hcju/isabelleps huggingface dataset:
coincide_hp_def: fixes \<alpha> :: "('d, 'e, 'f) hp"   and I :: "('d, 'e, 'f) interp"   and J :: "('d, 'e, 'f) interp" shows "coincide_hp \<alpha> I J = (\<forall>\<nu> \<nu>' \<mu> V. Iagree I J (SIGP \<alpha>) \<longrightarrow> Vagree \<nu> \<nu>' V \<longrightarrow> FVP \<alpha> \<subseteq> V \<longrightarrow> (\<nu>, \<mu>) \<in> prog_sem I \<alpha> \<longrightarrow> (\<exists>\<mu>'. (\<nu>', \<mu>') \<in> prog_sem J \<alpha> \<and> Vagree \<mu> \<mu>' (MBV \<alpha> \<union> V)))"
continuous_ident: fixes x :: "'c"   and S :: "'c set" shows "continuous (at x within S) (\<lambda>x. x)"
Posix_elims_3: fixes s :: "'b list"   and c :: "'b"   and v :: "'b val" assumes "s \<in> Atom c \<rightarrow> v" obtains "s = [c]" and "v = Atm c"
constructive_security_aux_obsf: shows "constructive_security_aux_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common adv"
in_homI: fixes f :: "'a2"   and a :: "'a2"   and b :: "'a2" assumes "A2.arr f"   and "A2.dom f = a"   and "A2.cod f = b" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>A\<^sub>2 b\<guillemotright>"
red_rtrancl_repE: fixes d :: "'a \<Rightarrow> nat"   and G :: "('a \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and r :: "'a \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "G \<subseteq> punit.dgrad_p_set d m"   and "finite G"   and "p \<in> punit.dgrad_p_set d m"   and "(punit.red G)\<^sup>*\<^sup>* p r" obtains q :: "('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'c" where "p = r + (\<Sum>g\<in>G. local.punit.mult_scalar (q g) g)" and "\<And>g. q g \<in> punit.dgrad_p_set d m" and "\<And>g. punit.lt (local.punit.mult_scalar (q g) g) \<preceq> punit.lt p"
NotNull: shows "\<not> (d = 0 \<and> e = 0 \<and> f = 0)"
cofinal_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "cofinal A r \<equiv> \<forall>a\<in>Field r. \<exists>b\<in>A. a \<noteq> b \<and> (a, b) \<in> r"
H_def': fixes xy :: "'v \<times> 'v" shows "ennreal (H xy) = lim (\<lambda>n. ennreal (h (k n) xy))"
hp_eq: shows "\<forall>x\<in>set ps. h x = g x"
inc_partial_density_operator_converge: fixes i :: "nat"   and j :: "nat" assumes "i \<in> {0..<dim}"   and "j \<in> {0..<dim}" shows "convergent (\<lambda>n. f n $$ (i, j))"
ty_dom_supp_2: fixes \<Gamma> :: "binding list" shows "supp (trm_dom \<Gamma>) = trm_dom \<Gamma>"
norm_le_square: fixes x :: "'b"   and a :: "real" shows "(norm x \<le> a) = (0 \<le> a \<and> x \<bullet> x \<le> a\<^sup>2)"
CRR_hyps_def: fixes p :: "real"   and M :: "bool stream measure"   and geom_proc :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and u :: "real"   and d :: "real"   and init :: "real"   and Mkt :: "(bool stream, 'b) discrete_market"   and r :: "real"   and risk_free_asset :: "'b"   and stk :: "'b" shows "CRR_hyps p M geom_proc u d init Mkt r risk_free_asset stk \<equiv> prob_grw p M geom_proc u d init \<and> rsk_free_asset Mkt r risk_free_asset \<and> CRR_hyps_axioms p geom_proc u d init Mkt risk_free_asset stk"
pt_set_inst: assumes "pt TYPE('a) TYPE('b)" shows "pt TYPE('a set) TYPE('b)"
evaluate_total: fixes env :: "v sem_env"   and s :: "'b state"   and e :: "exp0" shows "\<exists>s' r. evaluate True env s e (s', r)"
l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.known_ptr DocumentClass.type_wf get_child_nodes get_child_nodes_locs to_tree_order"
pref_profile_wf_axioms: shows "is_pref_profile R2"
to_vec_eq_iff: fixes M :: "('b, 'c) sq_matrix"   and N :: "('b, 'c) sq_matrix" shows "(to_vec M = to_vec N) = (M = N)"
