hcju/isabelleps huggingface dataset:
unrest_lit: fixes x :: "'b \<Longrightarrow> 'c"   and v :: "'d" shows "x \<sharp> \<guillemotleft>v\<guillemotright>"
chamber_morphism: fixes X :: "'c set set"   and Y :: "'d set set"   and f :: "'c \<Rightarrow> 'd" assumes "ChamberComplexIsomorphism X Y f" shows "ChamberComplexMorphism X Y f"
LtK_Tag: fixes X :: "ltkey"   and X' :: "tag" shows "LtK X \<noteq> Tag X'"
x1: fixes i :: "'a" assumes "i \<in> t" shows "is_orthogonal a i"
le_supE: fixes a :: "'c \<times> 'd"   and b :: "'c \<times> 'd"   and x :: "'c \<times> 'd" assumes "less_eq_prod (sup_prod a b) x" obtains "less_eq_prod a x" and "less_eq_prod b x"
elem_set_implies_set_eq_singleton: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "a \<in> local.set b" shows "local.set a = {a}"
simps_1: fixes x :: "'v"   and i :: "nat" shows "coef_hd (Var x) i = PVar (PCoef x i)"
step: fixes A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and l :: "'e"   and D :: "nat \<Rightarrow> nat \<Rightarrow> 'g DBMEntry"   and v :: "'f \<Rightarrow> nat"   and n :: "nat"   and l' :: "'e"   and D' :: "nat \<Rightarrow> nat \<Rightarrow> 'g DBMEntry"   and l'' :: "'e"   and D'' :: "nat \<Rightarrow> nat \<Rightarrow> 'g DBMEntry" assumes "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^bsub>v,n\<^esub> \<langle>l', D'\<rangle>"   and "A \<turnstile> \<langle>l', D'\<rangle> \<leadsto>*\<^bsub>v,n\<^esub> \<langle>l'', D''\<rangle>" shows "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>*\<^bsub>v,n\<^esub> \<langle>l'', D''\<rangle>"
assms_2: fixes x :: "'a" assumes "x \<in> set xs" shows "f x = g x"
tube_lemma: fixes K :: "'b set"   and W :: "('c \<times> 'b) set"   and x0 :: "'c" assumes "compact K"   and "open W"   and "{x0} \<times> K \<subseteq> W" shows "\<exists>X0. x0 \<in> X0 \<and> open X0 \<and> X0 \<times> K \<subseteq> W"
IH: assumes "finite (Xs - {Y})" shows "fresh_string (Xs - {Y}) (upChar Y) \<notin> Xs - {Y}"
distinct_36: fixes x4 :: "key"   and x3 :: "nat" shows "Key x4 \<noteq> Pwd x3"
ide_char: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "functor_category A B" shows "partial_magma.ide (concrete_category.COMP (Collect (functor A B)) (\<lambda>F G. Collect (natural_transformation A B F G)) (\<lambda>F G H \<tau> \<sigma>. vertical_composite.map A B \<sigma> \<tau>)) t = (t \<noteq> partial_magma.null (concrete_category.COMP (Collect (functor A B)) (\<lambda>F G. Collect (natural_transformation A B F G)) (\<lambda>F G H \<tau> \<sigma>. vertical_composite.map A B \<sigma> \<tau>)) \<and> functor A B (\<chi>.\<Phi>.Ya.Cop_S.Map t) \<and> concrete_category.Dom t = \<chi>.\<Phi>.Ya.Cop_S.Map t \<and> \<chi>.\<Phi>.Ya.Cop_S.Cod t = \<chi>.\<Phi>.Ya.Cop_S.Map t)"
elims_3: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b tree" assumes "class.linorder less_eq less"   and "\<not> linorder.heap less_eq x" obtains l :: "'b tree" and m :: "'b" and r :: "'b tree" where "x = \<langle>l, m, r\<rangle>" and "\<not> (Ball (set_tree l \<union> set_tree r) (less_eq m) \<and> linorder.heap less_eq l \<and> linorder.heap less_eq r)"
constr_sem_d_def: fixes \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" shows "\<I> \<Turnstile> \<langle>S,\<theta>\<rangle> \<equiv> \<theta> supports \<I> \<and> \<lbrakk>{}; S\<rbrakk>\<^sub>d \<I>"
