hcju/isabelleps huggingface dataset:
P1_security: fixes input\<^sub>1 :: "bool \<times> bool"   and \<sigma> :: "bool"   and x :: "'a \<times> 'a"   and y :: "bool" shows "R1 input\<^sub>1 \<sigma> = funct_OT_12 x y \<bind> (\<lambda>(s1, s2). S1 input\<^sub>1 s1)"
absorb1: fixes a :: "'b"   and b :: "'b" assumes "order.greater_eq b a" shows "inf a b = a"
sat_slice_iff: fixes v :: "'a list"   and S :: "'a list set"   and \<sigma> :: "(char list \<times> 'a list) trace"   and i :: "nat"   and \<phi> :: "'a MFOTL.formula" assumes "v \<in> S" shows "MFOTL.sat \<sigma> v i \<phi> = MFOTL.sat (map_\<Gamma> (\<lambda>D. D \<inter> relevant_events \<phi> S) \<sigma>) v i \<phi>"
unrestrict_effects: fixes v :: "bool"   and a :: "'b"   and S :: "'b literal set set" shows "(\<lambda>c. if {make_lit (\<not> v) a} \<union> c \<in> S then {make_lit (\<not> v) a} \<union> c else c) ` restrict_cnf_atom a v S \<subseteq> S"
assms: shows "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L"
comp_MkArr: fixes A :: "'a \<Rightarrow> 'b"   and B :: "'a \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and C :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b" assumes "arr (MkArr A B F)"   and "arr (MkArr B C G)" shows "MkArr B C G \<cdot> MkArr A B F = MkArr A C (vertical_composite.map (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G)"
assm: fixes s :: "'a formula set"   and n :: "nat" assumes "s \<subseteq> extender n S"   and "finite s" shows "sat s"
dense_ge_bounded: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and z :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.dense_linorder less_eq less"   and "less z x"   and "\<And>w. \<lbrakk>less z w; less w x\<rbrakk> \<Longrightarrow> less_eq y w" shows "less_eq y z"
ltl\<^sub>P_abs_rep: fixes \<phi> :: "'c ltl_prop_equiv_quotient" shows "Abs (Rep \<phi>) = \<phi>"
allocationInjectionsUnivervseProperty: fixes G :: "'d set"   and N :: "'c set" shows "allAllocations N G = injectionsUniverse \<inter> {a. Domain a \<subseteq> N \<and> Range a \<in> all_partitions G}"
gWlsInp_def: fixes MOD :: "('i, 'j, 'k, 'sort, 'l, 'm, 'n, 'o, 'p) Recursion.model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'n option" shows "gWlsInp MOD delta inp \<equiv> wlsOpS delta \<and> sameDom (arOf delta) inp \<and> liftAll2 (gWls MOD) (arOf delta) inp"
restricted_ideal_to_mono: fixes \<V> :: "nat set"   and I :: "int mpoly multiset"   and \<V>' :: "nat set"   and J :: "int mpoly multiset"   and \<U> :: "nat set" assumes "restricted_ideal_to\<^sub>I \<V> I \<subseteq> restricted_ideal_to\<^sub>I \<V>' J"   and "\<U> \<subseteq> \<V>" shows "restricted_ideal_to\<^sub>I \<U> I \<subseteq> restricted_ideal_to\<^sub>I \<U> J"
eval_lpoly_eq_dot_prod_split1: fixes x :: "rat vec"   and y :: "rat vec"   and c :: "rat vec"   and b :: "'a vec"   and X :: "(nat, rat) mapping" assumes "(x, y) = split_n_m_x (dim_vec c) (dim_vec b) X" shows "vec_to_lpoly c \<lbrace> \<langle>X\<rangle> \<rbrace> = c \<bullet> x"
h: shows "|~ \<langle>A\<rangle>_v = \<langle>B\<rangle>_w"
\<iota>_is_terminal_arr: shows "terminal_arr \<iota>"
b'_def: shows "b' \<equiv> \<Sum>j\<in>Basis. (if j = i then c else b \<bullet> j) *\<^sub>R j"
18: shows "e\<^sup>T \<le> top * e * - F"
