hcju/isabelleps huggingface dataset:
that: fixes b :: "'a"   and n\<^sub>b :: "nat"   and c :: "'a"   and n\<^sub>c :: "nat" assumes "b = f n\<^sub>b"   and "c = f n\<^sub>c"   and "[[b x c]]"   and "b \<in> Y"   and "c \<in> Y"   and "n\<^sub>b = n\<^sub>c - 1"   and "n\<^sub>c < card Y"   and "0 < n\<^sub>c"   and "\<not> (\<exists>k<card Y. [[f k x a\<^sub>n]] \<and> n\<^sub>b < k)"   and "\<not> (\<exists>k<n\<^sub>c. [[a\<^sub>0 x f k]])" shows "thesis"
is_natural_transformation: shows "natural_transformation J (\<cdot>) \<chi>f.A.map D' \<tau>o\<chi>f.map"
case_eq_if: fixes f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'e list \<Rightarrow> 'd"   and list :: "'e list" shows "(case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = (if list = [] then f1 else f2 (hd list) (tl list))"
INT_iff: fixes b :: "'c"   and B :: "'d \<Rightarrow> 'c set"   and A :: "'d set" shows "(b \<in> \<Inter> (B ` A)) = (\<forall>x\<in>A. b \<in> B x)"
d_equal_def: fixes d :: "'d"   and m :: "'id \<Rightarrow> 'val"   and m' :: "'id \<Rightarrow> 'val" shows "m =\<^bsub>d\<^esub> m' \<equiv> \<forall>x. DA x \<le> d \<longrightarrow> m x = m' x"
Ptl_edge: shows "v0 \<rightarrow> lhd (ltl P)"
summable_in_conv_radius: fixes z :: "'b"   and f :: "nat \<Rightarrow> 'b" assumes "ereal (norm z) < conv_radius f" shows "summable (\<lambda>n. f n * z ^ n)"
hyps_1: fixes p :: "'a poly" assumes "n = degree p"   and "p \<noteq> 0" shows "finite {x. poly p x = (0::'a)}"
X_def: shows "X = {x. le_vec 0 x \<and> x \<noteq> 0}"
hd_if_det: fixes b :: "expr"   and V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e1 :: "expr"   and g1 :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and e2 :: "expr"   and g2 :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal" assumes "randomfree b"   and "(V, V', \<Gamma>, if_dens_det \<delta> b True) \<turnstile>\<^sub>d e1 \<Rightarrow> g1"   and "(V, V', \<Gamma>, if_dens_det \<delta> b False) \<turnstile>\<^sub>d e2 \<Rightarrow> g2" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d IF b THEN e1 ELSE e2 \<Rightarrow> (\<lambda>\<rho> x. g1 \<rho> x + g2 \<rho> x)"
assms_5: shows "(G ===> (=)) inEdges1 inEdges2"
assms_5: shows "\<forall>i\<in>I. \<forall>x\<in>A i. card {i \<in> I. x \<in> A i} \<le> k"
frechet_derivative_sum: fixes I :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and a :: "'c" assumes "\<And>i. i \<in> I \<Longrightarrow> f i differentiable at a" shows "frechet_derivative (\<lambda>x. \<Sum>i\<in>I. f i x) (at a) = (\<lambda>x. \<Sum>i\<in>I. frechet_derivative (f i) (at a) x)"
product_lists_def: shows "product_lists \<equiv> rec_list [[]] (\<lambda>xs xss xssa. concat (map (\<lambda>x. map ((#) x) xssa) xs))"
less_imp_neq: fixes x :: "'t"   and y :: "'t" assumes "x \<prec>\<^sub>t y" shows "x \<noteq> y"
cont_ennreal_spmf: fixes x :: "'c" shows "cont lub_spmf (ord_spmf (=)) Sup (\<le>) (\<lambda>p. ennreal (spmf p x))"
sup1I2: fixes B :: "'a \<Rightarrow> bool"   and x :: "'a"   and A :: "'a \<Rightarrow> bool" assumes "B x" shows "(A \<squnion> B) x"
y: shows "y \<in> {x \<in> carrier G. h x \<in> I}"
set_sum_alt: fixes I :: "'c set"   and S :: "'c \<Rightarrow> 'd set" assumes "finite I" shows "sum S I = {sum s I |s. \<forall>i\<in>I. s i \<in> S i}"