all: shows "\<forall>Ds. Subobjs P C Ds \<and> (\<exists>Ts T m Bs ms. (\<exists>fs. class P (last Ds) = \<lfloor>(Bs, fs, ms)\<rfloor>) \<and> map_of ms M = \<lfloor>(Ts, T, m)\<rfloor>) \<longrightarrow> P,C \<turnstile> Cs \<sqsubseteq> Ds"
prems: shows "length (trace_parallel_plan_strips I (ops # \<pi>)) < length (ops # \<pi>) + 1"
hext_refl: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" shows "h \<unlhd> h"
abs_round_down_le: fixes f :: "real"   and e :: "int" shows "\<bar>f - round_down e f\<bar> \<le> 2 powr - real_of_int e"
rrsmult_range: fixes R :: "'a set"   and r :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "Ring1 R" shows "range (Ring1.rightreg_scalar_mult R r f) \<subseteq> {0::'b} \<union> range f"
intro: fixes G :: "'b pair_pre_digraph" assumes "pair_sym_digraph G"   and "pair_loopfree_digraph G" shows "pair_bidirected_digraph G"
val_Zp_mult: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp" shows "val_Zp (x \<otimes> y) = val_Zp x + val_Zp y"
quit_iteration: fixes l :: "'a list"   and p :: "'a node ref option"   and l' :: "'a list"   and it :: "'a node ref option" shows "os_is_it l p l' it \<Longrightarrow>\<^sub>A os_list l p * true"
g_def: fixes n :: "nat" shows "g n = c \<cdot>\<^sub>m (f n - f 0)"
ennreal_neg: fixes x :: "real" assumes "x \<le> 0" shows "ennreal x = 0"
simps: fixes Q_pos :: "(nat set \<times> 'a option list set) set"   and Q_neg :: "(nat set \<times> 'a option list set) set" shows "wrapperGenericJoin Q_pos Q_neg = (if (\<exists>(A, y)\<in>Q_pos. Set.is_empty y) \<or> (\<exists>(A, X)\<in>Q_neg. Set.is_empty A \<and> \<not> Set.is_empty X) then {} else let Q = Set.filter (\<lambda>(A, uu_). \<not> Set.is_empty A) Q_pos in if Set.is_empty Q then (\<Inter>(A, X)\<in>Q_pos. X) - (\<Union>(A, X)\<in>Q_neg. X) else let V = \<Union>(A, X)\<in>Q. A in Let (Set.filter (\<lambda>(A, uu_). A \<subseteq> V \<and> 1 \<le> card A) Q_neg) (genericJoin V Q))"
mupdate_refines: fixes \<sigma> :: "('V \<Rightarrow> 'W infty option) \<times> ('V \<Rightarrow> (('V \<times> 'W \<times> 'V) list \<times> 'W) option)"   and \<sigma>' :: "'V set \<times> ('V \<Rightarrow> ('V \<times> 'W \<times> 'V) list option)"   and wv :: "'W infty"   and v :: "'V"   and v' :: "'V" assumes "(\<sigma>, \<sigma>') \<in> br \<alpha>s dinvarm"   and "wv = mpath_weight' (snd \<sigma> v)"   and "v' = v" shows "mupdate v wv \<sigma> \<le> \<Down> (br \<alpha>s dinvarm) (update' v' \<sigma>')"
assertion_conjunctive: fixes x :: "'a" assumes "x \<in> assertion" shows "x \<in> conjunctive.conjunctive (\<sqinter>) (\<sqinter>) (*)"
flip_bit_eq_if: 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 uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_bit_operations.flip_bit one push_bit xor n a = (if bit a n then ring_bit_operations.unset_bit one push_bit and not else ring_bit_operations.set_bit one push_bit or) n a"
subresultant_smult_left: fixes c :: "'b"   and J :: "nat"   and f :: "'b poly"   and g :: "'b poly" assumes "c \<noteq> (0::'b)" shows "subresultant J (Polynomial.smult c f) g = Polynomial.smult (c ^ (degree g - J)) (subresultant J f g)"
preservation_no_await_plus: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set" assumes "\<langle>c, mds, mem\<rangle> \<leadsto>\<^sup>+ \<langle>c', mds', mem'\<rangle>"   and "\<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>',\<S>',P'"   and "no_await c" shows "no_await c' \<and> (\<exists>\<Gamma>'' \<S>'' P''. \<turnstile> \<Gamma>'',\<S>'',P'' {c'} \<Gamma>',\<S>',P' \<and> (tyenv_wellformed mds \<Gamma> \<S> P \<and> pred P mem \<and> (\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem \<le> dma mem x) \<longrightarrow> tyenv_wellformed mds' \<Gamma>'' \<S>'' P'' \<and> pred P'' mem' \<and> (\<forall>x\<in>dom \<Gamma>''. x \<notin> mds' AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma>'' x)) mem' \<le> dma mem' x)))"
