> 'a" assumes "Y = Y"   and "S = S"   and "\<And>x t. \<lbrakk>x \<in> Y; t \<in> S\<rbrakk> \<Longrightarrow> g t x = g t x" shows "ll_on_open S g Y"
Gcd_fin_dvd: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and A :: "'a set" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "a \<in> A" shows "dvd.dvd times (semiring_gcd.Gcd_fin one zero gcd A) a"
rel_compp: fixes R1 :: "'d \<Rightarrow> 'h \<Rightarrow> bool"   and S1 :: "'h \<Rightarrow> 'f \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'i \<Rightarrow> bool"   and S2 :: "'i \<Rightarrow> 'g \<Rightarrow> bool" shows "rel_nondetT' (R1 OO S1) (R2 OO S2) = rel_nondetT' R1 R2 OO rel_nondetT' S1 S2"
Rep_zero_matrix_def: fixes j :: "nat"   and i :: "nat" shows "Rep_matrix 0 j i = (0::'b)"
prod_power_distrib: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and n :: "nat" shows "prod f A ^ n = (\<Prod>x\<in>A. f x ^ n)"
diff_fun_space_eq: shows "diff_fun_space = {f. diff_fun k charts f} \<inter> {f. extensional0 carrier f}"
right_neutral: fixes a :: "'b \<Rightarrow> 'b" shows "a \<circ> id = a"
rotate_to_def: fixes vs :: "'a list"   and v :: "'a" shows "rotate_to vs v \<equiv> v # snd (splitAt v vs) @ fst (splitAt v vs)"
Confidentiality_Auth_A: fixes A :: "agent"   and authK :: "nat"   and Ta :: "nat"   and authTicket :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace> \<in> parts (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> kerbIV_gets"   and "\<not> expiredAK Ta evs"   and "A \<notin> bad"   and "evs \<in> kerbIV_gets" shows "Key authK \<notin> analz (knows Spy evs)"
simps_2: fixes A :: "real mat"   and m :: "real mat convnet"   and inputs :: "real vec list" shows "evaluate_net (Conv A m) inputs = A *\<^sub>v evaluate_net m inputs"
join_mono: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "compatible a b"   and "compatible c d"   and "a \<sqsubseteq> c"   and "b \<sqsubseteq> d" shows "a \<squnion> b \<sqsubseteq> c \<squnion> d"
col_ABC_ABD_2: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and D :: "Points" assumes "A \<noteq> B"   and "col A B C"   and "col A B D" shows "col A C D"
handshake_phase_inv_def: shows "handshake_phase_inv = (\<lambda>s. \<forall>x. (sys_ghost_hs_in_sync x s, sys_hs_type s, sys_ghost_hs_phase s, ghost_hs_phase (s (mutator x))) \<in> hp_step_rel \<and> (sys_hs_pending x s \<longrightarrow> \<not> sys_ghost_hs_in_sync x s))"
a: shows "R[a\<turnstile>c>b] = NotR x.N c"
mult_isor: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "\<lparr>x\<rparr>z\<lparr>y \<cdot> z\<rparr>"
S_inductive_Sup_in_Chain: fixes X :: "'a set" assumes "S_inductive_set D"   and "Chain D X"   and "ExPre (Iod D (insert (Algebra2.Sup D X) X)) (Algebra2.Sup D X)" shows "Algebra2.Sup D X \<in> X"
not_in_dom: fixes i :: "'a"   and x :: "'b"   and y :: "'b" assumes "i \<notin> dom" shows "\<not> R i x y"
coCallsLub_is_lub: fixes S :: "CoCalls set" shows "S <<| coCallsLub S"
ModuleEnd_axioms_def: fixes M :: "'a set"   and T :: "'a \<Rightarrow> 'a" shows "ModuleEnd_axioms M T \<equiv> T ` M \<subseteq> M"
min_simps_2: fixes b :: "'c"   and a :: "'c" assumes "b < a" shows "min a b = b"
list_frag_HOL4_def: fixes l :: "'b list"   and frag :: "'b list" shows "list_frag_HOL4 l frag \<equiv> \<exists>pfx sfx. pfx @ frag @ sfx = l"
IH: shows "evaluate_dec ck mn env s1 d (s2, Rval new_env) \<and> evaluate_decs ck mn (extend_dec_env new_env env) s2 ds (s3, r) \<and> (\<forall>extra count count' s' r'. clock s2 = count + extra \<and> clock s' = count' + extra \<and> (s3, r) = (s', r') \<and> ck = True \<longrightarrow> evaluate_decs ck mn (extend_dec_env new_env env) (update_clock (\<lambda>_. count) s2) ds (update_clock (\<lambda>_. count') s', r'))"
