hcju/isabelleps huggingface dataset:
card_1_singleton_iff: fixes A :: "'b set" shows "(card A = Suc 0) = (\<exists>x. A = {x})"
spr_simVal_def: shows "spr_simVal \<equiv> envVal \<circ> snd"
filterlim_inf: fixes f :: "'a \<Rightarrow> 'b"   and F2 :: "'b filter"   and F3 :: "'b filter"   and F1 :: "'a filter" shows "filterlim f (inf F2 F3) F1 = (filterlim f F2 F1 \<and> filterlim f F3 F1)"
apex_is_induced_by_cell: shows "h.is_induced_by_cell (src (tab\<^sub>0 h)) (tab\<^sub>0 h) h.tab"
sign_simps_33: fixes am :: "'b"   and bm :: "'b" shows "am * bm = bm * am"
set_mset_mset_hp: fixes h :: "'b hp" shows "set_mset (mset_hp h) = set_hp h"
list_rel_imp_same_length: fixes l :: "'a list"   and l' :: "'b list"   and R :: "('a \<times> 'b) set" assumes "(l, l') \<in> \<langle>R\<rangle>list_rel" shows "length l = length l'"
finite_atLeastZeroLessThan_integer: fixes u :: "integer" shows "finite {0..<u}"
unique_transversal_segment_intersection: fixes a :: "'a"   and b :: "'a"   and x :: "'a"   and u :: "'a" assumes "rev.transversal_segment a b"   and "{0..} \<subseteq> rev.existence_ivl0 x"   and "u \<in> rev.\<omega>_limit_set x \<inter> {a<--<b}" shows "rev.\<omega>_limit_set x \<inter> {a<--<b} = {u}"
Cons: shows "r \<down>m r|\<sigma>| = r \<down>l \<sigma>"
qFreshBinp_def: fixes xs :: "'varSort"   and x :: "'var"   and qbinp :: "'c \<Rightarrow> ('d, 'c, 'varSort, 'var, 'e) qAbs option" shows "qFreshBinp xs x qbinp \<equiv> liftAll (qFreshAbs xs x) qbinp"
cons_ap_list: fixes f :: "'e \<Rightarrow> 'd"   and fs :: "('e \<Rightarrow> 'd) list"   and xs :: "'e list" shows "(f # fs) \<diamondop> xs = map f xs @ fs \<diamondop> xs"
length_conv_length_it': fixes xs :: "'b list"   and acc :: "nat" shows "length xs + acc = length_it' acc xs"
wlp_soundness: fixes dims :: "nat list"   and S :: "com"   and P :: "complex mat" assumes "state_sig.well_com dims S"   and "state_sig.is_quantum_predicate dims P" shows "\<forall>\<rho>\<in>state_sig.density_states dims. trace (state_sig.wlp dims S P * \<rho>) = trace (P * state_sig.denote dims S \<rho>) + trace \<rho> - trace (state_sig.denote dims S \<rho>)"
hcomplex_surj: fixes z :: "complex star" shows "HComplex (hRe z) (hIm z) = z"
preserves_hseq: fixes \<mu> :: "'b"   and \<nu> :: "'b" assumes "B.arr \<mu> \<and> B.arr \<nu> \<and> src\<^sub>B \<mu> = trg\<^sub>B \<nu>" shows "C.arr (G \<mu>) \<and> C.arr (G \<nu>) \<and> src\<^sub>C (G \<mu>) = trg\<^sub>C (G \<nu>)"
closure_involutive_aux: fixes X :: "'d set"   and U :: "('d \<Rightarrow> real) set set"   and k :: "'d \<Rightarrow> nat" assumes "AlphaClosure X"   and "U \<subseteq> {region X I r |I r. valid_region X k I r}" shows "AlphaClosure.cla X k (\<Union> U) = \<Union> U"
tendsto_inner: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and b :: "'c" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x \<bullet> g x) \<longlongrightarrow> a \<bullet> b) F"
di_1: shows "case di of None \<Rightarrow> True | Some ddi \<Rightarrow> p_dport p \<in> prefix_to_wordset ddi"
empty_path: fixes n :: "'node" assumes "valid_node n" shows "n -[]\<rightarrow>* n"
star_step1: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "r x y" shows "star r x y"
