hcju/isabelleps huggingface dataset:
simps: fixes x :: "'a"   and l :: "'a"   and u :: "'a" shows "elem x (l, u) = (l < x \<and> x < u)"
unapp_app: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and u\<^sub>1 :: "'b"   and u\<^sub>2 :: "'b" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "unapp (app u\<^sub>1 u\<^sub>2) = Some (u\<^sub>1, u\<^sub>2)"
cases: fixes a1 :: "nat \<Rightarrow> classDef option"   and a2 :: "nat list"   and a3 :: "nat list" assumes "a1 \<turnstile>+ a2 <: a3" obtains CT :: "nat \<Rightarrow> classDef option" where "a1 = CT__" and "a2 = []" and "a3 = []"   | CT :: "nat \<Rightarrow> classDef option" and C0 :: "nat" and D0 :: "nat" and Cs :: "nat list" and Ds :: "nat list" where "a1 = CT" and "a2 = C0 # Cs" and "a3 = D0 # Ds" and "CT \<turnstile> C0 <: D0" and "CT \<turnstile>+ Cs <: Ds"
Re_sgn: fixes A :: "complex" assumes "is_real A"   and "A \<noteq> 0" shows "Re (sgn A) = sgn_bool (0 < Re A)"
parabolic_fixed_point_3: fixes f :: "'b \<Rightarrow> 'b" assumes "parabolic_isometry f" shows "additive_strength f (neutral_fixed_point f) = 0"
HIGH_IFF: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c1 :: "IMP"   and c2 :: "IMP"   and b :: "BExpr" assumes "G \<rhd> c1 : twiddle"   and "G \<rhd> c2 : twiddle" shows "G \<rhd> Iff b c1 c2 : twiddle"
idem: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "semilattice f" shows "f a a = a"
valid_split1: fixes fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme"   and c :: "'f list multiset"   and c' :: "'f list multiset" assumes "valid fg (c + c')" shows "valid fg c \<and> valid fg c' \<and> mon_c fg c \<inter> mon_c fg c' = {}"
integrate_insert_commute_def: fixes M :: "('a, 'b) message set"   and a :: "'a extended \<Rightarrow> 'a sort_key extended"   and s :: "('a, 'b option) insert_message list" shows "integrate_insert_commute M a s \<equiv> is_associated_string M s \<and> a_conditions (insert_messages M) a"
Transset_SUP: fixes A :: "'a set"   and B :: "'a \<Rightarrow> V" assumes "\<And>x. x \<in> A \<Longrightarrow> Transset (B x)" shows "Transset (\<Squnion> (B ` A))"
CX3h_irc: fixes h :: "H1" shows "irc (CX3h h)"
Union_least: fixes A :: "'a set set"   and C :: "'a set" assumes "\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C" shows "\<Union> A \<subseteq> C"
open_ball: fixes x :: "'a"   and e :: "real" shows "open (ball x e)"
invariant_iff_pos_and_neg_invariant: fixes M :: "'a set" shows "invariant M = (positively_invariant M \<and> negatively_invariant M)"
distinct_39: fixes x51 :: "CTypeId"   and x52 :: "nat" shows "objV x51 x52 \<noteq> nullV"
p_strict_mono: fixes x :: "real"   and y :: "real" assumes "x < y"   and "y \<le> other.t_stop" shows "other.p x < other.p y"
diff_numeral_Suc: fixes k :: "num"   and n :: "nat" shows "numeral k - Suc n = pred_numeral k - n"
inv_solve_right: fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "Group.group G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "(a = b \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> c) = (b = a \<otimes>\<^bsub>G\<^esub> c)"
supp_def: fixes x :: "'d" shows "supp x = {a. infinite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}}"
