hcju/isabelleps huggingface dataset:
reduced_word_append_reduce_contra1: fixes A :: "'a set"   and as :: "'a list"   and bs :: "'a list" assumes "\<not> reduced_word A as" shows "\<not> reduced_word A (as @ bs)"
IIA_def: fixes F :: "(indi \<Rightarrow> (alt \<times> alt) set) \<Rightarrow> (alt \<times> alt) set" shows "IIA F \<equiv> \<forall>P\<in>Prof. \<forall>P'\<in>Prof. \<forall>a b. (\<forall>i. (a <\<^bsub>P i\<^esub> b) = (a <\<^bsub>P' i\<^esub> b)) \<longrightarrow> (a <\<^bsub>F P\<^esub> b) = (a <\<^bsub>F P'\<^esub> b)"
wf_update: fixes i :: "'agent"   and Ri' :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "i \<in> agents'"   and "total_preorder_on alts' Ri'" shows "lowered.is_pref_profile (R(i := Ri'))"
path_image_reversepath: fixes g :: "real \<Rightarrow> 'c" shows "path_image (reversepath g) = path_image g"
poly_map_poly: fixes p :: "real poly"   and x :: "real" shows "poly (map_poly of_real p) (of_real x) = of_real (poly p x)"
vector_mult: fixes x :: "'a"   and y :: "'a" assumes "is_vector x"   and "is_vector y" shows "is_vector (x \<cdot> y)"
prec_count_def: fixes M :: "'a"   and u :: "'a"   and u' :: "'b"   and x :: "'c" shows "prec_count M u u' x = (u \<le> M)"
simps_16: fixes x61 :: "Unary_PCF.type"   and x62 :: "expr"   and x1 :: "char list \<times> Unary_PCF.type" shows "\<Lambda>\<langle>x61\<rangle> x62 \<noteq> \<langle>x1\<rangle>"
rev_cases: fixes xs :: "'b list" obtains "xs = []"   | ys :: "'b list" and y :: "'b" where "xs = ys @ [y]"
Diff_cover: fixes a :: "'a set"   and b :: "'a set" assumes "a \<in> M"   and "b \<in> M" shows "\<exists>C\<subseteq>M. finite C \<and> disjoint C \<and> a - b = \<Union> C"
power_Suc2: fixes a :: "'b list"   and n :: "nat" shows "a \<^sup>@ Suc n = a \<^sup>@ n \<cdot> a"
OpposedThinChamberComplexFoldings_axioms_def: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" shows "OpposedThinChamberComplexFoldings_axioms X f g C0 \<equiv> (SimplicialComplex.maxsimp X C0 \<and> C0 \<sim> g ` C0) \<and> C0 \<noteq> g ` C0 \<and> f ` g ` C0 = C0"
aug_bpp_closed: fixes z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e" assumes "assoc_bpp (aug_pm_set z i A) f"   and "x \<in> addition_set f (aug_pm_set z i A)"   and "y \<in> addition_set f (aug_pm_set z i A)" shows "x \<^sub>f+ y \<in> addition_set f (aug_pm_set z i A)"
dense_complement_subspace: fixes T :: "'a set"   and S :: "'a set" assumes "dim T < dim S"   and "subspace S" shows "closure (S - T) = S"
T_vrestr_0_2: fixes A :: "'a set" assumes "A \<in> sets M" shows "id --` A = A"
continuous_levelset_openin: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and a :: "'d" assumes "connected s"   and "continuous_on s f"   and "openin (top_of_set s) {x \<in> s. f x = a}"   and "\<exists>x\<in>s. f x = a" shows "\<forall>x\<in>s. f x = a"
memberdecl_is_static_def: fixes m :: "memberdecl" shows "is_static m = (case m of fdecl f \<Rightarrow> is_static f | mdecl m \<Rightarrow> is_static m)"
FL_Red_F_eq: fixes N :: "('a literal multiset \<times> label) set" shows "FL.Red_F_\<G> N = {C. \<forall>D\<in>\<G>_F (fst C). D \<in> G.Red_F (\<Union> (\<G>_F ` fst ` N)) \<or> (\<exists>E\<in>N. E \<sqsubset> C \<and> D \<in> \<G>_F (fst E))}"