leofD: fixes m :: "'c nres"   and m' :: "'c nres" assumes "nofail m"   and "m \<le>\<^sub>n m'" shows "m \<le> m'"
fcd: fixes x :: "complex" assumes "x \<in> interior S - k" shows "f field_differentiable at x"
nth_list_update: fixes i :: "nat"   and xs :: "'a list"   and x :: "'a"   and j :: "nat" assumes "i < length xs" shows "xs[i := x] ! j = (if i = j then x else xs ! j)"
coplanar_perm_15: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet C B D A"
game1_def: fixes \<A> :: "('x \<times> 'hash, 'x, 'hash) gpv" shows "game1 \<A> = key_gen \<bind> (\<lambda>key. exec_gpv (oracle_hash key) \<A> {} \<bind> (\<lambda>((m, h), L). return_spmf (h = hash key m \<and> m \<notin> L)))"
CL_def: fixes cl :: "'b potype" shows "CL cl \<equiv> PO cl \<and> CL_axioms cl"
PiE_eq_empty_iff: fixes I :: "'f set"   and F :: "'f \<Rightarrow> 'g set" shows "(Pi\<^sub>E I F = {}) = (\<exists>i\<in>I. F i = {})"
alpha_preserves_qGood: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "X #= X'" shows "qGood X = qGood X'"
map_ident: fixes p :: "'g spmf" shows "map_spmf (\<lambda>x. x) p = p"
fls_regpart_zero: shows "fls_regpart 0 = 0"
mult_div_mono_right: fixes c :: "real"   and a :: "real"   and b :: "real" assumes "0 \<le> c"   and "c \<noteq> 0"   and "inverse c * a \<le> b" shows "a \<le> c * b"
a1: shows "s' = snd (fst (sub_instr_sub1 instr_name result rs1_val operand2 s)) \<and> get_S (cpu_reg_val PSR s) = 0"
as2: shows "\<not> safe_distance_normal.collision_react a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta> {0..}"
I3: fixes e' :: "pexpr1"   and n :: "nat"   and e :: "pexpr" shows "P (PExpr1 e') n e"
validFrom_valid: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and tr :: "('a, 'b, 'c) trans list"   and s :: "'a" assumes "IO_Automaton.valid step tr \<and> srcOf (hd tr) = s" shows "IO_Automaton.validFrom step s tr"
m_ac_1: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "Group.comm_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"
FFreshBottom_1: fixes xvec :: "name list" shows "xvec \<sharp>* (\<bottom>\<^sub>F)"
similar_matD: fixes A :: "'a mat"   and B :: "'a mat" assumes "similar_mat A B" shows "\<exists>n P Q. {A, B, P, Q} \<subseteq> carrier_mat n n \<and> P * Q = 1\<^sub>m n \<and> Q * P = 1\<^sub>m n \<and> A = P * B * Q"
ord_option_reflI: fixes x :: "'c option"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>y. y \<in> set_option x \<Longrightarrow> ord y y" shows "ord_option ord x x"
z: shows "(tab(xs [\<mapsto>] ys)) vn = Some z"
mine: fixes nid :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "honest nid"   and "\<parallel>nid\<parallel>\<^bsub>t (Suc n)\<^esub>"   and "mining (\<sigma>\<^bsub>nid\<^esub>t (Suc n))" shows "\<parallel>nid\<parallel>\<^bsub>t n\<^esub>"