fact_prod_Suc: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "semiring_char_0.fact one plus zero n = semiring_1.of_nat one plus zero (prod Suc {0..<n})"
assoc_l_cancel: fixes a :: "'a"   and b :: "'a"   and b' :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "b' \<in> carrier G"   and "a \<otimes> b \<sim> a \<otimes> b'" shows "b \<sim> b'"
assert_all_code_def: shows "assert_all_code = AssertAllState.assert_all assert_all_state_code"
associatedI: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times a b"   and "dvd.dvd times b a" shows "normalize a = normalize b"
utp_theory_unital_zerol_axioms: shows "utp_theory_unital_zerol \<H> \<I>\<I>"
enm_def: fixes mac_alg :: "'c \<Rightarrow> 'a \<Rightarrow> 'b spmf" shows "macode.enm mac_alg \<equiv> CNV (\<lambda>bs m. if bs then Done ((), True) else Pause (Inl m) Done \<bind> (\<lambda>r. lift_spmf (mac_alg (projl r) m) \<bind> (\<lambda>a. Pause (Inr (a, m)) Done \<bind> (\<lambda>_. Done ((), True))))) False"
ab: shows "a / b = zerosign (of_bool (sign a \<noteq> sign b)) (IEEE.round To_nearest (valof a / valof b))"
lcoeff_nonzero_nonzero: fixes p :: "nat \<Rightarrow> 'a" assumes "deg R p = 0"   and "p \<noteq> \<zero>\<^bsub>P\<^esub>"   and "p \<in> carrier P" shows "coeff P p 0 \<noteq> \<zero>"
Ide_implies_Arr: fixes t :: "'a local.term" assumes "Ide t" shows "Arr t"
h3: shows "distinct (get_signs_R (reduce_system_R p (qs, matrix, subsets, signs)))"
hyps_2: fixes s :: "('a, 'b) xstate" assumes "\<Gamma>\<turnstile>strip_guards F c2 \<down> s" shows "\<Gamma>\<turnstile>c2 \<down> s"
Strongly_Secure_Assign: fixes x :: "'id"   and e :: "'exp"   and e' :: "'exp"   and d :: "'d" assumes "e \<equiv>\<^bsub>DA x\<^esub> e'" shows "[x:=e] \<approx>\<^bsub>d\<^esub> [x:=e']"
tllist_ord_trans: fixes xs :: "('c, 'b) tllist"   and ys :: "('c, 'b) tllist"   and zs :: "('c, 'b) tllist" assumes "local.tllist_ord xs ys"   and "local.tllist_ord ys zs" shows "local.tllist_ord xs zs"
classical_operator_def: fixes \<pi> :: "'c \<Rightarrow> 'd option" shows "classical_operator \<pi> = Let (\<lambda>t. case \<pi> (inv ket t) of None \<Rightarrow> 0 | Some i \<Rightarrow> ket i) (cblinfun_extension (range ket))"
p_le_1: fixes s :: "'s" assumes "s \<in> S" shows "p s \<le> 1"
collapse: fixes inference :: "'b inference" shows "Infer (prems_of inference) (concl_of inference) = inference"
row_space_def: fixes A :: "'c mat" shows "row_space A = (\<lambda>v. v \<^sub>v* A) ` carrier_vec (dim_row A)"
prems: shows "\<forall>v\<in>vars_lit (Neg x). \<sigma> v = \<tau> v"
IntE: fixes c :: "'ag"   and A :: "'ag set"   and B :: "'ag set" assumes "c \<in> A \<inter> B" obtains "c \<in> A" and "c \<in> B"
cospan_diagram_axioms: shows "cospan_diagram (\<cdot>) f0 f1"
z_ord: shows "tmf_c prec I f ord = Some (z ! ord)"
elims_3: fixes x :: "('b \<times> nat) tree"   and xa :: "('b \<times> nat) tree" assumes "\<not> balanced1 x xa" obtains t1 :: "('b \<times> nat) tree" and t2 :: "('b \<times> nat) tree" where "x = t1" and "xa = t2" and "\<not> 10 * (size_wbt t2 + 1) \<le> 25 * (size_wbt t1 + 1)"