pr_zero: shows "(\<lambda>x. 0) \<in> PrimRec1"
rangeI: fixes f :: "'g \<Rightarrow> 'f"   and x :: "'g" shows "f x \<in> range f"
inflation: fixes x :: "'a" assumes "x \<in> carrier \<X>" shows "x \<sqsubseteq>\<^bsub>\<X>\<^esub> \<pi>\<^sub>* (\<pi>\<^sup>* x)"
wf_lenlex: fixes r :: "('b \<times> 'b) set" assumes "wf r" shows "wf (lenlex r)"
le_fin_concat_start: fixes w :: "'item list"   and w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list" shows "(w @ w\<^sub>1 \<preceq>\<^sub>F w @ w\<^sub>2) = (w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2)"
simplex_map_id_gen: fixes p :: "nat"   and X :: "'c topology"   and c :: "(nat \<Rightarrow> real) \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'c" assumes "singular_simplex p X c"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = x" shows "simplex_map p f c = c"
cases: fixes a1 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and a2 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "a1 \<approx> a2" obtains \<R> :: "(((('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> (('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set" and lc\<^sub>1 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" and lc\<^sub>2 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" where "a1 = lc\<^sub>1" and "a2 = lc\<^sub>2" and "strong_low_bisim_mm \<R>" and "(lc\<^sub>1, lc\<^sub>2) \<in> \<R>"
unit_vec_eq: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < n" shows "(unit_vec n i = unit_vec n j) = (i = j)"
distinct_17: fixes x2 :: "'a basic_type"   and x3 :: "'a basic_type" shows "x2[1] \<noteq> x3[?]"
s_prod_isol: fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set"   and z :: "('a \<times> 'a set) set" assumes "x \<subseteq> y" shows "z \<cdot> x \<subseteq> z \<cdot> y"
no_smallest_element_skippedD: fixes E :: "'a set"   and F :: "'a set"   and e :: "'a"   and g :: "'b" assumes "no_smallest_element_skipped E F"   and "e \<in> carrier - E"   and "weight e < g"   and "indep (insert e (limi F g))" shows "e \<in> limi F g"
minus_iarray_def: fixes A :: "'c iarray"   and B :: "'c iarray" shows "A - B = (let length_A = IArray.length A; length_B = IArray.length B; n = max length_A length_B; A' = IArray.of_fun (\<lambda>a. if a < length_A then A !! a else (0::'c)) n; B' = IArray.of_fun (\<lambda>a. if a < length_B then B !! a else (0::'c)) n in IArray.of_fun (\<lambda>a. A' !! a - B' !! a) n)"
freshenLC_eq_iff: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and a :: "nat"   and v :: "'a"   and a' :: "nat"   and v' :: "'a" assumes "Abstract_Formulas freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP" shows "(freshenLC a v = freshenLC a' v') = (a = a' \<and> v = v')"
K: fixes x :: "'a" shows "norm (f x) \<le> norm x * K"
assms_1: shows "w\<^sub>1 =\<^sub>I w\<^sub>2"
cgenideal_ideal: fixes a :: "'a list" assumes "a \<in> carrier (K [X])" shows "ideal (PIdl\<^bsub>K [X]\<^esub> a) (K [X])"
locally_finite_on_closure_Union: fixes X :: "'c set"   and I :: "'d set"   and U :: "'d \<Rightarrow> 'c set" assumes "locally_finite_on X I U"   and "\<And>i. i \<in> I \<Longrightarrow> closure (U i) \<subseteq> X" shows "(\<Union>i\<in>I. closure (U i)) = closure (\<Union> (U ` I)) \<inter> X"
comp_id: fixes f :: "'c \<Rightarrow> 'd" shows "f \<circ> id = f"
