hcju/isabelleps huggingface dataset:
tNode_1: fixes p :: "ref" assumes "x \<in> set_of lt"   and "x \<noteq> Null"   and "high x \<noteq> Null"   and "Dag p low high lt" shows "high x \<in> set_of lt"
pow_zero: fixes b :: "'b"   and x :: "nat" assumes "(0::'b) < b"   and "b < (1::'b)" shows "(0::'b) < b ^ x"
sum_pos2: fixes I :: "'a set"   and i :: "'a"   and f :: "'a \<Rightarrow> 'b multiset" assumes "finite I"   and "i \<in> I"   and "{#} \<subset># f i"   and "\<And>i. i \<in> I \<Longrightarrow> {#} \<subseteq># f i" shows "{#} \<subset># comm_monoid_add.sum (+) {#} f I"
terminal_some_terminal: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "category_with_terminal_object C" shows "category.terminal C (category_with_terminal_object.some_terminal C)"
B_def: shows "B = (\<lambda>n. \<Union> (A ` {n..}))"
distinct_disc_1: fixes event :: "('a, 'b, 'c) event" assumes "isTrans event" shows "\<not> isSend event"
exp_ln_Gamma_series_complex: fixes n :: "nat"   and z :: "complex" assumes "0 < n"   and "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "exp (ln_Gamma_series z n) = Gamma_series z n"
wls_freshEnv_swapEnv_exchange1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and xs :: "'varSort"   and x2 :: "'var"   and x1 :: "'var" assumes "wlsEnv rho" shows "freshEnv xs x2 (rho &[x1 \<and> x2]_xs) = freshEnv xs x1 rho"
JF2map_simps: fixes u :: "'e \<Rightarrow> 'd"   and v :: "'e JF2" shows "dtor2 (JF2map u v) = F2map u (JF1map u) (JF2map u) (dtor2 v)"
valid_path_prefix: fixes P :: "'a llist"   and P' :: "'a llist" assumes "valid_path P"   and "lprefix P' P" shows "valid_path P'"
inject: fixes x1 :: "'c \<Rightarrow> 'd list"   and x2 :: "'f \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm set list"   and x3 :: "'c \<times> 'f \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qAbs set list"   and x4 :: "('c \<Rightarrow> 'd \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm set option) list"   and y1 :: "'c \<Rightarrow> 'd list"   and y2 :: "'f \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm set list"   and y3 :: "'c \<times> 'f \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qAbs set list"   and y4 :: "('c \<Rightarrow> 'd \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm set option) list" shows "(ParS x1 x2 x3 x4 = ParS y1 y2 y3 y4) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3 \<and> x4 = y4)"
statImpEnt: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and \<Psi> :: "'d"   and \<Psi>' :: "'d"   and \<Phi> :: "'e" assumes "assertionAux SCompose SImp SBottom SChanEq"   and "assertionAux.AssertionStatImp SImp \<Psi> \<Psi>'"   and "SImp \<Psi> \<Phi>" shows "SImp \<Psi>' \<Phi>"
\<V>_def: shows "\<V> \<equiv> {V. openin X V \<and> (\<exists>U\<in>\<U>. U = V \<inter> S)}"
disturbed_one_step_def: fixes t :: "nat \<Rightarrow> real"   and t1 :: "real"   and s :: "real \<Rightarrow> real \<Rightarrow> 'b \<Rightarrow> 'b"   and s0 :: "'b"   and x :: "real \<Rightarrow> 'b"   and incr :: "real \<Rightarrow> real \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "nat"   and B :: "real"   and L :: "real" shows "disturbed_one_step t t1 s s0 x incr p B L \<equiv> grid t \<and> disturbed_one_step_axioms t t1 s s0 x incr p B L"
