hcju/isabelleps huggingface dataset:
replicate_mset_eq_iff: fixes m :: "nat"   and a :: "'b"   and n :: "nat"   and b :: "'b" shows "(replicate_mset m a = replicate_mset n b) = (m = 0 \<and> n = 0 \<or> m = n \<and> a = b)"
scaleR_sum_right: fixes a :: "real"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "a *\<^sub>R sum f A = (\<Sum>x\<in>A. a *\<^sub>R f x)"
assms_1: shows "cri.InvGlobNonposImpRecordsNonpos (exchange_config c)"
sum_2: assumes "vars p2 \<subseteq> {..<n}" shows "(\<lambda>f. insertion f p2) \<in> borel_measurable (lborel_f n)"
var_not_in_eval2: fixes \<phi> :: "atom fm" shows "\<forall>L. freeIn 0 \<phi> \<longrightarrow> (\<exists>x. eval \<phi> (x # L)) = (\<forall>x. eval \<phi> (x # L))"
DC_eq: fixes K :: "nat"   and X :: "msg" shows "Decrypt K (Crypt K X) = X"
arev_dom: fixes a :: "'b" shows "(a \<in> arcs H) = (rev_H a \<noteq> a)"
language_def: shows "language \<equiv> {xs. nextl (dfa.init M) xs \<in> dfa.final M}"
CAS_\<tau>red1t_xt3: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and v :: "'addr val"   and D :: "String.literal"   and F :: "String.literal"   and v' :: "'addr val" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h (Val v\<bullet>compareAndSwap(D\<bullet>F, Val v', e), xs) (Val v\<bullet>compareAndSwap(D\<bullet>F, Val v', e'), xs')"
lt0: shows "unat a div 2 ^ n < 2 ^ (LENGTH('a) - n)"
star_of_simps_9: shows "star_of (0::'b) = 0"
load_tmps_distinct: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "distinct_load_tmps is"
weight_exec_gpv: fixes callee :: "'b \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'b) spmf"   and I :: "'b \<Rightarrow> bool"   and \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and s :: "'b" assumes "callee_invariant_on callee I \<I>"   and "\<And>s x. \<lbrakk>x \<in> outs_\<I> \<I>; I s\<rbrakk> \<Longrightarrow> lossless_spmf (callee s x)"   and "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "weight_gpv \<I> gpv \<le> weight_spmf (exec_gpv callee gpv s)"
greatest_def: fixes L :: "('b, 'c) gorder_scheme"   and g :: "'b"   and A :: "'b set" shows "greatest L g A = (A \<subseteq> carrier L \<and> g \<in> A \<and> (\<forall>x\<in>A. le L x g))"
pp_init_l_def: shows "pp_init_l \<equiv> (\<lambda>x. 0)(s := card V)"
multiplicity_times_same_power: fixes x :: "'a"   and p :: "'a"   and k :: "nat" assumes "x \<noteq> (0::'a)"   and "\<not> is_unit p"   and "p \<noteq> (0::'a)" shows "multiplicity p (p ^ k * x) = multiplicity p x + k"
fpxs_nth_compose_power: fixes r :: "rat"   and f :: "'b fpxs"   and n :: "rat" assumes "0 < r" shows "fpxs_nth (fpxs_compose_power f r) n = fpxs_nth f (n / r)"
qintersect: fixes Q :: "process set"   and Q' :: "process set" assumes "Q \<in> Quorum"   and "Q' \<in> Quorum" shows "Q \<inter> Q' \<noteq> {}"
chain_imp_chain': fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist" assumes "chain R xs" shows "chain' R xs"
simps: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and a1 :: "char list \<Rightarrow> ty option"   and a2 :: "expr list"   and a3 :: "ty list" shows "(P,a1 \<turnstile> a2 [::] a3) = ((\<exists>E. a1 = E \<and> a2 = [] \<and> a3 = []) \<or> (\<exists>E e T es Ts. a1 = E \<and> a2 = e # es \<and> a3 = T # Ts \<and> P,E \<turnstile> e :: T \<and> P,E \<turnstile> es [::] Ts))"
awalk_verts_arc1: fixes e :: "'a \<times> 'a"   and p :: "('a \<times> 'a) list"   and u :: "'a" assumes "e \<in> set p" shows "fst e \<in> set (pawalk_verts u p)"