cases: fixes a1 :: "nat \<Rightarrow> classDef option"   and a2 :: "nat \<Rightarrow> nat option"   and a3 :: "exp"   and a4 :: "nat" assumes "a1;a2 \<turnstile> a3 : a4" obtains \<Gamma> :: "nat \<Rightarrow> nat option" and x :: "nat" and C :: "nat" and CT :: "nat \<Rightarrow> classDef option" where "a1 = CT__" and "a2 = \<Gamma>__" and "a3 = Var x" and "a4 = C__" and "\<Gamma>__ x = Some C__"   | CT :: "nat \<Rightarrow> classDef option" and \<Gamma> :: "nat \<Rightarrow> nat option" and e0 :: "exp" and C0 :: "nat" and Cf :: "varDef list" and fi :: "nat" and fDef :: "varDef" and Ci :: "nat" where "a1 = CTa__" and "a2 = \<Gamma>'__" and "a3 = FieldProj e0__ fi" and "a4 = Ci" and "CTa__;\<Gamma>'__ \<turnstile> e0__ : C0__" and "fields(CTa__,C0__) = Cf" and "lookup Cf (\<lambda>fd. vdName fd = fi) = Some fDef" and "vdType fDef = Ci"   | CT :: "nat \<Rightarrow> classDef option" and \<Gamma> :: "nat \<Rightarrow> nat option" and e0 :: "exp" and C0 :: "nat" and m :: "nat" and Ds :: "nat list" and C :: "nat" and es :: "exp list" and Cs :: "nat list" where "a1 = CTb__" and "a2 = \<Gamma>''__" and "a3 = MethodInvk e0a__ m es__" and "a4 = Ca__" and "CTb__;\<Gamma>''__ \<turnstile> e0a__ : C0" and "mtype(CTb__,m,C0) = Ds__ \<rightarrow> Ca__" and "CTb__;\<Gamma>''__ \<turnstile>+ es__ : Cs__" and "CTb__ \<turnstile>+ Cs__ <: Ds__" and "length es__ = length Ds__"   | CT :: "nat \<Rightarrow> classDef option" and C :: "nat" and Df :: "varDef list" and es :: "exp list" and Ds :: "nat list" and \<Gamma> :: "nat \<Rightarrow> nat option" and Cs :: "nat list" where "a1 = CTc__" and "a2 = \<Gamma>'''__" and "a3 = New Cb__ es" and "a4 = Cb__" and "fields(CTc__,Cb__) = Df" and "length es = length Df" and "varDefs_types Df = Ds" and "CTc__;\<Gamma>'''__ \<turnstile>+ es : Cs" and "CTc__ \<turnstile>+ Cs <: Ds"   | CT :: "nat \<Rightarrow> classDef option" and \<Gamma> :: "nat \<Rightarrow> nat option" and e0 :: "exp" and D :: "nat" and C :: "nat" where "a1 = CTd__" and "a2 = \<Gamma>''''__" and "a3 = Cast Cc__ e0b__" and "a4 = Cc__" and "CTd__;\<Gamma>''''__ \<turnstile> e0b__ : D__" and "CTd__ \<turnstile> D__ <: Cc__"   | CT :: "nat \<Rightarrow> classDef option" and \<Gamma> :: "nat \<Rightarrow> nat option" and e0 :: "exp" and D :: "nat" and C :: "nat" where "a1 = CTe__" and "a2 = \<Gamma>'''''__" and "a3 = Cast Cd__ e0c__" and "a4 = Cd__" and "CTe__;\<Gamma>'''''__ \<turnstile> e0c__ : Da__" and "CTe__ \<turnstile> Cd__ <: Da__" and "Cd__ \<noteq> Da__"   | CT :: "nat \<Rightarrow> classDef option" and \<Gamma> :: "nat \<Rightarrow> nat option" and e0 :: "exp" and D :: "nat" and C :: "nat" where "a1 = CT" and "a2 = \<Gamma>" and "a3 = Cast C e0" and "a4 = C" and "CT;\<Gamma> \<turnstile> e0 : D" and "CT \<turnstile> C \<not><: D" and "CT \<turnstile> D \<not><: C"
pcr_mod_ring_to_int_mod_ring: shows "pcr_mod_ring = (\<lambda>x y. x = to_int_mod_ring y)"
cA: shows "char_poly_matrix A \<in> carrier_mat n n"
tuple_in'_keys: fixes as :: "'a option list" assumes "as \<in> Mapping.keys tuple_in'" shows "as \<in> Mapping.keys tuple_in"
is_nan_conv: fixes x :: "double" shows "Code_Double.is_nan x = (x = nan)"
degree_0_id: fixes p :: "'b poly" assumes "degree p = 0" shows "[:coeff p 0:] = p"
span_insertI: fixes p :: "'c"   and B :: "'c set"   and r :: "'c" assumes "p \<in> ideal B" shows "p \<in> ideal (insert r B)"
att_or_singleton: fixes x1 :: "'a attree"   and x :: "'a set \<times> 'a set" assumes "\<turnstile>[x1] \<oplus>\<^sub>\<or>\<^bsup>x\<^esup>" shows "\<turnstile>[] \<oplus>\<^sub>\<or>\<^bsup>(fst x - fst (attack x1), snd x)\<^esup>"
bisimE: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim> Q" shows "P \<leadsto>[bisim] Q"
