hcju/isabelleps huggingface dataset:
scaleR_right_distrib: fixes a :: "real"   and x :: "'d"   and y :: "'d" shows "a *\<^sub>R (x + y) = a *\<^sub>R x + a *\<^sub>R y"
hom_add: fixes x :: "int \<times> int"   and y :: "int \<times> int" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "\<phi> (x \<oplus>\<^bsub>R\<^esub> y) = \<phi> x \<oplus>\<^bsub>S\<^esub> \<phi> y"
sumE: fixes s :: "'b + 'c" obtains x :: "'b" where "s = Inl x"   | y :: "'c" where "s = Inr y"
assms_2: assumes "m < n \<and> x < y"   and "x = m"   and "y = n" shows "P (arr n) (arr m)"
iFROM_add_neg: fixes n :: "nat"   and k :: "nat" shows "[n\<dots>] \<oplus>- k = [n - k\<dots>]"
unit_in_vhom: fixes a :: "'a" assumes "obj a" shows "\<guillemotleft>\<i>[a] : a \<star> a \<Rightarrow> a\<guillemotright>"
hn_val_unit_conv_emp: fixes x :: "unit"   and y :: "unit" shows "hn_val unit_rel x y = emp"
coeff_powsum_mpoly: fixes X :: "nat set"   and k :: "nat"   and mon :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (powsum_mpoly X k) mon = (if infinite X \<or> k = 0 \<and> mon \<noteq> 0 then 0::'b else if k = 0 \<and> mon = 0 then of_nat (card X) else if finite X \<and> (\<exists>x\<in>X. mon = Poly_Mapping.single x k) then 1::'b else (0::'b))"
F_le: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and e :: "'v \<times> 'v" assumes "f \<in> Field leq" shows "f e \<le> f' e"
a1: shows "low_equal s1 s2 \<and> get_S (cpu_reg_val PSR s1) = 0 \<and> get_S (cpu_reg_val PSR s2) = 0 \<and> t1 = snd (fst (read_state_reg_instr instr s1)) \<and> t2 = snd (fst (read_state_reg_instr instr s2))"
mk_mat_def: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'd" shows "Matrix.mk_mat nr nc f \<equiv> \<lambda>(i, j). if i < nr \<and> j < nc then f (i, j) else undef_mat nr nc f (i, j)"
simps_1: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" shows "foldr f [] = id"
m\<tau>move_silentD: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "m\<tau>move1 s (t, ta) s'" shows "ta = \<lbrace>\<rbrace>"
qSwapped_preserves_qGood: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "(X, Y) \<in> qSwapped" shows "qGood Y = qGood X"
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"   and a2 :: "char list list" shows "Subobjs P a1 a2 = ((\<exists>C Cs. a1 = C \<and> a2 = Cs \<and> Subobjs\<^sub>R P C Cs) \<or> (\<exists>C C' D Cs. a1 = C \<and> a2 = Cs \<and> P \<turnstile> C \<preceq>\<^sup>* C' \<and> P \<turnstile> C' \<prec>\<^sub>S D \<and> Subobjs\<^sub>R P D Cs))"
gbinomial_Suc: fixes a :: "'a"   and k :: "nat" shows "a gchoose Suc k = (\<Prod>i = 0..k. a - of_nat i) div fact (Suc k)"
nat_pow_closed: fixes x :: "'c"   and n :: "nat" assumes "x \<in> carrier S" shows "x [^]\<^bsub>S\<^esub> n \<in> carrier S"
permute_minus_cancel_1: fixes p :: "perm"   and x :: "'a" shows "p \<bullet> - p \<bullet> x = x"
Bounded_mat_elements_mat: fixes A :: "'b mat"   and b :: "'b" shows "(A \<in> Bounded_mat b) = (\<forall>x\<in>elements_mat A. \<bar>x\<bar> \<le> b)"
input_unconstrained_aval_input_swap: fixes a :: "vname aexp"   and i :: "value list"   and r :: "nat \<Rightarrow>f value option"   and i' :: "value list" assumes "\<forall>i. \<not> aexp_constrains a (V (I i))" shows "aval a (join_ir i r) = aval a (join_ir i' r)"
set_4: fixes x21 :: "'b"   and x22 :: "('b, 'a) Term.term list" shows "fv (Fun x21 x22) = \<Union> (fv ` set x22)"
that: fixes c :: "nat \<Rightarrow> int" assumes "f = M.sumlist (map (\<lambda>i. of_int (c i) \<cdot>\<^sub>v fs ! i) [0..<length fs])" shows "thesis"