Diff_eq_empty_iff: fixes A :: "'a set"   and B :: "'a set" shows "(A - B = {}) = (A \<subseteq> B)"
S_inductive_maxl: assumes "S_inductive_set D"   and "carrier D \<noteq> {}" shows "\<exists>m. maximal m"
jipsen1r: fixes x :: "'a"   and y :: "'a" shows "x \<le> (y \<leftarrow> x) \<rightarrow> y"
trace_below: fixes d :: "'b \<Rightarrow> nat \<Rightarrow> nat"   and k :: "nat"   and xs :: "'b list" assumes "bounded d k" shows "\<forall>i<k. \<forall>n\<in>set (trace d i xs). n < k"
sim_while_cong_aux: fixes b :: "bexp"   and c :: "com"   and s :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int"   and c' :: "com" assumes "(WHILE b DO c, s) \<Rightarrow> t"   and "c \<sim> c'" shows "(WHILE b DO c', s) \<Rightarrow> t"
language_elim: fixes w :: "'c stream"   and A :: "('c, 'd) nba" assumes "w \<in> language A" obtains r :: "'d stream" and p :: "'d" where "p \<in> initial A" and "run A (w ||| r) p" and "infs (accepting A) (p ## r)"
prems_4: shows "(ac_, a'c_) \<in> R \<rightarrow> Rv"
chain_Diff: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b set" assumes "Complete_Partial_Order.chain ord A" shows "Complete_Partial_Order.chain ord (A - B)"
Ord_induct: fixes k :: "hf"   and P :: "hf \<Rightarrow> bool" assumes "Ord k"   and "\<And>x. \<lbrakk>Ord x; \<And>y. y \<^bold>\<in> x \<Longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P k"
finsum_finsum_swap: fixes A :: "'b set"   and B :: "'c set"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "finite A"   and "finite B"   and "\<And>a b. \<lbrakk>a \<in> A; b \<in> B\<rbrakk> \<Longrightarrow> g a b \<in> UNIV" shows "(\<Sum>a\<in>A. sum (g a) B) = (\<Sum>b\<in>B. \<Sum>a\<in>A. g a b)"
add: fixes b1 :: "('a, 'n) vec"   and b2 :: "('a, 'n) vec" shows "f (b1 + b2) = f b1 + f b2"
fmlookup_image_iff: fixes y :: "'c"   and m :: "('d, 'c) fmap"   and A :: "'d fset" shows "(y |\<in>| fmimage m A) = (\<exists>x. fmlookup m x = Some y \<and> x |\<in>| A)"
assms_9: shows "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"
is_reduced_GB_subsetI: fixes A :: "('t \<Rightarrow>\<^sub>0 'c) set"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "is_reduced_GB A"   and "is_Groebner_basis B"   and "is_monic_set B"   and "\<And>a b. \<lbrakk>a \<in> A; b \<in> B; a \<noteq> 0; b \<noteq> 0; a - b \<noteq> 0; lt (a - b) \<in> keys b; lt (a - b) \<prec>\<^sub>t lt b\<rbrakk> \<Longrightarrow> False"   and "pmdl A = pmdl B" shows "A \<subseteq> B"
filter_remove1: fixes Q :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "filter Q (remove1 x xs) = remove1 x (filter Q xs)"
bin_last_integer_code: fixes i :: "integer" shows "bin_last_integer i = (i AND 1 \<noteq> 0)"
image_respects_domain_permutation: fixes A :: "'c set"   and B :: "'d set" shows "(\<lambda>f. f ` A) respects domain_permutation A B"
leg0_commutes: shows "Cod_\<rho>.leg0 \<cdot> \<rho>.chine = Dom_\<rho>.leg0"
UP_DOWN: fixes t :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "t \<in> Univ" shows "UP (DOWN t) = t"
Tree\<^sub>\<alpha>_free_3: fixes tset\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha> set['a]"   and f :: "'d"   and \<alpha> :: "'c"   and t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>" shows "Conj\<^sub>\<alpha> tset\<^sub>\<alpha> \<noteq> Act\<^sub>\<alpha> f \<alpha> t\<^sub>\<alpha>"
adjoin_unique: fixes x :: "'a" assumes "x \<in> adjoin G H a" shows "\<exists>!z. z \<in> H \<times> {..<subgroup_indicator G H a} \<and> x = fst z \<otimes> a [^] snd z"