lexordp_append_left_rightI: fixes less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and us :: "'d list"   and xs :: "'d list"   and ys :: "'d list" assumes "less x y" shows "ord.lexordp less (us @ x # xs) (us @ y # ys)"
to_poly_mult_simp_1: fixes R :: "('c, 'd) ring_scheme"   and b :: "'c"   and f :: "nat \<Rightarrow> 'c" assumes "UP_cring R"   and "b \<in> carrier R"   and "f \<in> carrier (UP R)" shows "to_polynomial R b \<otimes>\<^bsub>UP R\<^esub> f = b \<odot>\<^bsub>UP R\<^esub> f"
eval_norm_eq_id: fixes L :: "('d multiset \<Rightarrow> 'c multiset \<Rightarrow> 'a) ring"   and i :: "'d"   and q :: "('d multiset \<Rightarrow> 'c multiset \<Rightarrow> 'a) list"   and a :: "'d multiset \<Rightarrow> 'c multiset \<Rightarrow> 'a" assumes "field L"   and "\<And>P. P \<in> carrier L \<Longrightarrow> Hom.carrier_coeff P"   and "\<And>P. P \<in> carrier L \<Longrightarrow> Hom.index_free P i"   and "\<zero>\<^bsub>L\<^esub> = Hom.indexed_const \<zero>\<^bsub>image_ring (eval_pmod q) (poly_ring L)\<^esub>"   and "q \<in> carrier (poly_ring L)"   and "0 < degree q"   and "a \<in> carrier L" shows "(Hom.eval_pmod L i q \<circ> ring.poly_of_const L) a = a"
Rep_fin_string: fixes x :: "fin_string" shows "Rep_fin_string x \<in> {s. length s \<le> MAX_STR}"
p_max_def: shows "ego2.p_max = ego2.p ego2.t_stop"
lfp_def: fixes f :: "'a \<Rightarrow> 'a" shows "complete_lattice.lfp Sup (\<lambda>x y. y \<le> x) f = \<Squnion> {u. u \<le> f u}"
lookup_single: fixes k :: "'d"   and v :: "'c"   and k' :: "'d" shows "lookup (Poly_Mapping.single k v) k' = (v when k = k')"
Infinitesimal_FreeUltrafilterNat_iff: fixes X :: "nat \<Rightarrow> 'a" shows "(star_n X \<in> Infinitesimal) = (\<forall>u>0. \<forall>\<^sub>F n in \<U>. norm (X n) < u)"
bounded_dist_le_SUP_dist: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and x :: "'d" assumes "bounded (range f)"   and "bounded (range g)" shows "dist (f x) (g x) \<le> (SUP x. dist (f x) (g x))"
s\<^sub>3_eq_s\<^sub>3': shows "s\<^sub>3 \<approx> s\<^sub>3'"
linear_inj_iff_eq_0: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "inj f = (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
open_empty: fixes "open" :: "'c set \<Rightarrow> bool" assumes "class.topological_space open" shows "open {}"
verit_sko_ex': fixes P :: "'d \<Rightarrow> bool"   and A :: "bool" assumes "P (SOME x. P x) = A" shows "(\<exists>x. P x) = A"
square_free_factorization_smult: fixes c :: "'a"   and p :: "'a poly"   and d :: "'a"   and bs :: "('a poly \<times> nat) list" assumes "c \<noteq> (0::'a)"   and "square_free_factorization p (d, bs)" shows "square_free_factorization (Polynomial.smult c p) (c * d, bs)"
leim: shows "(cmod (1 - z\<^sup>2) + (1 - Re (z\<^sup>2))) / 2 \<le> (Im z)\<^sup>2"
bounded_ball: fixes x :: "'c"   and e :: "real" shows "bounded (ball x e)"
up: shows "Matrix.upper_triangular (mat_delete (make_mat n k B) 0 0)"
foldl_prod_prod: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'a"   and as :: "'b list"   and g :: "'b \<Rightarrow> 'a"   and w :: "'a" shows "foldl (\<lambda>r x. r * f x) v as * foldl (\<lambda>r x. r * g x) w as = foldl (\<lambda>r x. r * f x * g x) (v * w) as"
getIJProperties_5: fixes V :: "nat set"   and I :: "nat set"   and J :: "nat set"   and Q_pos :: "(nat set \<times> 'a option list set) set"   and Q_neg :: "(nat set \<times> 'a option list set) set" assumes "2 \<le> card V"   and "(I, J) = getIJ Q_pos Q_neg V" shows "V = I \<union> J"
ldrop_infT: fixes t :: "'b llist"   and A :: "'b set"   and i :: "nat" assumes "t \<in> A\<^sup>\<omega>" shows "t \<up> i \<in> A\<^sup>\<omega>"
assms: shows "list_of_oalist_tc xs \<noteq> []"