post_def: shows "post = mat N N (\<lambda>(i, j). if i = j \<and> f i then 1 else 0)"
contour_integral_primitive_lemma: fixes a :: "real"   and b :: "real"   and S :: "complex set"   and f :: "complex \<Rightarrow> complex"   and f' :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex" assumes "a \<le> b"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_field_derivative f' x) (at x within S)"   and "g piecewise_differentiable_on {a..b}"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> S" shows "((\<lambda>x. f' (g x) * vector_derivative g (at x within {a..b})) has_integral f (g b) - f (g a)) {a..b}"
segmentp_segment_eq: fixes f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" shows "segmentp f a b = (\<lambda>x. x \<in> segment f a b)"
leq_refl: fixes x :: "'c llist" shows "x \<sqsubseteq> x"
comp_comp_fun_idem: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'h"   and g :: "'i \<Rightarrow> 'g" assumes "comp_fun_idem f" shows "comp_fun_idem (f \<circ> g)"
borel_measurable_det_Jacobian: fixes S :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec" assumes "S \<in> sets lebesgue"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)" shows "(\<lambda>x. det (matrix (f' x))) \<in> borel_measurable (lebesgue_on S)"
cyclic_split_segment: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and a :: "'c"   and b :: "'c" assumes "cyclic_on f S"   and "a \<in> S"   and "b \<in> S"   and "a \<noteq> b" shows "S = {a, b} \<union> segment f a b \<union> segment f b a"
ins3_Leaf: fixes n :: "nat"   and d :: "nat"   and x :: "'a" shows "ins3 n d x \<langle>\<rangle> = (if bal_i (n + 1) (d + 1) then Bal2 \<langle>\<langle>\<rangle>, x, \<langle>\<rangle>\<rangle> else Unbal2 \<langle>\<langle>\<rangle>, x, \<langle>\<rangle>\<rangle> 1 1)"
assms_1: shows "\<bar>u\<bar> \<in> Basis"
isDiagonal_diagonal_to_Smith_i: fixes A :: "(('d, 'e) vec, 'f) vec"   and xs :: "nat list"   and i :: "nat"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd" assumes "isDiagonal A"   and "\<forall>x. x \<in> set xs \<longrightarrow> x < min (nrows A) (ncols A)"   and "i < min (nrows A) (ncols A)" shows "isDiagonal (diagonal_to_Smith_i xs A i bezout)"
gen_dfsT_eq_assert: shows "gen_dfsT = gen_dfsT_assert"
nat_diff_distrib': fixes x :: "int"   and y :: "int" assumes "0 \<le> x"   and "0 \<le> y" shows "nat (x - y) = nat x - nat y"
face_of_Int_supporting_hyperplane_ge: fixes S :: "'b set"   and b :: "real"   and a :: "'b" assumes "convex S"   and "\<And>x. x \<in> S \<Longrightarrow> b \<le> a \<bullet> x" shows "S \<inter> {x. a \<bullet> x = b} face_of S"
ceiling_numeral: fixes v :: "num" shows "\<lceil>numeral v\<rceil> = numeral v"
data_DataOverride: fixes P :: "'b pdata"   and Q :: "'b data" assumes "PDataSpace P = Data.DataSpace Q" shows "(map OptionOverride (zip (PDataValue P) (DataValue Q)), Data.DataSpace Q) \<in> data"
circ_bot_edge_reparam_polar_circ_split: shows "reparam (rec_join [circle_bot_edge]) (rec_join [subcube (1 / 2) (3 / 4) (1, circle_polar), subcube (3 / 4) 1 (1, circle_polar)])"
left_sided_localp: fixes y :: "'a"   and x :: "'a" assumes "\<top> \<cdot> y = y" shows "x \<cdot> y \<le> y"