eventually_at_open_segment: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> {a<--<b}" shows "\<forall>\<^sub>F y in at x. (y - a) \<bullet> rot (a - b) = 0 \<longrightarrow> y \<in> {a<--<b}"
cindex_polyE_smult_1: fixes a :: "real"   and b :: "real"   and c :: "real"   and q :: "real poly"   and p :: "real poly" shows "cindex_polyE a b (smult c q) p = sgn c * cindex_polyE a b q p"
index_update_if_diff2: fixes n :: "nat"   and xs :: "'b list"   and x :: "'b"   and y :: "'b" assumes "n < length xs"   and "x \<noteq> xs ! n"   and "x \<noteq> y" shows "index (xs[n := y]) x = index xs x"
good_ruleset_fst: fixes r :: "'b rule"   and rs :: "'b rule list" assumes "good_ruleset (r # rs)" shows "good_ruleset [r]"
Loeb_Assumptions_axioms: shows "Loeb_Assumptions var trm fmla Var num FvarsT substT Fvars subst eql cnj imp all exi prv bprv enc S P"
1: shows "wls s (X #[x' \<and> x]_xs)"
conj_elim: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "(P \<and> Q) x" obtains "P x" and "Q x"
simps_2: fixes y :: "nat"   and z :: "nat" shows "f4 y z = (if y = z then 1 else 0)"
word_and_max: fixes x :: "'b word" shows "x AND - 1 = x"
induct: fixes x :: "msg"   and H :: "msg set"   and P :: "msg \<Rightarrow> bool" assumes "x \<in> analz H"   and "\<And>X. X \<in> H \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>\<langle>X, Y\<rangle> \<in> analz H; P \<langle>X, Y\<rangle>\<rbrakk> \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>\<langle>X, Y\<rangle> \<in> analz H; P \<langle>X, Y\<rangle>\<rbrakk> \<Longrightarrow> P Y"   and "\<And>X Y. \<lbrakk>Enc X Y \<in> analz H; P (Enc X Y); Y \<in> synth (analz H \<inter> Collect P)\<rbrakk> \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>Aenc X (pubK Y) \<in> analz H; P (Aenc X (pubK Y)); priK Y \<in> analz H; P (priK Y)\<rbrakk> \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>Aenc X (epubK Y) \<in> analz H; P (Aenc X (epubK Y)); epriK Y \<in> synth (analz H \<inter> Collect P)\<rbrakk> \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>Sign X (priK Y) \<in> analz H; P (Sign X (priK Y)); pubK Y \<in> analz H; P (pubK Y)\<rbrakk> \<Longrightarrow> P X" shows "P x"
lift_character_unchanged: fixes x :: "'a"   and \<chi>z :: "('a \<Rightarrow> complex) \<times> complex" assumes "x \<in> H'" shows "H'.lift_character \<chi>z x = fst \<chi>z x"
eq_onp_def: fixes R :: "'c \<Rightarrow> bool" shows "eq_onp R = (\<lambda>x y. R x \<and> x = y)"
assms_3: shows "borel_adapt_stoch_proc F (val_process Mkt p)"
iso_is_arr: fixes f :: "'a" assumes "A.iso f" shows "A.arr f"
tak_m2_def: shows "tak_m2 = (\<lambda>(x, y, z). nat (Max {x, y, z} - Min {x, y, z}))"
notStuck_SeqD1: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile> \<langle>Seq c1 c2,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}" shows "\<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"
fair_if_finite: assumes "lfinite Sts" shows "fair_state_seq (lmap state_of_wstate Sts)"
cval_add_def: fixes u :: "'d \<Rightarrow> 'e"   and d :: "'e" shows "u \<oplus> d = (\<lambda>x. u x + d)"
Cons_6: shows "literalFalse literal (elements (getM state))"
eq: shows "lst_conv (fst (fst (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))), snd (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))) @ snd \<delta>3), snd (fst (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))), snd (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))) @ snd \<delta>3) @ snd (conv_mirror \<gamma>3)) = fst (fst \<gamma>1''', map (Pair True) (snd \<gamma>1'''))"