vD_Some: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<in> vD rt" shows "\<exists>dsn dsk hops nhip pre. rt dip = Some (dsn, dsk, val, hops, nhip, pre)"
iT_Div_mono: fixes A :: "nat set"   and B :: "nat set"   and k :: "nat" assumes "A \<subseteq> B" shows "A \<oslash> k \<subseteq> B \<oslash> k"
simps_1: fixes xs :: "nat list" shows "eval_wellf Z xs \<down>= 0"
simps_2: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and accu :: "('b \<times> 'b) list"   and e :: "'b \<times> 'b"   and Es :: "('b \<times> 'b) list" shows "TopoS_Stateful_Policy_Algorithm.filter_compliant_stateful_ACS_accu G M accu (e # Es) = (if e \<notin> backflows (edges G) \<and> (\<forall>F\<in>get_offending_flows (TopoS_Composition_Theory.get_ACS M) (stateful_policy_to_network_graph \<lparr>hosts = nodes G, flows_fix = edges G, flows_state = set (e # accu)\<rparr>). F \<subseteq> backflows (set (e # accu))) then TopoS_Stateful_Policy_Algorithm.filter_compliant_stateful_ACS_accu G M (e # accu) Es else TopoS_Stateful_Policy_Algorithm.filter_compliant_stateful_ACS_accu G M accu Es)"
regular_trapC: shows "0 \<notin> (\<lambda>(x, y). (cx x y, cy x y)) ` trapC"
mult2_alt_emptyR: fixes X :: "'b multiset"   and b :: "bool"   and ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" shows "((X, {#}) \<in> mult2_alt b ns s) = (b \<and> X = {#})"
2: shows "class.nontriv TYPE('d) = (2 \<le> CARD('d))"
new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and new_document_ptr :: "(_) document_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_document_ptr, h')"   and "ptr \<noteq> cast new_document_ptr" shows "get ptr h = get ptr h'"
list_of_oalist_of_list: fixes xs :: "('a \<times> 'c) list \<times> 'o" shows "list_of_oalisty (oalist_of_listy xs) = sort_oalist_raw xs"
sum_defs_2: fixes a :: "hf" shows "HF.Inl a \<equiv> \<langle>0, a\<rangle>"
pfun_ext: fixes f :: "('a, 'b) pfun"   and g :: "('a, 'b) pfun" assumes "\<And>x y. ((x, y) \<in>\<^sub>p f) = ((x, y) \<in>\<^sub>p g)" shows "f = g"
quasi_inverse_simps_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "bicategory.equivalence_map V H \<a> \<i> src trg f" shows "bicategory.equivalence_map V H \<a> \<i> src trg (bicategory.some_quasi_inverse V H \<a> \<i> src trg f)"
inj_on_to_nat_on: fixes A :: "'d set" assumes "countable A" shows "inj_on (to_nat_on A) A"
assms: fixes x :: "'a"   and y :: "'a" assumes "x \<succeq>[relation] y" shows "f y \<le> f x"
distinct_29: fixes name :: "name"   and ccs :: "ccs"   and ccs' :: "ccs" shows "\<lparr>\<nu>name\<rparr>ccs \<noteq> !ccs'"
csv_hyp: shows "\<forall>r. consistent_sign_vec (qs1 @ qs2) r = consistent_sign_vec qs1 r @ consistent_sign_vec qs2 r"
is_pbqs: fixes p :: "'node"   and Q :: "'node set"   and p' :: "'node" assumes "quorum_of p Q"   and "p' \<in> Q" shows "quorum_of p' Q"
round_len_less: fixes p :: "nat"   and q :: "nat"   and t :: "nat \<times> nat list \<times> 'c list"   and r :: "nat"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd" assumes "bn_inv p q t"   and "r < q"   and "(r + (foldl (+) 0 (fst (snd t)) - bn_count (fst (snd t))) * p) mod q = 0" shows "(fst (round index key p q r t) + length (fst (snd (round index key p q r t)))) * q = (fst t + bn_count (fst (snd t))) * q + (foldl (+) 0 (fst (snd t)) - bn_count (fst (snd t))) * p + r"
non_strict_dominate_def: fixes n1 :: "nat"   and n2 :: "nat" shows "non_strict_dominate n1 n2 \<equiv> \<exists>pa. path_entry (g_E G) pa n2 \<and> n1 \<notin> set pa"
