hcju/isabelleps huggingface dataset:
next_send: fixes c :: "('a, 'b, 'c) configuration"   and i :: "nat"   and p :: "'a"   and q :: "'a"   and s :: "'b"   and s' :: "'b"   and msg :: "'c"   and c' :: "('a, 'b, 'c) configuration" shows "c \<turnstile> Send i p q s s' msg \<mapsto> c' = (can_occur (Send i p q s s' msg) c \<and> states c' p = s' \<and> (\<forall>r. r \<noteq> p \<longrightarrow> states c' r = states c r) \<and> msgs c' i = msgs c i @ [Msg msg] \<and> (\<forall>i'. i \<noteq> i' \<longrightarrow> msgs c' i' = msgs c i') \<and> no_snapshot_change c c')"
Nv1_inter_E2_empty: shows "N\<^bsub>\<V>1\<^esub> \<inter> E\<^bsub>ES2\<^esub> = {}"
prime_elem_def: fixes p :: "'a" shows "prime_elem p = (p \<noteq> (0::'a) \<and> \<not> p dvd (1::'a) \<and> (\<forall>a b. p dvd a * b \<longrightarrow> p dvd a \<or> p dvd b))"
absorb1: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b" shows "min a b = a"
sel_1: fixes x1 :: "bool"   and x2 :: "nat" shows "pos (Lit x1 x2) = x1"
put_M_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and setter :: "('ag \<Rightarrow> 'ag) \<Rightarrow> ('z, 'x, 'y, 'aa, 'ad option) RElement_scheme \<Rightarrow> ('z, 'x, 'y, 'aa, 'ad option) RElement_scheme"   and v :: "'ag" assumes "h \<turnstile> ok put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr setter v" shows "ptr |\<in>| element_ptr_kinds h"
vector_2_1: fixes x :: "'a"   and y :: "'a" shows "vector [x, y] $ (1::'b) = x"
inf_Int_eq: fixes R :: "'f set"   and S :: "'f set" shows "inf (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
finsum_vec: shows "finsum_vec TYPE('a) n = finsum V"
source_in: fixes x :: "'v" shows "\<not> edge \<Delta> x (source \<Delta>)"
Close: fixes i :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and A :: "'c set"   and n :: "nat" assumes "p at i in' set branch"   and "(\<^bold>\<not> p) at i in' set branch" shows "A, n \<turnstile> branch"
forget: fixes x :: "name"   and L :: "lam"   and P :: "lam" assumes "x \<sharp> L" shows "L[x::=P] = L"
hyps: assumes "k < ncols A" shows "Gram_Schmidt_upt_k A k = foldl Gram_Schmidt_column_k_efficient A [0..<Suc k]"
state_change_on_neither_2: fixes i :: "nat"   and t :: "nat" assumes "gap i (Suc t) = gap i t" shows "block i (Suc t) = e_snoc (block i t) 0"
block_complement_def: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and b :: "'a set" assumes "incidence_system point_set block_collection" shows "incidence_system.block_complement point_set b \<equiv> point_set - b"
lower_truncate_ivl: fixes p :: "nat"   and x :: "real interval" shows "lower (truncate_ivl p x) = truncate_down p (lower x)"
sup_absorb2: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x + y = y"
tObsC_tStep_eq_inv: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envInit :: "('b, 'e, 'f) BEState list"   and envAction :: "('b, 'e, 'f) BEState \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('b, 'e, 'f) BEState \<Rightarrow> ('b, 'e, 'f) BEState"   and envVal :: "('b, 'e, 'f) BEState \<Rightarrow> 'c \<Rightarrow> bool"   and agents :: "'b odlist"   and envObsC :: "'e \<Rightarrow> 'h"   and t' :: "('b, 'e, 'f) BEState Trace"   and t :: "('b, 'e, 'f) BEState Trace"   and s :: "('b, 'e, 'f) BEState" assumes "FiniteDetBroadcastEnvironment jkbp envInit envAction envTrans envVal agents envObsC"   and "FiniteDetBroadcastEnvironment.tObsC envObsC t' = FiniteDetBroadcastEnvironment.tObsC envObsC (t \<leadsto> s)" shows "\<exists>t'' s'. t' = t'' \<leadsto> s'"
fls_shift_eq0_iff: fixes m :: "int"   and f :: "'a fls" shows "(fls_shift m f = 0) = (f = 0)"
