hcju/isabelleps huggingface dataset:
equals_dom_if_value_is_ide: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and g :: "'ah" assumes "constant_transformation A B g"   and "partial_magma.ide B g" shows "constant_transformation.map A B g = constant_functor.map A B (partial_magma.dom B g)"
onotunicastT: fixes \<sigma>' :: "nat \<Rightarrow> 'e"   and i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and s\<^sub>i\<^sub>p :: "'e \<Rightarrow> nat"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "\<sigma>' i = \<sigma> i" shows "((\<sigma>, {l}unicast(s\<^sub>i\<^sub>p, s\<^sub>m\<^sub>s\<^sub>g) . p \<triangleright> q), \<not>unicast (s\<^sub>i\<^sub>p (\<sigma> i)), \<sigma>', q) \<in> oseqp_sos \<Gamma> i"
f_shrink_last_0: fixes xs :: "'b list" shows "xs \<div>\<^sub>l 0 = []"
that: fixes r :: "'a list"   and k :: "nat" assumes "primitive r"   and "k \<noteq> 0"   and "x = r \<^sup>@ k" shows "thesis"
closest_point_self: fixes x :: "'b"   and S :: "'b set" assumes "x \<in> S" shows "closest_point S x = x"
inj_on_lift: fixes I0 :: "'a set"   and P :: "'a set set"   and Q :: "'a set set"   and F :: "'a set \<Rightarrow> 'b set"   and J0 :: "'b set" assumes "part I0 P"   and "part I0 Q"   and "finer P Q"   and "inj_on F P"   and "part J0 (F ` P)"   and "{} \<notin> F ` P" shows "inj_on (lift P F) Q"
rel_sel: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c llist"   and b :: "'d llist" shows "llist_all2 R a b = (lnull a = lnull b \<and> (\<not> lnull a \<longrightarrow> \<not> lnull b \<longrightarrow> R (lhd a) (lhd b) \<and> llist_all2 R (ltl a) (ltl b)))"
assms_1: shows "a{b}.P \<longmapsto> c[d] \<prec> P'"
zero_ne_\<omega>\<^sub>z: shows "0 \<noteq> \<omega>\<^sub>z"
wf_bs_polysub: fixes bs :: "'b list"   and p :: "'b poly"   and q :: "'b poly" assumes "wf_bs bs p"   and "wf_bs bs q" shows "wf_bs bs (p -\<^sub>p q)"
prob_compl: fixes A :: "real set" assumes "A \<in> PZ.events" shows "PZ.prob (space Z - A) = 1 - PZ.prob A"
prime_imp_coprime_int: fixes p :: "int"   and n :: "int" assumes "prime p"   and "\<not> p dvd n" shows "coprime p n"
desargues_config_rot_CW: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "desargues_config A B C A' B' C' M N P R" shows "desargues_config C A B C' A' B' P M N R"
ivl_rel_internal: fixes S :: "('c \<times> 'd) set" shows "ivl_rel S = (S \<times>\<^sub>r S) O br set_of_ivl top"
choice_Basis_iff: fixes P :: "'a \<Rightarrow> real \<Rightarrow> bool" shows "(\<forall>i\<in>Basis. \<exists>x. P i x) = (\<exists>x. \<forall>i\<in>Basis. P i (inner x i))"
is_nextElem_a: fixes vs :: "'a list"   and a :: "'a"   and b :: "'a" assumes "is_nextElem vs a b" shows "a \<in> set vs"
assms_5: shows "r\<^sup>\<infinity> = r\<^sup>\<star>"
lexmax_set: fixes r :: "('a \<times> 'a) set"   and \<sigma> :: "'a list"   and S :: "'a set" assumes "trans r"   and "set_mset r|\<sigma>| \<subseteq> r \<down>s S" shows "set \<sigma> \<subseteq> r \<down>s S"
summable_geometric: fixes c :: "'c" assumes "norm c < 1" shows "summable ((^) c)"