Infinitesimal_hnorm_iff: fixes x :: "'a star" shows "(hnorm x \<in> Infinitesimal) = (x \<in> Infinitesimal)"
repeat_mset_block_point_rel: fixes b :: "'a set"   and n :: "nat"   and x :: "'a" assumes "b \<in># multiple_blocks n"   and "x \<in> b" shows "x \<in> \<V>"
substT_same_Var: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and zer :: "'b"   and suc :: "'b \<Rightarrow> 'b"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tms :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'b"   and x :: "'a" assumes "Syntax_Arith var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num zer suc pls tms"   and "s \<in> Syntax_Arith.atrm var Var num suc pls tms"   and "x \<in> var" shows "substT s (Var x) x = s"
openin_topology_generated_by_iff: fixes S :: "'b set set"   and s :: "'b set" shows "openin (topology_generated_by S) s = generate_topology_on S s"
is_associated_string_def: fixes M :: "('c, 'd) message set"   and s :: "('c, 'd option) insert_message list" shows "is_associated_string M s \<equiv> consistent M \<and> set s = to_woot_character M ` insert_messages M \<and> (\<forall>a. a_conditions (insert_messages M) a \<longrightarrow> sorted_wrt (<) (map a (ext_ids s)))"
set_integrable_def: fixes M :: "'d measure"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "set_integrable M A f \<equiv> integrable M (\<lambda>x. indicat_real A x *\<^sub>R f x)"
null_sets_UN': fixes I :: "'a set"   and N :: "'a \<Rightarrow> 'b set"   and M :: "'b measure" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> N i \<in> null_sets M" shows "\<Union> (N ` I) \<in> null_sets M"
ideduct_synth_mono: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and public :: "'a \<Rightarrow> bool"   and M :: "('a, 'b) Term.term set"   and t :: "('a, 'b) Term.term"   and M' :: "('a, 'b) Term.term set" assumes "intruder_model arity Ana"   and "intruder_model.intruder_synth arity public M t"   and "M \<subseteq> M'" shows "intruder_model.intruder_synth arity public M' t"
relations_zero_imp_pow_not_one: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A"   and "\<forall>f\<in>relations A. f a = 0" shows "\<forall>z. z \<noteq> 0 \<longrightarrow> a [^] z \<noteq> \<one>"
simps_2: fixes ParamUses :: "'h \<Rightarrow> 'g set list" shows "lift_ParamUses ParamUses NewEntry = []"
assms_13: shows "InvariantNoDecisionsWhenUnit (butlast (getF state)) (getM state) (currentLevel (getM state))"
simps_2: fixes p :: "char list"   and ts :: "fterm list" shows "hlit_of_flit (Neg p ts) = Neg p (hterms_of_fterms ts)"
one_eq_numeral_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "num" assumes "class.semiring_char_0 one times plus zero" shows "(one = numeral.numeral one plus n) = (num.One = n)"
lnth_list_of_stream: fixes xs :: "'b stream" shows "lnth (llist_of_stream xs) = (!!) xs"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "comm_semiring_1_hom hom"   and "unit_preserving_hom_axioms hom" shows "unit_preserving_hom hom"
assms: fixes r :: "char list"   and s :: "char list" shows "b r @ s = b (r @ s)"