qPsubstAbs_preserves_alphaAbs1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm option"   and A :: "('e, 'f, 'varSort, 'var, 'g) qAbs"   and B :: "('e, 'f, 'varSort, 'var, 'g) qAbs" assumes "qGoodEnv rho"   and "qGoodAbs A \<or> qGoodAbs B"   and "A $= B" shows "A $[[rho]] $= B $[[rho]]"
Strong_dlHPP_Bisimulation_def: fixes d :: "'d"   and PP :: "nat set"   and R :: "('com list \<times> 'com list) set" shows "SdlHPPB d PP R \<equiv> sym R \<and> trans R \<and> (\<forall>(V, V')\<in>R. length V = length V') \<and> (\<forall>(V, V')\<in>R. \<forall>i<length V. NDC d (V ! i) \<or> IDC d (V ! i) (htchLoc (pp (V ! i)))) \<and> (\<forall>(V, V')\<in>R. \<forall>i<length V. \<forall>m1 m1' m2 \<alpha> p. \<langle>V ! i,m1\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>p,m2\<rangle> \<and> m1 \<sim>\<^bsub>d,htchLocSet PP\<^esub> m1' \<longrightarrow> (\<exists>p' \<alpha>' m2'. \<langle>V' ! i,m1'\<rangle> \<rightarrow>\<lhd>\<alpha>'\<rhd> \<langle>p',m2'\<rangle> \<and> stepResultsinR p p' R \<and> (\<alpha>, \<alpha>') \<in> R \<and> dhequality_alternative d PP (pp (V ! i)) m2 m2'))"
in_mtx_nonzeroN: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<in> mtx_nonzero c" shows "u < N \<and> v < N"
fls_deriv_of_nat: fixes n :: "nat" shows "fls_deriv (of_nat n) = 0"
iso_set_sym: fixes h :: "'a \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme" assumes "h \<in> Group.iso G H" shows "inv_into (carrier G) h \<in> Group.iso H G"
simps_14: fixes bs :: "real list"   and p :: "fm" shows "Ifm bs (E p) = (\<exists>x. Ifm (x # bs) p)"
pred_equals_eq2: fixes R :: "('a \<times> 'b) set"   and S :: "('a \<times> 'b) set" shows "((\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> S)) = (R = S)"
minimal_antichain_member: fixes A :: "'a set"   and t :: "'a" assumes "finite A"   and "t \<in> A" shows "\<exists>t'. t' \<in> minimal_antichain A \<and> t' \<le> t"
list_emb_length: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "list_emb P xs ys" shows "length xs \<le> length ys"
ncols_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "ncols A = CARD('c)"
linear_simps_1: fixes uv :: "name" shows "linear (const uv) = True"
div_ceiling_def: fixes x :: "int"   and y :: "int" shows "div_ceiling x y = (let q = x div y in if q * y = x then q else q + 1)"
induct: fixes P :: "nat list \<Rightarrow> t list list \<Rightarrow> bool"   and a0 :: "nat list"   and a1 :: "t list list" assumes "\<And>lab_c. P [] lab_c"   and "\<And>i is lab_c. P (i # is) lab_c" shows "P a0 a1"
assms_3: fixes x :: "'a" assumes "x \<in> A" shows "f x \<le> g x"
residue_const: fixes c :: "complex"   and z :: "complex" shows "residue (\<lambda>_. c) z = 0"
twoCisTypeII_4: shows "twoC = (\<lambda>(y, x). ((1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b), (1 - x) * a + x * b))"
Tree\<^sub>\<alpha>_free_2: fixes tset\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha> set['e]"   and \<phi> :: "'f" shows "Conj\<^sub>\<alpha> tset\<^sub>\<alpha> \<noteq> Pred\<^sub>\<alpha> \<phi>"
l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and type_wf :: "(_) heap \<Rightarrow> bool" shows "l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_shadow_root get_shadow_root_locs get_host get_host_locs type_wf \<equiv> l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs \<and> l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_shadow_root get_shadow_root_locs get_host get_host_locs"
SN_on_conv_accp: fixes g :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "SN_on {(y, z). g z y} {x} = Wellfounded.accp g x"