profile_mono\<^sub>d_axioms_def: fixes g :: "'a \<Rightarrow> 'b" shows "profile_mono\<^sub>d_axioms g \<equiv> \<forall>x. x \<noteq> \<bottom> \<longrightarrow> x \<noteq> null \<longrightarrow> g x \<noteq> \<bottom>"
length_update_until: fixes pos :: "\<I>"   and I :: "bool"   and rel1 :: "'b option list set"   and rel2 :: "'b option list set"   and nt :: "nat"   and aux :: "(nat \<times> 'b option list set \<times> 'b option list set) list" shows "length (update_until pos I rel1 rel2 nt aux) = Suc (length aux)"
invc_baldR: fixes l :: "('c \<times> color) tree"   and r :: "('c \<times> color) tree"   and a :: "'c" assumes "invc l"   and "invc2 r"   and "color l = Black" shows "invc (baldR l a r)"
mbc_ex: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "\<exists>x. x \<in> actHn (t n) \<and> pow_cond t n (length (bc (\<sigma>\<^bsub>x\<^esub>t n)))"
pcr_CoCalls_def: shows "pcr_CoCalls \<equiv> rel_set (rel_prod (=) (=)) OO cr_CoCalls"
permutes_conv_has_dom: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set" shows "(f permutes S) = (bij f \<and> has_dom f S)"
u3: shows "\<parallel>r\<parallel> = \<parallel>s\<parallel>"
dec_interp_enc_Inl: fixes n :: "nat"   and FO :: "nat set"   and w :: "'a list"   and I :: "(nat + nat set) list"   and i :: "nat"   and p' :: "nat"   and p :: "nat" assumes "dec_interp n FO (enc (w, I)) ! i = Inl p'"   and "I ! i = Inl p"   and "i \<in> FO"   and "i < n"   and "length I = n"   and "p < length w"   and "0 < length w \<and> (\<forall>a\<in>set w. a \<in> set \<Sigma>) \<and> (\<forall>x\<in>set I. case x of Inl p \<Rightarrow> p < length w | Inr P \<Rightarrow> \<forall>p\<in>P. p < length w)" shows "p = p'"
mono: shows "\<forall>i j. i < j \<longrightarrow> f i < f j"
preserves_ide: fixes a :: "'a arrow_of_spans_data" assumes "ide a" shows "ide (L a)"
th0: shows "connected S = (\<nexists>e2 e1. closed e2 \<and> open e1 \<and> S \<subseteq> e1 \<union> - e2 \<and> e1 \<inter> - e2 \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> - e2 \<inter> S \<noteq> {})"
box_co: fixes a :: "'b"   and b :: "'b" shows "{x. a \<le> x \<and> x <e b} = {a..} \<inter> {x. x <e b}"
le_cases3: fixes x :: "'t"   and y :: "'t"   and z :: "'t" obtains "x \<preceq>\<^sub>t y" and "y \<preceq>\<^sub>t z"   | "y \<preceq>\<^sub>t x" and "x \<preceq>\<^sub>t z"   | "x \<preceq>\<^sub>t z" and "z \<preceq>\<^sub>t y"   | "z \<preceq>\<^sub>t y" and "y \<preceq>\<^sub>t x"   | "y \<preceq>\<^sub>t z" and "z \<preceq>\<^sub>t x"   | "z \<preceq>\<^sub>t x" and "x \<preceq>\<^sub>t y"
surj_pair: fixes p :: "'l \<times> 'm" shows "\<exists>x y. p = (x, y)"
brn_gt_0_L: fixes cl :: "('test, 'atom, 'choice) cmd list"   and c :: "('test, 'atom, 'choice) cmd" assumes "properL cl"   and "c \<in> set cl" shows "0 < brn c"
all_in_list_sublist2: fixes xs :: "('b, 'c) Combinators list"   and l :: "'b set list"   and P :: "('b, 'c) Combinators \<Rightarrow> bool" assumes "singleCombinators xs"   and "all_in_list xs l" shows "all_in_list (filter P xs) l"
lprefix_lnthD: fixes xs :: "'c llist"   and ys :: "'c llist"   and n :: "nat" assumes "xs \<sqsubseteq> ys"   and "enat n < llength xs" shows "lnth xs n = lnth ys n"
