hcju/isabelleps huggingface dataset:
nonzero_Ker_el_imp_n_inj: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b"   and g :: "'b" assumes "VectorSpaceEnd smult V T"   and "g \<in> V"   and "g \<noteq> (0::'b)"   and "T g = (0::'b)" shows "\<not> inj_on T V"
simps: fixes a :: "exp" shows "isval a = ((\<exists>n. a = ENat n) \<or> (\<exists>x e. a = ELam x e))"
map_leD: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option"   and k :: "'a"   and v :: "'b" assumes "m1 \<subseteq>\<^sub>m m2"   and "m1 k = Some v" shows "m2 k = Some v"
Weierstrass_m_test_ev: fixes A :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and M :: "nat \<Rightarrow> real" assumes "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>A. norm (f n x) \<le> M n"   and "summable M" shows "uniform_limit A (\<lambda>n x. \<Sum>i<n. f i x) (\<lambda>x. \<Sum>i. f i x) sequentially"
cos_times_cos: fixes w :: "'a"   and z :: "'a" shows "cos w * cos z = (cos (w - z) + cos (w + z)) / (2::'a)"
sum_single: fixes A :: "'b set"   and i :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "finite A"   and "\<And>j. \<lbrakk>j \<noteq> i; j \<in> A\<rbrakk> \<Longrightarrow> f j = (0::'c)" shows "sum f A = (if i \<in> A then f i else (0::'c))"
retractionE: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b" assumes "category C"   and "category.retraction C g" obtains f :: "'b" where "partial_magma.ide C (C g f)"
distinct_2: fixes x2 :: "nat"   and x1 :: "nat" shows "Intv x2 \<noteq> Const x1"
sgn_vec_def: fixes x :: "('a, 'b) vec" shows "sgn x = x /\<^sub>R norm x"
take_bit_nat_eq_self_iff: fixes n :: "nat"   and m :: "nat" shows "(take_bit n m = m) = (m < 2 ^ n)"
domSubset3: fixes x :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" shows "dom (Cp (DenyAll \<oplus> x)) = dom (Cp DenyAll)"
finite_primes_le: fixes x :: "real" shows "finite {p. prime p \<and> real p \<le> x}"
base: fixes f :: "'b \<Rightarrow> real"   and x :: "nat \<Rightarrow> real"   and A :: "nat \<Rightarrow> 'b set"   and S :: "nat set"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" assumes "f = (\<lambda>t. \<Sum>i\<in>S. x i * \<chi> (A i) t)"   and "\<forall>i\<in>S. A i \<in> measurable_sets M"   and "nonnegative x"   and "finite S"   and "\<forall>i\<in>S. \<forall>j\<in>S. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"   and "\<Union> (A ` S) = UNIV" shows "(\<Sum>i\<in>S. x i * Measure.measure M (A i)) \<in> sfis f M"
BC_3: shows "Q' * P' = 1\<^sub>m n"
measurable_embed_measure1: fixes g :: "'m \<Rightarrow> 'l"   and f :: "'k \<Rightarrow> 'm"   and M :: "'k measure"   and N :: "'l measure" assumes "(\<lambda>x. g (f x)) \<in> M \<rightarrow>\<^sub>M N" shows "g \<in> embed_measure M f \<rightarrow>\<^sub>M N"
P_not_null: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c" assumes "vm_path G P v0" shows "\<not> lnull P"
f0_2: shows "\<forall>i j. \<phi> (the (f\<^sub>0 i j)) = local.adverse 0 i j"
coboundedI1: fixes c :: "'c DBMEntry"   and a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "c \<preceq> a" shows "c \<preceq> linordered_monoid.max a b"
mktop_def: fixes p :: "alt \<Rightarrow> real"   and b :: "alt" shows "mktop p b \<equiv> p(b := Max (range p) + 1)"
induct: fixes P :: "D2 \<Rightarrow> bool"   and D2 :: "D2" assumes "P D1"   and "P D2" shows "P D2"