dom_phis'_codem: fixes nodes_of_phis :: "('val, ('node \<times> 'val) set) mapping"   and "next" :: "'node \<times> 'val"   and g :: "'g"   and next' :: "'val" assumes "\<And>ns. Mapping.lookup nodes_of_phis (snd next) = Some ns \<Longrightarrow> finite ns" shows "dom (Mapping.lookup (phis'_codem g next next' nodes_of_phis)) = dom (Mapping.lookup (phis g)) \<union> (case Mapping.lookup nodes_of_phis (snd next) of None \<Rightarrow> {} | Some x \<Rightarrow> id x) - {next}"
code: fixes g1 :: "'d \<Rightarrow> 'c"   and g2 :: "'d \<Rightarrow> 'd"   and a :: "'d" shows "unfold_stream g1 g2 a = g1 a ## unfold_stream g1 g2 (g2 a)"
alphaAbs_preserves_qGoodAbs: fixes A :: "('f, 'g, 'e, 'd, 'h) qAbs"   and A' :: "('f, 'g, 'e, 'd, 'h) qAbs" assumes "FixVars TYPE('d) TYPE('e)"   and "A $= A'" shows "FixVars.qGoodAbs A = FixVars.qGoodAbs A'"
card_psubset: fixes B :: "'b set"   and A :: "'b set" assumes "finite B"   and "A \<subseteq> B"   and "card A < card B" shows "A \<subset> B"
less_document_ptr_def: fixes x :: "(_) document_ptr"   and y :: "(_) document_ptr" shows "x < y \<equiv> x \<le> y \<and> \<not> y \<le> x"
a_add_kde: shows "a + complex_of_real kde \<in> closed_segment (a - complex_of_real kde) (a + complex_of_real e)"
bin_split_take: fixes n :: "nat"   and c :: "int"   and a :: "int"   and b :: "int"   and m :: "nat" assumes "bin_split n c = (a, b)" shows "bin_to_bl m a = take m (bin_to_bl (m + n) c)"
F1: fixes U :: "'b" shows "n_leaves (bt_map f (Br U t1 t2)) = n_leaves t1 + n_leaves t2"
Err_sup_Err2: fixes x :: "'c err"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b" shows "x \<squnion>\<^bsub>Err.sup f\<^esub> Err = Err"
is_class_def: fixes P :: "'a prog"   and C :: "String.literal" shows "is_class P C \<equiv> class P C \<noteq> None"
chamber_system_image: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ChamberComplexEndomorphism X f" shows "f \<turnstile> ChamberComplex.chamber_system X = ChamberComplex.chamber_system X \<inter> (f \<turnstile> X)"
sccs_finished: shows "\<Union> (sccs s) \<subseteq> dom (finished s)"
nat_mod: fixes n :: "nat"   and m :: "nat" assumes "n < m" shows "m mod n = (m - n) mod n"
nat_pow_eone: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier (UP R)" shows "[1] \<cdot>\<^bsub>UP R\<^esub> x = x"
rel_gpv_lossless_gpvD2: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and gpv :: "('f, 'h, 'j) gpv"   and gpv' :: "('g, 'i, 'j) gpv"   and \<I>' :: "('i, 'j) \<I>"   and \<I> :: "('h, 'j) \<I>" assumes "rel_gpv A C gpv gpv'"   and "lossless_gpv \<I>' gpv'"   and "rel_\<I> C (=) \<I> \<I>'" shows "lossless_gpv \<I> gpv"
borel_measurable_divide: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x / g x) \<in> borel_measurable M"
Picard_step_ivl_def: fixes X0 :: "'c set"   and t0 :: "real"   and h :: "real"   and X :: "'c set" shows "Picard_step_ivl X0 t0 h X = ASSERT (0 \<le> h) \<bind> (\<lambda>_. ASSERT (wd TYPE('c)) \<bind> (\<lambda>_. let H = lv_ivl [0] [h]; D = DIM('c); env = concat ` listset [list_of_eucl ` X0, H, list_of_eucl ` X] in approx_slp_spec (euler_incr_fas' D) D euler_incr_slp env \<bind> (\<lambda>env. case env of None \<Rightarrow> RETURN None | Some env \<Rightarrow> op_ivl_rep_of_set (eucl_of_list ` env) \<bind> (\<lambda>(l, u). ASSERT (l \<le> u) \<bind> (\<lambda>_. safe_set {l..u} \<bind> (\<lambda>s. if s then mk_safe ({l..u} ::: appr_rel) \<bind> (\<lambda>r. RETURN (Some (r ::: appr_rel))) else RETURN None))))))"
