hcju/isabelleps huggingface dataset:
twoChainVertDiv_of_itself: fixes C :: "(real \<times> real \<Rightarrow> real \<times> real) set" assumes "finite C"   and "\<forall>(k, \<gamma>)\<in>two_chain_boundary C. valid_path \<gamma>" shows "only_vertical_division (two_chain_boundary C) C"
continuous_on_open: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous_on S f = (\<forall>T. openin (top_of_set (f ` S)) T \<longrightarrow> openin (top_of_set S) (S \<inter> f -` T))"
Sup_least: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set"   and z :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq x z" shows "less_eq (Sup A) z"
mor_def: fixes B1 :: "'b set"   and B2 :: "'c set"   and s1 :: "'b \<Rightarrow> ('d, 'b, 'c) F1"   and s2 :: "'c \<Rightarrow> ('d, 'b, 'c) F2"   and B1' :: "'e set"   and B2' :: "'f set"   and s1' :: "'e \<Rightarrow> ('d, 'e, 'f) F1"   and s2' :: "'f \<Rightarrow> ('d, 'e, 'f) F2"   and f :: "'b \<Rightarrow> 'e"   and g :: "'c \<Rightarrow> 'f" shows "mor B1 B2 s1 s2 B1' B2' s1' s2' f g = (((\<forall>a\<in>B1. f a \<in> B1') \<and> (\<forall>a\<in>B2. g a \<in> B2')) \<and> (\<forall>z\<in>B1. F1map id f g (s1 z) = s1' (f z)) \<and> (\<forall>z\<in>B2. F2map id f g (s2 z) = s2' (g z)))"
assms_2: shows "h2 = \<langle>hs2, x2, \<langle>\<rangle>\<rangle>"
is_pole_tendsto: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "is_pole f x" shows "(inverse \<circ> f) \<midarrow>x\<rightarrow> (0::'c)"
neg_equal_0_iff_equal: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.group_add minus plus zero uminus" shows "(uminus a = zero) = (a = zero)"
rotate9: fixes A9 :: "fm"   and A1 :: "fm"   and A2 :: "fm"   and A3 :: "fm"   and A4 :: "fm"   and A5 :: "fm"   and A6 :: "fm"   and A7 :: "fm"   and A8 :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A9 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 H)))))))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 H)))))))) \<turnstile> B"
d1_5: fixes K1 :: "'a set"   and K2 :: "'a set" assumes "K1 \<in> p1"   and "K2 \<in> p1"   and "K1 \<noteq> K2" shows "interior K1 \<inter> interior K2 = {}"
unsafe_safe_eq_1: shows "OF_priority_match OF_match_fields_unsafe oft = OF_priority_match OF_match_fields_safe oft"
E_wf_1: fixes G :: "'b graph" assumes "wf_graph G" shows "fst ` edges G \<subseteq> nodes G"
UNIV_2: fixes A :: "'a set" shows "(A \<subseteq> UNIV) = True"
disjunctive_SUP_L_P: fixes R :: "'b \<Rightarrow> 'f \<Rightarrow> 'g"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'f"   and u :: "'a"   and i :: "'b" assumes "Disjunctive_fun R" shows "(R .. SUP_L_P P (pair u i)) = SUP_L_P (\<lambda>w. R .. P w) (pair u i)"
Collect_empty_eq_bot: fixes P :: "'d \<Rightarrow> bool" shows "(Collect P = {}) = (P = bot)"
degree_aform_err_powr_aform_err: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real"   and Z :: "(real \<times> real pdevs) \<times> real"   and Y :: "(real \<times> real pdevs) \<times> real"   and n :: "nat" assumes "powr_aform_err p X Z = Some Y"   and "degree_aform_err X \<le> n"   and "degree_aform_err Z \<le> n" shows "degree_aform_err Y \<le> n"