induct: fixes x1 :: "Core.term"   and x2 :: "Core.term"   and P :: "Core.term \<Rightarrow> Core.term \<Rightarrow> bool" assumes "x1 \<rightarrow>\<^sub>\<eta> x2"   and "\<And>s T. \<not> is_dependent s \<Longrightarrow> P (Abs T (s $ Bv 0)) (decr 0 s)"   and "\<And>s t u. \<lbrakk>s \<rightarrow>\<^sub>\<eta> t; P s t\<rbrakk> \<Longrightarrow> P (s $ u) (t $ u)"   and "\<And>s t u. \<lbrakk>s \<rightarrow>\<^sub>\<eta> t; P s t\<rbrakk> \<Longrightarrow> P (u $ s) (u $ t)"   and "\<And>s t T. \<lbrakk>s \<rightarrow>\<^sub>\<eta> t; P s t\<rbrakk> \<Longrightarrow> P (Abs T s) (Abs T t)" shows "P x1 x2"
modd: shows "\<forall>x k. P x = P (x - k * d)"
simps: fixes a :: "'b list"   and A :: "'b set" shows "(a \<in> lists A) = (a = [] \<or> (\<exists>aa l. a = aa # l \<and> aa \<in> A \<and> l \<in> lists A))"
ide_iso_in_hom: fixes a :: "'c" assumes "C.ide a" shows "\<guillemotleft>a : a \<cong>\<^sub>C a\<guillemotright>"
of_rat_1: shows "of_rat 1 = (1::'a)"
split_sel: fixes P :: "'g \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'i \<Rightarrow> 'g"   and prod :: "'h \<times> 'i" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (prod = (fst prod, snd prod) \<longrightarrow> P (f (fst prod) (snd prod)))"
precedesOrder_def: fixes list :: "'b list" shows "precedesOrder list \<equiv> {(a, b). precedes a b list \<and> a \<noteq> b}"
continuous_map_componentwise: fixes X :: "'b topology"   and Y :: "'c \<Rightarrow> 'd topology"   and I :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "continuous_map X (product_topology Y I) f = (f ` topspace X \<subseteq> extensional I \<and> (\<forall>k\<in>I. continuous_map X (Y k) (\<lambda>x. f x k)))"
hom_decomp_def: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "hom_decomp ps = (\<forall>hU\<in>set ps. homogeneous (fst hU))"
gram_schmidt_sub_span: fixes w :: "'a vec"   and us :: "'a vec list" assumes "w \<in> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us" shows "local.span (set ((adjuster n w us + w) # us)) = local.span (set (w # us))"
float_dense_in_real: fixes x :: "real"   and y :: "real" assumes "x < y" shows "\<exists>r\<in>float. x < r \<and> r < y"
bind_has_density': fixes M :: "'c measure"   and R :: "'d measure"   and N :: "'c measure"   and \<delta>M :: "'c \<Rightarrow> ennreal"   and f :: "'c \<Rightarrow> 'd measure"   and \<delta>f :: "'c \<Rightarrow> 'd \<Rightarrow> ennreal" assumes "sigma_finite_measure M"   and "sigma_finite_measure R"   and "space M \<noteq> {}"   and "has_density M N \<delta>M"   and "\<And>x. x \<in> space M \<Longrightarrow> has_density (f x) R (\<delta>f x)"   and "(\<lambda>(x, y). \<delta>f x y) \<in> borel_measurable (N \<Otimes>\<^sub>M R)"   and "f \<in> N \<rightarrow>\<^sub>M subprob_algebra R" shows "has_density (M \<bind> f) R (\<lambda>y. \<integral>\<^sup>+ x. \<delta>M x * \<delta>f x y \<partial>N)"
mat_row_first_dim_2: fixes A :: "'b mat"   and n :: "nat" shows "dim_col (mat_row_first A n) = dim_col A"
ideal_induct_homomorphism: fixes h :: "'b \<Rightarrow> 'b"   and B :: "'b set"   and p :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "\<And>x y. h (x + y) = h x + h y"   and "\<And>x y. h (x * y) = h x * h y"   and "\<And>x. h (h x) = h x"   and "B \<subseteq> range h"   and "p \<in> range h"   and "p \<in> ideal B"   and "P (0::'b)"   and "\<And>c b a. \<lbrakk>c \<in> range h; b \<in> B; P a; a \<in> range h\<rbrakk> \<Longrightarrow> P (c * b + a)" shows "P p"
