hcju/isabelleps huggingface dataset:
intro: fixes \<Gamma> :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set" assumes "\<And>\<gamma> N. \<lbrakk>\<gamma> \<in> \<Gamma>; concl_of \<gamma> \<in> N \<union> Rf N\<rbrakk> \<Longrightarrow> \<gamma> \<in> Ri N" shows "effective_redundancy_criterion_axioms \<Gamma> Rf Ri"
Compl_Diff_eq: fixes A :: "'c set"   and B :: "'c set" shows "- (A - B) = - A \<union> B"
sup_ge1: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y \<le> x"
mHom_ker_eq: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme"   and f :: "'e \<Rightarrow> 'i"   and a :: "'e"   and b :: "'e" assumes "R module M"   and "R module N"   and "f \<in> mHom R M N"   and "a \<in> carrier M"   and "b \<in> carrier M"   and "f a = f b" shows "a \<plusminus>\<^bsub>M\<^esub> -\<^sub>a\<^bsub>M\<^esub> b \<in> ker\<^bsub>M,N\<^esub> f"
push_bit_take_bit: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "push_bit m (take_bit n a) = take_bit (m + n) (push_bit m a)"
keys_to_list_def: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "pp_pm.punit.keys_to_list p = pp_pm.punit.pps_to_list (keys p)"
pupdate_def: shows "Expr.pupdate = snd \<circ> action"
low_eq_def: fixes mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" shows "mem\<^sub>1 =\<^sup>l mem\<^sub>2 \<equiv> \<forall>x. dma mem\<^sub>1 x = Low \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"
simps: fixes x :: "'a"   and y :: "'b" shows "rho x y = (SOME z. up_filter z = phi x \<sqinter> up_filter y)"
congruent_triangle_sas_20: fixes a1 :: "'b"   and b1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c1 :: "'b"   and c2 :: "'c" assumes "dist a1 b1 = dist a2 b2"   and "dist b1 c1 = dist b2 c2"   and "angle a1 b1 c1 = angle a2 b2 c2" shows "angle a1 b1 c1 = angle c2 b2 a2"
fundchamber_S_fundadjset: fixes s :: "'a permutation" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "s `\<rightarrow> C0 \<in> adjacentset C0 - {C0}"
well_def_signs_def: fixes num_polys :: "nat"   and sign_conds :: "rat list list" shows "well_def_signs num_polys sign_conds \<equiv> \<forall>signs\<in>set sign_conds. length signs = num_polys"
valid'_valid: fixes l :: "color list" assumes "valid' l" shows "valid l"
simps_6: fixes x :: "'e" shows "phantom_optionT (OptionT x) = {}"
odd_card_imp_not_empty: fixes A :: "'c set" assumes "odd (card A)" shows "A \<noteq> {}"
convert_list: fixes var :: "nat"   and L :: "atom list"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and L' :: "atomUni list"   and l :: "atom" assumes "convert_atom_list var L (xs' @ x # xs) = Some L'"   and "l \<in> set L" shows "\<exists>l'\<in>set L'. convert_atom var l (xs' @ x # xs) = Some l'"
right_plus_below_circ: fixes x :: "'a" shows "top \<sqinter> x \<le> top"
simps_2: fixes x :: "'c"   and l :: "'c list"   and p :: "'c node ref" shows "os_list (x # l) (Some p) = (\<exists>\<^sub>Aq. p \<mapsto>\<^sub>r Node x q * os_list l q)"
set_list_bind: fixes xs :: "'d list"   and f :: "'d \<Rightarrow> 'c list" shows "set (xs \<bind> f) = (\<Union>x\<in>set xs. set (f x))"
select_convs_5: fixes red :: "('d \<times> nat) rgraph"   and black :: "('d, 'e, 'f) lts"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and init_conf :: "('e, 'f) conf"   and confs :: "'d \<times> nat \<Rightarrow> ('e, 'f) conf"   and marked :: "'d \<times> nat \<Rightarrow> bool"   and strengthenings :: "'d \<times> nat \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool"   and more :: "'g" shows "confs \<lparr>red = red, black = black, subs = subs, init_conf = init_conf, confs = confs, marked = marked, strengthenings = strengthenings, \<dots> = more\<rparr> = confs"