letrec1: fixes funs :: "(char list \<times> char list \<times> exp0) list"   and ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and bv :: "'b state \<times> (v, v) result" assumes "allDistinct (map (\<lambda>x. case x of (x, y, z) \<Rightarrow> x) funs) \<and> evaluate ck (update_v (\<lambda>_. build_rec_env funs env (sem_env.v env)) env) s e bv" shows "evaluate ck env s (Letrec funs e) bv"
finite_span: fixes S :: "'e set" assumes "finite S"   and "S \<subseteq> carrier W" shows "W.module.span S = {W.module.lincomb a S |a. a \<in> S \<rightarrow> carrier K}"
comp_char: fixes g :: "'d \<times> 'd"   and f :: "'d \<times> 'd" shows "D.VxV.comp g f = (if D.seq (fst g) (fst f) \<and> D.seq (snd g) (snd f) then (fst g \<cdot>\<^sub>D fst f, snd g \<cdot>\<^sub>D snd f) else (D.null, D.null))"
that: fixes v' :: "'val" assumes "v' \<in> phiUses g m"   and "phiArg g v v'" shows "thesis"
unlabel_append: fixes A :: "('f \<times> 'e) list"   and B :: "('f \<times> 'e) list" shows "unlabel (A @ B) = unlabel A @ unlabel B"
cong_diff: 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 "A \<noteq> B"   and "Cong A B C D" shows "C \<noteq> D"
sorted_wrt_mono_rel: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs; P x y\<rbrakk> \<Longrightarrow> Q x y"   and "sorted_wrt P xs" shows "sorted_wrt Q xs"
comp_quasi_inverse_2: fixes f :: "'b" assumes "D.equivalence_map f" shows "f \<star>\<^sub>D f\<^sup>~\<^sup>D \<cong>\<^sub>D trg\<^sub>D f"
interval_integral_substitution_finite: fixes a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> 'b" assumes "a \<le> b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> (g has_real_derivative g' x) (at x within {a..b})"   and "continuous_on (g ` {a..b}) f"   and "continuous_on {a..b} g'" shows "LBINT x=ereal a..ereal b. g' x *\<^sub>R f (g x) = interval_lebesgue_integral lborel (ereal (g a)) (ereal (g b)) f"
simps_1: fixes less_eq :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and less :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and f :: "'m \<Rightarrow> 'l"   and x :: "'m" assumes "class.linorder less_eq less" shows "linorder.insort_key less_eq f x [] = [x]"
g: fixes j :: "nat" assumes "j < m" shows "map of_int_hom.vec_hom fs ! j \<in> Rn"
isCont_f: fixes x :: "real"   and n :: "nat" shows "isCont (\<lambda>s. f s n) x"
termFSwImorph_wlsInp: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) model_scheme" assumes "wlsInp delta inp"   and "termFSwImorph h hA MOD" shows "igWlsInp MOD delta (lift h inp)"
has_equalizers_def: shows "has_equalizers = (\<forall>f0 f1. par f0 f1 \<longrightarrow> (\<exists>e. has_as_equalizer f0 f1 e))"
Min_ge_iff: fixes A :: "'a Word.word set"   and x :: "'a Word.word" assumes "finite A"   and "A \<noteq> {}" shows "(x \<le>s signed.Min A) = (\<forall>a\<in>A. x \<le>s a)"
simps_2: fixes h :: "'b tree" shows "cost Del_min [h] = T_del_min h"
simps_2: fixes C :: "char list"   and e' :: "'a exp"   and Cs :: "char list list"   and e\<^sub>i :: "'a exp"   and e :: "'a exp" shows "init_switch (RI (C,e') ; Cs \<leftarrow> e\<^sub>i) e = RI (C,e') ; Cs \<leftarrow> e"
Inv_preserves_Ide: fixes t :: "'a local.term" assumes "Ide t" shows "Ide (Inv t)"
xor_one_left: fixes x :: "'b" shows "- (1::'b) XOR x = NOT x"
distrib': fixes I :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "finite {x \<in> I. g x \<noteq> (0::'c)}"   and "finite {x \<in> I. h x \<noteq> (0::'c)}" shows "sum' (\<lambda>i. g i + h i) I = sum' g I + sum' h I"