gfp_upperbound: 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 X :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less_eq X (f X)" shows "less_eq X (complete_lattice.gfp Sup less_eq f)"
triangle_left: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and f :: "'c"   and g :: "'c"   and \<eta> :: "'c"   and \<epsilon> :: "'c" assumes "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "V (H \<epsilon> f) (V (bicategory.\<a>' V H \<a> src trg f g f) (H f \<eta>)) = V (category.inv V (bicategory.lunit V H \<a> \<i> src trg f)) (bicategory.runit V H \<a> \<i> src f)"
brI: fixes a :: "'b"   and \<alpha> :: "'c \<Rightarrow> 'b"   and c :: "'c"   and I :: "'c \<Rightarrow> bool" assumes "a = \<alpha> c"   and "I c" shows "(c, a) \<in> br \<alpha> I"
closeSubstClosed: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list" assumes "eqvt Rel"   and "(\<Psi>, P, Q) \<in> closeSubst Rel" shows "(p \<bullet> \<Psi>, p \<bullet> P, p \<bullet> Q) \<in> closeSubst Rel"
base_grid: fixes p :: "(nat \<times> int) list"   and dm :: "nat"   and ds :: "nat set" assumes "p \<in> sparsegrid' dm" shows "grid (base ds p) ds \<subseteq> sparsegrid' dm"
less_mult_imp_div_less: fixes m :: "nat"   and i :: "nat"   and n :: "nat" assumes "m < i *n n" shows "m div n < i"
strict_monoD: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "class.order less_eq less"   and "order.strict_mono less f"   and "less x y" shows "f x < f y"
head_Var_reduction: fixes n :: "nat"   and rs :: "dB list"   and v :: "dB" assumes "Var n \<degree>\<degree> rs \<rightarrow>\<^sub>\<beta> v" shows "\<exists>ss. rs => ss \<and> v = Var n \<degree>\<degree> ss"
IH: fixes vs :: "v list"   and s :: "'a state" assumes "is_cupcake_all_env env"   and "list_all is_cupcake_exp es"   and "sequence_result ys = Rval vs" shows "\<exists>s'. Big_Step_Unclocked_Single.evaluate_list (Big_Step_Unclocked_Single.evaluate env) s es (s', Rval vs)"
no_initial_Marker: shows "\<forall>i. (\<exists>p q. channel i = Some (p, q)) \<longrightarrow> Marker \<notin> set (msgs init i)"
numgcd0: fixes t :: "Ferrack.num"   and bs :: "real list" assumes "numgcd t = 0" shows "Inum bs t = 0"
sample_vars_lookup: fixes V :: "'v set"   and x :: "'v"   and f :: "int \<Rightarrow> 'm" assumes "finite V"   and "x \<notin> V" shows "sample_vars X V (bind (lookup x) f) = bind (lookup x) (\<lambda>i. sample_vars X V (f i))"
finsum_not_depend: fixes A :: "'c set"   and N :: "'c set"   and f :: "'c \<Rightarrow> 'c" assumes "finite A"   and "A \<subseteq> N"   and "LinearCombinations.submodule R N (md N)"   and "f \<in> A \<rightarrow> N" shows "finsum (N.md N) f A = finsum (md N) f A"
simps_3: fixes x21 :: "'c"   and x22 :: "'c Tree"   and x23 :: "'c Tree" shows "T x21 x22 x23 \<noteq> E"
real_eq_0_iff_le_ge_0: fixes x :: "real" shows "x = 0 \<equiv> 0 \<le> x \<and> 0 \<le> - x"
ang_not_null_lg: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool"   and l :: "'p \<Rightarrow> 'p \<Rightarrow> bool"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "QCongA a"   and "QCong l"   and "a A B C"   and "l A B" shows "\<not> QCongNull l"
m_assoc: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "Group.monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"
point_measure_def: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ennreal" shows "point_measure A f = density (count_space A) f"
src_cod: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "trg (B.cod \<mu>) = trg \<mu>"
sqrt_rat: fixes x :: "rat" shows "set (sqrt_rat x) = {y. y * y = x}"
vwb_lens_iff_mwb_UNIV_src: fixes X :: "'e \<Longrightarrow> 'f" shows "vwb_lens X = (mwb_lens X \<and> \<S>\<^bsub>X\<^esub> = UNIV)"