space_measure_of: fixes A :: "'d set set"   and \<Omega> :: "'d set"   and \<mu> :: "'d set \<Rightarrow> ennreal" assumes "A \<subseteq> Pow \<Omega>" shows "space (measure_of \<Omega> A \<mu>) = \<Omega>"
increasing2: fixes as :: "'a list"   and bs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "increasing (as @ bs)"   and "x \<in> set as"   and "y \<in> set bs" shows "x \<le> y"
respection: shows "rel_respects_binary_pred (indRelLTPO TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))"
upd_nested_step_def: fixes d :: "'f"   and f :: "'f \<Rightarrow> 'f"   and x :: "'d \<times> 'e"   and m :: "('d, ('e, 'f) mapping) mapping" shows "upd_nested_step d f x m = (case x of (k, k') \<Rightarrow> case Mapping.lookup m k of None \<Rightarrow> Mapping.update k (Mapping.update k' d Mapping.empty) m | Some m' \<Rightarrow> case Mapping.lookup m' k' of None \<Rightarrow> Mapping.update k (Mapping.update k' d m') m | Some v \<Rightarrow> Mapping.update k (Mapping.update k' (f v) m') m)"
weakPsiCongTransitive: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<doteq> Q"   and "\<Psi> \<rhd> Q \<doteq> R" shows "\<Psi> \<rhd> P \<doteq> R"
assms: fixes x :: "nat" assumes "x < dim_row A * dim_col A" shows "f x = flatten_matrix A x"
simps_2: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> real"   and gs :: "('a \<Rightarrow> real) list" shows "landau_dominating_chain' F (f # gs) = (landau_function_family_pair F (powr_closure f) (prod_list (map powr_closure gs)) (\<lambda>x. f x powr 1) \<and> landau_dominating_chain' F gs)"
obeh_def: fixes S :: "('f, 'e) spec" shows "obeh S \<equiv> map (obs S) ` beh S"
smult_minus_1_back: fixes v :: "'c" assumes "v \<in> carrier M" shows "\<ominus>\<^bsub>M\<^esub> v = \<ominus>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> v"
card_sum_def: fixes a :: "nat"   and b :: "nat" shows "card_sum a b = (if a = 0 \<or> b = 0 then 0 else a + b)"
f1: fixes R' :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and gpv :: "('d, 'f, 'g) gpv"   and gpv' :: "('e, 'i, 'h) gpv"   and A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and A' :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "('d, 'f, ('d, 'f, 'g) rpv) generat"   and y :: "('e, 'i, ('e, 'i, 'h) rpv) generat"   and a :: "('d, 'f, 'g) rpv"   and b :: "('e, 'i, 'h) rpv"   and a' :: "'g"   and \<alpha> :: "'h"   and \<beta> :: "'g"   and b' :: "'h" assumes "??.Generative_Probabilistic_Value.foo R' results_gpv results_gpv gpv gpv' A A'"   and "x \<in> set_spmf (the_gpv gpv)"   and "y \<in> set_spmf (the_gpv gpv')"   and "a \<in> generat_conts x"   and "b \<in> generat_conts y"   and "R' a' \<alpha>"   and "R' \<beta> b'" shows "??.Generative_Probabilistic_Value.foo R' results_gpv results_gpv (a a') (b b') A A'"
isCont_divide: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "isCont f a"   and "isCont g a"   and "g a \<noteq> (0::'c)" shows "isCont (\<lambda>x. f x / g x) a"
e_props_1: shows "arc_to_ends G e = (u, v)"
a: shows "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"
inf_prefix_length_0_imp_hd_neq: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<noteq> []"   and "ys \<noteq> []"   and "inf_prefix_length xs ys = 0" shows "hd xs \<noteq> hd ys"