product_run_project: fixes \<iota>\<^sub>m :: "'b \<Rightarrow> 'a option"   and x :: "'b"   and q\<^sub>0 :: "'a"   and \<delta>\<^sub>m :: "'b \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'a"   and w :: "nat \<Rightarrow> 'c"   and i :: "nat" assumes "\<iota>\<^sub>m x = Some q\<^sub>0" shows "\<downharpoonleft>\<^sub>x (run\<^sub>t (\<Delta>\<^sub>\<times> \<delta>\<^sub>m) \<iota>\<^sub>m w i) = run\<^sub>t (\<delta>\<^sub>m x) q\<^sub>0 w i"
arctan_add: fixes x :: "real"   and y :: "real" assumes "\<bar>x\<bar> \<le> 1"   and "\<bar>y\<bar> < 1" shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
connex_cong: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and r' :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> r a b = r' a b" shows "connex A r = connex A r'"
assms_2: shows "return\<cdot>\<bottom> \<noteq> \<bottom>"
wlp_strengthen_modset: fixes \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "wlp \<pi> c Q s" shows "wlp \<pi> c (\<lambda>s'. Q s' \<and> modifies (lhsv \<pi> c) s' s) s"
abs_divide: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.field_abs_sgn divide inverse times one plus zero minus uminus abs sgn" shows "abs (divide a b) = divide (abs a) (abs b)"
simps_3: fixes s :: "'a \<Rightarrow> char list \<Rightarrow> char list"   and sep :: "char list \<Rightarrow> char list"   and x :: "'a"   and v :: "'a"   and va :: "'a list" shows "shows_sep s sep (x # v # va) = s x \<circ> sep \<circ> shows_sep s sep (v # va)"
ldeg_p_def: fixes R :: "('e, 'f) Ring_scheme"   and S :: "('e, 'g) Ring_scheme"   and X :: "'e"   and d :: "nat"   and p :: "'e" shows "ldeg_p R S X d p = polyn_expr R X d (scf_d R S X p (Suc d))"
a: shows "is_zero_row_upt_k a (Suc k) B"
all_orientable_characterisations_3: shows "(\<forall>x. orientable_1 x) = (\<forall>x. orientable_4 x)"
FV_Scon: fixes \<phi>s :: "('U, 'C, 'Out) Noninterference.aprop dfmla set" assumes "finite \<phi>s" shows "FV (Scon \<phi>s) = \<Union> (FV ` \<phi>s)"
hyp: fixes k :: "nat"   and cs :: "('b, 'a, 'c) com list" assumes "f k = (cs, css, Fault m)" shows "False"
ntrancl_on_0: fixes a :: "'c"   and b :: "'c"   and F :: "'c set"   and r :: "('c \<times> 'c) set" assumes "a = b"   and "a \<in> F" shows "ntrancl_onp F r 0 a b"
assms_2: shows "einterval a b = (\<Union>i. {l i..u i})"
ring_distribs_3: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "(a - b) * c = a * c - b * c"
le_add_diff_inverse: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and b :: "'d"   and a :: "'d" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq b a" shows "plus b (minus a b) = a"
enum_less: fixes a :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "a \<in> s"   and "i < n" shows "(s.enum i < a) = (s.enum (Suc i) \<le> a)"
fmranE: fixes y :: "'c"   and m :: "('d, 'c) fmap" assumes "y |\<in>| fmran m" obtains x :: "'d" where "m $$ x = Some y"
chop_def: fixes t :: "('a, 'b) tm" shows "chop t = apps (hd (args t)) (tl (args t))"
gtchr: shows "(\<Sum>s\<in>set (map (consistent_sign_vec_copr qs) (characterize_root_list_p p)) \<inter> {s. z I s = 1}. z I s * rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec_copr qs x = s}))) = rat_of_int (int (card {x. poly p x = 0 \<and> 0 < poly (prod_list (retrieve_polys qs I)) x}))"
simps_65: fixes v :: "int"   and va :: "MIR.num" shows "isrlfm (Dvd v va) = (isatom (Dvd v va) \<and> bound0 (Dvd v va))"