finprod_0: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "f \<in> {0} \<rightarrow> carrier (UP INTEG)" shows "finsum (UP INTEG) f {..0} = f 0"
chain_prefix_limit: fixes w :: "nat \<Rightarrow> 'a list"   and k :: "nat" assumes "chain w" shows "w k \<le>\<^sub>F\<^sub>I limit w"
existence_ivl_initial_time_iff: fixes x0 :: "'a" shows "(0 \<in> rev.existence_ivl0 x0) = (0 \<in> UNIV \<and> x0 \<in> X)"
set1_FGl_def: fixes x :: "(('p, 'q, 'r, 's, 't, 'u, 'v) G, 'p, 'w, 'r, 'x, 'y, 't, 'z, 'aa, 'ab) F" shows "set1_FGl x = \<Union> (set1_G ` set1_F x) \<union> set2_F x"
cinner_simps_5: fixes r :: "complex"   and x :: "'b"   and y :: "'b" shows "r *\<^sub>C x \<bullet>\<^sub>C y = cnj r * (x \<bullet>\<^sub>C y)"
Gets_B_knows_K: fixes B :: "agent"   and Tk :: "nat"   and A :: "agent"   and K :: "nat"   and Ta :: "nat"   and evs :: "event list" assumes "Gets B \<lbrace>Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace>, Crypt K \<lbrace>Agent A, Number Ta\<rbrace>\<rbrace> \<in> set evs"   and "evs \<in> bankerb_gets" shows "Key K \<in> analz (knows B evs)"
return_returns_result: fixes h :: "'a"   and x :: "'c"   and y :: "'c" shows "h \<turnstile> return x \<rightarrow>\<^sub>r y = (x = y)"
simps_7: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'c" shows "cIntensityF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cIntensityF"
set_intros_5: fixes yh :: "('d, 'e) Term.term"   and x22 :: "('d, 'e) Term.term list"   and yj :: "'e"   and x21 :: "'d"   and x23 :: "('d, 'e) ctxt"   and x24 :: "('d, 'e) Term.term list" assumes "yh \<in> set x22"   and "yj \<in> vars_term yh" shows "yj \<in> vars_ctxt (More x21 x22 x23 x24)"
content_gcd_primitive_1: fixes f :: "'b poly"   and g :: "'b poly" assumes "f \<noteq> 0" shows "content (gcd (primitive_part f) g) = (1::'b)"
A6: shows "\<forall>SQ TR. SQ \<in>S Q \<and> TR \<in>T R \<longrightarrow> False"
set_q_a: fixes n :: "nat" shows "set (map ((\<otimes>) a) q @ replicate n \<zero>) \<subseteq> carrier R"
iwf: fixes n :: "('form, 'rule) graph_node"   and p :: "'form"   and fc :: "('form, 'rule) graph_node \<Rightarrow> nat \<Rightarrow> 'subst \<Rightarrow> 'form fset \<times> 'form \<Rightarrow> bool"   and s :: "'subst"   and i :: "nat"   and \<Gamma> :: "'form fset"   and ants :: "('form, 'rule, 'subst, 'var) itnode rose_tree list"   and c :: "'form" assumes "n \<in> sset nodes"   and "Reg p |\<in>| outPorts n"   and "list_all2 (\<lambda>ip t. iwf fc t ((\<lambda>h. subst s (freshen i (labelsOut n h))) |`| hyps_for n ip |\<union>| \<Gamma> \<turnstile> subst s (freshen i (labelsIn n ip)))) (inPorts' n) ants"   and "fc n i s (\<Gamma> \<turnstile> c)"   and "c = subst s (freshen i p)" shows "iwf fc (INode n p i s ants) (\<Gamma> \<turnstile> c)"
complements_overlap: fixes x :: "i"   and y :: "i" assumes "CEMC"   and "x \<oplus> y \<noteq> u" shows "O (\<midarrow> x) (\<midarrow> y)"
continuousD: fixes F :: "ordinal \<Rightarrow> ordinal"   and f :: "nat \<Rightarrow> ordinal" assumes "continuous F" shows "F (oLimit f) = oLimit (\<lambda>n. F (f n))"
SUP_cong_simp: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b \<Rightarrow> 'c"   and D :: "'b \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c" assumes "A = B"   and "\<And>x. x \<in> B =simp=> C x = D x" shows "Sup (C ` A) = Sup (D ` B)"