simps_13: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b lambda \<Rightarrow> 'b lambda \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f3 :: "('b \<Rightarrow> 'b lambda \<times> 'a) \<Rightarrow> 'a"   and x1 :: "'b" shows "rec_lambda f1 f2 f3 (Var x1) = f1 x1"
conversion_rtrancl: fixes A :: "('b \<times> 'b) set" shows "(A\<^sup>\<leftrightarrow>\<^sup>*)\<^sup>* = A\<^sup>\<leftrightarrow>\<^sup>*"
prems_2: shows "InvariantCFalse (getConflictFlag state') (getM state') (getC state')"
that_1: fixes n :: "nat"   and st :: "'St" assumes "0 < n"   and "sl = map Idle (replicate n st)" shows "thesis"
assms_2: shows "fst a \<subseteq>\<^sub>f s"
inv_into_f_f: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and x :: "'b" assumes "inj_on f A"   and "x \<in> A" shows "inv_into A f (f x) = x"
phiDefNode_aux_is_join_node: fixes n :: "'node"   and g :: "'g"   and v :: "'var"   and un :: "'node list"   and m :: "'node" assumes "n \<in> phiDefNodes_aux g v un m" shows "length (predecessors g n) \<noteq> 1"
fderiv: fixes x :: "'c" assumes "x \<in> Csafe" shows "(ode has_derivative blinfun_apply (ode_d1 x)) (at x)"
range_reduce_aform_errE: fixes e :: "nat \<Rightarrow> real"   and x :: "real"   and X :: "(real \<times> real pdevs) \<times> real"   and p :: "nat"   and Y :: "(real \<times> real pdevs) \<times> real" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "x \<in> aform_err e X"   and "range_reduce_aform_err p X = Some Y" obtains n :: "int" where "x + real_of_int n * (2 * pi) \<in> aform_err e Y"
exl: shows "\<exists>l\<in>lotteries_on outcomes. pmf l x = 1"
tsubstT_trans: fixes ty :: "typ"   and \<rho>1 :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ"   and ty1 :: "typ"   and \<rho>2 :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ"   and ty2 :: "typ" assumes "tsubstT ty \<rho>1 = ty1"   and "tsubstT ty1 \<rho>2 = ty2" shows "tsubstT ty (\<lambda>idx s. case \<rho>1 idx s of Ty s Ts \<Rightarrow> Ty s (map (\<lambda>T. tsubstT T \<rho>2) Ts) | Tv idx' s' \<Rightarrow> \<rho>2 idx' s') = ty2"
hseq_char': fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "C.hseq \<nu> \<mu> = (\<nu> \<star>\<^sub>C \<mu> \<noteq> C.null)"
smap_singleton_def: fixes k :: "'c"   and vs :: "'d set" shows "{ k := vs}. = {}.(k := vs)"
single_row_inv: fixes j' :: "nat"   and j :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat"   and i' :: "nat" assumes "j' < j" shows "fwi m n k i' j i' j' = fwi m n k i' j' i' j'"
causal_program_history_def: fixes is\<^sub>s\<^sub>b :: "instr list"   and sb :: "'b memref list" shows "causal_program_history is\<^sub>s\<^sub>b sb \<equiv> \<forall>sb\<^sub>1 sb\<^sub>2. sb = sb\<^sub>1 @ sb\<^sub>2 \<longrightarrow> (\<exists>is. instrs sb\<^sub>2 @ is\<^sub>s\<^sub>b = is @ prog_instrs sb\<^sub>2)"
h_def: shows "h = (\<lambda>x. f (\<lambda>i. if i < r1 then 0 else if i < r2 then x else l))"
decs_unclocked_unchanged: fixes ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and ds :: "dec list"   and r :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_decs ck mn env s ds r" shows "\<forall>s' r'. ck = False \<and> r = (s', r') \<longrightarrow> r' \<noteq> Rerr (Rabort Rtimeout_error) \<and> clock s = clock s'"