list_trisect: fixes x :: "nat"   and lst :: "'a list" assumes "x < length lst" shows "[0..<length lst] = [0..<x] @ x # [Suc x..<length lst]"
simps_3: fixes master :: "clock"   and slave :: "clock" shows "\<lbrakk> master implies slave \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = {\<rho>. \<forall>n. hamlet (Rep_run \<rho> n master) \<longrightarrow> hamlet (Rep_run \<rho> n slave)}"
MakeFtor_def: fixes F :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) Functor_scheme" shows "MakeFtor F \<equiv> \<lparr>CatDom = CatDom F, CatCod = CatCod F, MapM = restrict ((##) F) (mor\<^bsub>CatDom F\<^esub>), \<dots> = Functor.more F\<rparr>"
nderiv_def: fixes i :: "nat"   and f :: "real \<Rightarrow> 'a"   and X :: "real set" shows "nderiv i f X = ((\<lambda>f x. vector_derivative f (at x within X)) ^^ i) f"
C'_sees: shows "P \<turnstile> C' sees M, b' :  Ts'\<rightarrow>T' = m' in D'"
assms: fixes k :: "'k" shows "proj_poly k p = proj_poly k q"
full_elims_2: fixes n :: "nat"   and l :: "'b tree234"   and p :: "'b"   and r :: "'b tree234" assumes "full n \<langle>l, p, r\<rangle>" obtains na :: "nat" where "n = Suc na" and "full na l" and "full na r"
real_sgn_eq: fixes x :: "real" shows "sgn x = x / \<bar>x\<bar>"
double_diff: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" assumes "A \<subseteq> B"   and "B \<subseteq> C" shows "B - (C - A) = A"
sblank_def: fixes P :: "'b prog"   and C :: "char list" shows "sblank P C \<equiv> init_sfields (isfields P C)"
is_nonterminal_item_nonterminal: fixes x :: "item" assumes "wellformed_item x" shows "is_nonterminal (item_nonterminal x)"
enabled_sdrop: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state"   and \<omega> :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream"   and n :: "nat" assumes "enabled cf \<omega>" shows "enabled ((cf ## \<omega>) !! n) (sdrop n \<omega>)"
a2: shows "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
open_segment_as_ball: fixes a :: "'a"   and b :: "'a" shows "open_segment a b = affine hull {a, b} \<inter> ball ((a + b) /\<^sub>R 2) (norm (b - a) / 2)"
derivablep_derivable_eq: fixes R :: "('b sequent list \<times> 'b sequent) set" shows "derivablep (\<lambda>x. x \<in> R) = (\<lambda>x xa. (x, xa) \<in> derivable R)"
lcf_eq: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "f (deg R f) = monom P (f (deg R f)) (deg R f) (deg R (monom P (f (deg R f)) (deg R f)))"
bnds_mult: fixes l :: "float"   and u :: "float"   and prec :: "nat"   and a1 :: "float"   and a2 :: "float"   and b1 :: "float"   and b2 :: "float"   and a :: "real"   and b :: "real" assumes "(l, u) = bnds_mult prec a1 a2 b1 b2"   and "a \<in> {real_of_float a1..real_of_float a2}"   and "b \<in> {real_of_float b1..real_of_float b2}" shows "a * b \<in> {real_of_float l..real_of_float u}"
strong_induct: fixes P :: "'a \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a"   and ty :: "ty" assumes "\<And>tvar z. P z (TVar tvar)"   and "\<And>ty1 ty2 z. \<lbrakk>\<And>z. P z ty1; \<And>z. P z ty2\<rbrakk> \<Longrightarrow> P z (ty1\<rightarrow>ty2)" shows "P z ty"
PO_s0g_dom_trans: shows "{s0g_dom} TS.trans s0g {> s0g_dom}"
disjoint_iff: fixes A :: "'k set"   and B :: "'k set" shows "(A \<inter> B = {}) = (\<forall>x. x \<in> A \<longrightarrow> x \<notin> B)"
\<sigma>3': shows "\<sigma>3' \<in> seq ars"