nfa_trans_def: fixes A :: "bool list bdd list \<times> bool list"   and qs :: "bool list"   and bs :: "bool list" shows "nfa_trans A qs bs = bdd_lookup (subsetbdd (fst A) qs (nfa_emptybdd (length qs))) bs"
admits_diagonal_imp_admits_triangular_1xn: fixes n :: "nat" assumes "Ball (carrier_mat 1 2) admits_diagonal_reduction" shows "Ball (carrier_mat 1 n) admits_triangular_reduction"
rep_eq: fixes p :: "'c pmf"   and s :: "'c set" assumes "set_pmf p \<inter> s \<noteq> {}" shows "measure_pmf (cond_pmf p s) = uniform_measure (measure_pmf p) s"
subspaceI: fixes S :: "('a \<Rightarrow>\<^sub>0 'b) set" assumes "0 \<in> S"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x + y \<in> S"   and "\<And>c x. x \<in> S \<Longrightarrow> c \<cdot> x \<in> S" shows "phull.subspace S"
Trans: shows "[a\<noteq>b]P \<longmapsto> Rs"
comp_eq_dest_lhs: fixes a :: "'f \<Rightarrow> 'e"   and b :: "'d \<Rightarrow> 'f"   and c :: "'d \<Rightarrow> 'e"   and v :: "'d" assumes "a \<circ> b = c" shows "a (b v) = c v"
replicate_0: fixes x :: "'d" shows "x\<^bsup>0\<^esup> = []"
i4: shows "ideal_generated {a1 * a1', b1 * a1', c1} = ideal_generated {1::'a}"
directional_derivative_is_derivation: fixes f :: "'c \<Rightarrow> real"   and k :: "enat"   and g :: "'c \<Rightarrow> real"   and a :: "'c"   and x :: "'c" assumes "f \<in> manifold_eucl.diff_fun_space k"   and "g \<in> manifold_eucl.diff_fun_space k"   and "k \<noteq> 0" shows "directional_derivative k a x (f * g) = f a * directional_derivative k a x g + g a * directional_derivative k a x f"
hd_sort_Min: fixes l :: "'a list" assumes "l \<noteq> []" shows "hd (sort l) = Min (set l)"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (branch_instr instr s1)) \<and> t2 = snd (fst (branch_instr instr s2))"
dD0k: fixes \<rho>' :: "complex mat"   and k :: "nat" assumes "\<rho>' \<in> density_states" shows "D0 k \<rho>' \<in> carrier_mat d d"
x_split: shows "ops = as @ [(xid, xr)] @ as1 \<and> (\<forall>i\<in>set (map fst as). i < xid) \<and> (\<forall>i\<in>set (map fst as1). xid < i)"
LCons_4: shows "\<not> lnull (lSup (snd ` Y))"
univ_poly_carrier_units: fixes R :: "('c, 'd) ring_scheme" assumes "field R" shows "Units (poly_ring R) = {[k] |k. k \<in> carrier R - {\<zero>\<^bsub>R\<^esub>}}"
intros_2: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event list"   and c'' :: "('p, 's, 'm) configuration" assumes "c \<turnstile> ev \<mapsto> c'"   and "trace c' t c''" shows "trace c (ev # t) c''"
assms_10: shows "b \<otimes> snd h [^] e' = g' [^] z'"
component_crit_def: fixes data :: "nat \<times> 'e"   and gs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and p :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and q :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g" shows "component_crit data gs bs hs p q = (component_of_term (lt (fst p)) \<noteq> component_of_term (lt (fst q)))"
exp_conj_comm: fixes a :: "'c \<Rightarrow> real"   and b :: "'c \<Rightarrow> real" shows "a && b = b && a"