get_tag_name_ptr_in_heap: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and children :: "char list" assumes "h \<turnstile> get_tag_name element_ptr \<rightarrow>\<^sub>r children" shows "element_ptr |\<in>| element_ptr_kinds h"
b: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y \<le> y"
cases: fixes a1 :: "'tp"   and a2 :: "('fsym, 'psym) lit list" assumes "a1 \<turnstile>2 a2" obtains \<sigma> :: "'tp" and c :: "('fsym, 'psym) lit list" where "a1 = \<sigma>'__" and "a2 = ca__" and "infTp \<sigma>'__"   | c :: "('fsym, 'psym) lit list" and \<sigma> :: "'tp" where "a1 = \<sigma>" and "a2 = c" and "\<And>l x. \<lbrakk>l \<in> set c; x \<in> nv2L l; tpOfV x = \<sigma>\<rbrakk> \<Longrightarrow> isGuard x (grdOf c l x)"
compute_float_down: fixes p :: "int"   and m :: "int"   and e :: "int" shows "float_down p (Float m e) = (if p + e < 0 then Float (div_twopow m (nat (- (p + e)))) (- p) else Float m e)"
res: shows "LLL_Impl.basis_reduction_add_rows_loop n (if round (fs.gs.\<mu> i j) = 0 then state else upd_fi_mu_state state i (vec n (\<lambda>ia. fs ! i $ ia - round (fs.gs.\<mu> i j) * fs ! j $ ia)) (IArray.of_fun (\<lambda>jj. if jj < j then dmu_ij_state state i jj - round (fs.gs.\<mu> i j) * dmu_ij_state state j jj else if jj = j then dmu_ij_state state i jj - d fs (Suc j) * round (fs.gs.\<mu> i j) else dmu_ij_state state i jj) i)) i j (map ((!) fs) (rev [0..<j])) = state'"
signed_0: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.signed one plus zero minus 0 = zero"
assms_6: shows "c2 \<sqsubseteq> c1"
hom_prod_mset: fixes X :: "int poly multiset" shows "of_int_poly (\<Prod>\<^sub># X) = \<Prod>\<^sub># (image_mset of_int_poly X)"
supp_perm_perm_eq: fixes x :: "'e"   and p :: "perm"   and q :: "perm" assumes "\<forall>a\<in>supp x. p \<bullet> a = q \<bullet> a" shows "p \<bullet> x = q \<bullet> x"
induced_map_meas: fixes A :: "'a set" assumes "A \<in> sets M" shows "induced_map A \<in> M \<rightarrow>\<^sub>M M"
ln_sqrt: fixes x :: "real" assumes "0 < x" shows "ln (sqrt x) = ln x / 2"
matchC_C_matchC_LC: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd" assumes "matchC_C theta c d" shows "matchC_LC theta c d"
Diagonalize_Tensor_Unity_Arr: fixes u :: "'c local.term" assumes "Arr u" shows "\<^bold>\<lfloor>\<^bold>\<I> \<^bold>\<otimes> u\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>"
HEndPhase1_blocksOf_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase1 s s' p"   and "p \<noteq> q" shows "blocksOf s' q \<subseteq> blocksOf s q"
Sim: fixes R :: "ccs"   and S :: "ccs" assumes "(R, S) \<in> Rel" shows "R \<leadsto>\<^sup>^<Rel> S"
tt_monomial: fixes c :: "'d"   and v :: "'a" assumes "c \<noteq> (0::'d)" shows "punit.tt (monomial c v) = v"
invar_def: fixes E :: "'b set set"   and C :: "'b set"   and F :: "'b set set" assumes "Edges E" shows "Edges.invar E C F = (F \<subseteq> E \<and> vertex_cover (E - F) C \<and> finite C \<and> (\<exists>M. Edges.inv_matching E C F M))"
smallstepsI_ps_emptyD_2: fixes e :: "Syntax.term"   and i :: "nat"   and \<pi> :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>[], e\<ggreater> I\<rightarrow>i \<lless>\<pi>, e'\<ggreater>" shows "\<lless>[], e\<ggreater> I\<rightarrow>i \<lless>[], e'\<ggreater>"