lunit'_naturality: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "(trg\<^sub>D \<mu> \<star>\<^sub>D \<mu>) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[D.dom \<mu>] = \<l>\<^sub>D\<^sup>-\<^sup>1[D.cod \<mu>] \<cdot>\<^sub>D \<mu>"
disj_eqvt: fixes pi :: "('a \<times> 'a) list"   and A :: "bool"   and B :: "bool" shows "pi \<bullet> (A \<or> B) = (pi \<bullet> A \<or> pi \<bullet> B)"
poincare_distance_formula_zero_sum: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc" shows "poincare_distance u 0\<^sub>h + poincare_distance 0\<^sub>h v = (let u' = cmod (to_complex u); v' = cmod (to_complex v) in arcosh (((1 + u'\<^sup>2) * (1 + v'\<^sup>2) + 4 * u' * v') / ((1 - u'\<^sup>2) * (1 - v'\<^sup>2))))"
TIP: fixes \<phi> :: "fm" assumes "\<phi> \<in> UNIV"   and "Fvars \<phi> = {}"   and "let PP = subst_fm (PfP (Var xx)) xx in eval_fm e0 (PP \<guillemotleft>\<phi>\<guillemotright>)" shows "{} \<turnstile> \<phi>"
count_diff: fixes M :: "'a multiset"   and N :: "'a multiset"   and a :: "'a" shows "count (M - N) a = count M a - count N a"
WT_resource_attach: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and conv :: "('i, 'j, 'k, 'l) converter"   and res :: "('k, 'l) resource" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "\<I>' \<turnstile>res res \<surd>" shows "\<I> \<turnstile>res conv \<rhd> res \<surd>"
F_LUB: fixes S :: "nat \<Rightarrow> 'b process" assumes "chain S" shows "F (lim_proc (range S)) = \<Inter> (F ` range S)"
keys_monom_mult_subset: fixes c :: "'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "keys (local.punit.monom_mult c t p) \<subseteq> local.punit.splus t ` keys p"
accept\<^sub>G\<^sub>R_simp2: fixes \<delta> :: "'d \<Rightarrow> 'e \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and \<alpha> :: "(('d \<times> 'e \<times> 'd) set \<times> ('d \<times> 'e \<times> 'd) set set) set"   and w :: "nat \<Rightarrow> 'e" shows "accept\<^sub>G\<^sub>R (\<delta>, q\<^sub>0, \<alpha>) w = (\<exists>P\<in>\<alpha>. accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 P w)"
mset_subset_eq_add_left: fixes A :: "'a multiset"   and B :: "'a multiset" shows "A \<subseteq># A + B"
sel_2: fixes x :: "complex"   and y :: "complex" shows "Im (x * y) = Re x * Im y + Im x * Re y"
leading_coeff_neq_0: fixes p :: "'b poly" assumes "p \<noteq> 0" shows "lead_coeff p \<noteq> (0::'b)"
bbox_def: fixes x :: "'a"   and y :: "'a" shows " |x] y = ad (x \<cdot> ad y)"
HIGH_WHILE: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and b :: "BExpr" assumes "G \<rhd> c : twiddle" shows "G \<rhd> While b c : twiddle"
suminf: fixes f :: "'c \<Rightarrow> 'd"   and X :: "nat \<Rightarrow> 'c" assumes "bounded_linear f"   and "summable X" shows "f (\<Sum>n. X n) = (\<Sum>n. f (X n))"
distinct_3: fixes x3 :: "'b" shows "dSUCCEEDi \<noteq> dRETURN x3"
AbstrLevels_S4opt_A41: fixes i :: "AbstrLevelsID" assumes "sS4opt \<in> AbstrLevel i" shows "sA41 \<notin> AbstrLevel i"
fair_def: fixes Ns :: "'a set llist" shows "fair Ns = (Inf_from (Liminf_llist Ns) \<subseteq> Sup_llist (lmap Red_I Ns))"
covering_space_exp_punctured_plane: shows "covering_space UNIV exp (- {0})"
snd_conv: fixes x1 :: "'f"   and x2 :: "'e" shows "snd (x1, x2) = x2"
1: shows "{EqOnUOut p p' u ou |u ou. (u, ou) \<in> GL \<times> UNIV} = (\<lambda>(u, ou). EqOnUOut p p' u ou) ` (GL \<times> UNIV)"