rotate_append: fixes l :: "'d list"   and q :: "'d list" shows "rotate (length l) (l @ q) = q @ l"
density_def: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" shows "density M f = measure_of (space M) (sets M) (\<lambda>A. \<integral>\<^sup>+ x. f x * indicator A x \<partial>M)"
OUT_plus_flow: fixes \<Delta> :: "('a, 'b) graph_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "\<And>e. e \<notin> \<^bold>E\<^bsub>\<Delta>\<^esub> \<Longrightarrow> f e = 0"   and "\<And>x y. edge \<Delta> x y \<Longrightarrow> g (y, x) \<le> f (x, y)"   and "d_IN g x \<noteq> \<top>" shows "d_OUT (f \<oplus>\<^bsub>\<Delta>\<^esub> g) x = d_OUT f x + (\<Sum>\<^sup>+ y. g (x, y) * indicator \<^bold>E\<^bsub>\<Delta>\<^esub> (x, y)) - (\<Sum>\<^sup>+ y. g (y, x) * indicator \<^bold>E\<^bsub>\<Delta>\<^esub> (x, y))"
right_translate_bij: fixes G :: "'a set"   and g :: "'a" assumes "Group G"   and "g \<in> G" shows "bij_betw (\<lambda>x. x + g) G G"
right_plus_below_circ: fixes x :: "'a" shows "x\<^sup>\<circ> * x \<le> x\<^sup>\<circ>"
suc_singl_pred: fixes a :: "'a"   and a' :: "'a" assumes "a \<in> Field r"   and "local.aboveS a \<noteq> {}"   and "(a', suc {a}) \<in> r"   and "a' \<noteq> suc {a}" shows "a' = a \<or> (a', a) \<in> r"
appl: shows "applicable C\<^sub>1' C\<^sub>2' L\<^sub>1' L\<^sub>2' \<sigma>"
unit_mult_left_cancel: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a dvd (1::'b)" shows "(a * b = a * c) = (b = c)"
1: shows "Map (src a) = EVAL (E.Src (Dom a))"
Init_def: shows "Init = {init_item r 0 |r. r \<in> \<RR> \<and> fst r = \<SS>}"
FCI_def: fixes \<Gamma> :: "'b Gamma"   and \<V> :: "'b V_rec"   and Tr :: "'b list set" shows "FCI \<Gamma> \<V> Tr \<equiv> \<forall>\<alpha> \<beta>. \<forall>c\<in>C\<^bsub>\<V>\<^esub> \<inter> \<Upsilon>\<^bsub>\<Gamma>\<^esub>. \<forall>v\<in>V\<^bsub>\<V>\<^esub> \<inter> \<nabla>\<^bsub>\<Gamma>\<^esub>. \<beta> @ [v] @ \<alpha> \<in> Tr \<and> \<alpha> \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<longrightarrow> (\<exists>\<alpha>' \<delta>'. set \<delta>' \<subseteq> N\<^bsub>\<V>\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>\<^esub> \<and> \<beta> @ [c] @ \<delta>' @ [v] @ \<alpha>' \<in> Tr \<and> \<alpha>' \<upharpoonleft> V\<^bsub>\<V>\<^esub> = \<alpha> \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> \<alpha>' \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [])"
PredicatePairFunctions2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<forall>x1 x2. \<exists>y1 y2. P x1 x2 y1 y2" obtains f1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" and f2 :: "'a \<Rightarrow> 'b \<Rightarrow> 'd" where "\<forall>x1 x2. \<exists>y1 y2. f1 x1 x2 = y1 \<and> f2 x1 x2 = y2 \<and> P x1 x2 (f1 x1 x2) (f2 x1 x2)"
binding_subst_fresh: fixes X :: "tyvrs"   and a :: "binding"   and P :: "ty"   and Y :: "tyvrs" assumes "X \<sharp> a"   and "X \<sharp> P" shows "X \<sharp> a[Y \<mapsto> P]\<^sub>b"
second_summand_overlap: fixes z :: "i"   and y :: "i"   and x :: "i" assumes "CEM"   and "O z y" shows "O z (x \<oplus> y)"
zero_closed: shows "\<zero>\<^bsub>UP INTEG\<^esub> \<in> carrier (UP INTEG)"
rbt_eqI: fixes t1 :: "('b, 'c) RBT.rbt"   and t2 :: "('b, 'c) RBT.rbt" assumes "rbt.impl_of t1 = rbt.impl_of t2" shows "t1 = t2"
Proj_on_own_range': fixes P :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a" assumes "P o\<^sub>C\<^sub>L P = P"   and "P = P*" shows "Proj (P *\<^sub>S \<top>) = P"
interior_of_def: fixes X :: "'a topology"   and S :: "'a set" shows "X interior_of S \<equiv> {x. \<exists>T. openin X T \<and> x \<in> T \<and> T \<subseteq> S}"