ifte_same: fixes a :: "'a"   and x :: "'a" shows "ifte a x x = x"
distinct_3: fixes x1 :: "res_crash"   and x3 :: "v list" shows "RSCrash x1 \<noteq> RSReturn x3"
cut: fixes G :: "'b fm list"   and p :: "'b fm"   and q :: "'b fm" assumes "\<turnstile> imply G p"   and "\<turnstile> imply (p # G) q" shows "\<turnstile> imply G q"
atom_fresh_perm: fixes x :: "name"   and y :: "name" assumes "x \<in> Vs"   and "y \<in> Vs" shows "atom x \<sharp> p \<bullet> y"
add_right_mono: fixes a :: "'d multiset"   and b :: "'d multiset"   and c :: "'d multiset" assumes "a \<subseteq># b" shows "a + c \<subseteq># b + c"
induces_bicategory: shows "bicategory (\<cdot>) (\<star>) \<alpha> some_unit some_src some_trg"
normalize_mult_normalize_right: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "normalize (times a (normalize b)) = normalize (times a b)"
nn_integral_snd_count_space: fixes f :: "'i \<times> 'h \<Rightarrow> ennreal" shows "(\<Sum>\<^sup>+ y. \<Sum>\<^sup>+ x. f (x, y)) = integral\<^sup>N (count_space UNIV) f"
intros_1: fixes t :: "trm"   and t' :: "trm"   and E :: "trm \<Rightarrow> trm" assumes "t \<longmapsto>\<^sub>c t'"   and "E \<in> ctxt" shows "E t \<longmapsto>\<^sub>c E t'"
composition_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> A" shows "addA a b \<in> A"
mapping_of_induct: fixes y :: "'a \<Rightarrow>\<^sub>0 'b"   and P :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool" assumes "y \<in> UNIV"   and "\<And>x. P (pp.mapping_of x)" shows "P y"
WTrt_binop_simps_13: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty"   and T :: "ty" shows "(P \<turnstile> T1\<guillemotleft>BinOr\<guillemotright>T2 : T) = (T1 = Boolean \<and> T2 = Boolean \<and> T = Boolean \<or> T1 = Integer \<and> T2 = Integer \<and> T = Integer)"
poly_altdef': fixes p :: "complex poly"   and k :: "nat"   and z :: "complex" assumes "degree p \<le> k" shows "poly p z = (\<Sum>i\<le>k. coeff p i * z ^ i)"
carrier_sum_group: fixes I :: "'f set"   and G :: "'f \<Rightarrow> ('g, 'h) monoid_scheme" assumes "\<And>i. i \<in> I \<Longrightarrow> Group.group (G i)" shows "carrier (sum_group I G) = {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}}"
braun_add_lo: fixes t :: "'b tree"   and x :: "'b" assumes "braun t" shows "braun (add_lo x t)"
key_gen_def: shows "key_gen = prf_key_gen \<bind> (\<lambda>k_prf. upf_key_gen \<bind> (\<lambda>k_upf. return_spmf (k_prf, k_upf)))"
contdd: shows "continuous_on (T - {a}) ((\<lambda>x. a + dd x *\<^sub>R x) \<circ> (\<lambda>z. z - a))"
gen_incl: fixes I :: "'a set"   and P :: "'a set set" assumes "I \<in> P" shows "gen P I \<subseteq> \<Union> P"
encode_problem_with_operator_interference_exclusion_has_model_then_also_partial_encodings_4: fixes \<A> :: "sat_plan_variable \<Rightarrow> bool"   and \<Pi> :: "'b strips_problem"   and t :: "nat" assumes "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<Pi> t" shows "\<A> \<Turnstile> encode_interfering_operator_exclusion \<Pi> t"
two_powr_minus_int_float: fixes i :: "int" shows "2 powr - real_of_int i \<in> float"
ordertype_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "ordertype A r \<equiv> set (ordermap A r ` A)"