assms_1: fixes c' :: "real" assumes "c' \<in> {l<..<c}" shows "\<forall>\<^sub>F x in F. c' * g x \<le> f x"
strategyproofness: shows "(pmf (sds R23) c + pmf (sds R23) a < pmf (sds R12) c + pmf (sds R12) a \<or> pmf (sds R23) c + (pmf (sds R23) a + pmf (sds R23) d) < pmf (sds R12) c + (pmf (sds R12) a + pmf (sds R12) d)) \<or> pmf (sds R23) c + pmf (sds R23) a = pmf (sds R12) c + pmf (sds R12) a \<and> pmf (sds R23) c + (pmf (sds R23) a + pmf (sds R23) d) = pmf (sds R12) c + (pmf (sds R12) a + pmf (sds R12) d)"
obv: shows "(reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))\<^sup>T \<in> carrier_mat (dim_row (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))\<^sup>T) (dim_col (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))\<^sup>T)"
more_temp_simps2_1: fixes F :: "'a \<Rightarrow> bool" shows "\<box>\<box>F \<equiv> \<box>F"
swap_cs_Trans_RecvMarker: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isTrans ev"   and "isRecvMarker ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'" shows "cs e i = cs e' i"
card_sccs_verts_dv: shows "card sccs_verts = Suc (card H.sccs_verts)"
collapse_2: fixes sum :: "'d + 'e" assumes "\<not> isl sum" shows "Inr (projr sum) = sum"
multiplicity_times_unit_left: fixes c :: "'a"   and p :: "'a"   and x :: "'a" assumes "is_unit c" shows "multiplicity (c * p) x = multiplicity p x"
vars_in_instance_in_range_term: fixes A :: "('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "vars_term (subst_atm_abbrev A \<sigma>) \<subseteq> \<Union> (vars_term ` range \<sigma>)"
IH_1: assumes "send\<langle>t\<rangle> \<in> set (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<sigma>)" shows "\<exists>t'. t = t' \<cdot> \<sigma> \<and> send\<langle>t'\<rangle> \<in> set S"
qty_single_def: fixes asset :: "'c"   and qt_proc :: "nat \<Rightarrow> 'd \<Rightarrow> real" shows "qty_single asset qt_proc = qty_empty(asset := qt_proc)"
InvariantNoDecisionsWhenConflictEnsuresCurrentLevelCl: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and clause :: "Literal list" assumes "InvariantNoDecisionsWhenConflict F M (currentLevel M)"   and "clause el F"   and "clauseFalse clause (elements M)"   and "uniq (elements M)"   and "0 < currentLevel M" shows "clause \<noteq> [] \<and> (let Cl = getLastAssertedLiteral (oppositeLiteralList clause) (elements M) in InvariantClCurrentLevel Cl M)"
cl: fixes ya :: "real \<times> real" assumes "ya \<in> set xs" shows "coll 0 ya (sum_list xs)"
Uint16_def: shows "Uint16 \<equiv> map_fun int_of_integer Abs_uint16 word_of_int"
summable_norm_cancel: fixes f :: "nat \<Rightarrow> 'c" assumes "summable (\<lambda>n. norm (f n))" shows "summable f"
front_tickFree_charn: fixes s :: "'b event list" shows "front_tickFree s = (s = [] \<or> (\<exists>a t. s = t @ [a] \<and> tickFree t))"
return_option_def: fixes return :: "'e option \<Rightarrow> 'f"   and x :: "'e" shows "return_option return x = OptionT (return (Some x))"
clsub1: shows "closed_segment (a + complex_of_real kde) (a + complex_of_real e) \<subseteq> closed_segment (a - complex_of_real d) (a + complex_of_real e)"
simps: fixes a :: "nat"   and F :: "nat set" shows "(a \<in> NatMultiples F) = ((\<exists>k. a = k \<and> k \<in> F) \<or> (\<exists>k m. a = k * m \<and> k \<in> F \<and> m \<in> NatMultiples F))"