prv_dsj_com: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla" shows "prv (eqv (dsj \<phi> \<chi>) (dsj \<chi> \<phi>))"
t2: fixes \<Gamma> :: "(name \<times> ty) list"   and t1 :: "lam"   and \<tau> :: "ty"   and \<sigma> :: "ty"   and t2 :: "lam" assumes "\<Gamma> \<turnstile> t1 : \<tau> \<rightarrow> \<sigma>"   and "\<Gamma> \<turnstile> t2 : \<tau>" shows "\<Gamma> \<turnstile> App t1 t2 : \<sigma>"
remdups_replicate: fixes n :: "nat"   and x :: "'a" shows "remdups (replicate n x) = (if n = 0 then [] else [x])"
analz_insert_freshK1: fixes evs :: "event list"   and K :: "nat"   and SesKey :: "nat" assumes "evs \<in> kerbV"   and "K \<in> authKeys evs \<union> range shrK"   and "SesKey \<notin> range shrK" shows "(Key K \<in> analz (insert (Key SesKey) (knows Spy evs))) = (K = SesKey \<or> Key K \<in> analz (knows Spy evs))"
assms_1: shows "P \<longmapsto> a<x> \<prec> P'"
evaluation_functor_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "evaluation_functor A B \<equiv> (category A \<and> category B) \<and> functor_category A B \<and> product_category (concrete_category.COMP (Collect (functor A B)) (\<lambda>F G. Collect (natural_transformation A B F G)) (\<lambda>F G H \<tau> \<sigma>. vertical_composite.map A B \<sigma> \<tau>)) A"
Pair_le: fixes a :: "'b"   and b :: "'c"   and c :: "'b"   and d :: "'c" shows "((a, b) \<le> (c, d)) = (a \<le> c \<and> b \<le> d)"
assm: shows "v=v1\<parallel>v2 \<and> len v1 ts c = ext v1 \<and> len v2 ts c = ext v2"
lunit_in_hom: fixes a :: "'c" assumes "C.ide a" shows "\<guillemotleft>\<l>\<^sub>C[a] : \<I>\<^sub>C \<otimes>\<^sub>C a \<rightarrow>\<^sub>C a\<guillemotright>"
tl_eq_c_tl: fixes x :: "nat" shows "tl (nat_to_list x) = nat_to_list (c_tl x)"
scaleRat_scaleRat: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "rat"   and b :: "rat"   and x :: "'a" assumes "class.rational_vector plus zero minus uminus scaleRat" shows "scaleRat a (scaleRat b x) = scaleRat (a * b) x"
inf_llist_inj: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "(inf_llist f = inf_llist g) = (f = g)"
height_balR2: fixes l :: "('c \<times> nat) tree"   and r :: "('c \<times> nat) tree"   and a :: "'c" assumes "avl l"   and "avl r"   and "height r \<noteq> height l + 2" shows "height (balR l a r) = 1 + max (height l) (height r)"
to_F_def: fixes \<iota>\<^sub>F\<^sub>L :: "('f \<times> 'l) inference" shows "q_lifting.to_F \<iota>\<^sub>F\<^sub>L = Infer (map fst (prems_of \<iota>\<^sub>F\<^sub>L)) (fst (concl_of \<iota>\<^sub>F\<^sub>L))"
sublist_drop: fixes n :: "nat"   and xs :: "'c list" shows "sublist (drop n xs) xs"
shiftr_div_2n: fixes w :: "'b word"   and n :: "nat" shows "uint (w >> n) = uint w div 2 ^ n"
max_repU1: fixes w1 :: "32 Word.word"   and x :: "real"   and w2 :: "32 Word.word"   and y :: "real" assumes "w1 \<equiv>\<^sub>U x"   and "w2 \<equiv>\<^sub>U y" shows "wmax w1 w2 \<equiv>\<^sub>U x"
substitutiont: fixes E :: "char list \<Rightarrow> 'a"   and F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and t :: "fterm"   and \<sigma> :: "char list \<Rightarrow> fterm" shows "eval\<^sub>t E F (t \<cdot>\<^sub>t \<sigma>) = eval\<^sub>t (evalsub E F \<sigma>) F t"