fract_expand_2: shows "1 = Fraction_Field.Fract (1::'b) (1::'b)"
Expr: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A" shows "Env\<turnstile> B \<guillemotright>\<langle>Expr e\<rangle>\<guillemotright> A"
minus_list_def: fixes xs :: "'b list"   and ys :: "'b list" shows "xs - ys = (if prefix ys xs then drop (length ys) xs else [])"
word_eq_iff_signed: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k"   and v :: "'l word"   and w :: "'l word" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(v = w) = (ring_1.signed one plus zero minus v = ring_1.signed one plus zero minus w)"
dord_pm_degD: fixes ord :: "('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool"   and s :: "'a \<Rightarrow>\<^sub>0 'c"   and t :: "'a \<Rightarrow>\<^sub>0 'c" assumes "dord_pm ord s t" shows "deg_pm s \<le> deg_pm t"
Ch_bilateral_substitutes: shows "\<forall>h. bilateral_substitutes_on UNIV (Ch h)"
inverts_mat_length_2: fixes A :: "'b mat"   and B :: "'b mat" assumes "square_mat A"   and "inverts_mat A B"   and "inverts_mat B A" shows "dim_col B = dim_col A"
measure_of_def: fixes \<Omega> :: "'d set"   and A :: "'d set set"   and \<mu> :: "'d set \<Rightarrow> ennreal" shows "measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>}, \<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)"
intros_2: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist"   and x :: "'b" assumes "chain R xs"   and "R x (lhd xs)" shows "chain R (LCons x xs)"
coincidence_sterm: fixes \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and \<nu>' :: "(real, 'd) vec \<times> (real, 'd) vec"   and \<theta> :: "('e, 'd) trm"   and I :: "('e, 'f, 'd) interp" assumes "Vagree \<nu> \<nu>' (FVT \<theta>)" shows "sterm_sem I \<theta> (fst \<nu>) = sterm_sem I \<theta> (fst \<nu>')"
axioms_2: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'e set"   and scale\<^sub>1 :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and scale\<^sub>2 :: "'f \<Rightarrow> 'e \<Rightarrow> 'e" assumes "vector_space_pair_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2" shows "vector_space_on U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2"
filter_for_biflows_correct: fixes E :: "('a \<times> 'a) list" shows "\<forall>(s, r)\<in>set (filter_for_biflows E). (r, s) \<in> set (filter_for_biflows E)"
f2r_def: fixes f :: "'a \<Rightarrow> 'b set" shows "\<R> f = {(x, y). s2p (f x) y}"
homE_1: fixes A :: "'a"   and B :: "'a"   and f :: "'b" assumes "A \<in> Ob"   and "B \<in> Ob"   and "f \<in> Hom A B" shows "Dom f = A"
zero_when: fixes zero :: "'b"   and P :: "bool" shows "zero.when zero zero P = zero"
step: fixes sa :: "'s"   and \<omega>' :: "'s stream" shows "reward_until X sa \<omega>' = (if sa \<in> X then 0 else \<rho> sa + \<iota> sa (shd \<omega>') + reward_until X (shd \<omega>') (stl \<omega>'))"
nonneg_bounded: fixes f :: "'d \<Rightarrow> 'e" assumes "bounded_linear f" shows "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