prod_state2_def: fixes u :: "complex Matrix.mat" shows "prod_state2 u \<equiv> if state 2 u then \<exists>v w. state 1 v \<and> state 1 w \<and> u = v \<Otimes> w else undefined"
closed_segment_subset: fixes x :: "'a"   and S :: "'a set"   and y :: "'a" assumes "x \<in> S"   and "y \<in> S"   and "convex S" shows "{x--y} \<subseteq> S"
sbelow_sctns_coll_impl_def: fixes ai :: "'b list list"   and sctnsi :: "real list sctn list" shows "sbelow_sctns_coll_impl ai sctnsi \<equiv> RETURN ai \<bind> (\<lambda>xa. FORWEAK_LIST xa (RETURN True) (\<lambda>xb. RETURN (sbelow_sctns_impl xb sctnsi)) (\<lambda>xb xc. RETURN (xb \<and> xc)))"
comp_null_2: fixes f :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" shows "S.VV.comp f S.VV.null = S.VV.null"
distinct_17: fixes x2 :: "nat"   and x4 :: "nat" shows "Number x2 \<noteq> Pan x4"
cases: fixes a1 :: "(name \<times> ty) list"   and a2 :: "trm"   and a3 :: "ty" assumes "a1 \<turnstile> a2 : a3" obtains \<Gamma> :: "(name \<times> ty) list" and x :: "name" and T :: "ty" where "a1 = \<Gamma>__" and "a2 = Var x__" and "a3 = T" and "valid \<Gamma>__" and "(x__, T) \<in> set \<Gamma>__"   | \<Gamma> :: "(name \<times> ty) list" and e\<^sub>1 :: "trm" and T\<^sub>1 :: "ty" and T\<^sub>2 :: "ty" and e\<^sub>2 :: "trm" where "a1 = \<Gamma>'__" and "a2 = App e\<^sub>1 e\<^sub>2" and "a3 = T\<^sub>2__" and "\<Gamma>'__ \<turnstile> e\<^sub>1 : T\<^sub>1__\<rightarrow>T\<^sub>2__" and "\<Gamma>'__ \<turnstile> e\<^sub>2 : T\<^sub>1__"   | x :: "name" and \<Gamma> :: "(name \<times> ty) list" and T\<^sub>1 :: "ty" and e :: "trm" and T\<^sub>2 :: "ty" where "a1 = \<Gamma>" and "a2 = Lam [x].e" and "a3 = T\<^sub>1\<rightarrow>T\<^sub>2" and "x \<sharp> \<Gamma>" and "(x, T\<^sub>1) # \<Gamma> \<turnstile> e : T\<^sub>2"
case_one_psi_total: fixes s :: "nat \<Rightarrow> nat option"   and j :: "nat"   and x :: "nat" assumes "total_cons s"   and "\<nexists>t. eval (total_cons.r_prefixes s) [t, j] \<up>"   and "0 < x" shows "eval (total_cons.r_psi s) [j, x] \<down>= 0 \<or> eval (total_cons.r_psi s) [j, x] \<down>= 1"
l6_4_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Out Bet P A B"   and "Tarski_neutral_dimensionless.Col Bet A P B" shows "\<not> Bet A P B"
Minus: shows "\<forall>x\<in>P. \<forall>y\<in>P. x - y \<in> P"
pdt_trans_in_pdt: fixes x :: "'n"   and y :: "'n" assumes "(x, y) \<in> pdt\<^sup>+" shows "(x, y) \<in> pdt"
cmem_empty: fixes P :: "heap \<Rightarrow> bool"   and update :: "'c \<Rightarrow> 'd \<Rightarrow> unit Heap"   and lookup :: "'c \<Rightarrow> 'd option Heap"   and empty :: "heap"   and dp :: "'c \<Rightarrow> 'd" assumes "dp_consistency_heap_empty P update lookup empty" shows "dp_consistency.cmem (heap_mem_defs.lookup' lookup) dp empty"
to_tree_order_get_root_node: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and to :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and root_ptr :: "(_) object_ptr"   and ptr'' :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.to_tree_order ptr \<rightarrow>\<^sub>r to"   and "ptr' \<in> set to"   and "h \<turnstile> Shadow_DOM.get_root_node ptr' \<rightarrow>\<^sub>r root_ptr"   and "ptr'' \<in> set to" shows "h \<turnstile> Shadow_DOM.get_root_node ptr'' \<rightarrow>\<^sub>r root_ptr"
logD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m Log]"   and "matches \<gamma> m p"   and "s = Undecided" shows "t = Undecided"
assms: shows "assoc_match (Ty a Ts) (Ty b Us) subs = Some subs'"