game2a_bad: shows "map_spmf snd game2_a = game2_a_bad"
extensibleI: fixes R2 :: "('b, 'c) labeled_graph"   and G :: "('b, 'd) labeled_graph"   and g :: "('c \<times> 'd) set"   and R1 :: "('b, 'c) labeled_graph"   and f :: "('c \<times> 'd) set" assumes "graph_homomorphism R2 G g"   and "agree_on R1 f g" shows "extensible (R1, R2) G f"
dagger_unfoldl_distr: fixes y :: "'a list set"   and x :: "'a list set" shows "y \<union> p_prod (p_prod x (p_star x)) y = p_prod (p_star x) y"
in_carrier_1: shows "local.eval (local.monom b (length p)) a \<in> carrier R"
ins_type_1: fixes t :: "'a bro"   and h :: "nat"   and x :: "'a" assumes "t \<in> B h" shows "ins x t \<in> Bp h"
le_Inf_iff: 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 bot :: "'c"   and top :: "'c"   and b :: "'c"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq b (Inf A) = (\<forall>a\<in>A. less_eq b a)"
fresh_1: fixes a :: "name"   and x2 :: "name"   and x1 :: "coname" shows "a \<sharp> Ax x2 x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
EQ2_Or: fixes f :: "atom fm"   and g :: "atom fm"   and xs :: "real list" shows "EQ2 (Or f g) xs = EQ2 f xs \<union> EQ2 g xs"
ltg: shows "lt (proj_poly_syz (length bs) g) = map_component (\<lambda>k. k - length bs) (lt g)"
widen_refl: fixes P :: "'a prog"   and T :: "ty" shows "subtype P T T"
tmmul_blt: fixes n :: "nat"   and t :: "tm"   and i :: "poly" assumes "tmboundslt n t" shows "tmboundslt n (tmmul t i)"
sup_cancel_left1: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "x \<squnion> a \<squnion> (- x \<squnion> b) = \<top>"
fi_map_Some: fixes u :: "'v"   and v :: "'u" shows "(fi_map u = Some v) = (u \<in> f ` V \<and> fi u = v)"
\<psi>_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool" assumes "grover_state n f" shows "grover_state.\<psi> n = Matrix.vec (grover_state.N n) (\<lambda>x. complex_of_real (1 / sqrt (real (grover_state.N n))))"
A: shows "\<forall>s\<in>set (toList ec). envObs a s = envObs a (tLast t)"
assms_3: fixes x :: "'a" assumes "x \<in> U"   and "P x" shows "x = a"
component_of_def: fixes g :: "'a ugraph"   and r :: "'a" shows "component_of g r \<equiv> ins_node r (restrict_nodes g (reachable_nodes g r))"
rsquarefree_def': fixes p :: "'a poly" shows "rsquarefree p = (p \<noteq> 0 \<and> (\<forall>a. order a p \<le> 1))"
expand_None: shows "(tab(xs [\<mapsto>] ys)) vn = None"
measure_bind: fixes f :: "'b stream \<Rightarrow> 'c measure"   and s :: "'b"   and N :: "'c measure"   and X :: "'c set" assumes "K2.T.random_variable s (subprob_algebra N) f"   and "X \<in> sets N" shows "Sigma_Algebra.measure (K2.T s \<bind> f) X = K2.T.expectation s (\<lambda>x. Sigma_Algebra.measure (f x) X)"
\<mu>_def: fixes G :: "('c, 'd) pre_digraph"   and f :: "'d \<Rightarrow> real"   and u :: "'c"   and v :: "'c" assumes "wf_digraph G" shows "wf_digraph.\<mu> G f u v \<equiv> INF p\<in>{p. pre_digraph.awalk G u p v}. ereal (wf_digraph.awalk_cost f p)"
that: fixes b :: "V" assumes "r\<lparr>a\<rparr> = b"   and "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r" shows "thesis"
s\<^sub>0: shows "s\<^sub>0 = (h\<^sub>0, l\<^sub>0(V := None))"