F: shows "foldl (+) 0 ns + Suc (Suc m) = n"
rep_eq: fixes x :: "('c, 'd) fmap"   and xa :: "('c, 'd) fmap" shows "(x \<subseteq>\<^sub>f xa) = (fmlookup x \<subseteq>\<^sub>m fmlookup xa)"
b: shows "Ide b \<and> Arr b \<and> Dom b = b \<and> Cod b = b \<and> ide \<lbrace>b\<rbrace> \<and> ide \<lbrace>\<^bold>\<lfloor>b\<^bold>\<rfloor>\<rbrace> \<and> \<lbrace>b\<^bold>\<down>\<rbrace> \<in> hom \<lbrace>b\<rbrace> \<lbrace>\<^bold>\<lfloor>b\<^bold>\<rfloor>\<rbrace>"
index_sl_four: shows "\<forall>i<4. i = 0 \<or> i = 1 \<or> i = 2 \<or> i = 3"
cs_def: shows "\<exists>c1 c2. cs = [$C ConstInt64 c1, $C ConstInt64 c2]"
succ_chain_tableau_equiv: fixes l :: "('i, 'a) state list"   and i :: "nat"   and j :: "nat"   and v :: "nat \<Rightarrow> 'a" assumes "rel_chain l {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}"   and "i < length l"   and "j < length l" shows "v \<Turnstile>\<^sub>t \<T> (l ! i) = v \<Turnstile>\<^sub>t \<T> (l ! j)"
zero_and_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "and zero a = zero"
trace_callee_eq_obsf_oracleI: fixes callee1 :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and callee2 :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) spmf"   and A :: "'b set"   and p :: "'a spmf"   and q :: "'d spmf" assumes "trace_callee_eq callee1 callee2 A p q" shows "trace_callee_eq (obsf_oracle callee1) (obsf_oracle callee2) A (TRY map_spmf OK p ELSE return_spmf Fault) (TRY map_spmf OK q ELSE return_spmf Fault)"
m_def: shows "m = geodesic_segment_param G x (Gromov_product_at x z y)"
denormal_mantissa_def: fixes x :: "('a, 'b) IEEE.float" shows "denormal_mantissa x = (- 1) ^ sign x * int (fraction x)"
cases: fixes a :: "'b list list" assumes "map_head_side a" obtains xs :: "'b list list" where "a = xs" and "\<And>x. x \<in> set xs \<Longrightarrow> x \<noteq> []"
expectation_Y_\<Delta>: shows "expectation (\<lambda>x. (Y x)\<^sup>2) = \<mu> + \<Delta>\<^sub>a"
partitioning1_empty0: fixes ts :: "'a set list"   and ss :: "'a set list" assumes "{} \<notin> set ts" shows "{} \<notin> set (partitioning1 ss ts)"
"1_2": shows "(P \<and>* P' \<and>* $ n) (a, b)"
app_vrangeI: fixes a :: "V"   and b :: "V"   and r :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" shows "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"
binomial_absorption: fixes k :: "nat"   and n :: "nat" shows "Suc k * (n choose Suc k) = n * (n - 1 choose k)"
coboundedI2: fixes b :: "'a Word.word"   and c :: "'a Word.word"   and a :: "'a Word.word" assumes "b \<le>s c" shows "signed.min a b \<le>s c"
that: fixes r :: "complex poly"   and rs :: "complex \<Rightarrow> complex poly" assumes "\<And>n. (fps_of_poly p / fps_of_poly q) $ n = coeff r n + (\<Sum>c | poly q' c = 0. poly (rs c) (of_nat n) * c ^ n)"   and "\<And>z. poly q' z = 0 \<Longrightarrow> degree (rs z) \<le> order z q' - 1" shows "thesis"
iso_reduced_homology_by_contractible: fixes X :: "'a topology"   and S :: "'a set"   and p :: "int" assumes "contractible_space (subtopology X S)"   and "topspace X \<inter> S \<noteq> {}" shows "hom_induced p X {} X S id \<in> Group.iso (reduced_homology_group p X) (relative_homology_group p X S)"