sel_2: fixes x1 :: "'b"   and x2 :: "'a"   and x3 :: "('b, 'a) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" shows "BsSkewBinomialHeapStruc.eprio (BsSkewBinomialHeapStruc.Element x1 x2 x3) = x2"
cases: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and X :: "'c set"   and gpv :: "('a, 'b, 'c) gpv" assumes "pred_gpv' P Q X gpv" obtains "\<And>x. x \<in> results_gpv (\<I>_uniform UNIV X) gpv \<Longrightarrow> P x" and "\<And>out. out \<in> outs_gpv (\<I>_uniform UNIV X) gpv \<Longrightarrow> Q out"
assms: shows "\<rho> f|` (- domA h) = \<rho>' f|` (- domA h)"
has_vector_derivative_id: fixes net :: "real filter" shows "((\<lambda>x. x) has_vector_derivative 1) net"
2: shows "openin (top_of_set S) {b. \<exists>T. openin (top_of_set S) T \<and> b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> \<not> Q x)}"
finfun_All_except_update: fixes A :: "'a list"   and f :: "('a, bool) finfun"   and a :: "'a"   and b :: "bool" shows "finfun_All_except A (finfun_update f a b) = ((a \<in> set A \<or> b) \<and> finfun_All_except (a # A) f)"
apply_power: fixes f :: "'a perm"   and n :: "nat" shows "(\<langle>$\<rangle>) (f ^ n) = (\<langle>$\<rangle>) f ^^ n"
no_grey_refs_not_rootD: fixes s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'h) Model.local_state"   and m :: "'e"   and r :: "'h" assumes "no_grey_refs s"   and "mut_m.reachable_snapshot_inv m s"   and "white r s" shows "r \<notin> roots (s (mutator m)) \<and> r \<notin> ghost_honorary_root (s (mutator m)) \<and> r \<notin> mut_m.tso_store_refs m s"
max1: fixes a :: "nat"   and x :: "nat"   and y :: "nat" assumes "a \<le> x" shows "a \<le> max x y"
le_approx2T: fixes P :: "'c process"   and Q :: "'c process"   and s :: "'c event list" assumes "P \<sqsubseteq> Q"   and "s \<notin> D P" shows "(s \<in> T Q) = (s \<in> T P)"
sequence_number_one_or_bigger: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, uu_). 1 \<le> sn \<xi>)"
hyps_3: shows "\<forall>s'. \<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow> Abrupt s' \<longrightarrow> \<Gamma>\<turnstile>c2 \<down> Normal s' \<and> (\<exists>t. \<Gamma>\<turnstile> \<langle>c2,Normal s'\<rangle> \<Rightarrow> t)"
subgroupE_3: fixes H :: "'a set set"   and a :: "'a set" assumes "subgroup H (G Mod N)"   and "a \<in> H" shows "inv\<^bsub>G Mod N\<^esub> a \<in> H"
emeasure_finite: fixes M :: "'b measure"   and A :: "'b set" assumes "finite_measure M" shows "emeasure M A \<noteq> top"
in_homI: fixes f :: "'a1"   and a :: "'a1"   and b :: "'a1" assumes "A1.arr f"   and "A1.dom f = a"   and "A1.cod f = b" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>A\<^sub>1 b\<guillemotright>"
convex_hull_subset_span: fixes s :: "'a set" shows "convex hull s \<subseteq> span s"
cmod_diff_le: fixes b :: "complex"   and c :: "complex" shows "cmod (b - c) \<le> sqrt (1 + (cmod b)\<^sup>2) + sqrt (1 + (cmod c)\<^sup>2)"
f1: shows "smap (\<lambda>n. x !! (n - Suc 0)) (fromN (Suc 0)) \<noteq> w"
diagonalize_2x2_JNF_works: fixes A :: "'b mat"   and P :: "'b mat"   and D :: "'b mat"   and Q :: "'b mat" assumes "A \<in> carrier_mat 2 2"   and "(P, D, Q) = diagonalize_2x2_JNF A" shows "D = P * A * Q \<and> invertible_mat P \<and> invertible_mat Q \<and> isDiagonal_mat D \<and> P \<in> carrier_mat 2 2 \<and> Q \<in> carrier_mat 2 2 \<and> D \<in> carrier_mat 2 2"