Gets_imp_knows_Spy: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> set_mr" shows "X \<in> knows Spy evs"
that: fixes c :: "'a" assumes "(a, c) \<in> A\<^sup>*"   and "(b, c) \<in> A\<^sup>*" shows "thesis"
SUP_outer_measure_of_incseq: fixes A :: "nat \<Rightarrow> 'c set"   and M :: "'c measure" assumes "\<And>n. A n \<subseteq> space M"   and "incseq A" shows "(SUP n. outer_measure_of M (A n)) = outer_measure_of M (\<Union> (range A))"
le_fin_length: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list" assumes "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2" shows "length w\<^sub>1 \<le> length w\<^sub>2"
unique_idx_append: fixes gs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'h"   and hs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data' :: "nat \<times> 'h"   and aux :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> 'g) list \<times> 'h" assumes "unique_idx gs data"   and "(hs, data') = add_indices aux data" shows "unique_idx (gs @ hs) data'"
map_3: fixes f :: "'e \<Rightarrow> 'd"   and x31 :: "'e rexp"   and x32 :: "'e rexp" shows "map_rexp f (Conc x31 x32) = Conc (map_rexp f x31) (map_rexp f x32)"
cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes ptr :: "(_) object_ptr" shows "cast ptr = (case ptr of object_ptr.Ext (Inl x) \<Rightarrow> Map.empty x | object_ptr.Ext (Inr (Inl document_ptr)) \<Rightarrow> Some document_ptr | object_ptr.Ext (Inr (Inr x)) \<Rightarrow> Map.empty x)"
diff_subset_eq_self: fixes M :: "'c multiset"   and N :: "'c multiset" shows "M - N \<subseteq># M"
8: fixes p :: "nat \<times> 'b" assumes "p \<in> gunodes A w"   and "gaccepting A p" shows "even (f p)"
collapse_2: fixes thread :: "thread" assumes "is_HOLD thread" shows "HOLD (ticket thread) = thread"
l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_tag_name set_tag_name_locs type_wf create_element known_ptr"
bdef: shows "b = real_of_rat (rat_of_nat b')"
image_inv_into_cancel: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and A' :: "'a set"   and B' :: "'a set" assumes "f ` A = A'"   and "B' \<subseteq> A'" shows "f ` inv_into A f ` B' = B'"
simps: fixes n :: "nat"   and k :: "nat"   and A :: "'b mat" shows "step_3_main n k A = (if n \<le> k then A else let B = ??.Jordan_Normal_Form_Existence.step_3_a (k - 1) k A; all_blocks = ??.Jordan_Normal_Form_Existence.identify_blocks B k; blocks = filter (\<lambda>block. B $$ (snd block, k) \<noteq> (0::'b)) all_blocks in Let (if blocks = [] then B else let (l_start, l) = ??.Jordan_Normal_Form_Existence.find_largest_block (hd blocks) (tl blocks); x = B $$ (l, k); C = ??.Jordan_Normal_Form_Existence.step_3_c x l k blocks B; D = mult_col_div_row (inverse x) k C; E = ??.Jordan_Normal_Form_Existence.swap_cols_rows_block (Suc l) k D in E) (step_3_main n (Suc k)))"
tag_eq_def: fixes tag :: "'c list \<Rightarrow> 'd" shows "=tag= \<equiv> {(x, y). tag x = tag y}"
Times_Star: fixes r :: "'a language" shows "Times (Star r) (Star r) = Star r"
in_read_tmps_conv: fixes t :: "nat"   and xs :: "'b memref list" shows "(t \<in> read_tmps xs) = (\<exists>volatile a v. Read\<^sub>s\<^sub>b volatile a t v \<in> set xs)"