deriv_sswSts_weighted_RP: shows "chain (\<leadsto>\<^sub>w) sswSts"
OclAnd_bot1: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'a state \<times> 'a state" assumes "X \<tau> \<noteq> false \<tau>" shows "(UML_Types.bot_class.bot and X) \<tau> = UML_Types.bot_class.bot \<tau>"
finite_or_infinite_iteration_axioms: shows "finite_or_infinite_iteration (;) nil"
AE_E_prob: fixes P :: "'s stream \<Rightarrow> bool"   and s :: "'s" assumes "almost_everywhere (T s) P" obtains S :: "'s stream set" where "S \<subseteq> {x \<in> space (T s). P x}" and "S \<in> T.events s" and "T.prob s S = 1"
Abs_swap1_1: fixes a :: "atom"   and x :: "'a"   and bs :: "atom set"   and b :: "atom" assumes "a \<notin> supp x - bs"   and "b \<notin> supp x - bs" shows "[bs]set. x = [(a \<rightleftharpoons> b) \<bullet> bs]set. (a \<rightleftharpoons> b) \<bullet> x"
measure_eqI: fixes M :: "'c measure"   and N :: "'c measure" assumes "sets M = sets N"   and "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A" shows "M = N"
rel_flip: fixes R :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b option"   and b :: "'c option" shows "rel_option R\<inverse>\<inverse> a b = rel_option R b a"
zero_eq_power2: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c" assumes "class.semiring_1_no_zero_divisors one times plus zero" shows "(power.power one times a 2 = zero) = (a = zero)"
Eq_Const: fixes c :: "event_data"   and d :: "event_data" shows "P (formula.Eq (Const c) (Const d))"
preserves_reflects_arr: fixes f :: "'a \<times> 'a \<times> 'a" shows "B.arr \<a>\<^sub>B[fst f, fst (snd f), snd (snd f)] = B.VVV.arr f"
div_normalize: fixes a :: "'a" shows "a div normalize a = unit_factor a"
a: shows "Cn {.\<not> (\<phi> .\<and>. \<psi>) .\<and>. .\<not> \<phi>} = Cn {.\<not> \<phi>}"
convergent_mult: fixes X :: "nat \<Rightarrow> 'a"   and Y :: "nat \<Rightarrow> 'a" assumes "convergent X"   and "convergent Y" shows "convergent (\<lambda>n. X n * Y n)"
1: shows "local.map (b, f) = S.mkArr (local.set (b, C.dom f)) (local.set (b, C.cod f)) (\<phi> (b, C.cod f) \<circ> (\<lambda>h. f \<cdot> h \<cdot> b) \<circ> \<psi> (b, C.dom f))"
qSwap_qSwapped: fixes X :: "('a, 'b, 'c, 'd, 'e) qTerm"   and zs :: "'c"   and x :: "'d"   and y :: "'d" shows "(X, X #[[x \<and> y]]_zs) \<in> qSwapped"
negated_normalized_folded_ports_normalized_n_primitive: fixes disc :: "'b common_primitive \<Rightarrow> bool"   and C :: "ipt_l4_ports \<Rightarrow> 'b common_primitive"   and pts :: "ipt_l4_ports list"   and x :: "'b common_primitive match_expr"   and sel :: "'b common_primitive \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> bool" assumes "\<forall>a. \<not> disc (C a)"   and "(\<forall>a. \<not> disc (Prot a)) \<or> pts = []"   and "x \<in> set (normalize_match (andfold_MatchExp (map (l4_ports_negate_one C) pts)))" shows "normalized_n_primitive (disc, sel) f x"
trivial_def: fixes x :: "'c set" shows "trivial x = (x \<subseteq> {the_elem x})"
bin_lcp_def: fixes u\<^sub>0 :: "'b list"   and u\<^sub>1 :: "'b list" assumes "binary_code u\<^sub>0 u\<^sub>1" shows "binary_code.\<alpha> u\<^sub>0 u\<^sub>1 = u\<^sub>0 \<cdot> u\<^sub>1 \<and>\<^sub>p u\<^sub>1 \<cdot> u\<^sub>0"
shift_simps_1: fixes xs :: "'c list"   and s :: "'c stream" shows "shd (xs @- s) = (if xs = [] then shd s else hd xs)"
line_extension_def: fixes K :: "'c set"   and a :: "'c"   and E :: "'c set" shows "S.line_extension K a E = K #>\<^bsub>S\<^esub> a <+>\<^bsub>S\<^esub> E"
segment_face_cycle_x_x_eq: fixes x :: "'b" shows "segment face_cycle_succ x x = face_cycle_set x - {x}"