Goal: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "x \<sharp> P"   and "hnf P" shows "depth P' < depth P"
mult_sheaf_spec_in_sheaf_spec: fixes s :: "('a \<times> 'a) set set \<Rightarrow> (('a \<times> 'a) set \<times> ('a \<times> 'a) set) set"   and U :: "('a \<times> 'a) set set set"   and t :: "('a \<times> 'a) set set \<Rightarrow> (('a \<times> 'a) set \<times> ('a \<times> 'a) set) set" assumes "s \<in> local.local.sheaf_spec U"   and "t \<in> local.local.sheaf_spec U"   and "U \<subseteq> local.local.spectrum" shows "local.local.mult_sheaf_spec U s t \<in> local.local.sheaf_spec U"
rcis_def: fixes r :: "real"   and a :: "real" shows "rcis r a = cor r * cis a"
eventually_happens': fixes F :: "'b filter"   and P :: "'b \<Rightarrow> bool" assumes "F \<noteq> bot"   and "eventually P F" shows "\<exists>x. P x"
renumber_aux: fixes a :: "'a"   and f :: "nat \<Rightarrow> 'a"   and r :: "('a \<times> 'a) set"   and b :: "'a" assumes "\<forall>i. (a, f i) \<in> r\<^sup>* \<and> (f i, f (Suc i)) \<in> r"   and "(a, b) \<in> r\<^sup>*" shows "b = f 0 \<longrightarrow> (\<exists>f. f 0 = a \<and> (\<forall>i. (f i, f (Suc i)) \<in> r))"
monoD: fixes f :: "'a \<Rightarrow> 'c"   and y :: "'a"   and x :: "'a" assumes "order.mono (\<lambda>x y. y \<le> x) f"   and "y \<le> x" shows "f x \<le> f y"
abs_eq: fixes xa :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and x :: "'a \<Rightarrow> 'c option" shows "Mapping.map_values xa (mapping.Mapping x) = mapping.Mapping (\<lambda>xb. map_option (xa xb) (x xb))"
le_fin_ind'': fixes a :: "'item"   and w :: "'item list"   and b :: "'item" assumes "[a] \<preceq>\<^sub>F w"   and "[b] \<preceq>\<^sub>F w"   and "a \<noteq> b" shows "ind a b"
cotr_init: fixes c :: "('a, 'b, 'c) configuration" shows "cotrace c LNil"
bind_pmf_cong: fixes p :: "'c pmf"   and q :: "'c pmf"   and f :: "'c \<Rightarrow> 'd pmf"   and g :: "'c \<Rightarrow> 'd pmf" assumes "p = q"   and "\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x" shows "p \<bind> f = q \<bind> g"
n_val_n_pow: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and n :: "ant" assumes "valuation K v"   and "x \<in> carrier (Vr K v)"   and "0 \<le> n" shows "(n \<le> n_val K v x) = (x \<in> vp K v\<^bsup> Vr K v n\<^esup>)"
elims_2: fixes x :: "'b extended"   and xa :: "'b extended" assumes "x \<le> xa" obtains xb :: "'b" and y :: "'b" where "x = Fin xb" and "xa = Fin y" and "xb \<le> y"   | uu_ :: "'b extended" where "x = uu_" and "xa = \<infinity>"   | v :: "'b" where "x = -\<infinity>" and "xa = Fin v"   | "x = -\<infinity>" and "xa = -\<infinity>"
assms_1: shows "(\<box>(\<lambda>s. P s \<longrightarrow> Q s)) \<sigma>"
elementLevelLeqCurrentLevel: fixes a :: "'b"   and M :: "('b \<times> bool) list" shows "elementLevel a M \<le> currentLevel M"
homeomorphism_def: fixes s :: "'b set"   and t :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" shows "homeomorphism s t f g = ((\<forall>x\<in>s. g (f x) = x) \<and> f ` s = t \<and> continuous_on s f \<and> (\<forall>y\<in>t. f (g y) = y) \<and> g ` t = s \<and> continuous_on t g)"