abs_summable_on_restrict: fixes A :: "'d set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "A \<subseteq> B" shows "(f abs_summable_on A) = ((\<lambda>x. indicat_real A x *\<^sub>R f x) abs_summable_on B)"
netmask_def': fixes I :: "nat set"   and \<sigma> :: "nat \<Rightarrow> 'd"   and \<zeta> :: "'e" shows "netmask I (\<sigma>, \<zeta>) = (\<lambda>i. if i \<in> I then Some (\<sigma> i) else None, \<zeta>)"
IS: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and x' :: "'c" assumes "\<And>x x'. (x, x') \<in> R \<Longrightarrow> f x \<le> M x'"   and "(x, x') \<in> R" shows "body f x \<le> M x'"
Cons_nth_drop_Suc: fixes i :: "nat"   and xs :: "'c list" assumes "i < |xs|" shows "xs ! i # drop (Suc i) xs = drop i xs"
powr_mono': fixes a :: "real"   and b :: "real"   and x :: "real" assumes "a \<le> b"   and "0 \<le> x"   and "x \<le> 1" shows "x powr b \<le> x powr a"
prems_1: shows "timpls_transformable_to TI (Fun f T) t"
cblinfun_image_ccspan: fixes A :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and G :: "'c set" shows "A *\<^sub>S ccspan G = ccspan ((*\<^sub>V) A ` G)"
ind_cpa_def: shows "ind_cpa \<equiv> \<lambda>uu. case uu of (\<A>1, \<A>2) \<Rightarrow> TRY key_gen \<bind> (\<lambda>(pk, sk). \<A>1 pk \<bind> (\<lambda>((m0, m1), \<sigma>). assert_spmf (valid_plains m0 m1) \<bind> (\<lambda>_. coin_spmf \<bind> (\<lambda>b. aencrypt pk (if b then m0 else m1) \<bind> (\<lambda>cipher. \<A>2 cipher \<sigma> \<bind> (\<lambda>b'. return_spmf (b = b'))))))) ELSE coin_spmf"
less_eq_rexp_def: fixes r :: "'b rexp"   and s :: "'b rexp" shows "r \<le> s \<equiv> le_rexp r s"
List_conv_islist_list: fixes h :: "'b \<Rightarrow> 'b ref"   and p :: "'b ref"   and as :: "'b list" shows "List h p as = (islist h p \<and> as = list h p)"
congruent_triangle_sss_9: fixes a1 :: "'b"   and b1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c1 :: "'b"   and c2 :: "'c" assumes "dist a1 b1 = dist a2 b2"   and "dist b1 c1 = dist b2 c2"   and "dist a1 c1 = dist a2 c2" shows "dist b1 c1 = dist c2 b2"
TriSumA_def: 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"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.TriSumA Bet Cong A B C D E F \<equiv> \<exists>G H I. Tarski_neutral_dimensionless.SumA Bet Cong A B C B C A G H I \<and> Tarski_neutral_dimensionless.SumA Bet Cong G H I C A B D E F"
P: shows "real (4 * N) * C / e \<le> real P"
rel_symp: fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "symp R1"   and "symp R2" shows "symp (rel_prod R1 R2)"
2: shows "cmod (char M t - (\<Sum>k\<le>2. (\<i> * complex_of_real t) ^ k * complex_of_real (expectation (\<lambda>x. x ^ k)) / fact k)) \<le> t\<^sup>2 * expectation power2"
assms_4: shows "A' B' C' A' B' C' SumA D E F"
f_def: shows "f \<equiv> float_of (real_of_int m * 2 powr real_of_int e)"
word_of_nat_eq_0_iff: fixes n :: "nat" shows "(word_of_nat n = 0) = (2 ^ LENGTH('b) dvd n)"
Y_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and f :: "'a" assumes "yoneda_functor C S \<phi>" shows "yoneda_functor.Y C S \<phi> f \<equiv> Cop_S.Map (curried_functor'.map (dual_category.comp C) C S (hom_functor.map C S \<phi>) f)"