take_Suc_conv_app_nth: fixes i :: "nat"   and xs :: "'a list" assumes "i < length xs" shows "take (Suc i) xs = take i xs @ [xs ! i]"
intro: fixes X :: "'a set"   and is_open :: "'a set \<Rightarrow> bool"   and U :: "'a set"   and I :: "real set"   and C :: "real \<Rightarrow> 'a set" assumes "topological_space X is_open"   and "cover_of_subset X U I C"   and "open_cover_of_subset_axioms is_open I C" shows "open_cover_of_subset X is_open U I C"
map2_map_map: fixes h :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and f :: "'g \<Rightarrow> 'e"   and xs :: "'g list"   and g :: "'g \<Rightarrow> 'f" shows "Map2.map2 h (map f xs) (map g xs) = map (\<lambda>x. h (f x) (g x)) xs"
\<phi>'\<psi>': shows "meta_adjunction C D F' F.G \<phi>' \<psi>'"
AudioPlayer_CTRL_States_def: shows "AudioPlayer_CTRL_States = {''Off'', ''On''}"
diff_fun_space_add: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "c_manifold charts k"   and "f \<in> c_manifold.diff_fun_space charts k"   and "g \<in> c_manifold.diff_fun_space charts k" shows "f + g \<in> c_manifold.diff_fun_space charts k"
fwd_bc: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and nid :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "active nid (t n)" shows "pout (cmp nid (t n)) = bc (cmp nid (t n))"
genset_order2: fixes S :: "'b set"   and s :: "'b" assumes "PreCoxeterSystem S"   and "s \<in> S" shows "add_order s = 2"
min_le2: fixes w :: "'a"   and y :: "'a" assumes "w \<in> S"   and "y \<in> S" shows "le (min_for w y) w"
pm_eq: shows "poincare_map p2 x0 = poincare_map p2 (flow0 x0 h)"
lower_arctan_float_interval: fixes p :: "nat"   and x :: "float interval" shows "lower (arctan_float_interval p x) = lb_arctan p (lower x)"
While: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and b :: "state \<Rightarrow> bool"   and c :: "com" assumes "C \<turnstile> {\<lambda>z s. P z s \<and> b s} c {P}" shows "C \<turnstile> {P} WHILE b DO c {\<lambda>z s. P z s \<and> \<not> b s}"
rsd_winners_finite: fixes agents' :: "'agent list"   and alts' :: "'alt set" assumes "set agents' \<subseteq> agents"   and "finite alts"   and "alts' \<subseteq> alts" shows "finite (rsd_winners R alts' agents')"
cinner_commute: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and x :: "'c"   and y :: "'c" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner x y = cnj (cinner y x)"
intro: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<And>a. f a a = a" shows "semilattice_axioms f"
arith_sum4: fixes t :: "nat"   and k :: "nat"   and d :: "nat" shows "t + 3 + k + d = Suc (t + 2 + k + d)"
simps_2: fixes t :: "ty"   and e :: "expr" shows "constVal (New t[e]) = None"
external_equiv_internal_invariant: fixes S :: "('b, 'a) spec"   and P :: "'b set"   and Q :: "'a set" assumes "obs S ` P = Q"   and "observable (obs S) P" shows "(oreach S \<subseteq> Q) = (reach S \<subseteq> P)"
range_col_a_r: fixes x :: "nat" assumes "x < k + 1" shows "col (a_r x) < k"
success_not_cond_effect: fixes h :: "heap" assumes "success p h"   and "\<not> cond p h" shows "effect p h h False"
insert_3: fixes \<iota>\<^sub>m :: "'a \<Rightarrow> 'b option" assumes "K = dom \<iota>\<^sub>m"   and "\<And>x. x \<in> dom \<iota>\<^sub>m \<Longrightarrow> finite (reach \<Sigma> (\<delta>\<^sub>m x) (the (\<iota>\<^sub>m x)))" shows "finite (reach \<Sigma> (\<Delta>\<^sub>\<times> \<delta>\<^sub>m) \<iota>\<^sub>m)"
insert_ops_memb_ref_older: fixes op_list :: "('a \<times> 'a option) list"   and oid :: "'a"   and ref :: "'a" assumes "insert_ops op_list"   and "(oid, Some ref) \<in> set op_list" shows "ref < oid"