eventually_in_Qinf: fixes D :: "'a literal multiset"   and C :: "'a literal multiset"   and Gs :: "'a literal multiset set llist" assumes "chain (\<leadsto>) Sts"   and "D \<in> N_of_state (Sup_state Sts) \<union> P_of_state (Sup_state Sts) \<union> Q_of_state (Sup_state Sts)"   and "subsumes D C"   and "\<forall>E\<in>{E \<in> N_of_state (Sup_state Sts) \<union> P_of_state (Sup_state Sts) \<union> Q_of_state (Sup_state Sts). subsumes E C}. \<not> strictly_subsumes E D"   and "fair_state_seq Sts"   and "Gs = lmap (\<lambda>St. grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)) Sts"   and "C \<in> Liminf_llist Gs - sr.Rf (Liminf_llist Gs)"   and "is_ground_cls C" shows "\<exists>D' \<sigma>'. D' \<in> Q_of_state (Liminf_state Sts) \<and> D' \<cdot> \<sigma>' = C \<and> is_ground_subst \<sigma>'"
zig_zig: fixes lb :: "'b tree"   and x :: "'b"   and lb1 :: "'b tree"   and lb2 :: "'b tree"   and lx :: "'b tree"   and rx :: "'b tree"   and u :: "'b"   and b :: "'b"   and rb :: "'b tree"   and a :: "'b"   and ra :: "'b tree" assumes "lb \<noteq> \<langle>\<rangle>"   and "real (T_splay x lb) + \<Phi> lb1 + \<Phi> lb2 - \<Phi> lb \<le> 2 * \<phi> lb - 3 * \<phi> \<langle>lx, x, rx\<rangle> + 1"   and "size lb = size lb1 + size lb2 + 1"   and "\<langle>lx, x, rx\<rangle> \<in> subtrees lb" shows "real (T_splay x lb) + \<Phi> \<langle>lb1, u, \<langle>lb2, b, \<langle>rb, a, ra\<rangle>\<rangle>\<rangle> - \<Phi> \<langle>\<langle>lb, b, rb\<rangle>, a, ra\<rangle> \<le> 3 * (\<phi> \<langle>\<langle>lb, b, rb\<rangle>, a, ra\<rangle> - \<phi> \<langle>lx, x, rx\<rangle>)"
update_minus_monomial: fixes k :: "'a"   and i :: "'b"   and m :: "'a \<Rightarrow>\<^sub>0 'b" shows "Poly_Mapping.update k i (m - monomial i k) = Poly_Mapping.update k i m"
relhoare_conj_right_cartesian: fixes pre :: "('e \<times> 'f) set"   and Ra :: "('e \<times> 'e) set"   and I :: "'e set"   and Rc :: "('f \<times> 'f) set"   and J :: "'f set"   and post :: "('e \<times> 'f) set" assumes "{Domain pre} Ra {> I}"   and "{Range pre} Rc {> J}"   and "{pre} Ra, Rc {> post}" shows "{pre} Ra, Rc {> post \<inter> I \<times> J}"
measurable_pair_measureI: fixes f :: "'c \<Rightarrow> 'd \<times> 'e"   and M :: "'c measure"   and M1 :: "'d measure"   and M2 :: "'e measure" assumes "f \<in> space M \<rightarrow> space M1 \<times> space M2"   and "\<And>A B. \<lbrakk>A \<in> sets M1; B \<in> sets M2\<rbrakk> \<Longrightarrow> f -` (A \<times> B) \<inter> space M \<in> sets M" shows "f \<in> M \<rightarrow>\<^sub>M M1 \<Otimes>\<^sub>M M2"
neg_a: shows "\<nexists>!\<alpha>. \<alpha> \<in> {0..1} \<and> q \<approx>[\<R>] mix_pmf \<alpha> p r"
rel_advantage_def: fixes \<A> :: "'pub_input \<Rightarrow> 'witness spmf" shows "rel_advantage \<A> \<equiv> spmf (rel_game \<A>) True"
even_unset_bit_iff: fixes m :: "nat"   and a :: "'a" shows "even (unset_bit m a) = (even a \<or> m = 0)"
T4_1: fixes \<phi> :: "'b ltlr" shows "create_gba \<phi> \<le> SPEC (\<lambda>\<A>. gba \<A> \<and> finite (g_V \<A>) \<and> (\<forall>\<xi>. gba.accept \<A> \<xi> = \<xi> \<Turnstile>\<^sub>r \<phi>) \<and> nds_invars (g_V \<A>))"