Poly_Mapping_def: fixes istate :: "'a istate" shows "Poly_Mapping istate = (case istate of IState x1 x2 x3 x4 x5 \<Rightarrow> x4)"
that: fixes \<eta> :: "'s"   and \<eta>s :: "'s list"   and \<eta>2 :: "'s"   and CAs0 :: "'a literal multiset list"   and DA0 :: "'a literal multiset"   and AAs0 :: "'a multiset list"   and As0 :: "'a list"   and \<tau> :: "'s"   and E0 :: "'a literal multiset" assumes "is_ground_subst \<eta>"   and "is_ground_subst_list \<eta>s"   and "is_ground_subst \<eta>2"   and "ord_resolve_rename S CAs0 DA0 AAs0 As0 \<tau> E0"   and "CAs0 \<cdot>\<cdot>cl \<eta>s = CAs"   and "DA0 \<cdot> \<eta> = DA"   and "E0 \<cdot> \<eta>2 = E"   and "{DA0} \<union> set CAs0 \<subseteq> M"   and "length CAs0 = length CAs"   and "length \<eta>s = length CAs" shows "thesis"
val_equiv_axiom2: fixes v :: "'a \<Rightarrow> ant"   and v' :: "'a \<Rightarrow> ant" assumes "valuation K v"   and "valuation K v'"   and "v_equiv K v v'" shows "v_equiv K v' v"
division_of_subset: fixes p :: "'c set set"   and q :: "'c set set" assumes "p division_of \<Union> p"   and "q \<subseteq> p" shows "q division_of \<Union> q"
monad_reader_writerT: fixes return :: "'a \<times> 'b list \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b list \<Rightarrow> 'c) \<Rightarrow> 'c"   and ask :: "('d \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_reader return bind ask" shows "monad_reader (return_writer return) (bind_writer return bind) (ask_writer ask)"
nor_sim_2: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H \<^bold>\<not> (F \<^bold>\<or> G)" shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H \<^bold>\<not> G"
a_get_attribute_def: fixes ptr :: "(_) element_ptr"   and k :: "char list" shows "local.a_get_attribute ptr k = Heap_Error_Monad.bind (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr attrs) (\<lambda>m. return (fmlookup m k))"
deduct: shows "grounding_of_clss (N_of_state St' \<union> P_of_state St' \<union> Q_of_state St') - grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St) \<subseteq> concls_of (sr_ext.inferences_from (grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)))"
vector_conv_covector: fixes v :: "'a" shows "(v \<sqinter> top = v) = (top \<sqinter> id v = id v)"
simps_1: fixes p :: "('a, 'b) form" shows "compl (Neg p) = p"
shows_foldr_append: fixes xs :: "'b list"   and showx :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a list"   and r :: "'a list"   and s :: "'a list" assumes "\<And>r s. \<forall>x\<in>set xs. showx x (r @ s) = showx x r @ s" shows "foldr showx xs (r @ s) = foldr showx xs r @ s"
induct: fixes x1 :: "nat \<Rightarrow> classDef option"   and x2 :: "nat list"   and x3 :: "nat list"   and P :: "(nat \<Rightarrow> classDef option) \<Rightarrow> nat list \<Rightarrow> nat list \<Rightarrow> bool" assumes "x1 \<turnstile>+ x2 <: x3"   and "\<And>CT. P CT [] []"   and "\<And>CT C0 D0 Cs Ds. \<lbrakk>CT \<turnstile> C0 <: D0; CT \<turnstile>+ Cs <: Ds; P CT Cs Ds\<rbrakk> \<Longrightarrow> P CT (C0 # Cs) (D0 # Ds)" shows "P x1 x2 x3"
eval_fds_uminus: fixes f :: "'b fds"   and s :: "'b" assumes "fds_converges f s" shows "eval_fds (- f) s = - eval_fds f s"
fref_param1: fixes R :: "('b \<times> 'e) set"   and S :: "('d \<times> 'f) set" shows "R \<rightarrow> S = R \<rightarrow>\<^sub>f S"