ideal_nprod_incTr: fixes A :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "ideal R A" shows "(\<forall>i\<le>n. f i \<in> carrier R) \<and> (\<exists>l\<le>n. f l \<in> A) \<longrightarrow> e\<Pi>\<^bsub>R,n\<^esub> f \<in> A"
iterAbs_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'c) model" shows "iterAbs MOD \<equiv> check \<circ> iterAbsSTR (errMOD MOD)"
bernoulli_denom_nonzero: fixes n :: "nat" shows "bernoulli_denom n \<noteq> 0"
triangle_left_iff_right: shows "((\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) = \<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f]) = ((g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) = \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g])"
lleq_trans: fixes x :: "char list"   and y :: "char list"   and z :: "char list" assumes "lleq_charlist x y"   and "lleq_charlist y z" shows "lleq_charlist x z"
h: shows "steps0 (Suc 0, Bk \<up> l, <lm>) tp stp = (0, Bk \<up> m, Oc \<up> rs @ Bk \<up> n)"
gso_carrier: fixes i :: "nat" assumes "i < length (map of_int_hom.vec_hom fs')" shows "gs2.gso i \<in> Rn"
map_int_g: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and Tl :: "'fsym trm list" shows "map (Ik.int \<xi>) Tl = map (GE.int \<xi> \<circ> gT) Tl"
AA7: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" assumes "|~ F \<longrightarrow> G" shows "\<turnstile> \<diamond>\<langle>F\<rangle>_v \<longrightarrow> \<diamond>\<langle>G\<rangle>_v"
subset_was_path: fixes G :: "('c, 'd) graph"   and H :: "('c, 'd) graph"   and x :: "'c"   and p :: "('c \<times> 'd \<times> 'c) list"   and y :: "'c" assumes "valid_graph G"   and "is_path_undir H x p y"   and "edges H \<subseteq> edges G"   and "nodes H \<subseteq> nodes G" shows "is_path_undir G x p y"
divides_prod_r: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a divides b"   and "a \<in> carrier G"   and "c \<in> carrier G" shows "a divides b \<otimes> c"
wqo_on_imp_transp_on: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "wqo_on P A" shows "transp_on P A"
simps_5: fixes las :: "lock_action list" shows "must_acquire_lock (ReleaseAcquire # las) = must_acquire_lock las"
to_F_def: fixes Bot_F :: "'a set"   and Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and entails_G :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Inf_G :: "'b inference set"   and Red_I_G :: "'b set \<Rightarrow> 'b inference set"   and Red_F_G :: "'b set \<Rightarrow> 'b set"   and \<G>_F :: "'a \<Rightarrow> 'b set"   and \<G>_I :: "'a inference \<Rightarrow> 'b inference set option"   and Prec_F :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Inf_FL :: "('a \<times> 'c) inference set"   and \<iota>\<^sub>F\<^sub>L :: "('a \<times> 'c) inference" assumes "labeled_tiebreaker_lifting Bot_F Inf_F Bot_G entails_G Inf_G Red_I_G Red_F_G \<G>_F \<G>_I Prec_F Inf_FL" shows "labeled_tiebreaker_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))"
adhA: shows "\<forall>x\<in>carrier. x adh A \<longrightarrow> x \<in> A"
it_to_it_genord_correct: fixes it :: "('a list \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a list \<Rightarrow> 'a list) \<Rightarrow> 'a list \<Rightarrow> 'a list"   and S :: "'a set"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "set_iterator_genord it S R" shows "set_iterator_genord (it_to_it it) S R"
in_space_S: fixes x :: "real \<times> 'a" shows "x \<in> space S"
plus_right_id: fixes x :: "'a" shows "x + zer = x"
card_Suc_Diff1: fixes A :: "'e set"   and x :: "'e" assumes "finite A"   and "x \<in> A" shows "Suc (card (A - {x})) = card A"