y_def: shows "y \<noteq> x \<and> y \<noteq> z \<and> y \<in> X"
split: fixes P :: "'j \<Rightarrow> bool"   and f1 :: "'j"   and f2 :: "'k \<Rightarrow> 'j"   and option :: "'k option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (\<forall>x2. option = Some x2 \<longrightarrow> P (f2 x2)))"
fold_map_append: fixes f :: "'b \<Rightarrow> 'a Heap"   and xs :: "'b list"   and ys :: "'b list" shows "Heap_Monad.fold_map f (xs @ ys) = Heap_Monad.fold_map f xs \<bind> (\<lambda>xs. Heap_Monad.fold_map f ys \<bind> (\<lambda>ys. return (xs @ ys)))"
S_insert_mod: fixes s :: "'s"   and x :: "'a" assumes "S s" shows "S (insert_mod x s)"
fail_option_def: fixes return :: "'e option \<Rightarrow> 'f" shows "fail_option return = OptionT (return None)"
iT_Mult_iprev2_if: fixes n :: "nat"   and I :: "nat set"   and k :: "nat" shows "iprev n (I \<otimes> k) = (if n mod k = 0 then iprev (n div k) I * k else n)"
reachable1_awalk: fixes u :: "'a"   and v :: "'a" shows "(u \<rightarrow>\<^sup>+ v) = (\<exists>p. awalk u p v \<and> p \<noteq> [])"
P_LCons: shows "P' = LCons (P $ n) (ltl P')"
listsetI: fixes ys :: "'b list"   and xs :: "'b set list" assumes "length ys = length xs"   and "\<And>i. i < length xs \<Longrightarrow> ys ! i \<in> xs ! i" shows "ys \<in> listset xs"
view_partition_rel_ipurge: fixes P :: "'a process"   and D :: "'a \<Rightarrow> 'b"   and I :: "('b \<times> 'b) set" shows "view_partition P D (rel_ipurge P I D)"
rel_funD2: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c"   and x :: "'a" assumes "rel_fun A B f g"   and "A x x" shows "B (f x) (g x)"
Ord_omega_sum_aux: fixes k :: "nat"   and ms :: "nat list" shows "Ord (omega_sum_aux k ms)"
sees_field_i_i_i_o_o_def: shows "sees_field_i_i_i_o_o \<equiv> \<lambda>x xa xb. pred.Pred (\<lambda>(xc, xd). x \<turnstile> xa sees xb:xc in xd)"
lookup_cases: fixes y :: "'b \<Rightarrow> 'c" assumes "y \<in> {f. finite {x. f x \<noteq> (0::'c)}}" obtains x :: "'b \<Rightarrow>\<^sub>0 'c" where "y = lookup x"
Fin: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and c1 :: "stmt"   and Q :: "'b assn"   and c2 :: "stmt"   and R :: "'b assn" assumes "G,A\<turnstile>{Normal P} .c1. {Q}"   and "\<forall>x. G,A\<turnstile>{Q \<and>. (\<lambda>s. x = fst s) ;. abupd Map.empty} .c2. {abupd (abrupt_if (x \<noteq> None) x) .; R}" shows "G,A\<turnstile>{Normal P} .c1 Finally c2. {R}"
lead_coeff_prod: fixes f :: "'c \<Rightarrow> 'b poly"   and A :: "'c set" shows "lead_coeff (prod f A) = (\<Prod>x\<in>A. lead_coeff (f x))"
exhaust: fixes y :: "val" obtains x1 :: "nat" where "y = VNat x1"   | x2 :: "(val \<times> val) list" where "y = VFun x2"
intro: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set" assumes "order_pair S NS"   and "SN_ars S" shows "SN_order_pair S NS"
w2: shows "w2 \<in> set (awalk_verts u p)"
dg_prod_Cod_app_in_Obj: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Arr\<rparr>" shows "dg_prod I \<AA>\<lparr>Cod\<rparr>\<lparr>f\<rparr> \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Obj\<rparr>"
id: fixes y :: "(real, 'n) vec" assumes "y \<in> T" shows "h' y \<circ> g' (h y) = id"