redT_preserves_lock_thread_ok: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s -t\<triangleright>ta\<rightarrow> s'"   and "lock_thread_ok (locks s) (thr s)" shows "lock_thread_ok (locks s') (thr s')"
\<chi>_def: shows "\<chi> = principal_char (G\<lparr>carrier := {\<one>}\<rparr>)"
shortest_path_pos_cost_pred_def: fixes G :: "('a, 'b) pre_digraph"   and dist :: "'a \<Rightarrow> ereal"   and c :: "'b \<Rightarrow> real"   and s :: "'a"   and num :: "'a \<Rightarrow> enat"   and pred :: "'a \<Rightarrow> 'b option" shows "shortest_path_pos_cost_pred G dist c s num pred \<equiv> basic_just_sp_pred G dist c s num pred \<and> shortest_path_pos_cost_pred_axioms G dist c s num"
2: shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H F \<^bold>\<rightarrow> H"
funion_iff: fixes c :: "'b"   and A :: "'b fset"   and B :: "'b fset" shows "(c |\<in>| A |\<union>| B) = (c |\<in>| A \<or> c |\<in>| B)"
subset_singletonD: fixes A :: "'c set"   and x :: "'c" assumes "A \<subseteq> {x}" shows "A = {} \<or> A = {x}"
strict_boundedE: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subset># b \<inter># c" obtains "a \<subset># b" and "a \<subset># c"
RHS1: shows "Cring_Poly.compose R (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) q) r = Cring_Poly.compose R (to_polynomial R a \<otimes>\<^bsub>P\<^esub> q [^]\<^bsub>P\<^esub> n) r"
LPDs_safe_fv_regex: fixes mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and ms :: "mregex" assumes "safe_regex Futu Strict (from_mregex mr \<phi>s)"   and "ms \<in> LPDs mr" shows "fv_regex (from_mregex ms \<phi>s) = fv_regex (from_mregex mr \<phi>s)"
dioph_succs_def: fixes n :: "nat"   and ks :: "int list"   and m :: "int" shows "dioph_succs n ks m = List.map_filter (\<lambda>xs. if eval_dioph ks xs mod 2 = m mod 2 then Some ((m - eval_dioph ks xs) div 2) else None) (mk_nat_vecs n)"
lfinite_LCons: fixes x :: "'b"   and xs :: "'b llist" shows "lfinite (LCons x xs) = lfinite xs"
SubstForm_quot: fixes e :: "(name, hf) finfun"   and i :: "name"   and t :: "tm"   and A :: "fm" shows "SubstForm \<lbrakk>\<guillemotleft>Var i\<guillemotright>\<rbrakk>e \<lbrakk>\<guillemotleft>t\<guillemotright>\<rbrakk>e \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e \<lbrakk>\<guillemotleft>A(i::=t)\<guillemotright>\<rbrakk>e"
Units_l_inv_ex: fixes x :: "nat" assumes "x \<in> Units R" shows "\<exists>y\<in>carrier R. y \<otimes>\<^bsub>R\<^esub> x = \<one>\<^bsub>R\<^esub>"
A12_DAcc_level1: shows "DAcc level1 sA12 = {}"
simps_3: fixes x :: "'a"   and y :: "'a" shows "Num_arc_delta x * Num_arc_delta y = Num_arc_delta (x + y)"
split_cond: fixes A :: "bool \<Rightarrow> bool"   and P :: "'b \<times> 'b \<Rightarrow> bool"   and b :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool"   and x :: "'b \<times> 'b" shows "A ((P \<triangleleft> b \<triangleright>  Q) x) = ((b x \<longrightarrow> A (P x)) \<and> (\<not> b x \<longrightarrow> A (Q x)))"
assms: shows "common_boundary_sudivision_exists one_chain1 one_chain2"
union_of_def: fixes P :: "'c set set \<Rightarrow> bool"   and Q :: "'c set \<Rightarrow> bool" shows "P union_of Q \<equiv> \<lambda>S. \<exists>\<U>. P \<U> \<and> \<U> \<subseteq> Collect Q \<and> \<Union> \<U> = S"