intro: fixes \<gamma> :: "'a \<Rightarrow> int set"   and num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "Val_semilattice \<gamma> num' plus'" shows "Gamma_semilattice num' plus' \<gamma>"
atrm_Var: fixes x :: "'var" assumes "x \<in> var" shows "Var x \<in> atrm"
functor_axioms: shows "functor (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>[\<^sub>A\<^sub>2,\<^sub>B\<^sub>]) F"
minus_one_mult_self: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and n :: "nat" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "times (power.power one times (uminus one) n) (power.power one times (uminus one) n) = one"
properL_notEmp: fixes cl :: "('a, 'b, 'c) cmd list" assumes "properL cl" shows "cl \<noteq> []"
asIMOD_igAbs3: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and s :: "'sort"   and X :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b"   and xs :: "'varSort"   and y :: "'var"   and x :: "'var" assumes "sWlsDisj SEM"   and "igWls (asIMOD SEM) s X" shows "igAbs (asIMOD SEM) xs y (igSubst (asIMOD SEM) xs (igVar (asIMOD SEM) xs y) x X) = (\<lambda>val. if sWlsVal SEM val then sAbs xs (\<lambda>sX. if sWls SEM (asSort xs) sX then igSubst (asIMOD SEM) xs (igVar (asIMOD SEM) xs y) x X (val (y := sX)_xs) else sDummy SEM s) else undefined)"
inv_ordermap_mono_eq: fixes M :: "'a set"   and r :: "('a \<times> 'a) set"   and \<alpha> :: "V"   and \<beta> :: "V" assumes "inv_into M (ordermap M r) \<alpha> = inv_into M (ordermap M r) \<beta>"   and "\<alpha> \<in> elts (ordertype M r)"   and "\<beta> \<in> elts (ordertype M r)" shows "\<alpha> = \<beta>"
insert_before_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and node :: "(_) node_ptr"   and reference_child :: "(_) node_ptr option" assumes "h \<turnstile> ok insert_before ptr node reference_child" shows "ptr |\<in>| object_ptr_kinds h"
Seq2: fixes P0 :: "'b com"   and s :: "'b"   and P2 :: "'b com"   and t :: "'b"   and P1 :: "'b com" assumes "(Some P0, s) -c\<rightarrow> (Some P2, t)" shows "(Some (Seq P0 P1), s) -c\<rightarrow> (Some (Seq P2 P1), t)"
simps_10: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_tree R Leaf Leaf"
Red_F_\<G>_q_def: fixes q :: "'q"   and N :: "('f \<times> 'l) set" shows "Red_F_\<G>_q q N = {C. \<forall>D\<in>\<G>_F_L_q q C. D \<in> Red_F_q q (\<Union> (\<G>_F_L_q q ` N)) \<or> (\<exists>E\<in>N. E \<sqsubset> C \<and> D \<in> \<G>_F_L_q q E)}"
assms: shows "(succl, succ) \<in> \<langle>Id\<rangle>list_set_rel"
weakTransitionClosed: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list" assumes "\<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'" shows "p \<bullet> \<Psi> : p \<bullet> Q \<rhd> p \<bullet> P \<Longrightarrow>p \<bullet> \<alpha> \<prec> p \<bullet> P'"
member_of_to_member_in: fixes G :: "prog"   and m :: "qtname \<times> memberdecl"   and C :: "qtname" assumes "G \<turnstile> m member_of C" shows "G \<turnstile> m member_in C"
le_infE: fixes x :: "'b multiset"   and a :: "'b multiset"   and b :: "'b multiset" assumes "x \<subseteq># a \<inter># b" obtains "x \<subseteq># a" and "x \<subseteq># b"
\<iota>\<chi>: shows "\<iota> \<chi> = \<Phi>.FUN a (\<phi> (a, a) a)"