comp_char: fixes g :: "'c local.term set"   and f :: "'c local.term set" shows "\<F>\<^sub>SC.comp g f = (if \<F>\<^sub>SC.arr f \<and> \<F>\<^sub>SC.arr g \<and> \<F>C.seq g f then g \<cdot> f else \<F>C.null)"
upto_rec1: fixes i :: "int"   and j :: "int" assumes "i \<le> j" shows "[i..j] = i # [i + 1..j]"
rel_simps_25: fixes n :: "num" shows "(1::'a) \<le> numeral n"
infsetsum_cong: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and B :: "'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x"   and "A = B" shows "infsetsum f A = infsetsum g B"
hastyE_2: fixes xm :: "ExVar"   and em :: "Ex"   and evm :: "ExVar \<Rightarrow> Val option"   and u :: "Ty"   and t :: "Ty" assumes "v_clos \<langle>xm , em , evm\<rangle> hasty u -> t" obtains te :: "ExVar \<Rightarrow> Ty option" where "te \<turnstile> fn xm => em ===> u -> t" and "dom evm = dom te" and "\<forall>x. x \<in> dom te \<longrightarrow> the (evm x) hasty the (te x)"
prod_decode_inverse: fixes n :: "nat" shows "prod_encode (prod_decode n) = n"
word_ao_absorbs_1: fixes x :: "'b word"   and y :: "'b word" shows "x AND (y OR x) = x"
flush_simulates_filter_writes: fixes m :: "nat \<Rightarrow> nat"   and sb :: "'b memref list"   and \<O> :: "'c"   and \<R> :: "'d"   and \<S> :: "'e"   and m' :: "nat \<Rightarrow> nat"   and sb' :: "'b memref list"   and \<O>' :: "'c"   and \<R>' :: "'d"   and \<S>' :: "'e"   and sb\<^sub>h :: "'b memref list"   and \<O>\<^sub>h :: "nat set"   and \<R>\<^sub>h :: "nat \<Rightarrow> bool option"   and \<S>\<^sub>h :: "nat \<Rightarrow> bool option" assumes "(m, sb, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>w (m', sb', \<O>', \<R>', \<S>')"   and "sb = filter is_Write\<^sub>s\<^sub>b sb\<^sub>h" shows "\<exists>sb\<^sub>h' \<O>\<^sub>h' \<R>\<^sub>h' \<S>\<^sub>h'. (m, sb\<^sub>h, \<O>\<^sub>h, \<R>\<^sub>h, \<S>\<^sub>h) \<rightarrow>\<^sub>f\<^sup>* (m', sb\<^sub>h', \<O>\<^sub>h', \<R>\<^sub>h', \<S>\<^sub>h') \<and> sb' = filter is_Write\<^sub>s\<^sub>b sb\<^sub>h' \<and> (sb' = [] \<longrightarrow> sb\<^sub>h' = [])"
update_convs_1: fixes break_status' :: "bool \<Rightarrow> bool"   and break_status :: "bool"   and return_status :: "bool"   and more :: "'a" shows "break_status_update break_status' \<lparr>break_status = break_status, return_status = return_status, \<dots> = more\<rparr> = \<lparr>break_status = break_status' break_status, return_status = return_status, \<dots> = more\<rparr>"
simps_1: fixes bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and t :: "'a"   and x :: "'b"   and m :: "'c"   and ta :: "('a, 'd, 'e) new_thread_action" shows "nta_bisim bisim (NewThread t x m) ta = (\<exists>x' m'. ta = NewThread t x' m' \<and> bisim t (x, m) (x', m'))"
real_of_int_floor_gt_diff_one: fixes r :: "real" shows "r - 1 < real_of_int \<lfloor>r\<rfloor>"
simps: fixes a :: "intv" shows "isIntv a = (\<exists>uu_. a = Intv uu_)"
AbstrLevels_S1_A12: fixes i :: "AbstrLevelsID" assumes "sS1 \<in> AbstrLevel i" shows "sA12 \<notin> AbstrLevel i"
list_of_dlist: fixes x :: "'d dlist" shows "list_of_dlist x \<in> {xs. distinct xs}"
game1'_eq: shows "ind_cca'.game \<A> = map_spmf fst (game01' oracle_decrypt1' \<A>)"
simps_2: fixes n :: "nat" shows "similar' (Suc n) = similar'_step (similar' n)"
list_emb_suffix: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" assumes "list_emb P xs ys"   and "suffix ys zs" shows "list_emb P xs zs"