weakly_Pareto_optimal_for_men_stable: fixes \<mu> :: "('m \<times> 'w) set" assumes "weakly_Pareto_optimal_for_men \<mu>" shows "stable \<mu>"
wf_mstate_msteps_stateless: fixes \<pi> :: "(char list \<times> 'a list) prefix"   and R :: "'a list set"   and st :: "'a mstate"   and v :: "'a option list"   and \<pi>' :: "(char list \<times> 'a list) prefix"   and i :: "nat" assumes "wf_mstate \<phi> \<pi> R st"   and "mem_restr R v"   and "\<pi> \<le> \<pi>'" shows "((i, v) \<in> msteps_stateless (pdrop (plen \<pi>) \<pi>') st) = ((i, v) \<in> M \<pi>' - M \<pi>)"
ex_crossing_at_alt: fixes Q :: "'a set"   and a :: "'a" assumes "Q \<in> \<P>"   and "a \<in> Q" shows "\<exists>ac c. path ac a c \<and> ac \<noteq> Q \<and> c \<notin> Q"
filterlim_sup: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'c filter"   and F1 :: "'b filter"   and F2 :: "'b filter" assumes "filterlim f F F1"   and "filterlim f F F2" shows "filterlim f F (sup F1 F2)"
ctendsto_componentwise1: fixes b :: "'b \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'c"   and a :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'c"   and F :: "'b filter" assumes "\<And>j. j \<in> CBasis \<Longrightarrow> ((\<lambda>n. cblinfun_apply (b n) j) \<longlongrightarrow> cblinfun_apply a j) F" shows "(b \<longlongrightarrow> a) F"
assms_1: shows "(t, s) \<in> timpl_closure' c"
isometry_def: fixes U :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "isometry U = (U* o\<^sub>C\<^sub>L U = id_cblinfun)"
ACI_set_eq: fixes r :: "'d rexp"   and s :: "'d rexp" assumes "r ~ s" shows "set_rexp r = set_rexp s"
analz_monotone: fixes c :: "msg"   and G :: "msg set"   and H :: "msg set" assumes "c \<in> analz G"   and "G \<subseteq> H" shows "c \<in> analz H"
plossless_restrict_converter: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and conv :: "('i, 'j, 'k, 'l) converter"   and A :: "'i set" assumes "plossless_converter \<I> \<I>' conv"   and "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "outs_\<I> \<I> \<subseteq> A" shows "plossless_converter \<I> \<I>' (restrict_converter A \<I>' conv)"
takeWhile_eq_Nil_iff: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "(takeWhile P xs = []) = (xs = [] \<or> \<not> P (hd xs))"
typ_ok_code: fixes \<Theta> :: "exetheory"   and ty :: "typ" assumes "exe_wf_theory' \<Theta>" shows "typ_ok (translate_theory \<Theta>) ty = exetyp_ok \<Theta> ty"
space_pair_measure: fixes A :: "'d measure"   and B :: "'e measure" shows "space (A \<Otimes>\<^sub>M B) = space A \<times> space B"
homeomorphic_frontiers_same_dimension: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T"   and "closed S"   and "closed T"   and "DIM('a) = DIM('b)" shows "frontier S homeomorphic frontier T"
dagger_unfoldl: fixes x :: "'b \<times> 'c" shows "less_eq_prod (sup_prod one_prod (times_prod' x (prod.qstar x))) (prod.qstar x)"
of_int_diff: fixes w :: "int"   and z :: "int" shows "of_int (w - z) = of_int w - of_int z"
contains_propagates_patoms: fixes n :: "nat"   and p :: "nat"   and vs :: "nat list"   and q :: "nat" assumes "infinite (deriv s)"   and "contains f n (0, PAtom p vs)" shows "contains f (n + q) (0, PAtom p vs)"