ground_SMP_disjointI_aux3: fixes f :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and Q :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term \<Rightarrow> bool"   and R :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term \<Rightarrow> bool"   and C :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and A :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and B :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set" assumes "f \<equiv> \<lambda>M. {t \<cdot> \<delta> |t \<delta>. t \<in> M \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> fv (t \<cdot> \<delta>) = {}}"   and "Q \<equiv> intruder_synth' public arity {}"   and "R \<equiv> \<lambda>t. \<exists>u\<in>C. is_wt_instance_of_cond \<Gamma> t u"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s A"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s B"   and "fv\<^sub>s\<^sub>e\<^sub>t A \<inter> fv\<^sub>s\<^sub>e\<^sub>t B = {}"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s C"   and "\<forall>t\<in>A. \<forall>s\<in>B. (\<exists>\<delta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> Unifier \<delta> t s) \<longrightarrow> Q t \<and> Q s \<or> R t \<and> R s" shows "f A \<inter> f B \<subseteq> f C \<union> {m. {} \<turnstile>\<^sub>c m}"
not_prime_1: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "\<not> normalization_semidom.prime zero times one normalize one"
isomorphic_to_identity_is_equivalence: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "natural_isomorphism (\<cdot>) (\<cdot>) L local.map \<phi>" shows "equivalence_functor (\<cdot>) (\<cdot>) L"
p1_def: shows "p1 = quadratic_part_1 var a b d (Leq p)"
invariant_def: fixes A :: "('e, 'f) ioa"   and P :: "'e \<Rightarrow> bool" shows "invariant A P \<equiv> \<forall>s. reachable A s \<longrightarrow> P s"
fimageI: fixes v :: "'b set"   and u :: "'a set"   and F :: "'a set set" assumes "v \<subseteq> carrier\<^bsub>T\<^esub>"   and "u \<in> F"   and "f ` u \<subseteq> v" shows "v \<in> fimage F"
b: shows "\<langle>\<V> s'\<rangle> \<Turnstile>\<^sub>b \<B> s' \<parallel> - lvars (\<T> s')"
rel_funD: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g"   and x :: "'d"   and y :: "'e" assumes "rel_fun A B f g"   and "A x y" shows "B (f x) (g y)"
simps_2: fixes v :: "'c match_expr"   and va :: "'c match_expr list" shows "add_match_MatchAnd_foldr (v # va) = add_match (MatchAnd_foldr (v # va))"
r_1: shows "T, \<Gamma> \<Rightarrow>\<^sub>n"
simps: fixes n :: "Integer"   and xs :: "['b]" shows "Data_List.drop\<cdot>n\<cdot>xs = If le\<cdot>n\<cdot>0 then xs else (case xs of [] \<Rightarrow> [] | y : ys \<Rightarrow> Data_List.drop\<cdot>(n - 1)\<cdot>ys)"
distr_Stream_subprob: fixes I :: "'s pmf" shows "measure_pmf.random_variable I (subprob_algebra (stream_space (count_space UNIV))) (\<lambda>s. distr (T s) (stream_space (count_space UNIV)) ((##) s))"
mon_ww_unconc: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and ww1 :: "('c, 'f) label list list"   and ww2 :: "('c, 'f) label list list" shows "mon_ww fg (ww1 @ ww2) = mon_ww fg ww1 \<union> mon_ww fg ww2"
trivial_FinGroupRep: fixes smult :: "('a, 'g) aezfun \<Rightarrow> 'b \<Rightarrow> 'b" assumes "of_nat (card G) \<noteq> (0::'a)"   and "\<forall>a\<in>group_ring. smult a (0::'b) = (0::'b)" shows "FinGroupRepresentation G smult 0"
abs_of_nonneg: fixes abs :: "'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 less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq zero a" shows "abs a = a"
rep_eq: fixes x :: "integer" shows "rat_of_integer x = Rat.of_int (int_of_integer x)"
closedin_discrete_topology: fixes U :: "'b set"   and S :: "'b set" shows "closedin (discrete_topology U) S = (S \<subseteq> U)"
main: fixes p :: "nat fm" shows "valid p = (\<turnstile> p)"