distributed_nn_integral: fixes M :: "'b measure"   and N :: "'c measure"   and X :: "'b \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> ennreal"   and g :: "'c \<Rightarrow> ennreal" assumes "distributed M N X f"   and "g \<in> borel_measurable N" shows "\<integral>\<^sup>+ x. f x * g x \<partial>N = \<integral>\<^sup>+ x. g (X x) \<partial>M"
bind_spmf_of_set: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b spmf" assumes "finite A"   and "A \<noteq> {}" shows "spmf_of_set A \<bind> f = pmf_of_set A \<bind> f"
agree_on_subset: fixes f :: "('e \<times> 'f) set"   and g :: "('e \<times> 'f) set"   and G :: "('g, 'e) labeled_graph" assumes "f \<subseteq> g"   and "vertices G \<subseteq> Domain f"   and "univalent g" shows "agree_on G f g"
invertible_is_bijective: fixes \<alpha> :: "'a \<Rightarrow> 'a" assumes "\<alpha> \<in> S \<rightarrow>\<^sub>E S" shows "invertible \<alpha> = bij_betw \<alpha> S S"
sgn_ell2_def: fixes x :: "'c ell2" shows "sgn x = x /\<^sub>R norm x"
largest_contained_prefix_subset_s1D: fixes r :: "'b wordinterval"   and s :: "'b prefix_match"   and u :: "'b wordinterval" assumes "wordinterval_CIDR_split1 r = (Some s, u)" shows "wordinterval_to_set (prefix_to_wordinterval s) \<subseteq> wordinterval_to_set r"
intpow_plus: fixes x :: "'b"   and m :: "int"   and n :: "int" assumes "(0::'b) < x" shows "x ^\<^sub>Z (m + n) = x ^\<^sub>Z m * x ^\<^sub>Z n"
nm1: shows "netgmap sr n1 = (\<sigma>\<^sub>1, \<zeta>\<^sub>1)"
endofunctor_def: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'c \<Rightarrow> 'c" shows "endofunctor A F \<equiv> functor A A F"
wit: shows "Q1 * A0 * P2 \<in> carrier_mat n m"
augmenting_edge_no_swap: fixes p :: "(nat \<times> nat) list" assumes "isAugmentingPath p" shows "set p \<inter> (set p)\<inverse> = {}"
that: fixes x :: "'a"   and y :: "'b" assumes "m = OptionT x"   and "m' = OptionT y"   and "M x y" shows "thesis"
left_total_alt_def: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "left_total R = ((=) \<le> R OO R\<inverse>\<inverse>)"
pairwise_compat: fixes c1 :: "('a, 'b) chart"   and c2 :: "('a, 'b) chart" assumes "c1 \<in> charts"   and "c2 \<in> charts" shows "l-smooth_compat c1 c2"
subcomplex_image: shows "order.greater_eq X (f \<turnstile> X) \<and> SimplicialComplex (f \<turnstile> X)"
CSP1_is_R1_H1_b: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "(P = (R \<circ> R1 \<circ> H1 \<circ> H2) P) = (P = (R \<circ> CSP1 \<circ> H2) P)"
filter_mset_def: shows "filter_mset \<equiv> map_fun id (map_fun count Abs_multiset) (\<lambda>P M x. if P x then M x else 0)"
oneD_gauge_integral_lesbegue_integrable: shows "integrable lborel (\<lambda>x. integral UNIV (\<lambda>y. f (x, y)))"
Nil_is_append_conv: fixes xs :: "'f list"   and ys :: "'f list" shows "([] = xs @ ys) = (xs = [] \<and> ys = [])"
nn_integral_monotone_convergence_INF_AE': fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "\<And>i. AE x in M. f (Suc i) x \<le> f i x"   and "\<And>i. f i \<in> borel_measurable M"   and "integral\<^sup>N M (f 0) < \<infinity>" shows "\<integral>\<^sup>+ x. (INF i. f i x) \<partial>M = (INF i. integral\<^sup>N M (f i))"
intros_2: fixes R :: "('b \<times> 'b) set"   and xs :: "'b list"   and x :: "'b" assumes "Linorder_Relations.sorted_wrt R xs"   and "\<And>y. y \<in> set xs \<Longrightarrow> (x, y) \<in> R" shows "Linorder_Relations.sorted_wrt R (x # xs)"