rbt_product_def: fixes f :: "'c \<Rightarrow> 'f \<Rightarrow> 'd \<Rightarrow> 'g \<Rightarrow> 'e"   and rbt1 :: "('c, 'f) rbt"   and rbt2 :: "('d, 'g) rbt" shows "rbt_product f rbt1 rbt2 = rbtreeify (alist_product f (RBT_Impl.entries rbt1) (RBT_Impl.entries rbt2))"
cross3_def: fixes a :: "(real, 3) vec"   and b :: "(real, 3) vec" shows "a \<times> b \<equiv> vector [a $ 2 * b $ 3 - a $ 3 * b $ 2, a $ 3 * b $ 1 - a $ 1 * b $ 3, a $ 1 * b $ 2 - a $ 2 * b $ 1]"
not_in_affected_iff_orbit_eq_singleton: fixes a :: "'b"   and f :: "'b perm" shows "(a \<notin> affected f) = (orbit f a = {a})"
mult1_def_lemma: fixes M :: "'b multiset"   and M0 :: "'b multiset"   and a :: "'b"   and N :: "'b multiset"   and K :: "'b multiset"   and r :: "('b \<times> 'b) set" assumes "M = M0 + {#a#} \<and> N = M0 + K \<and> (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r)" shows "(N, M) \<in> mult1 r"
lcm_0_left: fixes a :: "'a" shows "lcm (0::'a) a = (0::'a)"
arr_dom_iff_arr: fixes f :: "'a" shows "Right_a.arr (Right_a.dom f) = Right_a.arr f"
FINITE_ALL_DISTINCT_LISTS_i: fixes P :: "'a set" assumes "finite P" shows "{p. distinct p \<and> set p \<subseteq> P} = {[]} \<union> (\<Union>e\<in>P. {e # p0 |p0. distinct p0 \<and> set p0 \<subseteq> P - {e}})"
id: shows "lhs (eq_for_lvar (\<T> s') x\<^sub>i) = x\<^sub>i"
start_state_def: fixes f :: "String.literal \<Rightarrow> String.literal \<Rightarrow> ty list \<Rightarrow> ty \<Rightarrow> 'd \<Rightarrow> 'addr val list \<Rightarrow> 'c"   and P :: "'d prog"   and C :: "String.literal"   and M :: "String.literal"   and vs :: "'addr val list" shows "start_state f P C M vs \<equiv> let (D, Ts, T, m) = method P C M in (K$ None, ([start_tid \<mapsto> (f D M Ts T (the m) vs, no_wait_locks)], start_heap), Map.empty, {})"
set_eqI: fixes A :: "'d set"   and R :: "('d, 'e) eq_object_scheme"   and B :: "'d set" assumes "\<And>a. a \<in> A \<Longrightarrow> a .\<in>\<^bsub>R\<^esub> B"   and "\<And>b. b \<in> B \<Longrightarrow> b .\<in>\<^bsub>R\<^esub> A" shows "A {.=}\<^bsub>R\<^esub> B"
vars_mset_def: fixes M :: "(('d, 'c) Term.term \<times> ('e, 'c) Term.term) multiset" shows "vars_mset M = \<Union> (set_mset {#vars_term (fst r) \<union> vars_term (snd r). r \<in># M#})"
Nil_right_tensor: fixes xs :: "wall" shows "basic [] \<otimes> xs = xs"
equivalence_cancel_right: fixes e :: "'d"   and \<mu> :: "'d"   and \<mu>' :: "'d" assumes "D.equivalence_map e"   and "D.par \<mu> \<mu>'"   and "src\<^sub>D \<mu> = trg\<^sub>D e"   and "\<mu> \<star>\<^sub>D e = \<mu>' \<star>\<^sub>D e" shows "\<mu> = \<mu>'"
exhaust_sel: fixes exesignature :: "exesignature" obtains "exesignature = ExeSignature (execonst_type_of exesignature) (exetyp_arity_of exesignature) (exesorts exesignature)"
co_c: shows "connected (path_image c - {pathstart c, pathfinish c})"
D: shows "U \<in> key_sets (Pwd n) (crypts (Log -` spied s'))"
ex_min_if_finite: fixes S :: "'c set" assumes "finite S"   and "S \<noteq> {}" shows "\<exists>m\<in>S. \<not> (\<exists>x\<in>S. x < m)"
decel: fixes a :: "real"   and v :: "real" assumes "braking_movement a v" shows "a < 0"
lhd_ldropn: fixes n :: "nat"   and xs :: "'a llist" assumes "enat n < llength xs" shows "lhd (ldropn n xs) = lnth xs n"