neville_aitken_impl: fixes xs :: "('b \<times> 'b \<times> 'b poly) list"   and x :: "nat \<Rightarrow> 'b"   and f :: "nat \<Rightarrow> 'b"   and k :: "nat" assumes "xs = map (\<lambda>i. (x i, x i, [:f i:])) [0..<Suc k]" shows "neville_aitken_impl_main xs = neville_aitken x f k"
jonsson2b: shows "(\<forall>x y. x \<lhd> y = x \<cdot> ((1::'a) \<lhd> y)) = (\<forall>x y. x \<cdot> y = x \<lhd> ((1::'a) \<lhd> y))"
cardSuc_least: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "Card_order r"   and "Card_order r'"   and "r <o r'" shows "cardSuc r \<le>o r'"
IH: shows "Matrix.row K i = Matrix.row (addrow (- div_op (H $$ (x, 0)) (H $$ (0, 0))) x 0 K_init) i"
2: shows "{0..<x} \<times> {0..<y} = (\<lambda>ia. (ia div y, ia mod y)) ` {0..<x * y}"
collapse: fixes llist :: "'c llist" assumes "llist \<noteq> LNil" shows "LCons (un_LCons1 llist) (un_LCons2 llist) = llist"
dg_Obj_vsubset_Vset: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "\<CC>\<lparr>Obj\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
theFTOne_length_1: fixes cl :: "('d, 'e, 'f) cmd list"   and dl :: "('g, 'h, 'i) cmd list"   and n :: "nat" assumes "length cl = length dl"   and "n \<in> theFTOne cl dl" shows "n < length cl"
indRelLSTPO_impl_SRel_and_TRel_preserve_barbs_1: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_preserves_binary_pred (indRelLSTPO SRel TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))" shows "rel_preserves_binary_pred SRel (\<lambda>a. hasBarb a SWB)"
split_list: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs"
distinct_5: fixes x2 :: "xname"   and x3 :: "tnam" shows "SXcpt' x2 \<noteq> TName x3"
fminusD2: fixes c :: "'a"   and A :: "'a fset"   and B :: "'a fset"   and P :: "bool" assumes "c |\<in>| A |-| B"   and "c |\<in>| B" shows "P"
alt_option_def: fixes alt :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and m1 :: "('e, 'f) optionT"   and m2 :: "('e, 'f) optionT" shows "alt_option alt m1 m2 = OptionT (alt (run_option m1) (run_option m2))"
genPrefix_prefix_trans: fixes x :: "'a list"   and y :: "'a list"   and r :: "('a \<times> 'a) set"   and z :: "'a list" assumes "(x, y) \<in> genPrefix r"   and "y \<le> z" shows "(x, z) \<in> genPrefix r"
assms_2: shows "NTCod \<eta>1 = NTCod \<eta>2"
Vr_ring: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "Ring (Vr K v)"
add_pCons: fixes a :: "'a"   and p :: "'a poly"   and b :: "'a"   and q :: "'a poly" shows "pCons a p + pCons b q = pCons (a + b) (p + q)"
locally_finite_on_closedin_Union_closure: fixes X :: "'c set"   and I :: "'d set"   and U :: "'d \<Rightarrow> 'c set" assumes "locally_finite_on X I U"   and "\<And>i. i \<in> I \<Longrightarrow> closure (U i) \<subseteq> X" shows "closedin (top_of_set X) (\<Union>i\<in>I. closure (U i))"
mk_lossless_def: fixes p :: "'c spmf" shows "mk_lossless p = scale_spmf (inverse (weight_spmf p)) p"
less_asym': fixes a :: "'a"   and b :: "'a"   and P :: "bool" assumes "a < b"   and "b < a" shows "P"