fsubset_strict: fixes x2 :: "'b fset"   and x1 :: "'b fset" assumes "x2 |\<subset>| x1" shows "\<exists>e. e |\<in>| x1 \<and> e |\<notin>| x2"
genPrefix_Nil: fixes xs :: "'a list"   and r :: "('a \<times> 'a) set" shows "((xs, []) \<in> genPrefix r) = (xs = [])"
extends_tgt_out_edges: fixes e :: "'d edge"   and g :: "('d, 'e) rgraph_scheme"   and g' :: "('d, 'e) rgraph_scheme" assumes "ArcExt.extends g e g'" shows "out_edges g' (tgt e) = {}"
transp_on_list_emb: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "transp_on P A" shows "transp_on (list_emb P) (lists A)"
par_strict_not_col_3: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B ParStrict C D" shows "\<not> Col C D A"
changes_poly_at_def: fixes ps :: "'b poly list"   and a :: "'b" shows "changes_poly_at ps a = changes (map (\<lambda>p. poly p a) ps)"
Lt_lt_conv: fixes comp :: "'c \<Rightarrow> 'c \<Rightarrow> order"   and x :: "'c"   and y :: "'c" assumes "comparator comp" shows "(comp x y = Lt) = lt_of_comp comp x y"
induced_fsmult_conv_fsmult_1ddh: fixes f :: "('f, 'g) aezfun \<Rightarrow> 'v"   and h :: "'g"   and r :: "'f" assumes "f \<in> indV"   and "h \<in> H" shows "(r \<currency>\<currency> f) ((1::'f) \<delta>\<delta> h) = r \<sharp>\<cdot> f ((1::'f) \<delta>\<delta> h)"
wf_digraphI_app_iso: fixes G :: "('c, 'd) pre_digraph"   and hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "wf_digraph G"   and "pre_digraph.digraph_isomorphism G hom" shows "wf_digraph (app_iso hom G)"
intersects_sctns_spec_clw_def: fixes R :: "'c set"   and sctns :: "'c sctn set" shows "intersects_sctns_spec_clw R sctns = sets_of_coll ((R ::: clw_rel appr_rel) ::: clw_rel appr_rel) \<bind> (\<lambda>Rs. FORWEAK Rs (RETURN False) (\<lambda>R. intersects_sctns R sctns) (\<lambda>a b. RETURN (a \<or> b)))"
id: fixes i :: "nat"   and xs :: "NominalSequents.var list" shows "([],  \<LM> At i xs  \<RM> \<Rightarrow>* \<LM> At i xs  \<RM>) \<in> Ax"
B_eq_Gauss: shows "B = Gauss_Jordan_in_ij A (0::'rows) (mod_type_class.from_nat k)"
Par1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "bn \<alpha> \<sharp>* Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* \<alpha>" shows "\<Psi> \<rhd> P \<parallel> Q \<longmapsto> \<alpha> \<prec> P' \<parallel> Q"
update_apply: fixes destr :: "'a \<Rightarrow> 'b"   and constr :: "'c \<Rightarrow> 'a"   and n :: "'d"   and f :: "'b \<Rightarrow> 'c"   and s :: "'d \<Rightarrow> 'a"   and x :: "'d" shows "update destr constr n f s x = (if x = n then constr (f (destr (s n))) else s x)"
diff_chain_at: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and x :: "'b"   and g :: "'c \<Rightarrow> 'd"   and g' :: "'c \<Rightarrow> 'd" assumes "(f has_derivative f') (at x)"   and "(g has_derivative g') (at (f x))" shows "(g \<circ> f has_derivative g' \<circ> f') (at x)"
set_bits_int_unfold': fixes f :: "nat \<Rightarrow> bool" shows "set_bits f = (if \<exists>n. \<forall>n'\<ge>n. \<not> f n' then let n = LEAST n. \<forall>n'\<ge>n. \<not> f n' in horner_sum of_bool 2 (map f [0..<n]) else if \<exists>n. \<forall>n'\<ge>n. f n' then let n = LEAST n. \<forall>n'\<ge>n. f n' in signed_take_bit n (horner_sum of_bool 2 (map f [0..<n] @ [True])) else 0)"
singleton_insert_inj_eq: fixes b :: "'g"   and a :: "'g"   and A :: "'g set" shows "({b} = insert a A) = (a = b \<and> A \<subseteq> {b})"
add_eq_times_2_in_float: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "a + b = c * 2"   and "is_float a"   and "is_float b" shows "is_float c"