rec_sign_def: shows "rec_sign = CN rec_minus [constn 1, CN rec_minus [constn 1, recf.Id 1 0]]"
closest_point_lipschitz: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "convex S"   and "closed S"   and "S \<noteq> {}" shows "dist (closest_point S x) (closest_point S y) \<le> dist x y"
reflexion_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and P :: "'b eclause"   and C :: "'b eclause"   and \<sigma> :: "('b \<times> 'b trm) list"   and k :: "inferences"   and C' :: "'b literal set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.reflexion trm_ord sel filter_trms P C \<sigma> k C' = (\<exists>L1 t s Cl_P Cl_C trms_C. basic_superposition.eligible_literal trm_ord sel L1 P \<sigma> \<and> L1 \<in> cl_ecl P \<and> Cl_C = cl_ecl C \<and> Cl_P = cl_ecl P \<and> basic_superposition.orient_lit_inst trm_ord L1 t s neg \<sigma> \<and> ck_unifier t s \<sigma> k \<and> C = Ecl Cl_C trms_C \<and> trms_C = basic_superposition.get_trms filter_trms Cl_C (basic_superposition.dom_trms trm_ord Cl_C (subst_set (trms_ecl P \<union> {t}) \<sigma>)) k \<and> Cl_C = subst_cl C' \<sigma> \<and> C' = Cl_P - {L1})"
simps_2: fixes y :: "'a word" shows "proper_interval None (Some y) = (y \<noteq> 0)"
hyp2_axiom1: shows "\<forall>a b. a b \<congruent>\<^sub>K b a"
vertex_bipartite_web_1: fixes x :: "'v" shows "vertex (bipartite_web_of \<Gamma>) (Inl x) = (vertex \<Gamma> x \<and> x \<notin> B \<Gamma>)"
reflect_poly_reflect_poly: fixes p :: "'a poly" assumes "coeff p 0 \<noteq> (0::'a)" shows "reflect_poly (reflect_poly p) = p"
SeqGuard: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and \<Theta> :: "'a \<Rightarrow> ('b, 'a, 'c) ann list option"   and F :: "'c set"   and P :: "'b set"   and g :: "'b set"   and a :: "('b, 'a, 'c) ann"   and c :: "('b, 'a, 'c) com"   and Q :: "'b set"   and A :: "'b set"   and f :: "'c"   and r :: "'b set" assumes "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> (P \<inter> g) a c Q, A"   and "P \<inter> - g \<noteq> {} \<Longrightarrow> f \<in> F" shows "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P (AnnRec r a) (Guard f g c) Q, A"
id_rsp: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "(R ===> R) id id"
conv': shows "\<forall>x\<in>s. (\<lambda>k. f (real k) x) \<longlonglongrightarrow> g x"
gSwapPresGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gSwapPresGWls MOD \<equiv> \<forall>zs z1 z2 s X' X. wls s X' \<and> gWls MOD s X \<longrightarrow> gWls MOD s (gSwap MOD zs z1 z2 X' X)"
new_character_data_is_character_data_ptr: fixes h :: "(_) heap"   and new_character_data_ptr :: "(_) character_data_ptr" assumes "h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr" shows "is_character_data_ptr\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr"
1: fixes g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "g \<in> G" shows "\<exists>q. g = (\<Sum>f\<in>set fs. q f * f) \<and> (\<forall>f. q f \<in> P[X] \<and> poly_deg (q f * f) \<le> b)"
"2_2": shows "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) (x # xs)"
Q_def: shows "Q \<equiv> \<lambda>ys. \<forall>y\<in>set ys. \<exists>x\<in>set xs. P x y"