a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes node_ptr :: "(_) node_ptr"   and uu :: "unit" shows "local.CD.a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr uu = Heap_Error_Monad.bind (get_root_node_si (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)) (\<lambda>root. case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r root of None \<Rightarrow> Heap_Error_Monad.bind document_ptr_kinds_M (\<lambda>ptrs. Heap_Error_Monad.bind (filter_M (\<lambda>document_ptr. Heap_Error_Monad.bind (get_disconnected_nodes document_ptr) (\<lambda>disconnected_nodes. return (root \<in> cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ` set disconnected_nodes))) ptrs) (\<lambda>candidates. return (hd candidates))) | Some document_ptr \<Rightarrow> return document_ptr)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) C.map FG.map \<epsilon>'.map"
stm2d1: shows "stm2.d1 = 2 ^ Suc m"
low_high_exchange_dag: fixes lt :: "dag"   and low :: "ref \<Rightarrow> ref"   and lowa :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and higha :: "ref \<Rightarrow> ref"   and rep :: "ref \<Rightarrow> ref"   and rt :: "dag"   and q :: "ref" assumes "\<forall>pt. pt \<notin> set_of lt \<longrightarrow> low pt = lowa pt \<and> high pt = higha pt"   and "\<forall>pt\<in>set_of lt. lowa pt = (rep \<propto> low) pt \<and> higha pt = (rep \<propto> high) pt"   and "\<forall>pt\<in>set_of rt. rep pt = pt"   and "Dag q (rep \<propto> low) (rep \<propto> high) rt" shows "Dag q (rep \<propto> lowa) (rep \<propto> higha) rt"
numeric_binop_type_det: fixes op :: "numeric_binop"   and \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type"   and \<rho>\<^sub>1 :: "'a OCL_Types.type"   and \<rho>\<^sub>2 :: "'a OCL_Types.type" assumes "numeric_binop_type op \<tau> \<sigma> \<rho>\<^sub>1"   and "numeric_binop_type op \<tau> \<sigma> \<rho>\<^sub>2" shows "\<rho>\<^sub>1 = \<rho>\<^sub>2"
him: shows "h ` (T - \<xi> ` K) \<subseteq> rel_frontier U"
tendsto_null_power: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and n :: "nat" assumes "(f \<longlongrightarrow> (0::'b)) F"   and "0 < n" shows "((\<lambda>x. f x ^ n) \<longlongrightarrow> (0::'b)) F"
sees_field_decl_above: fixes P :: "'a prog"   and C :: "String.literal"   and F :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and D :: "String.literal" assumes "P \<turnstile> C sees F:T (fm) in D" shows "P \<turnstile> C \<preceq>\<^sup>* D"
E20: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<diamond>\<box>F \<longrightarrow> \<box>\<diamond>F"
a1: shows "fst i = ctrl_type RETT \<and> get_ET (cpu_reg_val PSR s) \<noteq> 1 \<and> get_S (cpu_reg_val PSR s) \<noteq> 0 \<and> get_WIM_bit (nat ((uint (get_CWP (cpu_reg_val PSR s)) + 1) mod NWINDOWS)) (cpu_reg_val WIM s) = 0 \<and> get_addr (snd i) s AND 3 = 0"
encode_kind_2: fixes m :: "nat"   and n :: "nat" shows "encode_kind (encode (recf.Id m n)) = 2"
maxR_code: fixes x :: "'a"   and y :: "'a" shows "local.MaxR.maxR x y = (if (x, y) \<in> r then y else x)"
simps: fixes xs :: "'a list"   and x :: "'a"   and y :: "'a" shows "pad xs x y = (if xs = [] then [x, x] else if last xs = x then xs @ [x] else xs @ [y])"
injf_max_mem_set: fixes E :: "'a set"   and n :: "nat" assumes "E \<noteq> {}"   and "\<forall>x. \<exists>y\<in>E. x < y" shows "injf_max n E \<in> E"