bisim1AAssFail: fixes P :: "'b prog"   and h :: "'heap"   and a :: "'addr expr1"   and i :: "'addr expr1"   and e :: "'addr expr1"   and ad :: "'addr"   and xs :: "'addr val list"   and v' :: "'addr val"   and v :: "'addr val"   and v'' :: "'addr val" shows "P,a\<lfloor>i\<rceil> := e,h \<turnstile> (Throw ad, xs) \<leftrightarrow> ([v', v, v''], xs, length (compE2 a) + length (compE2 i) + length (compE2 e), \<lfloor>ad\<rfloor>)"
eval_monom_1: fixes \<alpha> :: "'d \<Rightarrow> 'c" shows "eval_monom \<alpha> 1 = (1::'c)"
pred_rel_simps_2: fixes s :: "('a, 'e) simple_state_scheme"   and s' :: "('a, 'e) simple_state_scheme" assumes "(s, s') \<in> rel" shows "is_empty_stack_impl s = is_empty_stack_impl s'"
jump_poly_def: fixes q :: "real poly"   and p :: "real poly"   and x :: "real" shows "jump_poly q p x \<equiv> if p \<noteq> 0 \<and> q \<noteq> 0 \<and> odd (order x p - order x q) then if sign_r_pos (q * p) x then 1 else - 1 else 0"
norm_18: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "(\<lambda>s. \<not> (\<diamond>P) s) = \<box>(\<lambda>s. \<not> P s)"
length_mapWithIndex: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> 'a"   and xs :: "'b list" shows "length (mapWithIndex f xs) = length xs"
x2lt: shows "\<forall>x2<x1. sign_vec qs x1 \<noteq> sign_vec qs x2 \<longrightarrow> (\<exists>r<x1. x2 \<le> r \<and> (\<exists>q\<in>set qs. rpoly q r = 0))"
f: fixes x :: "'c" assumes "x \<in> T" shows "f differentiable at x"
simps_5: fixes b :: "bexp"   and c :: "com" shows "gen (WHILE b DO c) = vars b \<union> gen c"
k_2: shows "\<forall>x\<in>{0..1} - k. g differentiable at x within {0..1}"
WT_converter_parallel_converter2: fixes \<I>1 :: "('y, 'z) \<I>"   and \<I>2 :: "('aa, 'ab) \<I>"   and conv1 :: "('y, 'z, 'aa, 'ab) converter"   and \<I>1' :: "('ac, 'ad) \<I>"   and \<I>2' :: "('ae, 'af) \<I>"   and conv2 :: "('ac, 'ad, 'ae, 'af) converter" assumes "\<I>1, \<I>2 \<turnstile>\<^sub>C conv1 \<surd>"   and "\<I>1', \<I>2' \<turnstile>\<^sub>C conv2 \<surd>" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>1', \<I>2 \<oplus>\<^sub>\<I> \<I>2' \<turnstile>\<^sub>C conv1 |\<^sub>= conv2 \<surd>"
dlist_eqI: fixes dxs :: "'b dlist"   and dys :: "'b dlist" assumes "list_of_dlist dxs = list_of_dlist dys" shows "dxs = dys"
left_unique_rel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "left_unique R" shows "left_unique (list_all2 R)"
check: shows "check_poly_strict_mono_discrete (\<succ>) p v"
L_def: shows "L \<equiv> linepath (- complex_of_real R) (complex_of_real R)"
eij_def: fixes a :: "nat list"   and b :: "nat list"   and i :: "nat"   and j :: "nat" shows "eij a b i j = lcm (a ! i) (b ! j) div b ! j"
superposition_is_sound: fixes P1 :: "'a eclause"   and P2 :: "'a eclause"   and C :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list"   and k :: "inferences"   and C' :: "'a literal set" assumes "finite (cl_ecl P1)"   and "finite (cl_ecl P2)"   and "superposition P1 P2 C \<sigma> k C'" shows "set_entails_clause {cl_ecl P1, cl_ecl P2} (cl_ecl C)"
not_evaluate_dec_timeout: fixes mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and d :: "dec" assumes "\<forall>r. \<not> evaluate_dec False mn env s d r" shows "\<exists>r. evaluate_dec True mn env s d r \<and> snd r = Rerr (Rabort Rtimeout_error)"
isNotDSource_EmptyAcc: fixes i :: "AbstrLevelsID"   and S :: "CSet" assumes "isNotDSource i S" shows "Acc i S = {}"