construct_witness_correct: fixes \<delta> :: "('d, 'e) ta_rule set"   and Q :: "'d \<Rightarrow> 'e tree option"   and r :: "('d, 'e) ta_rule" assumes "witness_prop \<delta> Q"   and "r \<in> \<delta>"   and "set (rhsq r) \<subseteq> dom Q" shows "accs \<delta> (construct_witness Q r) (lhs r)"
NotR: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "F, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> \<^bold>\<not> F, \<Delta> \<down> Suc n"
assms_2: shows "a \<in>\<^sub>\<circ> Vset \<alpha>"
hom_uminus: fixes x :: "int" shows "of_int (- x) = - of_int x"
integral_pmf_of_set: fixes S :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "S \<noteq> {}"   and "finite S" shows "measure_pmf.expectation (pmf_of_set S) f = sum f S / real (card S)"
correctCompositionVAR_def: fixes S :: "CSet" shows "correctCompositionVAR S \<equiv> \<forall>C\<in>subcomp S. \<forall>v\<in>VAR C. v \<in> VAR S"
group_BijGroup: fixes S :: "'c set" shows "Group.group (BijGroup S)"
renaming_preserves_literal_order: fixes L1 :: "'at Literal"   and L2 :: "'at Literal"   and A :: "'at set" assumes "literal_ordering L1 L2" shows "literal_ordering (rename_literal A L1) (rename_literal A L2)"
finite_wfrestrictedvars\<^sub>s\<^sub>t\<^sub>p: fixes x :: "('d, 'c) strand_step" shows "finite (wfrestrictedvars\<^sub>s\<^sub>t\<^sub>p x)"
FBinds: fixes x :: "'a \<Rightarrow> Vars.var" shows "(\<lambda>xa. (FBinds (\<Delta> @ \<Gamma>)\<cdot>(ae \<squnion> Aheap \<Delta> e\<cdot>a)) (x xa)) = (\<lambda>xa. (FBinds \<Gamma>\<cdot>ae) (x xa) \<otimes>\<otimes> (FBinds \<Delta>\<cdot>(Aheap \<Delta> e\<cdot>a)) (x xa))"
is_singletonI: fixes x :: "'e" shows "is_singleton {x}"
G: fixes g :: "'t \<Rightarrow>\<^sub>0 'b" assumes "g \<in> G" shows "g \<in> pmdl F \<and> g \<in> dgrad_p_set d m \<and> g \<noteq> 0"
mirror_unfold: fixes x :: "'a"   and l :: "'a tree"   and r :: "'a tree" shows "mirror (Node x l r) = Node x (mirror r) (mirror l)"
convexD_alt: fixes s :: "'c set"   and a :: "'c"   and b :: "'c"   and u :: "real" assumes "convex s"   and "a \<in> s"   and "b \<in> s"   and "0 \<le> u"   and "u \<le> 1" shows "(1 - u) *\<^sub>R a + u *\<^sub>R b \<in> s"
is_prefix_eq_proper_prefix: fixes a :: "'b list"   and b :: "'b list" shows "is_prefix a b = (a = b \<or> is_proper_prefix a b)"
condition1_part1: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and a :: "'f"   and b :: "'f" assumes "echelon_form_upt_k A k"   and "is_zero_row_upt_k a (Suc k) A"   and "a < b"   and "\<forall>m\<ge>mod_type_class.from_nat (if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1). A $ m $ mod_type_class.from_nat k = (0::'d)" shows "is_zero_row_upt_k b (Suc k) A"
UN_iff: fixes b :: "'k"   and B :: "'l \<Rightarrow> 'k set"   and A :: "'l set" shows "(b \<in> \<Union> (B ` A)) = (\<exists>x\<in>A. b \<in> B x)"
combine_t_wise_balance: fixes \<V> :: "'b set"   and \<B> :: "'b set multiset"   and \<t> :: "int"   and \<Lambda>\<^sub>t :: "int"   and \<B>' :: "'b set multiset"   and \<Lambda>\<^sub>t' :: "int" assumes "two_t_wise_eq_points \<V> \<B> \<t> \<Lambda>\<^sub>t \<B>' \<Lambda>\<^sub>t'" shows "t_wise_balance (\<V> \<union> \<V>) (\<B> + \<B>') \<t> (\<Lambda>\<^sub>t + \<Lambda>\<^sub>t')"
Gcd_fin_eq_Gcd: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "finite A" shows "semiring_gcd.Gcd_fin one zero gcd A = Gcd A"