baEA: fixes n :: "nat"   and c :: "'bid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'BB"   and \<phi> :: "'BB \<Rightarrow> bool" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "pb.eval c t t' n [\<phi>]\<^sub>b" shows "\<phi> (\<sigma>\<^bsub>c\<^esub>t \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>)"
prj_simps_4: shows "C.dom \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>1\<^sub>1 = HH\<mu>H\<nu>\<pi>\<rho>.chine"
connected: fixes x0 :: "'c" shows "connected (existence_ivl0 x0)"
replicate_append_same: fixes i :: "nat"   and x :: "'c" shows "replicate i x @ [x] = x # replicate i x"
simps: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and n :: "nat"   and r :: "('b \<times> nat) tree" shows "get_min \<langle>l, (a, n), r\<rangle> = a"
le_approx_lemma_T: fixes P :: "'a process"   and Q :: "'a process" assumes "P \<sqsubseteq> Q" shows "T Q \<subseteq> T P"
eff\<^sub>i_Load: fixes n :: "nat"   and P :: "'b prog"   and ST :: "ty list"   and LT :: "ty err list" shows "eff\<^sub>i (Load n, P, ST, LT) = (ok_val (LT ! n) # ST, LT)"
degp: shows "MPoly_Type.degree p var = 1 \<or> MPoly_Type.degree p var = 2"
SeqThrow\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e\<^sub>0 :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e :: "nat exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e\<^sub>1 :: "nat exp" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>0,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e\<^sub>0;; e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>1\<rangle>"
quote_all_empty: fixes p :: "perm" shows "quote_all p {} = {}"
simps: fixes a :: "hmultiset" shows "local.well_base\<^sub>h a = (\<exists>M. a = M \<and> Multiset.Ball (hmsetmset M) local.well_base\<^sub>h \<and> local.well_base (hmsetmset M))"
cases: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>"   and cnv :: "('h, 'i, 'j, 'k) converter"   and a :: "('h \<Rightarrow> 'j) \<times> ('k \<Rightarrow> 'i)" assumes "wiring \<I> \<I>' cnv a" obtains f :: "'h \<Rightarrow> 'j" and g :: "'k \<Rightarrow> 'i" where "a = (f, g)" and "\<I>,\<I>' \<turnstile>\<^sub>C cnv \<sim> map_converter id id f g 1\<^sub>C" and "\<I>, \<I>' \<turnstile>\<^sub>C cnv \<surd>"
stabiliser_cosets_equivalent: fixes g :: "'a"   and h :: "'a"   and x :: "'b" assumes "g \<in> carrier G"   and "h \<in> carrier G" shows "(g \<odot> x = h \<odot> x) = (g <# stabiliser x = h <# stabiliser x)"
memFreshChain_1: fixes p :: "'e"   and l :: "'e list"   and xvec :: "name list" assumes "p mem l"   and "xvec \<sharp>* l" shows "xvec \<sharp>* p"
fin: fixes x :: "real" assumes "x \<in> cbox a b" shows "f x \<in> cbox c d"
measurable_restrict_countable: fixes X :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and N :: "'d measure" assumes "countable X"   and "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"   and "\<And>x. x \<in> X \<Longrightarrow> f x \<in> space N"   and "f \<in> restrict_space M (- X) \<rightarrow>\<^sub>M N" shows "f \<in> M \<rightarrow>\<^sub>M N"
Rep_N_inject: fixes x :: "'b N"   and y :: "'b N" shows "(Rep_N x = Rep_N y) = (x = y)"