swap_Snapshot_Recv: 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 "isSnapshot ev"   and "isRecv ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
OUT_g_A: fixes x :: "'v" assumes "x \<in> A \<Gamma>" shows "d_OUT g x = d_OUT f x + d_IN j \<langle>x\<rangle> - d_OUT j \<langle>x\<rangle>"
comp_injective_below_complement: fixes y :: "'a"   and x :: "'a" assumes "injective y" shows "- x * y \<le> - (x * y)"
vector_mult_component: fixes x :: "('a, 'b) vec"   and y :: "('a, 'b) vec"   and i :: "'b" shows "(x * y) $ i = x $ i * y $ i"
max_fun_diff_alt: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" shows "s.max_fun_diff f g = s.maxim ((rs.SUPP f \<union> rs.SUPP g) \<inter> {a. f a \<noteq> g a})"
S6MClkReply_enabled: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> S6 rmhist p \<longrightarrow> Enabled (<MClkReply memCh crCh cst p>_(c p))"
get_ancestors_reads: fixes h :: "(_) heap"   and node_ptr :: "(_) object_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h" shows "reads Shadow_DOM.get_ancestors_locs (Shadow_DOM.get_ancestors node_ptr) h h'"
Always_LeadsTo_weaken: fixes F :: "'c program"   and C :: "'c set"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> Always C"   and "F \<in> A \<longmapsto>w A'"   and "C \<inter> B \<subseteq> A"   and "C \<inter> A' \<subseteq> B'" shows "F \<in> B \<longmapsto>w B'"
finite_two_chain_horizontal_boundary: fixes two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set" assumes "finite two_chain" shows "finite (two_chain_horizontal_boundary two_chain)"
synIfSC_def: fixes e :: "('c, 'd) Expr"   and c1 :: "(('c, 'd) Expr, 'c) MWLfCom"   and c2 :: "(('c, 'd) Expr, 'c) MWLfCom" shows "synIfSC e c1 c2 \<equiv> \<forall>d. \<not> e \<equiv>\<^bsub>d\<^esub> e \<longrightarrow> [c1] \<approx>\<^bsub>d\<^esub> [c2]"
update_convs_1: fixes c_work' :: "('a \<Rightarrow> 'b zmultiset) \<Rightarrow> 'a \<Rightarrow> 'b zmultiset"   and c_work :: "'a \<Rightarrow> 'b zmultiset"   and c_pts :: "'a \<Rightarrow> 'b zmultiset"   and c_imp :: "'a \<Rightarrow> 'b zmultiset"   and more :: "'c" shows "c_work_update c_work' \<lparr>c_work = c_work, c_pts = c_pts, c_imp = c_imp, \<dots> = more\<rparr> = \<lparr>c_work = c_work' c_work, c_pts = c_pts, c_imp = c_imp, \<dots> = more\<rparr>"
ApplyExplainUIPTermination: fixes state :: "State"   and F0 :: "Literal list list" assumes "InvariantUniq (getM state)"   and "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) (set (getQ state))"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantClCurrentLevel (getCl state) (getM state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantCnCharacterization (getCn state) (getC state) (getM state)"   and "InvariantCEntailed (getConflictFlag state) F0 (getC state)"   and "InvariantEquivalentZL (getF state) (getM state) F0"   and "getConflictFlag state"   and "0 < currentLevel (getM state)" shows "applyExplainUIP_dom state"
distinct_269: fixes x12a :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Exp x12a \<noteq> x151 ^\<^sub>e x152"