phi_None: fixes pc :: "nat" assumes "pc < length ins"   and "c ! pc = \<bottom>" shows "\<tau>s ! pc = wtl (take pc ins) c 0 s\<^sub>0"
fps_lr_inverse_gp_ring1_2: shows "fps_right_inverse (Abs_fps (\<lambda>n. 1::'a)) (1::'a) = Abs_fps (\<lambda>n. if n = 0 then 1::'a else if n = 1 then - (1::'a) else (0::'a))"
Spy_dont_know_TTPKey_parts: fixes evs :: "event list"   and b :: "keymode" assumes "evs \<in> certified_mail" shows "Key (privateKey b TTP) \<notin> parts (knows Spy evs)"
Inf_mono: fixes B :: "('b \<times> 'c) set"   and A :: "('b \<times> 'c) set" assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. less_eq_prod a b" shows "less_eq_prod (Inf_prod A) (Inf_prod B)"
Rep_hf_inject: fixes x :: "hf"   and y :: "hf" shows "(Rep_hf x = Rep_hf y) = (x = y)"
at_most_one_marker: fixes t :: "('a, 'b, 'c) event list"   and cid :: "nat"   and p :: "'a"   and q :: "'a"   and i :: "nat" assumes "trace init t final"   and "channel cid = Some (p, q)" shows "Marker \<notin> set (msgs (S t i) cid) \<or> (\<exists>!j. j < length (msgs (S t i) cid) \<and> msgs (S t i) cid ! j = Marker)"
gen_new_Addr_def: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and n :: "nat" shows "gen_new_Addr h n \<equiv> if \<exists>a\<ge>n. h a = None then \<lfloor>LEAST a. n \<le> a \<and> h a = None\<rfloor> else None"
subset_Pow_Union: fixes A :: "'b set set" shows "order.greater_eq (Pow (\<Union> A)) A"
not_None_eq: fixes x :: "'a option" shows "(x \<noteq> None) = (\<exists>y. x = pure y)"
frame_thms_3: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and y :: "'b" shows "hn_ctxt R x y \<Longrightarrow>\<^sub>t hn_invalid R x y"
comp_def: fixes f :: "'ai \<Rightarrow> 'ah"   and g :: "'ag \<Rightarrow> 'ai" shows "f \<circ> g = (\<lambda>x. f (g x))"
nonzero_inverse_scaleR_distrib: fixes a :: "real"   and x :: "'a" assumes "a \<noteq> 0"   and "x \<noteq> (0::'a)" shows "inverse (a *\<^sub>R x) = inverse x /\<^sub>R a"
imageE: fixes b :: "'h"   and f :: "'i \<Rightarrow> 'h"   and A :: "'i set" assumes "b \<in> f ` A" obtains x :: "'i" where "b = f x" and "x \<in> A"
step_Normal_elim_cases_9: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "'b \<Rightarrow> ('b, 'a, 'c) com"   and s :: "'b"   and u :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (DynCom c, Normal s) \<rightarrow> u" obtains "u = (c s, Normal s)"
inj_cball: shows "inj_on (\<lambda>x. norm x *\<^sub>R surf (proj x)) (cball (0::'a) 1 \<inter> affine hull S)"
simps: fixes I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and S :: "'b literal set set" shows "validate_clause_set I S = (\<forall>C. C \<in> S \<longrightarrow> validate_clause I C)"
may_lock_unlock_lock_lock_lock_id: fixes l :: "('b \<times> nat) option"   and t :: "'b" assumes "may_lock l t" shows "unlock_lock (lock_lock l t) = l"
countable_finite: fixes S :: "'c set" assumes "finite S" shows "countable S"
that: fixes p :: "'a \<Rightarrow> 'a" assumes "p permutes A"   and "\<forall>x\<in>A. f x = f' (p x)" shows "thesis"
eq2: assumes "r1 x y" shows "map f (merge_wrt r1 xs (y # ys)) = merge_wrt r2 (map f xs) (map f (y # ys))"
set_rotate: fixes n :: "nat"   and xs :: "'b list" shows "set (rotate n xs) = set xs"
cinner_gt_zero_iff: fixes x :: "'a" shows "(0 < cinner x x) = (x \<noteq> (0::'a))"