transition_def: fixes stateA :: "State"   and stateB :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set" shows "transition stateA stateB F0 decisionVars \<equiv> appliedDecide stateA stateB decisionVars \<or> appliedUnitPropagate stateA stateB F0 decisionVars \<or> appliedConflict stateA stateB \<or> appliedExplain stateA stateB \<or> appliedLearn stateA stateB \<or> appliedBackjump stateA stateB"
degree_le_fresh_index: fixes a :: "'a \<times> 'a pdevs"   and A :: "('a \<times> 'a pdevs) list" assumes "a \<in> set A" shows "degree_aform a \<le> fresh_index_aforms A"
rep_eq: shows "comp_fun_idem_apply sup_cfi = sup"
connected_open_path_connected: fixes S :: "'a set" assumes "open S"   and "connected S" shows "path_connected S"
splitFace_new_f21: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and newVs :: "nat list"   and f12 :: "face"   and f21 :: "face"   and newGraph :: "graph" assumes "pre_splitFace g ram1 ram2 oldF newVs"   and "minGraphProps g"   and "(f12, f21, newGraph) = splitFace g ram1 ram2 oldF newVs" shows "f21 \<notin> \<F> g"
evaluation_map_axioms: shows "Coherence.evaluation_map (\<cdot>\<^sub>C) src\<^sub>C trg\<^sub>C (\<cdot>) (\<star>) \<a> \<i> src trg E"
float_lt: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" assumes "is_finite a"   and "is_finite b" shows "(a < b) = (valof a < valof b)"
countable_card_le_natLeq: fixes A :: "'a set" shows "countable A = (|A| \<le>o natLeq)"
fin_2: fixes n :: "'b" assumes "n \<in> N" shows "finite (B1 n)"
assms_1: fixes t :: "'a"   and s :: "'a" assumes "(t, s) \<in> r" shows "(f t, f s) \<in> r"
measure_pair_spmf_times: fixes p :: "'h spmf"   and q :: "'i spmf"   and A :: "'h set"   and B :: "'i set" shows "Sigma_Algebra.measure (measure_spmf (pair_spmf p q)) (A \<times> B) = Sigma_Algebra.measure (measure_spmf p) A * Sigma_Algebra.measure (measure_spmf q) B"
sqrt_differentiable: fixes f :: "'b \<Rightarrow> real"   and x :: "'b"   and S :: "'b set" assumes "f differentiable at x within S"   and "f x \<noteq> 0" shows "(\<lambda>x. sqrt (f x)) differentiable at x within S"
a_def: shows "a = f' \<turnstile> folding_g.\<C>"
applicableUnitPropagate_def: fixes state :: "State" shows "applicableUnitPropagate state \<equiv> \<exists>state'. appliedUnitPropagate state state'"
iIN_subset: fixes n :: "nat"   and n' :: "nat"   and d' :: "nat"   and d :: "nat" assumes "n \<le> n'"   and "n' + d' \<le> n + d" shows "[n'\<dots>,d'] \<subseteq> [n\<dots>,d]"
hermitean_ortho_eigenvecs_1: fixes H :: "complex \<times> complex \<times> complex \<times> complex"   and k1 :: "complex"   and v1 :: "complex \<times> complex"   and k2 :: "complex"   and v2 :: "complex \<times> complex" assumes "hermitean H"   and "eigenpair k1 v1 H"   and "eigenpair k2 v2 H"   and "k1 \<noteq> k2" shows "vec_cnj v2 *\<^sub>v\<^sub>v v1 = 0"
trace_rest'_parametric: fixes S :: "'a \<Rightarrow> 'h \<Rightarrow> bool"   and IA :: "'c \<Rightarrow> 'i \<Rightarrow> bool"   and IU :: "'d \<Rightarrow> 'j \<Rightarrow> bool"   and M :: "'g \<Rightarrow> 'k \<Rightarrow> bool" shows "rel_fun (rel_rest' S (=) IA IU (=) (=) M) (rel_fun (rel_spmf S) (rel_fun (list_all2 (rel_sum (rel_prod IA (=)) (rel_prod IU (=)))) (rel_prod (rel_fun IA (=)) (rel_fun IU (=))))) trace_rest' trace_rest'"