prefix_fininf_le_inf_transp: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action stream"   and w\<^sub>3 :: "'action stream" assumes "w\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>2"   and "w\<^sub>2 \<preceq>\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>3"
jumpF_times: fixes f :: "real \<Rightarrow> real"   and c :: "real"   and F :: "real filter"   and g :: "real \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "c \<noteq> 0"   and "F \<noteq> bot" shows "jumpF (\<lambda>x. f x * g x) F = sgn c * jumpF g F"
T_subprob': shows "T \<in> count_space UNIV \<rightarrow>\<^sub>M subprob_algebra (stream_space (count_space UNIV))"
idiff_eq_conv_enat: fixes x :: "enat"   and y :: "enat"   and z :: "enat" shows "(x - y = z) = (if y < x then x = y + z else if x \<noteq> \<infinity> then z = enat 0 else z = \<infinity>)"
assms_2: shows "appliedDecide stateA stateB decisionVars \<or> appliedUnitPropagate stateA stateB F0 \<or> appliedBacktrack stateA stateB F0"
b_assn_subtyping_match_eqA: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and y :: "'b"   and P :: "'a \<Rightarrow> bool"   and P' :: "'a \<Rightarrow> bool" assumes "\<lbrakk>vassn_tag (hn_ctxt A x y); P x\<rbrakk> \<Longrightarrow> P' x" shows "hn_ctxt (b_assn A P) x y \<Longrightarrow>\<^sub>t hn_ctxt (b_assn A P') x y"
icard_empty_iff: fixes A :: "'a set" shows "(icard A = 0) = (A = {})"
bounded_def: fixes A :: "(nat \<times> 'b) set" shows "bounded A = (\<exists>N. boundedBy N A)"
R: shows "(w, w') \<in> (\<Union> (range (relations M)))\<^sup>*"
main: fixes i :: "nat"   and j :: "nat" assumes "i < degree f'"   and "j < degree f'" shows "R (mat_of_rows_list (degree f') (map (\<lambda>cs. cs @ replicate (degree f' - length cs) (arith_ops_record.zero ff_ops)) (power_polys_i ff_ops (power_poly_f_mod_i ff_ops (\<lambda>v. mod_field_poly_i ff_ops v f) [arith_ops_record.zero ff_ops, arith_ops_record.one ff_ops] (nat p)) f [arith_ops_record.one ff_ops] (degree f'))) $$ (i, j)) (mat_of_rows_list (degree f') (map (\<lambda>cs. coeffs cs @ replicate (degree f' - length (coeffs cs)) 0) (power_polys (power_poly_f_mod f' [:0, 1:] (nat p)) f' 1 (degree f'))) $$ (i, j))"
proj_in_iff: fixes P :: "'d \<Rightarrow> bool"   and \<phi> :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" assumes "equivP P \<phi>" shows "(P /// \<phi>) (proj \<phi> x) = P x"
linear_def: fixes f :: "'a \<Rightarrow> 'b" shows "linear f \<equiv> Vector_Spaces.linear (*\<^sub>R) (*\<^sub>R) f"
not_less_zero: fixes n :: "'a multiset" shows "\<not> n \<subset># {#}"
ring_hom_ringI2: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g" assumes "ring R"   and "ring S"   and "h \<in> ring_hom R S" shows "ring_hom_ring R S h"
assms_2: shows "A \<subseteq>\<^sub>\<circ> Vset \<alpha>"
BSD2: shows "BSD \<V>2 Tr\<^bsub>ES2\<^esub>"
pmf_of_multiset_impl_def: fixes A :: "'b multiset" shows "pmf_of_multiset_impl A = mapping_of_pmf (pmf_of_multiset A)"
eq2: shows "eval_dioph ks (map (\<lambda>x. x div 2) xs) = (l - eval_dioph ks (map (\<lambda>x. x mod 2) xs)) div 2"