of_int_le_0_iff: 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 z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (ring_1.of_int one plus zero minus z) zero = (z \<le> 0)"
divmod_nat_def: fixes m :: "nat"   and n :: "nat" shows "divmod_nat m n = (m div n, m mod n)"
cdj: shows "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j"
bmod_assoc_weak_aux: fixes f :: "'a \<Rightarrow> 'c"   and u :: "'a"   and g :: "'a \<Rightarrow> 'c"   and h :: "'a \<Rightarrow> 'c"   and y :: "'a" shows "f u \<cdot> \<Squnion>{g v \<cdot> h z |v z. \<rho> y v z} = \<Squnion>{f u \<cdot> g v \<cdot> h z |v z. \<rho> y v z}"
simps_1: fixes Chn :: "'b"   and Dom :: "'b span_data"   and Cod :: "'b span_data"   and more :: "'c" shows "Chn \<lparr>Chn = Chn, Dom = Dom, Cod = Cod, \<dots> = more\<rparr> = Chn"
strict_subset_subsumption_redundant_clss: fixes D :: "'a literal multiset"   and \<sigma> :: "'s"   and C :: "'a literal multiset"   and CC :: "'a literal multiset set" assumes "D \<cdot> \<sigma> \<subset># C"   and "is_ground_subst \<sigma>"   and "D \<in> CC" shows "C \<in> sr.Rf (grounding_of_clss CC)"
function_factors_left: fixes g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'c" shows "(\<forall>x y. g x = g y \<longrightarrow> f x = f y) = (\<exists>h. f = h \<circ> g)"
elims: fixes x :: "nat"   and y :: "hf" assumes "htuple x = y" obtains "x = 0" and "y = \<langle>0, 0\<rangle>"   | k :: "nat" where "x = Suc k" and "y = \<langle>0, htuple k\<rangle>"
pure_split: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.pure P = (P = (\<lambda>s. True) \<or> P = (\<lambda>s. False))"
map_key_PP_zero: shows "Poly_Mapping.map_key PP 0 = 0"
vfinite_vcard_omega_iff: fixes A :: "V" shows "vfinite A = (vcard A \<in>\<^sub>\<circ> \<omega>)"
zero_less_diff': fixes i :: "nat"   and j :: "nat" assumes "i < j" shows "j - i \<noteq> 0"
simps: fixes sinvar :: "'a graph \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"   and defbot :: "'b"   and receiver_violation :: "bool"   and nP :: "'a \<Rightarrow> 'b" shows "new_configured_SecurityInvariant (sinvar, defbot, receiver_violation, nP) = (if SecurityInvariant sinvar defbot receiver_violation then Some \<lparr>c_sinvar = \<lambda>G. sinvar G nP, c_offending_flows = \<lambda>G. SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP, c_isIFS = receiver_violation\<rparr> else None)"
fresh_at_base_permute_iff: fixes p :: "perm"   and a :: "'b"   and x :: "'a" shows "atom (p \<bullet> a) \<sharp> p \<bullet> x = atom a \<sharp> x"
newton_seq_fact7: fixes p :: "int"   and f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and n :: "nat" assumes "hensel p f a" shows "hensel.newton_seq p f a (Suc n) \<ominus>\<^bsub>padic_int p\<^esub> hensel.newton_seq p f a n = \<ominus>\<^bsub>padic_int p\<^esub> padic_integers.divide p (UP_cring.to_fun (padic_int p) f (hensel.newton_seq p f a n)) (UP_cring.to_fun (padic_int p) (UP_cring.pderiv (padic_int p) f) (hensel.newton_seq p f a n))"
a1: shows "M \<in> Cignorabimus \<alpha> V"
assms_1: shows "(f \<longlongrightarrow> l) F"
assms_3: fixes x :: "'a" assumes "x \<in> carrier G" shows "\<chi> x = \<chi>' x"
not0: shows "0 \<notin> norm ` comb' ` sphere 0 1"
assms: shows "it = foldli l \<and> it' = foldli l"
Vinv1_finite_vote_set: fixes s :: "'a v_state_scheme"   and Q :: "process set" assumes "s \<in> Vinv1" shows "finite (vote_set (votes s) Q)"
split_domI: fixes U :: "'x set"   and S :: "('x \<Rightarrow>\<^sub>0 nat) set"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "finite X"   and "U \<subseteq> X"   and "finite S" shows "split_dom TYPE('b) (t, U, S)"