m_closed: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme"   and x :: "'e"   and y :: "'e" assumes "submonoid H G"   and "x \<in> H"   and "y \<in> H" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> H"
rep_eq: fixes x :: "('a, 'b) ffun"   and xa :: "('a, 'b) ffun" shows "(x \<subseteq>\<^sub>f xa) = (pfun_of x \<subseteq>\<^sub>p pfun_of xa)"
dimensionI: fixes Us :: "'a list"   and E :: "'a set" assumes "local.independent K Us"   and "Span K Us = E" shows "dimension (length Us) K E"
linorder_lattice_inf_le2: fixes x :: "'a"   and y :: "'a" shows "inf x y \<le> y"
Inf_lower2: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and u :: "'a"   and A :: "'a set"   and v :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "u \<in> A"   and "less_eq u v" shows "less_eq (Inf A) v"
erase_mat_def: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and i :: "'c"   and j :: "'b" shows "erase_mat A i j = (\<chi>i' j'. if i' = i \<or> j' = j then 0::'a else A $h i' $h j')"
TA_def: fixes x :: "('a, 'n) vec" shows "TA x = A *v x"
permute_flip_cancel: fixes a :: "'b"   and b :: "'b"   and x :: "'a" shows "(a \<leftrightarrow> b) \<bullet> (a \<leftrightarrow> b) \<bullet> x = x"
power_diff_power_eq: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and a :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "a \<noteq> zero" shows "divide (power.power one times a m) (power.power one times a n) = (if n \<le> m then power.power one times a (m - n) else divide one (power.power one times a (n - m)))"
landau_function_family_chain_axioms: shows "landau_function_family_chain F gs get_fun"
induct: fixes P :: "rat poly \<Rightarrow> rat poly \<Rightarrow> bool"   and a0 :: "rat poly"   and a1 :: "rat poly" assumes "\<And>p q. (degree q \<noteq> 0 \<Longrightarrow> P q (- (p mod q))) \<Longrightarrow> P p q" shows "P a0 a1"
subgroup_in_rcosets: assumes "Group.group G" shows "H \<in> rcosets\<^bsub>G\<^esub> H"
inject: fixes x1 :: "'d"   and x2 :: "'d stream"   and y1 :: "'d"   and y2 :: "'d stream" shows "(x1 ## x2 = y1 ## y2) = (x1 = y1 \<and> x2 = y2)"
case_prodE2: fixes Q :: "'a \<Rightarrow> bool"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and z :: "'b \<times> 'c" assumes "Q (case z of (a, b) \<Rightarrow> P a b)" obtains x :: "'b" and y :: "'c" where "z = (x, y)" and "Q (P x y)"
when_1: fixes P :: "bool"   and a :: "'a" assumes "P" shows "(a when P) = a"
proper_uprod_simps: fixes x :: "'a"   and y :: "'a" shows "proper_uprod (Upair x y) = (x \<noteq> y)"
inj_on_id2: fixes A :: "'a set" shows "inj_on (\<lambda>x. x) A"
returns_heap_def: fixes h :: "'d"   and p :: "('d, 'e, 'f) prog"   and h' :: "'d" shows "h \<turnstile> p \<rightarrow>\<^sub>h h' = (case h \<turnstile> p of Inl x \<Rightarrow> False | Inr (x, h'') \<Rightarrow> h' = h'')"
chamberD_simplex: fixes x :: "'b set" assumes "codomain.chamber x" shows "x \<in> Y"
not_eq_order_implies_strict: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "a \<noteq> b"   and "a \<preceq> b" shows "a \<prec> b"
parts_mono: fixes H :: "msg set"   and H' :: "msg set" assumes "H \<subseteq> H'" shows "parts H \<subseteq> parts H'"
continuous_on_real_range: fixes s :: "'a set"   and f :: "'a \<Rightarrow> real" shows "continuous_on s f = (\<forall>x\<in>s. \<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. norm (x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e)"
is_LT_def: shows "is_LT \<equiv> Ordering_case\<cdot>TT\<cdot>FF\<cdot>FF"