A: shows "pr_index_enumerator n1 m = pr_index_enumerator n2 m"
constructive_security_obsf'I: fixes real_resource :: "nat \<Rightarrow> ('m + 'n, 'o + 'p) resource"   and ideal_resource :: "nat \<Rightarrow> ('q + 'n, 'r + 'p) resource"   and sim :: "nat \<Rightarrow> ('m, 'o, 'q, 'r) converter"   and \<I>_real :: "nat \<Rightarrow> ('m, 'o) \<I>"   and \<I>_ideal :: "nat \<Rightarrow> ('q, 'r) \<I>"   and \<I>_common :: "nat \<Rightarrow> ('n, 'p) \<I>"   and \<A> :: "nat \<Rightarrow> ('m + 'n, 'o + 'p) distinguisher_obsf"   and adv :: "nat \<Rightarrow> real" assumes "\<And>\<eta>. constructive_security_obsf (real_resource \<eta>) (ideal_resource \<eta>) (sim \<eta>) (\<I>_real \<eta>) (\<I>_ideal \<eta>) (\<I>_common \<eta>) (\<A> \<eta>) (adv \<eta>)"   and "(\<And>\<eta>. exception_\<I> (\<I>_real \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta>) \<turnstile>g \<A> \<eta> \<surd>) \<Longrightarrow> Negligible.negligible adv" shows "constructive_security_obsf' real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common \<A>"
image_uminus_atLeastAtMost: 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 x :: "'a"   and y :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "uminus ` ord.atLeastAtMost less_eq x y = ord.atLeastAtMost less_eq (uminus y) (uminus x)"
o_apply: fixes f :: "'l \<Rightarrow> 'k"   and g :: "'m \<Rightarrow> 'l"   and x :: "'m" shows "(f \<circ> g) x = f (g x)"
hyp1: shows "(\<Sum>s\<leftarrow>signs. z_R I s * rat_of_nat (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s})) = (\<Sum>s\<in>set signs. z_R I s * rat_of_nat (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s}))"
nn_integral_zero': fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a measure" assumes "AE x in M. f x = 0" shows "integral\<^sup>N M f = 0"
independent_Union_directed: fixes C :: "'b set set" assumes "\<And>c d. \<lbrakk>c \<in> C; d \<in> C\<rbrakk> \<Longrightarrow> c \<subseteq> d \<or> d \<subseteq> c"   and "\<And>c. c \<in> C \<Longrightarrow> independent c" shows "independent (\<Union> C)"
cases: fixes x :: "'a val" obtains a :: "'a" where "x = Addr a"   | "x = Unit"   | "x = Null"   | v :: "bool" where "x = Bool v__"   | v :: "32 word" where "x = Intg v"
singletonD: fixes b :: "'a"   and a :: "'a" assumes "s2p (\<eta> a) b" shows "b = a"
unrest_all_in: fixes y :: "'a \<Longrightarrow> 'b"   and x :: "'c \<Longrightarrow> 'b"   and P :: "'b upred" assumes "mwb_lens y"   and "x \<subseteq>\<^sub>L y" shows "x \<sharp> (\<forall> y \<bullet> P)"
preserves_cod: fixes f :: "'j" assumes "J.arr f" shows "C.cod (A.map f) = A.map (J.cod f)"
interfree_def: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and Ps :: "(('e, 'd, 'f) ann \<times> 'e set \<times> 'e set) list"   and Ts :: "('e, 'd, 'f) com list" shows "interfree \<Gamma> \<Theta> F Ps Ts \<equiv> \<forall>i j. i < length Ts \<and> j < length Ts \<and> i \<noteq> j \<longrightarrow> interfree_aux \<Gamma> \<Theta> F (Ts ! i, Ps ! i, Ts ! j, fst (Ps ! j))"
F: shows "u \<bullet> \<bar>u\<bar> = - 1"
simps_1: fixes x :: "variable"   and \<theta> :: "trm" shows "Assigno x (Aterm \<theta>) = Agame (x := \<theta>)"
singleton_0: shows "singleton_pdevs (0::'a) = zero_pdevs"