ideal_inc_one: fixes I :: "'a set" assumes "ideal R I"   and "1\<^sub>r \<in> I" shows "I = carrier R"
take_update_swap: fixes m :: "nat"   and xs :: "'a list"   and n :: "nat"   and x :: "'a" shows "take m (xs[n := x]) = (take m xs)[n := x]"
map_cong0: fixes x :: "'e \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> 'f"   and g :: "'d \<Rightarrow> 'f" assumes "\<And>z. z \<in> range x \<Longrightarrow> f z = g z" shows "f \<circ> x = g \<circ> x"
assms: shows "FB \<equiv> \<lambda>Bb Cc. four_block_mat Bb (0\<^sub>m (dim_row Bb) (dim_col Cc)) (0\<^sub>m (dim_row Cc) (dim_col Bb)) Cc"
cases: fixes x :: "'a extC" obtains "x = extC.Bot"   | uu_ :: "'a" where "x = extC.Val uu_"   | "x = extC.Top"
assms_3: shows "R2 \<in> rcosets stabilizer G \<phi> x"
c': shows "conv_abscissa f < ereal c' \<and> c' < c - 1"
less_eq_language_def: fixes r :: "'a language"   and s :: "'a language" shows "(r \<le> s) = (Plus r s = s)"
cinner_sup_norm: fixes \<psi> :: "'b" shows "norm \<psi> = (SUP \<phi>. cmod \<langle>\<phi>, \<psi>\<rangle> / norm \<phi>)"
bis_imp: fixes c1 :: "('test, 'atom, 'choice) cmd"   and c2 :: "('test, 'atom, 'choice) cmd" assumes "c1 \<approx>s c2" shows "c1 \<approx>01 c2"
wf_NFOR: fixes \<Phi> :: "('a, 'k) aformula set"   and idx :: "'n" assumes "finite \<Phi>" shows "local.wf idx (NFOR \<Phi>) = (\<forall>\<phi>\<in>\<Phi>. local.wf idx \<phi>)"
in_gamma_update: fixes s :: "char list \<Rightarrow> int"   and S :: "char list \<Rightarrow> 'av"   and i :: "int"   and a :: "'av"   and x :: "char list" assumes "s \<in> \<gamma>\<^sub>s S"   and "i \<in> \<gamma> a" shows "s(x := i) \<in> \<gamma>\<^sub>s (S(x := a))"
OAlist_tc_sorted_domain_def: fixes xs :: "('b, 'c) oalist_tc" shows "OAlist_tc_sorted_domain xs = map fst (list_of_oalist_tc xs)"
BasicFF: fixes G :: "('a, 'b) form list" shows "\<stileturn> \<bottom> # G"
frontier_Int: fixes S :: "'b set"   and T :: "'b set" shows "frontier (S \<inter> T) = closure (S \<inter> T) \<inter> (frontier S \<union> frontier T)"
that: fixes P :: "nat"   and Q :: "nat" assumes "N = P * Q"   and "P \<noteq> Q"   and "prime P"   and "prime Q"   and "coprime e ((P - 1) * (Q - 1))"   and "d = nat (fst (bezw e ((P - 1) * (Q - 1))) mod int ((P - 1) * (Q - 1)))" shows "thesis"
dimension_def: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and Basis :: "'d set" assumes "finite_dimensional_vector_space scale Basis" shows "finite_dimensional_vector_space.dimension Basis = card Basis"
intros_22: fixes i :: "nat"   and \<C> :: "t_context"   and tf :: "tf" assumes "i < length (func_t \<C>)"   and "func_t \<C> ! i = tf" shows "\<C> \<turnstile> [Call i] : tf"
notinf_fin: fixes x :: "'b llist" shows "(x \<notin> UNIV\<^sup>\<omega>) = (x \<in> UNIV\<^sup>\<star>)"
eqvtI: fixes P :: "pi"   and Q :: "pi"   and perm :: "(name \<times> name) list" assumes "P \<sim>\<^sup>s Q" shows "perm \<bullet> P \<sim>\<^sup>s perm \<bullet> Q"
consequence_graph_def: fixes Rs :: "(('b, 'c) labeled_graph \<times> ('b, 'c) labeled_graph) set"   and G :: "('b, 'd) labeled_graph" shows "consequence_graph Rs G \<equiv> graph G \<and> (\<forall>R\<in>Rs. subgraph (fst R) (snd R) \<and> maintained R G)"
cltn2_left_inverse: fixes A :: "cltn2" shows "cltn2_compose (cltn2_inverse A) A = cltn2_id"