simps_1: shows "UP.eval [] = (\<lambda>_. \<zero>\<^bsub>K [X]\<^esub>)"
uw\<theta>: shows "ide w \<and> \<guillemotleft>\<theta> : f \<star> w \<Rightarrow> u\<guillemotright>"
bisimClosed: fixes P :: "pi"   and Q :: "pi"   and p :: "(name \<times> name) list" assumes "P \<sim> Q" shows "p \<bullet> P \<sim> p \<bullet> Q"
exhaust: fixes y :: "Def" obtains "y = SimpleDef"   | "y = PhiDef"
assms_3: fixes x :: "'c" shows "f (p x) (q x) = x"
ERETURN_refine: fixes xi :: "'c"   and x :: "'d"   and R :: "('c \<times> 'd) set"   and E :: "('e \<times> 'f) set" assumes "(xi, x) \<in> R" shows "ERETURN xi \<le> \<Down>\<^sub>E E R (ERETURN x)"
redsp_reds_eq: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "redsp P = reds' P"
vintersection_complement: fixes I :: "V"   and J :: "V" shows "vdisjnt (I -\<^sub>\<circ> J) J"
wffp_State_State: fixes sl1 :: "('St, 'U, 'C) state list"   and st1 :: "'St"   and u1 :: "'U"   and c1 :: "'C"   and st2 :: "'St"   and u2 :: "'U"   and c2 :: "'C"   and sl2 :: "('St, 'U, 'C) state list" assumes "wffp (sl1 @ [State st1 u1 c1] @ [State st2 u2 c2] @ sl2)" shows "st2 = do st1 u2 c2"
foundation23: fixes \<tau> :: "'d state \<times> 'd state"   and P :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(\<tau> \<Turnstile> P) = (\<tau> \<Turnstile> \<lambda>_. P \<tau>)"
N_ge_0: fixes n :: "nat"   and fs :: "'a vec list" assumes "gram_schmidt_fs_lin_indpt n fs"   and "0 < length fs" shows "(0::'a) \<le> gram_schmidt_fs_lin_indpt.N fs"
abs_eq: fixes x :: "((real, 'a) vec, 'a) vec" shows "(*\<^sub>V) (to_mtx x) = (*v) x"
map_sel_2: fixes a :: "'a msllist"   and f :: "'a \<Rightarrow> 'b" assumes "a \<noteq> MSLNil" shows "un_MSLCons2 (mslmap f a) = mslmap f (un_MSLCons2 a)"
eq_upto_seq_inconsist_def: fixes P :: "'c prog" shows "eq_upto_seq_inconsist P = foldl_list_all2 (\<lambda>ob ob' vs. mrw_value P vs ob) (\<lambda>ob ob' vs. case ob of NormalAction (ReadMem ad al v) \<Rightarrow> \<exists>b. vs (ad, al) = \<lfloor>(v, b)\<rfloor> | NormalAction _ \<Rightarrow> True | _ \<Rightarrow> True) (\<lambda>ob ob' vs. if case ob of NormalAction (ReadMem ad al v) \<Rightarrow> \<exists>b. vs (ad, al) = \<lfloor>(v, b)\<rfloor> | NormalAction _ \<Rightarrow> True | _ \<Rightarrow> True then ob = ob' else ob \<approx> ob')"
binary_endofunctor_axioms: shows "binary_endofunctor (\<cdot>\<^sub>D) T\<^sub>D"
in_span_delete: fixes a :: "'d"   and S :: "'d set"   and b :: "'d" assumes "a \<in> cspan S"   and "a \<notin> cspan (S - {b})" shows "b \<in> cspan (insert a (S - {b}))"
solution_simplex'_def: shows "solution_simplex' = solution_cs solution_nsc_code from_ns"
hyps: shows "\<Psi> \<rhd> P \<sim> Q"
assms_1: shows "T = get_trms C (dom_trms (subst_cl D \<sigma>) (subst_set E \<sigma>)) Ground"
Characters_simps_3: fixes G :: "('c, 'd) monoid_scheme" shows "\<one>\<^bsub>Characters G\<^esub> = principal_char G"
correct_def: fixes key_gen :: "('a \<times> 'b) spmf"   and commit :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'e) spmf"   and verify :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and valid_msg :: "'c \<Rightarrow> bool" shows "abstract_commitment.correct key_gen commit verify valid_msg \<equiv> \<forall>m. valid_msg m \<longrightarrow> spmf (abstract_commitment.correct_game key_gen commit verify m) True = 1"
measurable_space: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and A :: "'b measure"   and x :: "'a" assumes "f \<in> M \<rightarrow>\<^sub>M A"   and "x \<in> space M" shows "f x \<in> space A"