chainI: fixes C :: "'g set set"   and A :: "'g set set" assumes "C \<subseteq> A"   and "\<And>x y. \<lbrakk>x \<in> C; y \<in> C\<rbrakk> \<Longrightarrow> (\<subset>)\<^sup>=\<^sup>= x y \<or> (\<subset>)\<^sup>=\<^sup>= y x" shows "subset.chain A C"
exp_cf7_upper_bound_pos: fixes x :: "real" assumes "0 \<le> x"   and "0 < numer_cf7 (- x)" shows "exp x \<le> exp_cf7 x"
ordered_keys_def: fixes m :: "('a, 'b) mapping" shows "Mapping.ordered_keys m = (if finite (Mapping.keys m) then sorted_list_of_set (Mapping.keys m) else [])"
unsimp_context: fixes \<Gamma> :: "'a set" shows "\<Gamma> = {} \<union> \<Gamma>"
dimension_independent: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list" assumes "ring R"   and "ring.independent R K Us" shows "ring.dimension R (length Us) K (ring.Span R K Us)"
exhaust_sel: fixes list :: "'a list" obtains "list = []"   | "list = hd list # tl list"
PLUS_eq_Zero: fixes xs :: "'a rexp list" shows "(PLUS xs = Zero) = (xs = [] \<or> xs = [Zero])"
distinct_113: fixes x51 :: "expr"   and x52 :: "bop"   and x53 :: "expr"   and x121 :: "expr"   and x122 :: "expr" shows "x51 \<guillemotleft>x52\<guillemotright> x53 \<noteq> x121;; x122"
map_acom_Assign: fixes f :: "'c \<Rightarrow> 'b"   and c :: "'c acom"   and x :: "char list"   and e :: "aexp"   and S' :: "'b" shows "(ACom.map_acom f c = x ::= e {S'}) = (\<exists>S. c = x ::= e {S} \<and> S' = f S)"
mat_default_def: fixes d :: "'c"   and n :: "nat" shows "default\<^sub>m d n = mat n n (\<lambda>(i, j). if i = j then d else (0::'c))"
rbt_minus_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t1 :: "('b, 'c) rbt"   and t2 :: "('b, 'c) rbt" shows "ord.rbt_minus less t1 t2 = (let h1 = bheight t1; h2 = bheight t2 in if 2 * h1 \<le> h2 then rbtreeify (fold_rev (\<lambda>k v kvs. if ord.rbt_lookup less t2 k = None then (k, v) # kvs else kvs) t1 []) else RBT_Impl.fold (\<lambda>k v. ord.rbt_delete less k) t2 t1)"
nonneg_fg: fixes x :: "(real, 'n) vec" assumes "x \<in> S" shows "0 \<le> f (g x)"
delete_type: fixes t :: "('c \<times> 'd) bro"   and h :: "nat"   and x :: "'c" assumes "t \<in> B h" shows "delete x t \<in> B h \<union> B (h - 1)"
prob_space_imp_sigma_finite: fixes M :: "'a measure" assumes "prob_space M" shows "sigma_finite_measure M"
Inherited: fixes G :: "prog"   and m :: "qtname \<times> memberdecl"   and C :: "qtname"   and S :: "qtname" assumes "G \<turnstile> m inheritable_in pid C"   and "G\<turnstile> memberid m undeclared_in C"   and "G\<turnstile>C\<prec>\<^sub>C1S"   and "G \<turnstile> Class S accessible_in pid C"   and "G \<turnstile> m member_of S" shows "G \<turnstile> m member_of C"
multiset_sum_of_Suc_f_monotone: fixes N :: "'d multiset"   and M :: "'d multiset"   and f :: "'d \<Rightarrow> nat" assumes "N \<subset># M" shows "(\<Sum>x\<in>#N. Suc (f x)) < (\<Sum>x\<in>#M. Suc (f x))"
Un_iff: fixes c :: "'e"   and A :: "'e set"   and B :: "'e set" shows "(c \<in> A \<union> B) = (c \<in> A \<or> c \<in> B)"
remdups_adj_Cons: fixes x :: "'c"   and xs :: "'c list" shows "remdups_adj (x # xs) = (case remdups_adj xs of [] \<Rightarrow> [x] | y # xs \<Rightarrow> if x = y then y # xs else x # y # xs)"