aext_var: fixes x :: "'d \<Longrightarrow> 'f"   and a :: "'f \<Longrightarrow> 'e" shows "x \<oplus>\<^sub>p a = x ;\<^sub>L a"
order_power: fixes p :: "'a poly"   and c :: "'a"   and n :: "nat" assumes "p \<noteq> 0" shows "order c (p ^ n) = n * order c p"
1: shows "convergent_subseq s n n = residue (p ^ n) (acc_point s n)"
1: shows "Right_a.iso \<phi> \<and> \<phi> \<in> Right_a.hom (f \<star> a) (g \<star> a)"
inside_Gromov_distance_approx: fixes C :: "ereal" assumes "C < \<infinity>" shows "\<exists>e>0. \<forall>x y. extended_Gromov_distance (to_Gromov_completion basepoint) x \<le> C \<longrightarrow> dist x y \<le> e \<longrightarrow> esqrt (extended_Gromov_distance x y) \<le> 2 * ereal (dist x y)"
ap_Nil: fixes fs :: "('d \<Rightarrow> 'c) list" shows "fs \<diamondop> [] = []"
lookup_default': fixes x :: "'d"   and d :: "'c"   and m :: "('d, 'c) mapping"   and y :: "'d" shows "Mapping.lookup (Mapping.default x d m) y = (if x = y then Some (Mapping.lookup_default d m x) else Mapping.lookup m y)"
uMult_commute: fixes x :: "nat discr"   and y :: "nat discr" shows "x *\<^sub># y = y *\<^sub># x"
const_type_of_lookup_code: fixes \<Sigma> :: "exesignature"   and n :: "String.literal" shows "const_type (translate_signature \<Sigma>) n = lookup (\<lambda>k. k = n) (execonst_type_of \<Sigma>)"
simps_295: fixes f1 :: "String.literal \<Rightarrow> trm list \<Rightarrow> 'a"   and f2 :: "String.literal \<Rightarrow> formula \<Rightarrow> formula \<Rightarrow> 'a"   and f3 :: "trm \<Rightarrow> trm \<Rightarrow> 'a"   and f4 :: "trm \<Rightarrow> trm \<Rightarrow> 'a"   and f5 :: "trm \<Rightarrow> trm \<Rightarrow> 'a"   and f6 :: "formula \<Rightarrow> 'a"   and f7 :: "formula \<Rightarrow> formula \<Rightarrow> 'a"   and f8 :: "formula \<Rightarrow> formula \<Rightarrow> 'a"   and f9 :: "formula list \<Rightarrow> 'a"   and f10 :: "formula \<Rightarrow> 'a"   and f11 :: "nat \<Rightarrow> agg_type \<times> event_data \<Rightarrow> nat \<Rightarrow> trm \<Rightarrow> formula \<Rightarrow> 'a"   and f12 :: "\<I> \<Rightarrow> formula \<Rightarrow> 'a"   and f13 :: "\<I> \<Rightarrow> formula \<Rightarrow> 'a"   and f14 :: "formula \<Rightarrow> \<I> \<Rightarrow> formula \<Rightarrow> 'a"   and f15 :: "formula \<Rightarrow> \<I> \<Rightarrow> formula \<Rightarrow> 'a"   and f16 :: "\<I> \<Rightarrow> formula Regex.regex \<Rightarrow> 'a"   and f17 :: "\<I> \<Rightarrow> formula Regex.regex \<Rightarrow> 'a"   and x6 :: "formula" shows "(case Neg x6 of Pred x xa \<Rightarrow> f1 x xa | formula.Let x xa xb \<Rightarrow> f2 x xa xb | formula.Eq x xa \<Rightarrow> f3 x xa | Less x xa \<Rightarrow> f4 x xa | LessEq x xa \<Rightarrow> f5 x xa | Neg x \<Rightarrow> f6 x | Or x xa \<Rightarrow> f7 x xa | And x xa \<Rightarrow> f8 x xa | Ands x \<Rightarrow> f9 x | Exists x \<Rightarrow> f10 x | Agg x xa xb xc xd \<Rightarrow> f11 x xa xb xc xd | Prev x xa \<Rightarrow> f12 x xa | Next x xa \<Rightarrow> f13 x xa | Since x xa xb \<Rightarrow> f14 x xa xb | Until x xa xb \<Rightarrow> f15 x xa xb | MatchF x xa \<Rightarrow> f16 x xa | MatchP x xa \<Rightarrow> f17 x xa) = f6 x6"
u_antimono: fixes x :: "real"   and y :: "real" assumes "x \<le> y"   and "local.t_stop \<le> x" shows "u y \<le> u x"