exp_conj_std_split: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "\<guillemotleft> \<lambda>s. P s \<and> Q s \<guillemotright> = \<guillemotleft> P \<guillemotright> && \<guillemotleft> Q \<guillemotright>"
has_fields_rec: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and D :: "char list"   and fs :: "(char list \<times> ty) list"   and ms :: "(char list \<times> ty list \<times> ty \<times> 'b) list"   and FDTs :: "((char list \<times> char list) \<times> ty) list"   and FDTs' :: "((char list \<times> char list) \<times> ty) list" assumes "class P C = \<lfloor>(D, fs, ms)\<rfloor>"   and "C \<noteq> Object"   and "P \<turnstile> D has_fields FDTs"   and "FDTs' = map (\<lambda>(F, T). ((F, C), T)) fs @ FDTs" shows "P \<turnstile> C has_fields FDTs'"
is_norm_1_def: fixes z :: "complex" shows "is_norm_1 z = ((Re z)\<^sup>2 + (Im z)\<^sup>2 = 1)"
wf_inst_imp_inst_ok: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and l :: "(variable \<times> String.literal set) list"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ" assumes "wf_theory \<Theta>"   and "distinct l"   and "\<forall>(v, S)\<in>set l. typ_ok \<Theta> (Tv v S)"   and "wf_inst \<Theta> \<rho>" shows "inst_ok \<Theta> (map (\<lambda>(v, S). ((v, S), \<rho> v S)) l)"
tl_append: fixes xs :: "'c list"   and ys :: "'c list" shows "tl (xs @ ys) = (case xs of [] \<Rightarrow> tl ys | z # zs \<Rightarrow> zs @ ys)"
assms_5: shows "F \<le> forest_components (F \<sqinter> w)"
euclidean_representation_sum': fixes b :: "'c"   and f :: "'c \<Rightarrow> real" shows "(b = (\<Sum>i\<in>Basis. f i *\<^sub>R i)) = (\<forall>i\<in>Basis. f i = b \<bullet> i)"
MWLsSteps_det_cases_2: fixes x :: "'d"   and \<iota> :: "nat"   and e :: "'c"   and m :: "'d \<Rightarrow> 'e"   and \<alpha> :: "('c, 'd) MWLsCom list"   and p :: "('c, 'd) MWLsCom option"   and m' :: "'d \<Rightarrow> 'e"   and E :: "'c \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and BMap :: "'e \<Rightarrow> bool" assumes "((x:=\<^bsub>\<iota>\<^esub> e, m), \<alpha>, p, m') \<in> MWLs_semantics.MWLsSteps_det E BMap" obtains "\<alpha> = []" and "p = None" and "m' = m(x := E e m)"
list_eq_setD: fixes xs :: "'c list"   and ys :: "'c list" assumes "xs = ys" shows "set xs = set ys"
assms: shows "X \<subseteq> N \<times> (Pow G - {{}})"
low_proc_0\<^sub>2_def: fixes s\<^sub>L :: "stmt" shows "low_proc_0\<^sub>2 s\<^sub>L \<equiv> \<forall>\<sigma> u. match \<sigma> \<Gamma>\<^sub>0 \<and> the (\<sigma> ''lowIn'') = 0 \<longrightarrow> (\<exists>\<sigma>'. s\<^sub>L \<rhd> \<sigma> \<leadsto> Some \<sigma>' \<and> the (\<sigma>' ''lowOut'') = u)"
Riemann_lebesgue_sin: fixes f :: "real \<Rightarrow> real" assumes "f absolutely_integrable_on {- pi..pi}" shows "(\<lambda>n. LINT x|lebesgue_on {- pi..pi}. sin (real n * x) * f x) \<longlonglongrightarrow> 0"
set_s: shows "set s = to_woot_character M ` insert_messages M"
intro: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and final1 :: "'g \<Rightarrow> bool"   and final2 :: "'i \<Rightarrow> bool" assumes "bisimulation trsys1 trsys2 bisim tlsim"   and "bisimulation_final_axioms bisim final1 final2" shows "bisimulation_final trsys1 trsys2 bisim tlsim final1 final2"
strict_sorted_iff: fixes l :: "'a list" shows "strict_sorted l = (sorted l \<and> distinct l)"
one_im_sheaf_def: shows "one_im_sheaf \<equiv> \<lambda>V. one_sheaf_spec (identity Spec \<^sup>\<inverse> Spec V)"
u: fixes m :: "nat"   and n :: "nat" shows "\<exists>a. a \<in> sfis (u m n) M"