wfp_on_strict_suffix: fixes A :: "'b list set" shows "wfp_on strict_suffix A"
stirling_row_def: fixes n :: "nat" shows "stirling_row n = map (stirling n) [0..<Suc n]"
B_card_eq: shows "card B = nat ((int p - 1) div 2)"
IH: fixes f :: "complex fps" assumes "0 < fps_conv_radius f" shows "fps_nth f n = (deriv ^^ n) (eval_fps f) 0 / fact n"
product_idempotence: fixes x :: "i" shows "x \<otimes> x = x"
assms: shows "\<forall>i>n. a $ i = (0::'a)"
If: fixes G :: "'a triple set"   and P :: "'a assn"   and b :: "state \<Rightarrow> bool"   and c :: "com"   and Q :: "'a assn"   and d :: "com" assumes "G|-{P &> b}. c .{Q}"   and "G|-{P &> Not \<circ> b}. d .{Q}" shows "G|-{P}. IF b THEN c ELSE d .{Q}"
inv_obj_obj: fixes n :: "nat" shows "inv obj (obj n) = n"
poly_mapping_sum_monomials: fixes p :: "'a \<Rightarrow>\<^sub>0 'b" shows "(\<Sum>a\<in>keys p. monomial (lookup p a) a) = p"
effect_shiftr_f: fixes a :: "'b array"   and i :: "nat ref"   and h :: "heap"   and h' :: "heap"   and u :: "unit" assumes "effect (shiftr_f a i) h h' u" shows "Ref.get h' i = Ref.get h i - 1 \<and> Array.get h' a = (Array.get h a)[Ref.get h i := Array.get h a ! (Ref.get h i - 1)]"
Ghost: fixes m :: "nat \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and rs :: "'b memref list"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" shows "(m, Ghost\<^sub>s\<^sub>b A L R W # rs, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>f (m, rs, \<O> \<union> A - R, augment_rels (dom \<S>) R \<R>, \<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L)"
app_ass: shows "lin_ts_mmuaux aux = ts # tss @ tss'"
set_mkIde: fixes A :: "'s set" assumes "A \<subseteq> Univ" shows "local.set (mkIde A) = A"
push_down_state_rank_tokens: fixes q :: "'a ltl_prop_equiv_quotient"   and n :: "nat"   and i :: "nat" assumes "\<UU>.state_rank q n = Some i" shows "\<UU>.configuration q n \<noteq> {}"
ap_state_rev_rev_B: fixes f :: "('c, 'e \<Rightarrow> 'd) state dual"   and x :: "('c, 'e) state dual" shows "B f \<diamondop> B x = B (pure (\<lambda>x f. f x) \<diamondop> x \<diamondop> f)"
zero_closed: fixes H :: "'e set"   and G :: "('e, 'f) ring_scheme" assumes "additive_subgroup H G" shows "\<zero>\<^bsub>G\<^esub> \<in> H"
match_4: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a"   and h :: "'a \<times> 'a" assumes "VoV.seq g f"   and "VoV.seq h g" shows "VoV.seq h (VoV.comp g f)"
liminf_obtain: fixes u :: "nat \<Rightarrow> 'a"   and c :: "'a"   and N :: "nat" assumes "liminf u < c" shows "\<exists>n\<ge>N. u n < c"
type_definition_bit1: shows "type_definition Rep_bit1 Abs_bit1 {0..<1 + 2 * int CARD('a)}"
order_N_def: fixes a :: "'b N"   and b :: "'b N" shows "a \<le> b \<equiv> a l\<rightarrow> b = 1"
map_le_def: fixes m\<^sub>1 :: "'f \<Rightarrow> 'g option"   and m\<^sub>2 :: "'f \<Rightarrow> 'g option" shows "(m\<^sub>1 \<subseteq>\<^sub>m m\<^sub>2) = (\<forall>a\<in>dom m\<^sub>1. m\<^sub>1 a = m\<^sub>2 a)"
divides_cong_r: fixes x :: "'a list"   and y :: "'a list"   and y' :: "'a list" assumes "x divides\<^bsub>poly_ring R\<^esub> y"   and "y \<sim>\<^bsub>poly_ring R\<^esub> y'"   and "x \<in> carrier (poly_ring R)" shows "x divides\<^bsub>poly_ring R\<^esub> y'"
distinct_38: fixes x5 :: "nat"   and x4 :: "nat" shows "Key x5 \<noteq> Pan x4"