trm_ord_subst: shows "\<forall>s x y. (x, y) \<in> trm_ord \<longrightarrow> (x \<lhd> s, y \<lhd> s) \<in> trm_ord"
ins_neq_Leaf: fixes k :: "'a"   and p :: "'b"   and t :: "('a \<times> 'b) tree" shows "ins k p t \<noteq> \<langle>\<rangle>"
nn_integral_cmult: fixes f :: "'h \<Rightarrow> ennreal"   and M :: "'h measure"   and c :: "ennreal" assumes "f \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. c * f x \<partial>M = c * integral\<^sup>N M f"
f_sum: shows "(\<Sum>v\<in>T. f v *\<^sub>R v) = (0::'a)"
name_of_nat_nat_of_name: fixes i :: "name" shows "name_of_nat (nat_of_name i) = i"
isometry_on_subset: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and Y :: "'b set" assumes "isometry_on X f"   and "Y \<subseteq> X" shows "isometry_on Y f"
meta_cong_bigtheta: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<Theta>[F](f) \<equiv> \<Theta>[F](g)" shows "O[F](f) \<equiv> O[F](g)"
zip_replicate1: fixes n :: "nat"   and x :: "'a"   and ys :: "'b list" shows "zip (replicate n x) ys = map (Pair x) (take n ys)"
B: fixes x :: "real" assumes "x \<in> S" shows "norm (f' x - f' x0) \<le> B"
step_nonneg: fixes i :: "nat"   and x :: "real" assumes "i < k"   and "x\<^sub>1 \<le> x" shows "0 \<le> bs ! i * x + (hs ! i) x"
star_zero_def: shows "0 \<equiv> star_of (0::'c)"
intros: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "d_OUT f x = 0" shows "x \<in> SINK f"
spr_knowledge_def: fixes envObs :: "'e \<Rightarrow> ('e, 'f, 'g) BEState \<Rightarrow> 'h \<times> 'g option"   and cec :: "(('e, 'f, 'g) BEState \<times> ('e, 'f, 'g) BEState) odlist" shows "spr_knowledge envObs cec \<equiv> \<lambda>a s. fromList (concat (map (\<lambda>s'. if (s, s') \<in> spr_repRels envObs a then [s'] else []) (toList cec)))"
elims: fixes x :: "'b kdt"   and y :: "nat" assumes "height x = y" obtains uu_ :: "(real, 'b) vec" where "x = Leaf uu_" and "y = 0"   | uv_ :: "'b" and uw_ :: "real" and l :: "'b kdt" and r :: "'b kdt" where "x = Node uv_ uw_ l r" and "y = max (height l) (height r) + 1"
simps: fixes v1 :: "'a"   and p1 :: "'b"   and v2 :: "'a"   and p2 :: "'c" shows "Pre_Port_Graph.edge_begin ((v1, p1), v2, p2) = v1"
split0_npoly_2: fixes t :: "tm" assumes "allpolys isnpoly t" shows "allpolys isnpoly (snd (split0 t))"
ap_list_def: fixes fs :: "('b \<Rightarrow> 'a) list"   and xs :: "'b list" shows "fs \<diamondop> xs = List.bind fs (\<lambda>f. List.bind xs (\<lambda>x. [f x]))"
cosine_law_triangle: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "(dist b c)\<^sup>2 = (dist a b)\<^sup>2 + (dist a c)\<^sup>2 - 2 * dist a b * dist a c * cos (angle b a c)"
gen_lucas_lehmer_sequence''_bounds: fixes a :: "int"   and p :: "nat"   and n :: "nat" assumes "0 \<le> a"   and "a < 2 ^ p + 5"   and "2 < p" shows "gen_lucas_lehmer_sequence'' p a n \<in> {0..<2 ^ p + 5}"
S3: fixes x :: "i"   and y :: "i" shows "(E (dom x) \<^bold>\<and> \<^bold>\<not> (\<^bold>\<not> (E (cod y)) \<^bold>\<or> dom x \<noteq> cod y) \<^bold>\<leftarrow> E (x \<cdot> y)) \<^bold>\<and> (E (x \<cdot> y) \<^bold>\<leftarrow> E (dom x) \<^bold>\<and> \<^bold>\<not> (\<^bold>\<not> (E (cod y)) \<^bold>\<or> dom x \<noteq> cod y))"