residual_network_sel_4: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "sink (RES.residual_network f) = sink (residual_network f)"
Standard_hnorm: fixes x :: "'a star" assumes "x \<in> Standard" shows "hnorm x \<in> Standard"
uw\<theta>_axioms: shows "uw\<theta> (\<cdot>) (\<star>) \<a> \<i> src trg r \<rho> f g u w \<theta>"
pred_True: shows "pred_dual (\<lambda>_. True) = (\<lambda>_. True)"
correctCompositionDiffLevelsSYSTEM_def: shows "correctCompositionDiffLevelsSYSTEM \<equiv> \<forall>S. correctCompositionDiffLevels S"
adv_P2_game_def: fixes m1 :: "'msg1"   and m2 :: "'msg2"   and D :: "'view2 \<Rightarrow> bool spmf" shows "adv_P2_game m1 m2 D = \<bar>2 * spmf (P2_game_alt m1 m2 D) True - 1\<bar>"
HARootStates_Range_ChiPlus2: fixes S :: "'a"   and A :: "('a, 'b, 'c) hierauto"   and x :: "'a" assumes "S \<in> States (HARoot A)" shows "(x, S) \<notin> ChiPlus A"
finite_field_ops32_def: fixes p :: "uint32" shows "finite_field_ops32 p \<equiv> Arith_Ops_Record 0 1 (plus_p32 p) (mult_p32 p) (minus_p32 p) (uminus_p32 p) (divide_p32 p) (inverse_p32 p) (\<lambda>x y. if y = 0 then x else 0) (\<lambda>x. if x = 0 then 0 else 1) (\<lambda>x. x) uint32_of_int int_of_uint32 (\<lambda>x. 0 \<le> x \<and> x < p)"
verticesFrom_distinct: fixes f :: "face"   and v :: "nat" assumes "distinct (vertices f)"   and "v \<in> \<V> f" shows "distinct (verticesFrom f v)"
Unless_def: fixes A :: "'c set"   and B :: "'c set" shows "A Unless B \<equiv> A - B Co A \<union> B"
from_hma\<^sub>m_def: fixes a :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.from_hma\<^sub>m a = Matrix.mat CARD('c) CARD('b) (\<lambda>(i, j). a $h mod_type_class.from_nat i $h mod_type_class.from_nat j)"
of_fun_nth: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'd" assumes "i < n" shows "IArray.of_fun f n !! i = f i"
that: fixes U :: "'a set"   and s :: "'b" assumes "U \<in> I"   and "s \<in> \<FF> U"   and "X = \<lfloor>U, s\<rfloor>" shows "thesis"
inf_left_commute: fixes x :: "'a multiset"   and y :: "'a multiset"   and z :: "'a multiset" shows "x \<inter># (y \<inter># z) = y \<inter># (x \<inter># z)"
wf_Integer_ge_less_than: fixes d :: "int" shows "wf (Integer_ge_less_than d)"
elemlcos_carrier: fixes a :: "'a"   and a' :: "'a" assumes "Group.group G"   and "a \<in> carrier G"   and "a' \<in> a <# H" shows "a' \<in> carrier G"
evaln_elim_cases_14: fixes G :: "prog"   and s :: "st"   and va :: "Term.var"   and n :: "nat"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1l (Acc va)\<succ>\<midarrow>n\<rightarrow> (v, s')" obtains v :: "val" and f :: "val \<Rightarrow> state \<Rightarrow> state" where "vb__ = In1 v" and "G\<turnstile>Norm s \<midarrow>va=\<succ>(v, f)\<midarrow>n\<rightarrow> s'"
E_of_succ_def: fixes succ :: "'c \<Rightarrow> 'd set" shows "E_of_succ succ \<equiv> {(u, v). v \<in> succ u}"
eq_imp_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" shows "eq_imp f g \<equiv> \<forall>s s'. (\<forall>x. f x s = f x s') \<longrightarrow> g s = g s'"
F0: fixes i :: "nat" assumes "i \<noteq> 0" shows "(F i = 0) = (k < i)"
eq_trans: fixes a :: "Arity"   and v1 :: "Value"   and v2 :: "Value"   and v3 :: "Value" assumes "eq a v1 v2"   and "eq a v2 v3" shows "eq a v1 v3"