wf_graph_add_subset_edges: fixes V :: "'b set"   and E :: "('b \<times> 'b) set"   and E' :: "('b \<times> 'b) set" assumes "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "E' \<subseteq> E" shows "wf_graph \<lparr>nodes = V, edges = E \<union> E'\<rparr>"
X_to_d: fixes d :: "nat" shows "X^\<^bsup>R d\<^esup> = polyn_expr R X d (ext_cf S d (C\<^sub>0 S))"
assms: shows "\<LM> A  \<RM> = \<Gamma> \<oplus> B"
simps_26: fixes v :: "'a rexp"   and va :: "'a rexp"   and vf :: "'a rexp" shows "local.le_rexp (Plus v va) (Star vf) = False"
Q_unfolded: shows "partition_without elem P = (\<lambda>X. X - {elem}) ` P - {{}}"
Pd': shows "\<forall>d. d \<noteq> Xd x \<longrightarrow> Pd' d = Pd d"
runit_char_2: fixes a :: "'a" assumes "ide a" shows "\<r>[a] \<otimes> \<I> = (a \<otimes> \<iota>) \<cdot> \<a>[a, \<I>, \<I>]"
extremum_strict: fixes a :: "'b" shows "\<not> \<top> < a"
unique_CryptKey: fixes B :: "agent"   and A :: "agent"   and SesKey :: "nat"   and T :: "msg"   and evs :: "event list"   and B' :: "agent"   and A' :: "agent"   and T' :: "msg" assumes "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key SesKey, T\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK B') \<lbrace>Agent A', Agent B', Key SesKey, T'\<rbrace> \<in> parts (knows Spy evs)"   and "Key SesKey \<notin> analz (knows Spy evs)"   and "evs \<in> kerbIV" shows "A = A' \<and> B = B' \<and> T = T'"
ennreal_tendsto_cmult: fixes c :: "ennreal"   and f :: "'b \<Rightarrow> ennreal"   and x :: "ennreal"   and F :: "'b filter" assumes "c < \<top>"   and "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. c * f x) \<longlongrightarrow> c * x) F"
impl_def: shows "impl = Sublists_Foldr_Impl local.create_subseqs local.next_subseqs"
Call: fixes G :: "'b triple set"   and P :: "'b assn"   and pn :: "pname"   and Q :: "'b assn"   and s' :: "state"   and X :: "vname"   and a :: "state \<Rightarrow> nat" assumes "G|-{P}. BODY pn .{\<lambda>Z s. Q Z (setlocs s (loc s')[X::=s<Res>])}" shows "G|-{\<lambda>Z s. s' = s \<and> P Z (setlocs s newlocs[Loc Arg::=a s])}. X:=CALL pn(a) .{Q}"
hull_def: fixes S :: "'b set \<Rightarrow> bool"   and s :: "'b set" shows "S hull s = \<Inter> {t. S t \<and> s \<subseteq> t}"
gen_le_dfp: fixes y :: "complex" assumes "y \<in> cball a r" shows "cmod (deriv f y) * (r - cmod (y - a)) / (r - cmod (p - a)) \<le> cmod (deriv f p)"
assms_1: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"
simps_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c" assumes "class.linorder less_eq less" shows "linorder.insort_key less_eq f x [] = [x]"
\<alpha>_ide_simp: fixes a :: "'c local.term set"   and b :: "'c local.term set"   and c :: "'c local.term set" assumes "ide a"   and "ide b"   and "ide c" shows "assoc a b c = \<a>[a, b, c]"
l6_6: fixes P :: "'p"   and A :: "'p"   and B :: "'p" assumes "P Out A B" shows "P Out B A"
e: shows "singular_chain p (subtopology X (topspace X \<inter> T)) e"
n_circ_3: fixes k :: "nat" shows "pathstart (n_circ k) = p + complex_of_real (h p)"
fs_int_mu_d_Z: fixes j :: "nat"   and ii :: "nat" assumes "j \<le> ii"   and "ii < length fs" shows "rat_of_int (fs'.d fs (Suc j)) * fs.gs.\<mu> ii j \<in> \<int>"
monom_mult_dist_right: fixes c :: "'b"   and t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'b"   and q :: "'a \<Rightarrow>\<^sub>0 'b" shows "MPoly_Type_Class.punit.monom_mult c t (p + q) = MPoly_Type_Class.punit.monom_mult c t p + MPoly_Type_Class.punit.monom_mult c t q"
harm_le: fixes n :: "nat" assumes "1 \<le> n" shows "harm n \<le> ln (real n) + 1"