perm_swap_permutes2: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and S :: "'c set" assumes "f permutes (x \<rightleftharpoons>\<^sub>F y) ` S" shows "perm_swap x y f permutes S"
strict_suffix_def: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs <s ys) = (xs \<le>s ys \<and> xs \<noteq> ys)"
refinesI: fixes R :: "('a \<times> 'b) set"   and Sa :: "('a, 'c) spec"   and Sc :: "('b, 'd) spec"   and pi :: "'d \<Rightarrow> 'c" assumes "PO_refines R Sa Sc"   and "obs_consistent R pi Sa Sc" shows "refines R pi Sa Sc"
bi_unique_rel_set_lemma: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and X :: "'c set"   and Y :: "'d set" assumes "bi_unique R"   and "rel_set R X Y" obtains f :: "'c \<Rightarrow> 'd" where "Y = f ` X" and "inj_on f X" and "\<forall>x\<in>X. R x (f x)"
strict_mono_sets_def: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd set" shows "strict_mono_sets A f \<equiv> \<forall>x\<in>A. \<forall>y\<in>A. x < y \<longrightarrow> f x \<lless> f y"
ln_Gamma_complex_LIMSEQ: fixes z :: "complex" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "ln_Gamma_series z \<longlonglongrightarrow> ln_Gamma z"
b: shows "[\<lparr>F,y\<^sup>P\<rparr> \<^bold>\<equiv> \<lbrace>y\<^sup>P,H\<rbrace> in v]"
true_clss_subclause: fixes C :: "'b literal multiset"   and D :: "'b literal multiset"   and I :: "'b set" assumes "C \<subseteq># D"   and "I \<TTurnstile>s {C}" shows "I \<TTurnstile>s {D}"
option_bindE: fixes x :: "'e option"   and f :: "'e \<Rightarrow> 'd option"   and a :: "'d" assumes "x \<bind> f = Some a" obtains x' :: "'e" where "x = Some x'" and "f x' = Some a"
eq_diff_eq: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> V"   and "y \<in> V"   and "z \<in> V" shows "(x = z - y) = (x + y = z)"
q_def: shows "q = p - monom_mult (lookup p (t \<oplus> lt f) / lc f) t f"
not_distinct_decomp_min_prefix: fixes ws :: "'c list" assumes "\<not> distinct ws" shows "\<exists>xs ys zs y. ws = xs @ y # ys @ y # zs \<and> distinct xs \<and> y \<notin> set xs \<and> y \<notin> set ys"
mult_cancel_left1: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and c :: "'e"   and b :: "'e" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(c = times c b) = (c = zero \<or> b = one)"
get_element_by_id_is_strongly_dom_component_safe_step: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and idd :: "char list"   and result :: "(_) element_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_element_by_id ptr' idd \<rightarrow>\<^sub>r Some result" shows "(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r result \<in> set c) = (ptr' \<in> set c)"
lcm_eq_1_iff: 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 b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "(lcm a b = one) = (dvd.dvd times a one \<and> dvd.dvd times b one)"
asym_iff: fixes R :: "('a \<times> 'a) set" shows "asym R = (\<forall>x y. (x, y) \<in> R \<longrightarrow> (y, x) \<notin> R)"
uabv: fixes m :: "nat"   and n :: "nat" shows "u \<le> a (real m / 2 ^ n) \<and> a (real m / 2 ^ n) \<le> b (real m / 2 ^ n) \<and> b (real m / 2 ^ n) \<le> v"
intros: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and \<Delta> :: "('a, 'b) network_scheme" assumes "\<And>e. f e \<le> capacity \<Delta> e"   and "\<And>x. \<lbrakk>x \<noteq> source \<Delta>; x \<noteq> sink \<Delta>\<rbrakk> \<Longrightarrow> KIR f x" shows "flow \<Delta> f"
simps: fixes i :: "nat"   and j :: "nat" shows "h2 i j = (if i = j then One else if i + 1 = j then Two else Zero)"