D: shows "0 < offs_num (length ns) (x # xs) index key mi ma j"
genPrefix_iff_nth: fixes xs :: "'a list"   and ys :: "'a list"   and r :: "('a \<times> 'a) set" shows "((xs, ys) \<in> genPrefix r) = (length xs \<le> length ys \<and> (\<forall>i<length xs. (xs ! i, ys ! i) \<in> r))"
gFreshClsRev_recAll_reflFreshAll: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) Recursion.model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.presWlsAll Delta (FixSyn.rec Delta MOD) (FixSyn.recAbs Delta MOD) MOD"   and "FixSyn.presCons Delta (FixSyn.rec Delta MOD) (FixSyn.recAbs Delta MOD) MOD"   and "FixSyn.presFreshAll Delta (FixSyn.rec Delta MOD) (FixSyn.recAbs Delta MOD) MOD"   and "FixSyn.gFreshClsRev Delta MOD" shows "FixSyn.reflFreshAll Delta (FixSyn.rec Delta MOD) (FixSyn.recAbs Delta MOD) MOD"
min_term_def: shows "punit.min_term \<equiv> fst (0::'a, the_min)"
prems: shows "M, g \<Turnstile>\<^sub>\<Theta> (ps, a) # branch"
l_set_child_nodes_get_disconnected_nodes_axioms: shows "l_set_child_nodes_get_disconnected_nodes DocumentClass.type_wf set_child_nodes set_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_subst: fixes F :: "(('d, 'e) Term.term \<times> ('d, 'e) Term.term) list"   and \<theta> :: "'e \<Rightarrow> ('d, 'c) Term.term" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta>) = fv\<^sub>s\<^sub>e\<^sub>t (\<theta> ` fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F)"
a_eq_minus_1: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "\<forall>a. plus a one = zero \<longrightarrow> a = uminus one"
Embed_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "Embed t = (\<lambda>ab. t)"
functor_axioms: shows "functor (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) L"
fun_upd_apply: fixes f :: "'g \<Rightarrow> 'f"   and x :: "'g"   and y :: "'f"   and z :: "'g" shows "(f(x := y)) z = (if z = x then y else f z)"
len: fixes j :: "nat"   and v :: "nat" shows "eval len [j, v, i, e] \<down>= e_length e"
strand_sem_induct: fixes P :: "('fun, 'var) Term.term set \<Rightarrow> ('fun, 'var) strand_step list \<Rightarrow> bool"   and a0 :: "('fun, 'var) Term.term set"   and a1 :: "('fun, 'var) strand_step list" assumes "\<And>M. P M []"   and "\<And>M t S. (\<And>x. P M S) \<Longrightarrow> P M (send\<langle>t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>M t S. (\<And>x. P (insert (t \<cdot> x) M) S) \<Longrightarrow> P M (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>M uu_ t t' S. (\<And>x. P M S) \<Longrightarrow> P M (\<langle>uu_: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>M X F S. (\<And>x. P M S) \<Longrightarrow> P M (\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S)" shows "P a0 a1"
ideal_points_poincare_line_moebius: fixes M :: "moebius"   and u :: "complex_homo"   and v :: "complex_homo"   and i1 :: "complex_homo"   and i2 :: "complex_homo" assumes "unit_disc_fix M"   and "u \<in> unit_disc"   and "v \<in> unit_disc"   and "u \<noteq> v"   and "ideal_points (poincare_line u v) = {i1, i2}" shows "ideal_points (poincare_line (moebius_pt M u) (moebius_pt M v)) = {moebius_pt M i1, moebius_pt M i2}"
freshInp_def: fixes xs :: "'b"   and x :: "'c"   and inp :: "'d \<Rightarrow> ('e, 'f, 'b, 'c, 'g) qTerm set option" shows "freshInp xs x inp \<equiv> liftAll (fresh xs x) inp"