eval_fds_norm_mult_le: fixes f :: "'a fds"   and s :: "real"   and g :: "'a fds" assumes "fds_abs_converges (fds_norm f) s"   and "fds_abs_converges (fds_norm g) s" shows "eval_fds (fds_norm (f * g)) s \<le> eval_fds (fds_norm f) s * eval_fds (fds_norm g) s"
KRP_graph_aux_def: shows "KRP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>KRP_aux y v x x'. x1 = (v, x, x') \<and> x2 = SyntaxN.Ex y (HRP x (Var y) AND SubstFormP v (Var y) x x') \<and> atom y \<sharp> (v, x, x'))"
rel_mset_size: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and M :: "'c multiset"   and N :: "'d multiset" assumes "rel_mset R M N" shows "size M = size N"
simps_2: fixes K :: "'b set"   and n :: "nat" shows "all_formulas_of_size K (Suc n) = (let af = \<Union> (set (map (all_formulas_of_size K) [0..<Suc n])) in (\<Union>F\<in>af. (\<Union>G\<in>af. if size F + size G \<le> Suc n then {F \<^bold>\<and> G, F \<^bold>\<or> G, F \<^bold>\<rightarrow> G} else {}) \<union> (if size F \<le> Suc n then {\<^bold>\<not> F} else {})) \<union> af)"
merge_id_def: fixes A :: "'e cset"   and f :: "'e \<Rightarrow> 'd cset Monomorphic_Monad.id" shows "merge_id A f = return_id (cUnion (cimage (extract \<circ> f) A))"
fmlookup_add: fixes m :: "'d \<rightharpoonup>\<^sub>f 'c"   and n :: "'d \<rightharpoonup>\<^sub>f 'c"   and x :: "'d" shows "fmlookup (m ++\<^sub>f n) x = (if x |\<in>| fmdom n then fmlookup n x else fmlookup m x)"
clear_signs: shows "signs_smash (get_signs (calculate_data p qs1)) (get_signs (calculate_data p qs2)) = get_signs (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))"
liftWT_def: shows "liftWT = (\<Lambda> m. WriterT\<cdot>(fmap\<cdot>(Writer\<cdot>mempty)\<cdot>m))"
r_append_zeros_def: shows "r_append_zeros \<equiv> r_swap (Pr 1 (recf.Id 1 0) (Cn 3 r_snoc [recf.Id 3 1, r_constn 2 0]))"
idc: shows "charpoly (HMA_Connect.map_matrix complex_of_real A) = prod_root_unity ks * map_poly complex_of_real f"
Rep_mod_ring_mod: fixes a :: "'a mod_ring" shows "Rep_mod_ring a mod int CARD('a) = Rep_mod_ring a"
list_insert_def: fixes equal :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" shows "equal_base.list_insert equal x xs = (if equal_base.list_member equal xs x then xs else x # xs)"
assms: shows "0 < Re z \<and> Re z < pi \<or> Re z = 0 \<and> 0 \<le> Im z \<or> Re z = pi \<and> Im z \<le> 0"
Quotient_uexpr: shows "Quotient (=) mk\<^sub>e Rep_uexpr cr_uexpr"
A: fixes Z :: "'a set" shows "(xs @ zs, Z) \<in> failures P \<longrightarrow> (xs @ y # ipurge_tr I D (D y) zs, ipurge_ref I D (D y) zs Z) \<in> failures P"
dbl_simps_3: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl plus one = numeral.numeral one plus (num.Bit0 num.One)"
E_def: shows "E \<equiv> {(u, v). c (u, v) \<noteq> 0}"
simps_2: fixes c :: "color"   and l :: "('a, 'b) rbt"   and c' :: "color"   and x :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "paint c (Node l c' x v r) = Node l c x v r"
errMOD_igFreshIGAbs1STR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igAbsIPresIGWls MOD"   and "igFreshIGAbs1 MOD" shows "igFreshIGAbs1STR (errMOD MOD)"
functor_def: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd" shows "functor A B F \<equiv> category A \<and> category B \<and> functor_axioms A B F"
\<gamma>_inf: fixes iv1 :: "ivl"   and iv2 :: "ivl" shows "\<gamma>_ivl (iv1 \<sqinter> iv2) = \<gamma>_ivl iv1 \<inter> \<gamma>_ivl iv2"
prj1_simps_2: fixes f :: "'a"   and g :: "'a" assumes "cospan f g" shows "local.dom \<p>\<^sub>1[f, g] = f \<down>\<down> g"