mdinit_refines: shows "mdinit \<le> \<Down> (br \<alpha>s dinvarm) dinit"
list_minus: fixes l1 :: "'c list"   and l2 :: "'c list" shows "l1 - l2 \<equiv> the (list_diff l1 l2)"
uncurry0_apply: fixes c :: "'c"   and x :: "unit" shows "uncurry0 c x = c"
1: shows "stateSpace ss domain = (\<forall>s. s \<in> ss \<longrightarrow> fmdom' s = domain)"
homeomorphic_closed_intervals_real: fixes a :: "real"   and b :: "real"   and c :: "real"   and d :: "real" assumes "a < b"   and "c < d" shows "{a..b} homeomorphic {c..d}"
reach_uw: shows "u \<rightarrow>\<^sup>*\<^bsub>Digraph_Component.union G H\<^esub> w"
disjointAddSubset: fixes ts :: "'b set set"   and a :: "'b set" assumes "disjoint ts" shows "disjoint (addSubsetSet a ts)"
Z_car: shows "X = {vec_first x n |x. x \<in> Z}"
induct: fixes P :: "'b list list \<Rightarrow> bool"   and a0 :: "'b list list" assumes "P []"   and "\<And>xs. P [xs]"   and "\<And>xs ys zss. P zss \<Longrightarrow> P (xs # ys # zss)" shows "P a0"
cardSuc_ordLeq: fixes r :: "('d \<times> 'd) set" assumes "Card_order r" shows "r \<le>o cardSuc r"
symvars_of_adapt_aexp: fixes e :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd"   and s :: "'c \<Rightarrow> nat" shows "Aexp.vars (adapt_aexp e s) \<subseteq> symvars s"
Sym: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> Rel" shows "(Q, P) \<in> Rel"
Max_in: fixes A :: "'b word set" assumes "finite A"   and "A \<noteq> {}" shows "signed.Max A \<in> A"
wf_\<delta>x: shows "wf\<^sub>t\<^sub>r\<^sub>m (\<delta> x)"
mult: fixes h1 :: "'a \<Rightarrow> real"   and h2 :: "'a \<Rightarrow> real" assumes "h1 \<in> G"   and "h2 \<in> G" shows "(\<lambda>x. h1 x * h2 x) \<in> G"
subadditive_converges_ereal: fixes u :: "nat \<Rightarrow> real" assumes "Fekete.subadditive u" shows "(\<lambda>m. ereal (u m / real m)) \<longlonglongrightarrow> \<Sqinter> {ereal (u n / real n) |n. 0 < n}"
cdf_is_right_cont: fixes a :: "real"   and k :: "nat" shows "continuous (at_right a) (cdf (\<mu> k))"
fine_def: fixes d :: "'d \<Rightarrow> 'e set"   and s :: "('d \<times> 'e set) set" shows "(d fine s) = (\<forall>(x, k)\<in>s. k \<subseteq> d x)"
eq_iff_diff_eq_0: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "(a = b) = (minus a b = zero)"
const_map_is_None: fixes y :: "'c"   and A :: "'d set"   and x :: "'d" shows "(const_map y A x = None) = (x \<notin> A)"
PO_refines_def: fixes R :: "('a \<times> 'b) set"   and Ta :: "('a, 'c) TS_scheme"   and Tc :: "('b, 'd) TS_scheme" shows "PO_refines R Ta Tc \<equiv> init Tc \<subseteq> R `` init Ta \<and> {R} TS.trans Ta, TS.trans Tc {> R}"
matrix_equation_R: fixes p :: "real poly"   and signs :: "rat list list"   and qs :: "real poly list"   and subsets :: "(nat list \<times> nat list) list" assumes "p \<noteq> 0"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set signs"   and "all_list_constr_R subsets (length qs)" shows "satisfy_equation_R p qs subsets signs"
f_expr: shows "f = (monom P \<one> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r"