not_is_poincare_line_unit_circle: shows "\<not> is_poincare_line unit_circle"
Zfun_ssubst: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<forall>\<^sub>F x in F. f x = g x"   and "Zfun g F" shows "Zfun f F"
ext_inject: fixes prism_match :: "'d \<Rightarrow> 'c option"   and prism_build :: "'c \<Rightarrow> 'd"   and more :: "'e"   and prism_match' :: "'d \<Rightarrow> 'c option"   and prism_build' :: "'c \<Rightarrow> 'd"   and more' :: "'e" shows "(\<lparr>prism_match = prism_match, prism_build = prism_build, \<dots> = more\<rparr> = \<lparr>prism_match = prism_match', prism_build = prism_build', \<dots> = more'\<rparr>) = (prism_match = prism_match' \<and> prism_build = prism_build' \<and> more = more')"
uconnectedV_refl: fixes E' :: "'a uprod set" assumes "E' \<subseteq> E" shows "refl_on verts (Restr (uconnected E') verts)"
powr_log_cancel: fixes a :: "real"   and x :: "real" assumes "0 < a"   and "a \<noteq> 1"   and "0 < x" shows "a powr log a x = x"
assert_cs_unsat: fixes cs :: "('a \<times> constraint) list"   and J :: "'a set"   and s :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state"   and j :: "'a"   and I :: "'a list" assumes "invariant_simplex' cs J s"   and "assert_simplex' j s = Inl I" shows "set I \<subseteq> insert j J \<and> minimal_unsat_core (set I) cs"
normNum_Nadd: fixes x :: "int \<times> int"   and y :: "int \<times> int" shows "normNum (x +\<^sub>N y) = x +\<^sub>N y"
simps: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset" assumes "ground_resolution_with_selection S" shows "ground_resolution_with_selection.production S N C = {A. C \<in> N \<and> C \<noteq> {#} \<and> Max_mset C = Pos A \<and> \<not> \<Union> (ground_resolution_with_selection.production S N ` {D. D < C}) \<Turnstile> C \<and> S C = {#}}"
r1: shows "right (Abs_real_int (left r1, right r3)) = right r3"
scalar_prod_def: fixes v :: "'c vec"   and w :: "'c vec" shows "v \<bullet> w \<equiv> \<Sum>i = 0..<dim_vec w. v $ i * w $ i"
BndQQ: shows "QQ \<subseteq> Bounded_vec DBnd"
fmdom'_clearjunk0: fixes xs :: "('c, 'd) fmap" shows "fmdom' (clearjunk0 xs) = fmdom' xs - {x. fmlookup xs x = Some (0::'d)}"
rotate2_def: fixes x :: "(real, 2) vec" shows "rotate2 x \<equiv> vector [- x $ 2, x $ 1]"
simps_5: fixes r :: "'a pfm"   and p :: "'a pfm"   and q :: "'a pfm" shows "reduce' r (p \<^bold>\<longrightarrow>\<^sub>! q) = (reduce' r p \<^bold>\<longrightarrow>\<^sub>! reduce' r q)"
mahler_measure_poly_0: shows "mahler_measure_poly 0 = 0"
times_Quantity_ext_def: fixes x :: "('c, 'd, 'e) Quantity_scheme"   and y :: "('c, 'd, 'e) Quantity_scheme" shows "x \<cdot> y = \<lparr>mag = mag x \<cdot> mag y, dim = Quantity.dim x \<cdot> Quantity.dim y, \<dots> = Quantity.more x \<cdot> Quantity.more y\<rparr>"
app_not_fst: fixes A :: "'b trm"   and B :: "'b trm"   and P :: "'b trm" shows "App A B \<noteq> Fst P"
AE_iff_measurable: fixes N :: "'c set"   and M :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "N \<in> sets M"   and "{x \<in> space M. \<not> P x} = N" shows "almost_everywhere M P = (emeasure M N = 0)"
src_trg: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "src (trg \<mu>) = trg \<mu>"