STCalWB_STCal: shows "Calculus (STCalWB SWB TWB) = STCal Source Target"
simps_1: fixes f :: "'b \<Rightarrow> 'a"   and bs :: "'a list" shows "map_tailrec_rev f [] bs = bs"
uminus_eq_mat: fixes A :: "'c mat"   and B :: "'c mat" shows "(- A = - B) = (A = B)"
subset_of: shows "subset_of A (domain_permutation A X `` {f}) = f ` A"
continuous_on_imp_closedin: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set" assumes "continuous_on S f"   and "closedin (top_of_set (f ` S)) T" shows "closedin (top_of_set S) (S \<inter> f -` T)"
approximating_semantics_iff_fun_good_ruleset: fixes rs :: "'c rule list"   and \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and s :: "state"   and t :: "state" assumes "good_ruleset rs" shows "\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow>\<^sub>\<alpha> t = (approximating_bigstep_fun \<gamma> p rs s = t)"
ket_zero_def: shows "ket_zero = Matrix.vec 2 (\<lambda>k. if k = 0 then 1 else 0)"
rcostOp: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and a :: "'c"   and b :: "'c" assumes "Ring R"   and "ideal R I"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "rcostOp R I (a \<uplus>\<^bsub>R\<^esub> I) (b \<uplus>\<^bsub>R\<^esub> I) = a \<cdot>\<^sub>r\<^bsub>R\<^esub> b \<uplus>\<^bsub>R\<^esub> I"
x_pow_n: fixes n :: "nat" shows "monom (1::'b) 1 ^ n = monom (1::'b) n"
d6: fixes T :: "'a set"   and v :: "'a"   and a :: "'a" assumes "T \<in> \<C>"   and "v \<in> T"   and "a \<in> S" shows "dist a (\<A> T) \<le> 6 * dist a v"
basis_iff_basis_in: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and B :: "'b set" assumes "indep_system carrier indep" shows "indep_system.basis carrier indep B = indep_system.basis_in indep carrier B"
rcp_specD5: fixes rcp :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" assumes "rcp_spec rcp"   and "is_Groebner_basis (fst ` set gs)"   and "unique_idx (gs @ bs) data" shows "fst ` set (fst (rcp gs bs ps sps data)) \<subseteq> pmdl (args_to_set (gs, bs, sps))"
move_stability_res: fixes ts :: "traffic"   and c :: "cars"   and ts' :: "traffic"   and v :: "view" shows "ts \<^bold>\<midarrow>r( c ) \<^bold>\<rightarrow> ts' \<longrightarrow> move ts ts' v = v"
sum_vS_wS: shows "(\<Sum>n\<in>theFTOne cl dl. 1 / real (length cl)) = (\<Sum>n\<in>theFTOne dl cl. 1 / real (length dl))"
le_F_adm: fixes u :: "'a \<Rightarrow> 'b process"   and v :: "'a \<Rightarrow> 'b process" assumes "cont u"   and "monofun v" shows "adm (\<lambda>x. u x \<sqsubseteq>\<^sub>F v x)"
eq_finfun_All_ext: fixes f :: "'a \<Rightarrow>f 'b"   and g :: "'a \<Rightarrow>f 'b" shows "(f = g) = finfun_All ((\<lambda>(x, y). x = y) \<circ>$ ($f, g$))"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_append: fixes F :: "(('d, 'c) Term.term \<times> ('e, 'c) Term.term) list"   and G :: "(('d, 'c) Term.term \<times> ('e, 'c) Term.term) list" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F @ G) = fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<union> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s G"
disjoint_sets_emeasure_Cesaro_tendsto_zero: fixes A :: "nat \<Rightarrow> 'a set"   and P :: "'a measure" assumes "\<And>n. A n \<in> sets M"   and "disjoint_family A"   and "absolutely_continuous M P"   and "sets P = sets M"   and "emeasure P (space M) \<noteq> \<infinity>" shows "(\<lambda>n. (\<Sum>i<n. emeasure P (space M \<inter> (T ^^ i) -` A i)) / of_nat n) \<longlonglongrightarrow> 0"