cases: fixes a1 :: "'f list"   and a2 :: "'f list multiset"   and a3 :: "('g, 'h) label"   and a4 :: "'f list"   and a5 :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme" assumes "((a1, a2), a3, a4, a5) \<in> trss fg" obtains u :: "'f" and a :: "'h" and v :: "'f" and r :: "'f list" and c :: "'f list multiset" where "a1 = u__ # r__" and "a2 = cb__" and "a3 = LBase a" and "a4 = v__ # r__" and "a5 = cb__" and "(u__, Base a, v__) \<in> edges fg"   | u :: "'f" and p :: "'g" and v :: "'f" and c :: "'f list multiset" and r :: "'f list" where "a1 = ua__ # ra__" and "a2 = cc__" and "a3 = LCall p__" and "a4 = entry fg p__ # va__ # ra__" and "a5 = cc__" and "(ua__, Call p__, va__) \<in> edges fg" and "mon fg p__ \<inter> mon_c fg cc__ = {}"   | p :: "'g" and r :: "'f list" and c :: "'f list multiset" where "a1 = return fg pa__ # rb__" and "a2 = cd__" and "a3 = LRet" and "a4 = rb__" and "a5 = cd__"   | u :: "'f" and p :: "'g" and v :: "'f" and r :: "'f list" and c :: "'f list multiset" where "a1 = u # r" and "a2 = c" and "a3 = LSpawn p" and "a4 = v # r" and "a5 = add_mset [entry fg p] c" and "(u, Spawn p, v) \<in> edges fg"
WT: shows "\<I>1 \<turnstile>g gpv \<surd>"
obj_simps_3: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "horizontal_homs V src trg"   and "horizontal_homs.obj V src a" shows "trg a = a"
r_shrink_recfn: fixes f :: "recf" assumes "recfn 2 f" shows "recfn 1 (r_shrink f)"
simply_connected_eq_contractible_path: fixes S :: "'c set" shows "simply_connected S = (path_connected S \<and> (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p \<longrightarrow> homotopic_paths S p (linepath (pathstart p) (pathstart p))))"
disc_3: fixes x1 :: "'a" shows "\<not> isState (Idle x1)"
cases: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist" assumes "a \<in> read_actions E" obtains aa :: "nat" and ad :: "'b" and al :: "addr_loc" and v :: "'b val" where "a = aa" and "aa \<in> actions E" and "action_obs E aa = NormalAction (ReadMem ad al v)"
assms: shows "\<psi> \<in> \<R>\<^sup>2"
simps_3: fixes \<phi> :: "'b ltln" shows "restricted_subformulas (X\<^sub>n \<phi>) = restricted_subformulas \<phi>"
vsv_vimageI2: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> xs" shows "xs\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> xs"
B_A_step: fixes a' :: "'b"   and b' :: "'b"   and a :: "'a" assumes "B a' b'"   and "sim a a'"   and "PA a"   and "PB a'" shows "\<exists>b. A a b \<and> sim b b'"
induct: fixes P :: "'b tree \<Rightarrow> bool"   and a0 :: "'b tree" assumes "\<And>hsx x hsy y hs. P \<langle>hsx, x, \<langle>hsy, y, hs\<rangle>\<rangle>"   and "P \<langle>\<rangle>"   and "\<And>v va. P \<langle>v, va, \<langle>\<rangle>\<rangle>" shows "P a0"
fold_mset_fusion: fixes f :: "'k \<Rightarrow> 'l \<Rightarrow> 'l"   and g :: "'k \<Rightarrow> 'm \<Rightarrow> 'm"   and h :: "'m \<Rightarrow> 'l"   and w :: "'m"   and A :: "'k multiset" assumes "comp_fun_commute f"   and "comp_fun_commute g"   and "\<And>x y. h (g x y) = f x (h y)" shows "h (fold_mset g w A) = fold_mset f (h w) A"
semiring_normalization_rules_33: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "power.power one times x 1 = x"
prod_belowI: fixes p :: "'b \<times> 'c"   and q :: "'b \<times> 'c" assumes "fst p \<sqsubseteq> fst q"   and "snd p \<sqsubseteq> snd q" shows "p \<sqsubseteq> q"