ss_adversary_def: fixes y :: "'a"   and c1 :: "'a \<times> nat \<times> nat \<times> nat"   and c2 :: "'a \<times> nat \<times> nat \<times> nat" shows "ss_adversary y c1 c2 = (let (a, e, z1, z2) = c1; (a', e', z1', z2') = c2 in return_spmf (if e' < e then nat ((int z1 - int z1') * Number_Theory_Aux.inverse (e - e') (order \<G>) mod int (order \<G>)) else nat ((int z1' - int z1) * Number_Theory_Aux.inverse (e' - e) (order \<G>) mod int (order \<G>)), if e' < e then nat ((int z2 - int z2') * Number_Theory_Aux.inverse (e - e') (order \<G>) mod int (order \<G>)) else nat ((int z2' - int z2) * Number_Theory_Aux.inverse (e' - e) (order \<G>) mod int (order \<G>))))"
simps: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and a :: "('a, 'b, 'c) trans list" shows "IO_Automaton.valid2 step a = ((\<exists>trn. a = [trn] \<and> IO_Automaton.validTrans step trn) \<or> (\<exists>tr trn. a = tr ## trn \<and> IO_Automaton.valid2 step tr \<and> tgtOf (last tr) = srcOf trn \<and> IO_Automaton.validTrans step trn))"
pred_un: fixes P :: "'BExp set"   and P' :: "'BExp set"   and mem :: "'Var \<Rightarrow> 'Val" shows "pred (P \<union> P') mem = (pred P mem \<and> pred P' mem)"
case_2: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Neg of Pos \<Rightarrow> f1 | Neg \<Rightarrow> f2) = f2"
trans_diff_Id: fixes r :: "('a \<times> 'a) set" assumes "trans r"   and "antisym r" shows "trans (r - Id)"
round_2: fixes prf_key_gen :: "'a spmf"   and prf_fun :: "'a \<Rightarrow> bool list \<Rightarrow> bool list"   and prf_domain :: "bool list set"   and prf_dlen :: "nat"   and prf_clen :: "nat"   and upf_key_gen :: "'b spmf"   and upf_fun :: "'b \<Rightarrow> bool list \<Rightarrow> 'c"   and \<A> :: "(bool, bool list \<times> bool list + bool list \<times> bool list \<times> 'c, (bool list \<times> bool list \<times> 'c) option + bool list option) gpv" assumes "simple_cipher prf_key_gen prf_fun prf_domain prf_dlen prf_clen upf_key_gen" shows "\<bar>spmf (IND_CCA2_sym.ind_cca.game (simple_cipher.key_gen prf_key_gen upf_key_gen) (simple_cipher.encrypt prf_fun prf_domain upf_fun) (\<lambda>_. Map.empty) (\<lambda>m. length m = prf_clen) \<A>) True - spmf (simple_cipher.game2 prf_key_gen prf_domain prf_clen upf_key_gen upf_fun \<A>) True\<bar> = prf.advantage prf_key_gen prf_fun (spmf_of_set (nlists UNIV prf_clen)) (simple_cipher.reduction_prf prf_domain prf_clen upf_key_gen upf_fun \<A>)"
primeI: fixes p :: "'c"   and G :: "('c, 'd) monoid_scheme" assumes "p \<notin> Units G"   and "\<And>a b. \<lbrakk>a \<in> carrier G; b \<in> carrier G; p divides\<^bsub>G\<^esub> a \<otimes>\<^bsub>G\<^esub> b\<rbrakk> \<Longrightarrow> p divides\<^bsub>G\<^esub> a \<or> p divides\<^bsub>G\<^esub> b" shows "prime G p"
valid_def: fixes implem :: "chan \<Rightarrow> msg" shows "basic_implem.valid implem \<equiv> {implem (Chan x A B M) |x A B M. M \<in> payload}"
simps_2: fixes a :: "('a, 'b) Node" shows "FingerTreeStruc.viewLn (Single a) = Some (a, Empty)"
in_progress_Some: fixes r :: "Runs.var set option"   and x :: "Runs.var" assumes "in_progress r x" shows "\<exists>x. r = Some x"
tensor_mat_positive: fixes m1 :: "complex mat"   and m2 :: "complex mat" assumes "m1 \<in> carrier_mat d1 d1"   and "m2 \<in> carrier_mat d2 d2"   and "positive m1"   and "positive m2" shows "positive (tensor_mat m1 m2)"
mod_diff1_eq_if: fixes a :: "nat"   and b :: "nat"   and m :: "nat" assumes "a \<le> b" shows "(b - a) mod m = (if a mod m \<le> b mod m then b mod m - a mod m else m + b mod m - a mod m)"
c_def': shows "c = insertion (nth_default 0 (xs' @ 0 # xs)) (isolate_variable_sparse p var 0)"