irreducible_m_dvdm_prod_list_connect: fixes m :: "int"   and a :: "int poly"   and xs :: "int poly list" assumes "poly_mod_prime_type TYPE('d) m"   and "poly_mod.irreducible_m m a"   and "poly_mod.dvdm m a (prod_list xs)" shows "\<exists>b\<in>set xs. poly_mod.dvdm m a b"
result: shows "pEquivalence_Checking.closure as ([(R, S)], []) = Some ([], ps)"
one_local_ring_at_def: shows "one_local_ring_at \<equiv> cq.one_rel"
left_total_rel_gpv: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "left_total A"   and "left_total B" shows "left_total (rel_gpv A B)"
Par2B_2: fixes Q :: "pi"   and u :: "name"   and Q'' :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi"   and P :: "pi" assumes "Q \<Longrightarrow>\<^sub>lu in Q''\<rightarrow>a<x> \<prec> Q'"   and "x \<sharp> P" shows "P \<parallel> Q \<Longrightarrow>\<^sub>lu in P \<parallel> Q''\<rightarrow>a<x> \<prec> P \<parallel> Q'"
T_F_spec: fixes t :: "'c event list"   and P :: "'c process" shows "((t, {}) \<in> F P) = (t \<in> T P)"
star_sum_unfold: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<star> = x\<^sup>\<star> + x\<^sup>\<star> \<cdot> y \<cdot> (x + y)\<^sup>\<star>"
InvariantsClAfterApplyExplain: fixes state :: "State"   and F0 :: "Literal list list" assumes "InvariantUniq (getM state)"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantClCurrentLevel (getCl state) (getM state)"   and "InvariantCEntailed (getConflictFlag state) F0 (getC state)"   and "InvariantCnCharacterization (getCn state) (getC state) (getM state)"   and "InvariantEquivalentZL (getF state) (getM state) F0"   and "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) (set (getQ state))"   and "getCn state \<noteq> 1"   and "getConflictFlag state"   and "0 < currentLevel (getM state)" shows "let state' = applyExplain (getCl state) state in InvariantCFalse (getConflictFlag state') (getM state') (getC state') \<and> InvariantCEntailed (getConflictFlag state') F0 (getC state') \<and> InvariantClCharacterization (getCl state') (getC state') (getM state') \<and> InvariantClCurrentLevel (getCl state') (getM state') \<and> InvariantCnCharacterization (getCn state') (getC state') (getM state') \<and> InvariantUniqC (getC state')"
hyps: fixes l :: "nat" assumes "n < l" shows "map_pmf (\<lambda>y. y ! n) (bv l) = bernoulli_pmf (5 / 10)"
altts: shows "altts \<equiv> \<lambda>n. alternating_list n t s"
coprime_right_2_iff_odd: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "algebraic_semidom.coprime times one a (numeral.numeral one plus (num.Bit0 num.One)) = (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
assms_1: shows "\<phi>\<^sub>s \<in> reachable g \<phi>"
finite_UNIV: assumes "class.finite TYPE('d)" shows "finite UNIV"
rel_1: shows "0 \<le> int_of_uint64 x"
power_add: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and m :: "nat"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (m + n) = times (power.power one times a m) (power.power one times a n)"
filter_is_subset: fixes P :: "'e \<Rightarrow> bool"   and xs :: "'e list" shows "set (filter P xs) \<subseteq> set xs"
de_morgan4: fixes x :: "'a"   and y :: "'a" shows "!(t x \<oplus> t y) = !x \<cdot> !y"
under_empty: fixes a :: "'c"   and r :: "('c \<times> 'c) set" assumes "a \<notin> Field r" shows "under r a = {}"