refl_on_converse: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "refl_on A (r\<inverse>) = refl_on A r"
quasinorm_equivalent_def: fixes N1 :: "'b quasinorm"   and N2 :: "'b quasinorm" shows "N1 =\<^sub>N N2 = (N1 \<subseteq>\<^sub>N N2 \<and> N2 \<subseteq>\<^sub>N N1)"
reachable_union: fixes mS :: "('b \<times> 'b) set"   and aS :: "'b ref set"   and bS :: "'b ref set" shows "reachable mS aS \<union> reachable mS bS = reachable mS (aS \<union> bS)"
monom_list_subst_inv: fixes \<sigma> :: "'c \<Rightarrow> ('d monom \<times> 'e) list"   and m :: "('c \<times> nat) list" assumes "\<And>x. poly_inv (\<sigma> x)" shows "poly_inv (monom_list_subst \<sigma> m)"
cINF_le_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'b"   and x :: "'b" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_below less_eq (f ` A)" shows "less_eq (Inf (f ` A)) x = (\<forall>y. less x y \<longrightarrow> (\<exists>i\<in>A. less (f i) y))"
sorted_nonzero_root_list_set_def: fixes p :: "(real \<times> real \<times> real) set" shows "sorted_nonzero_root_list_set p \<equiv> sorted_list_of_set (nonzero_root_set p)"
count_mset: fixes xs :: "'c list"   and x :: "'c" shows "count (mset xs) x = length (filter ((=) x) xs)"
lift_to_add2: fixes z :: "'a"   and q :: "'a"   and s :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<oplus> q = Some s"   and "x \<oplus> y = Some q" shows "Some z ++ Some x ++ Some y = Some s"
simps_4: fixes l :: "('b \<times> nat) option"   and t :: "'b" shows "upd_lock l t ReleaseAcquire = release_all l t"
comp_null_2: fixes f :: "'a" shows "C f A.null = A.null"
g: shows "\<guillemotleft>g : a \<rightarrow>\<^sub>A a'\<guillemotright> \<and> F g = g'"
inject_1: fixes x1 :: "('c, 'd) atm"   and y1 :: "('c, 'd) atm" shows "(Pos x1 = Pos y1) = (x1 = y1)"
bound: fixes \<E> :: "'a set set" assumes "\<E> \<subseteq> \<D>"   and "finite \<E>" shows "Sigma_Algebra.measure M (\<Union> \<E>) \<le> B"
degree_def: fixes p :: "'a poly" shows "degree p = (LEAST n. \<forall>i>n. coeff p i = (0::'a))"
g_le_f: fixes x :: "'a"   and y :: "'a" assumes "edge \<Delta> x y" shows "g (y, x) \<le> f (x, y)"
run_put_option: fixes s :: "'s"   and m :: "('a, 'm) optionT" shows "run_option (local.put_option s m) = put s (run_option m)"
assms_1: shows "falsifies\<^sub>l (ds @ d) l"
fls_regpart_fls_X: shows "fls_regpart fls_X = fps_X"
parallel_reduction_is_beta_reduction: fixes A :: "'a trm"   and B :: "'a trm" assumes "A >> B" shows "A \<rightarrow>\<beta>\<^sup>* B"
rem_effectless_works_13_i: fixes x :: "'c"   and l :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "ListMem x l"   and "list_all P l" shows "P x"
dual_orthocomplemented_lattice: shows "class.orthocomplemented_lattice (\<lambda>x y. x \<squnion> - y) uminus (\<squnion>) (\<lambda>x y. y \<le> x) (\<lambda>x y. y < x) (\<sqinter>) \<top> \<bottom>"
some_betw2: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and X :: "'a set"   and j :: "nat" assumes "[f[a .. b ..  c]X]"   and "j < card X"   and "0 < j"   and "f j \<noteq> b" shows "[[a b f j]] \<or> [[a f j b]]"
rotate1_is_Nil_conv: fixes xs :: "'a list" shows "(rotate1 xs = []) = (xs = [])"