hom_lead_coeff: fixes p :: "'b poly" shows "lead_coeff (map_poly to_fract p) = to_fract (lead_coeff p)"
canon_ang_diff: fixes \<alpha> :: "real"   and \<beta> :: "real" shows "\<downharpoonright>\<alpha> - \<beta>\<downharpoonleft> = \<downharpoonright>\<downharpoonright>\<alpha>\<downharpoonleft> - \<downharpoonright>\<beta>\<downharpoonleft>\<downharpoonleft>"
is_Sup_uniq: fixes A :: "'b set"   and sup :: "'b"   and sup' :: "'b" assumes "is_Sup A sup"   and "is_Sup A sup'" shows "sup = sup'"
i_extension_def: fixes f :: "'e \<Rightarrow> 'd"   and R :: "('e, 'f) Ring_scheme"   and R' :: "('d, 'g) Ring_scheme"   and I :: "'e set" shows "i_extension f R R' I = sum_mult R' (f ` I) (carrier R')"
comp_fun_idem_axioms: shows "comp_fun_idem (\<lambda>x yo. yo \<bind> m x)"
fact_ge_1: fixes n :: "nat" shows "(1::'c) \<le> fact n"
mult_less_0_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less (times a b) zero = (less zero a \<and> less b zero \<or> less a zero \<and> less zero b)"
supd_init_obj: fixes G :: "prog"   and oi :: "obj_tag"   and r :: "oref"   and s :: "state" shows "supd (init_obj G oi r) s = (fst s, init_obj G oi r (snd s))"
enc_atom_lang_Arbitrary_Except_True: fixes r :: "nat"   and I :: "('b + 'b set) list"   and p :: "'b"   and n :: "nat"   and a :: "'a" assumes "r < length I"   and "case I ! r of Inl p' \<Rightarrow> p = p' | Inr P \<Rightarrow> p \<in> P"   and "length I = n"   and "a \<in> set \<Sigma>" shows "[enc_atom I p a] \<in> lang n (Atom (Arbitrary_Except r True))"
lex_2: shows "Counterclockwise_2D_Arbitrary.lex p r"
config_rand_distinct: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and init :: "'c list"   and qs :: "'c list" shows "\<forall>x\<in>set_pmf (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs). distinct (fst x) = distinct init"
simps_2: fixes e :: "'a exp"   and es :: "'a exp list"   and A :: "'a set option" shows "\<D>s (e # es) A = (\<D> e A \<and> \<D>s es (A \<squnion> \<A> e))"
image_group_def: fixes f :: "'d \<Rightarrow> 'c"   and G :: "('d, 'e) monoid_scheme" shows "image_group f G \<equiv> \<lparr>carrier = f ` carrier G, mult = \<lambda>a b. f (inv_into (carrier G) f a \<otimes>\<^bsub>G\<^esub> inv_into (carrier G) f b), one = f \<one>\<^bsub>G\<^esub>\<rparr>"
A: shows "g_add_dj m1 m2 = iterate_add_to_map m1 update_dj (iteratei m2)"
inv_correspondence_into: fixes x :: "'a permutation" assumes "x \<in> pG" shows "\<ii>\<pp> x \<in> G"
Fun_2: shows "\<forall>x\<in>vars\<^sub>t (Fun f ts). \<sigma>\<^sub>1 x = \<sigma>\<^sub>2 x"
non_gen_dec: fixes u :: "'b list"   and G :: "'b list set" assumes "u \<in> \<langle>G\<rangle>"   and "u \<notin> G" shows "Dec G u \<noteq> [u]"
F_def: fixes g :: "'d" shows "F g = (if D.arr g then initial_arrow_to_functor.the_ext (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G (Fo (D.dom g)) (\<eta>o (D.dom g)) (Fo (D.cod g)) (\<eta>o (D.cod g) \<cdot>\<^sub>D g) else C.null)"
P_mod_whole: fixes p :: "'a" assumes "p \<in> carrier R" shows "P_mod R S X (carrier S) p"
boolean_is_huntington: shows "class.huntington_algebra uminus (\<sqinter>) (\<squnion>) \<bottom> \<top>"