case_1: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'b"   and f3 :: "'b"   and x1 :: "'c" shows "(case flat_complete_lattice.Value x1 of flat_complete_lattice.Value x \<Rightarrow> f1 x | flat_complete_lattice.Bot \<Rightarrow> f2 | flat_complete_lattice.Top \<Rightarrow> f3) = f1 x1"
wt_elim_BinOp: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and binop :: "binop"   and e1 :: "expr"   and e2 :: "expr"   and T :: "tys"   and b :: "bool" assumes "E,dt\<Turnstile>In1l (BinOp binop e1 e2)\<Colon>T" obtains T1 :: "ty" and T2 :: "ty" and T3 :: "tys" where "E,dt\<Turnstile>e1\<Colon>-T1" and "E,dt\<Turnstile>e2\<Colon>-T2" and "wt_binop (prg E) binop T1 T2" and "E,dt\<Turnstile>(if b then In1l e2 else In1r Skip)\<Colon>T3" and "T = Inl (PrimT (binop_type binop))"
assms_3: shows "`\<not> b \<and> p \<Rightarrow> post`"
kbp_def: shows "kbp \<equiv> [\<lparr>guard = \<^bold>K\<^sub>Robot Kprop InGoal, action = Halt\<rparr>, \<lparr>guard = Knot \<^bold>K\<^sub>Robot Kprop InGoal, action = NOP\<rparr>]"
chain: fixes C :: "nat \<Rightarrow> 'a set" assumes "chain C" shows "chain (limit f \<circ> C)"
update_inf: fixes u :: "'a"   and y :: "'a"   and z :: "'a"   and x :: "'a" assumes "u \<le> y" shows "(x[y\<longmapsto>z]) \<sqinter> u = z\<^sup>T \<sqinter> u"
rel_with_target_impl_TRel_preserves_barbs: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_preserves_binary_pred Rel (\<lambda>a. hasBarb a (STCalWB SWB TWB))"   and "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel" shows "rel_preserves_binary_pred TRel (\<lambda>a. hasBarb a TWB)"
distinct_3: fixes x1 :: "xcpt"   and x3 :: "error" shows "Xcpt x1 \<noteq> Error x3"
1: shows "bprv (imp (all yy (neg (PPf (Var yy) \<langle>\<phi>R\<rangle>))) (neg (exi yy (PPf (Var yy) \<langle>\<phi>R\<rangle>))))"
StrongEq_L_subst3_rev: fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> 'b"   and y :: "'a state \<times> 'a state \<Rightarrow> 'b"   and P :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> x \<triangleq> y"   and "cp P" shows "(\<tau> \<Turnstile> P x) = (\<tau> \<Turnstile> P y)"
fun_upd_restrict: fixes m :: "'a \<Rightarrow> 'b option"   and D :: "'a set"   and x :: "'a"   and y :: "'b option" shows "(m |` D)(x := y) = (m |` (D - {x}))(x := y)"
add_le_imp_le_right: fixes a :: "'k multiset"   and c :: "'k multiset"   and b :: "'k multiset" assumes "a + c \<subseteq># b + c" shows "a \<subseteq># b"
pre: shows "pre @ [Broadcast (r, Store e mo r)] prefix of k"
f0: shows "(Matrix.mat (2 ^ n) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> 1 / sqrt 2 ^ n)) \<Otimes> Matrix.mat 2 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2))) $$ (i, j) = complex_of_real (1 / sqrt 2 ^ n) * Matrix.mat 2 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2)) $$ (i mod 2, j)"
connected_awalkE: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "wf_digraph G"   and "Digraph_Component.connected G"   and "u \<in> verts G"   and "v \<in> verts G" obtains p :: "('c \<times> 'c) list" where "pre_digraph.awalk (with_proj (mk_symmetric G)) u p v"
extr_fake: fixes X :: "chan"   and bad :: "agent set"   and IK :: "msg set"   and chan :: "chan set"   and IK' :: "msg set" assumes "X \<in> fake bad IK chan" shows "extr bad IK' {X} \<subseteq> IK \<union> extr bad IK' chan"
simps: fixes a1 :: "'b Heap"   and a2 :: "heap option"   and a3 :: "heap option"   and a4 :: "'b" shows "run a1 a2 a3 a4 = ((\<exists>c r. a1 = c \<and> a2 = None \<and> a3 = None \<and> a4 = r) \<or> (\<exists>c h r. a1 = c \<and> a2 = Some h \<and> a3 = None \<and> a4 = r \<and> execute c h = None) \<or> (\<exists>c h r h'. a1 = c \<and> a2 = Some h \<and> a3 = Some h' \<and> a4 = r \<and> execute c h = Some (r, h')))"