tinc_correct_pre: fixes ly :: "nat list"   and ap :: "abc_inst list"   and as :: "nat"   and lm :: "nat list"   and n :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and lm' :: "nat list"   and f :: "nat \<Rightarrow> nat \<times> cell list \<times> cell list"   and P :: "nat \<times> cell list \<times> cell list \<Rightarrow> bool"   and Q :: "nat \<times> cell list \<times> cell list \<Rightarrow> bool" assumes "ly = layout_of ap"   and "inv_locate_a (as, lm) (n, l, r) ires"   and "lm' = abc_lm_s lm n (Suc (abc_lm_v lm n))"   and "f = steps0 (Suc 0, l, r) tinc_b"   and "P = (\<lambda>(s, l, r). s = 10)"   and "Q = (\<lambda>(s, l, r). inc_inv n (as, lm) (s, l, r) ires)" shows "\<exists>stp. P (f stp) \<and> Q (f stp)"
ltl_lmap: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b llist" shows "ltl (lmap f xs) = lmap f (ltl xs)"
shadow_root_ptr_kinds_def: fixes heap :: "(_) heap" shows "shadow_root_ptr_kinds heap = the |`| cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r |`| ffilter is_shadow_root_ptr_kind (object_ptr_kinds heap)"
add_tvarsT'_tvsT_pre: fixes T :: "typ"   and acc :: "(variable \<times> String.literal set) list" shows "set (add_tvarsT' T acc) = set acc \<union> tvsT T"
bi_unique_alt_def: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "bi_unique A = (left_unique A \<and> right_unique A)"
FL_bisimilarity_implies_equivalence_Act: fixes f :: "'effect"   and F :: "'effect fs_set"   and \<alpha> :: "'act"   and x :: "('a, 'pred, 'act, 'effect) formula"   and P :: "'state"   and Q :: "'state" assumes "f \<in>\<^sub>f\<^sub>s F"   and "bn \<alpha> \<sharp>* (F, f)"   and "x \<in> \<A>[L (\<alpha>, F, f)]"   and "\<And>P Q. P \<sim>\<cdot>[L (\<alpha>, F, f)] Q \<Longrightarrow> P \<Turnstile> x = Q \<Turnstile> x"   and "P \<sim>\<cdot>[F] Q"   and "P \<Turnstile> Act f \<alpha> x" shows "Q \<Turnstile> Act f \<alpha> x"
zeroes_ni_generate': fixes a :: "nat list"   and b :: "nat list"   and A :: "nat"   and B :: "nat" shows "(zeroes (length a), zeroes (length b)) \<notin> set (generate' A B a b)"
adj_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" shows "f \<stileturn> g = (\<forall>x y. (f x \<le> y) = (x \<le> g y))"
state_id: shows "dmu_ij_state state i (i - 1) = d\<mu> fs i (i - 1)"
P_deadend_v0_LCons: assumes "deadend v0" shows "P = LCons v0 LNil"
0: shows "to_fun f (a \<ominus> b) = to_fun f a \<ominus> deriv f a \<otimes> b \<oplus> c \<otimes> b [^] 2"
limsup_INF_SUP: fixes f :: "nat \<Rightarrow> 'b" shows "limsup f = (INF n. Sup (f ` {n..}))"
distinct_append_swap: fixes xs :: "'a list"   and ys :: "'a list" assumes "distinct (xs @ ys)" shows "distinct (ys @ xs)"
delete_point_blocks_wf: fixes b :: "'a set"   and p :: "'a" assumes "b \<in># del_point_blocks p" shows "b \<subseteq> \<V> - {p}"
Ring_Congruence_def: shows "Ring_Congruence = {(a, b). a \<in> R \<and> b \<in> R \<and> a - b \<in> I}"
finite_dom_graph: fixes f :: "'c \<Rightarrow> 'd option" assumes "finite (dom f)" shows "finite (map_graph f)"
map_sel_2: fixes f :: "'e \<Rightarrow> 'd"   and a :: "'e stream" shows "stl (smap f a) = smap f (stl a)"
prems: shows "\<forall>xa\<in>set (x # xs). cnt xa (x # xs) \<le> 1"
singleton_lst_proj_2: fixes l :: "'d"   and l' :: "'d"   and a :: "'e" assumes "l \<noteq> l'" shows "proj_unl l' [(ln l, a)] = []"
neg: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "module_hom s1 s2 f" shows "f (- x) = - f x"
invertible_left_inverse2: fixes u :: "'a"   and v :: "'a" assumes "invertible u"   and "u \<in> M"   and "v \<in> M" shows "inverse u \<cdot> (u \<cdot> v) = v"