appendr_correct_1: fixes s :: "'queue"   and x :: "'t" assumes "queue_invar s" shows "queue_\<alpha> (queue_enqueue x s) = queue_\<alpha> s @ [x]"
remove_child_children_subset: fixes h :: "(_) heap"   and parent :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and children' :: "(_) node_ptr list" assumes "h \<turnstile> remove_child parent child \<rightarrow>\<^sub>h h'"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children'"   and "DocumentClass.known_ptrs h"   and "DocumentClass.type_wf h" shows "set children' \<subseteq> set children"
Abs_game_inverse: fixes y :: "ZF" assumes "y \<in> game" shows "Rep_game (Abs_game y) = y"
convert_TA_initial_inject: fixes ta :: "('a, 'b, 'c, 'd, 'e, 'f) thread_action"   and ta' :: "('a, 'b, 'c, 'd, 'e, 'f) thread_action" shows "(convert_TA_initial ta = convert_TA_initial ta') = (ta = ta')"
size_mset_set: fixes A :: "'a set" shows "size (mset_set A) = card A"
lr_3: shows "(s \<circ> r) \<longlonglongrightarrow> l"
case_swap: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'b"   and p :: "'d \<times> 'c" shows "(case prod.swap p of (y, x) \<Rightarrow> f x y) = (case p of (x, y) \<Rightarrow> f x y)"
find_path1_gen_correct: fixes E :: "('b \<times> 'b) set"   and u0 :: "'b"   and P :: "'b \<Rightarrow> bool" shows "find_path1_gen E u0 P \<le> find_path1 E u0 P"
sclose_sopen_eq_t: fixes s :: "char list"   and t :: "sterm"   and p :: "char list"   and n :: "nat" assumes "s \<notin> FV t"   and "p \<notin> FV t"   and "s \<noteq> p" shows "{n \<leftarrow> [s,p]} {n \<rightarrow> [Fvar s,Fvar p]} t = t"
deterministic_step_def: fixes S :: "'b \<times> 'c \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'c) pmf" shows "deterministic_step S = (\<forall>i is q. card (set_pmf (S (i, is) q)) = 1)"
outstanding_non_volatile_refs_conv: fixes sb :: "'b memref list" shows "outstanding_refs (Not \<circ> is_volatile) sb = outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b sb \<union> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b sb"
igWlsInp_def: fixes MOD :: "('c, 'd, 'e, 'sort, 'f, 'g, 'h, 'i, 'j) Iteration.model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'h option" shows "igWlsInp MOD delta inp \<equiv> wlsOpS delta \<and> sameDom (arOf delta) inp \<and> liftAll2 (igWls MOD) (arOf delta) inp"
in_set_inl_lex: fixes x :: "real \<times> real"   and xs :: "(real \<times> real) pdevs" assumes "x \<in> set (inl xs)" shows "Counterclockwise_2D_Arbitrary.lex x 0"
sq_norm_fs_via_sum_mu_gso: fixes i :: "nat" assumes "i < length fs" shows "rat_of_int \<parallel>fs ! i\<parallel>\<^sup>2 = (\<Sum>j\<leftarrow>[0..<Suc i]. (fs.gs.\<mu> i j)\<^sup>2 * \<parallel>fs.gs.gso j\<parallel>\<^sup>2)"
equal_fset_def: fixes A :: "'a fset"   and B :: "'a fset" shows "equal_class.equal A B = (A |\<subseteq>| B \<and> B |\<subseteq>| A)"
r_cancel_one: fixes G :: "('f, 'g) monoid_scheme"   and x :: "'f"   and a :: "'f" assumes "Group.group G"   and "x \<in> carrier G"   and "a \<in> carrier G" shows "(a \<otimes>\<^bsub>G\<^esub> x = x) = (a = \<one>\<^bsub>G\<^esub>)"
D_def: fixes fs :: "int vec list" shows "D fs = nat (prod (d fs) {..<m})"