universal_domain_def: fixes A :: "'a set"   and Is :: "'b set"   and P :: "'b \<Rightarrow> ('a \<times> 'a) set" shows "universal_domain A Is P \<equiv> profile A Is P"
T_small: fixes a :: "'a"   and b :: "real" assumes "a \<in> S"   and "b \<in> T" shows "X + b *\<^sub>R (a - X) \<in> S"
simps_19: shows "cospan_shape.comp cospan_shape.AT cospan_shape.TT = cospan_shape.Null"
transC_MtransC: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and cf :: "('a, 'c) com \<times> 'b"   and cf' :: "('a, 'c) com \<times> 'b" assumes "PL.transC tval aval cf cf'" shows "PL.MtransC tval aval cf cf'"
rep_eq: fixes x :: "unit2"   and xa :: "unit2" shows "(x \<le> xa) = (Rep_unit2 x \<le> Rep_unit2 xa)"
snd_eq_Range: fixes R :: "('d \<times> 'c) set" shows "snd ` R = Range R"
nth_least_eq: fixes n :: "nat"   and A :: "'b set"   and B :: "'b set" assumes "enat n < esize A"   and "enat n < esize B"   and "\<And>i. \<lbrakk>i \<le> nth_least A n; i \<le> nth_least B n\<rbrakk> \<Longrightarrow> (i \<in> A) = (i \<in> B)" shows "nth_least A n = nth_least B n"
CESem_simps_no_tick_3: fixes x :: "Vars.var"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" shows "\<N>\<lbrakk> Var x \<rbrakk>\<^bsub>\<rho>\<^esub> \<sqsubseteq> \<rho> x"
assms_2: shows "io_targets M q1 (vs @ xs) = {q3}"
wlog_rotation_to_positive_x_axis: fixes u :: "complex_homo"   and P :: "complex_homo \<Rightarrow> bool" assumes "u \<in> unit_disc"   and "u \<noteq> 0\<^sub>h"   and "\<And>\<phi> u. \<lbrakk>u \<in> unit_disc; u \<noteq> 0\<^sub>h; P (moebius_pt (moebius_rotation \<phi>) u)\<rbrakk> \<Longrightarrow> P u"   and "\<And>x. \<lbrakk>is_real x; 0 < Re x; Re x < 1\<rbrakk> \<Longrightarrow> P (of_complex x)" shows "P u"
range_vars_def: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "range_vars \<sigma> = \<Union> (vars_term ` subst_range \<sigma>)"
0: shows "while_option (\<lambda>(ws, s). ws \<noteq> []) (\<lambda>(ws, s). case ws of x # ws' \<Rightarrow> (succs x @ ws', f x s)) (ws, s) = Some (ws', s')"
norm_mult: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.real_normed_div_algebra divide inverse one times plus zero minus uminus scaleR dist norm sgn uniformity open" shows "norm (times x y) = norm x * norm y"
pairwise_mp: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a stream"   and Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "PTA_Reachability.pairwise P xs"   and "\<And>x y. \<lbrakk>x \<in> sset xs; y \<in> sset xs; P x y\<rbrakk> \<Longrightarrow> Q x y" shows "PTA_Reachability.pairwise Q xs"
nonempty_def: fixes X :: "'e set" shows "nonempty X = (X \<noteq> {})"
UNIF1_subst_domain_range_vars_Int: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset"   and E' :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "UNIF1 \<sigma> E E'" shows "subst_domain \<sigma> \<inter> range_vars \<sigma> = {}"
evalOP_deterministic: fixes P :: "db"   and V :: "db"   and V' :: "db" assumes "P \<Down> V"   and "P \<Down> V'" shows "V = V'"
characteristic_formula_def: fixes A :: "('b \<Rightarrow> bool) set"   and s :: "'b" shows "characteristic_formula A s \<equiv> AND {atom\<^sub>p(p) |p. p \<in> A \<and> p s} and\<^sub>p AND {not\<^sub>p atom\<^sub>p(p) |p. p \<in> A \<and> \<not> p s}"
sm: fixes x :: "'a" assumes "norm (x - \<xi>) < r" shows "(\<lambda>n. a n * (x - \<xi>) ^ n) sums f x"
is_unit_left_imp_coprime: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" shows "algebraic_semidom.coprime times one a b"