hcju/isabelleps huggingface dataset:
Zfun_imp_Zfun: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'c"   and K :: "real" assumes "Zfun f F"   and "\<forall>\<^sub>F x in F. norm (g x) \<le> norm (f x) * K" shows "Zfun g F"
VCG_step_inv_stable: fixes sys :: "('h, 'i, 'g, 'j, 'k, 'l) pre_system_scheme"   and I :: "('h, 'i, 'g, 'j, 'k) system_state \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> ('h, 'i, 'g, 'j, 'k) system_state" assumes "\<And>p. \<forall>(c, aft)\<in>vcg_fragments (PGMs sys p). PGMs sys, p, aft \<turnstile> \<lbrace>I\<rbrace> c"   and "prerun sys \<sigma>" shows "(\<box>(\<lambda>s. \<lceil>I\<rceil> s \<longrightarrow> (\<circle>\<lceil>I\<rceil>) s)) \<sigma>"
integrable_on_superset: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and t :: "'a set" assumes "f integrable_on S"   and "\<And>x. x \<notin> S \<Longrightarrow> f x = (0::'b)"   and "S \<subseteq> t" shows "f integrable_on t"
lens_obs_eq_def: fixes X :: "'d \<Longrightarrow> 'e"   and s\<^sub>1 :: "'e"   and s\<^sub>2 :: "'e" shows "(s\<^sub>1 \<simeq>\<^bsub>X\<^esub> s\<^sub>2) = (s\<^sub>1 = put\<^bsub>X\<^esub> s\<^sub>2 (get\<^bsub>X\<^esub> s\<^sub>1))"
summable_iff_convergent: fixes f :: "nat \<Rightarrow> 'b" shows "summable f = convergent (\<lambda>n. sum f {..<n})"
function_mult_comm: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F"   and "y \<in> carrier F" shows "x \<otimes>\<^bsub>F\<^esub> y = y \<otimes>\<^bsub>F\<^esub> x"
nullhomotopic_from_contractible: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set" assumes "continuous_on S f"   and "f ` S \<subseteq> T"   and "contractible S" obtains c :: "'c" where "homotopic_with_canon (\<lambda>h. True) S T f (\<lambda>x. c)"
inverse_transformations_axioms: shows "inverse_transformations A B G H \<tau> \<tau>'"
simps_2: fixes n :: "nat"   and ps :: "(real, 'b) vec list"   and p :: "(real, 'b) vec"   and k :: "'b"   and v :: "real"   and l :: "'b kdt"   and r :: "'b kdt" shows "nearest_nbors n ps p (Node k v l r) = (if p $ k \<le> v then let candidates = nearest_nbors n ps p l in if length candidates = n \<and> dist p (last candidates) \<le> dist v (p $ k) then candidates else nearest_nbors n candidates p r else let candidates = nearest_nbors n ps p r in if length candidates = n \<and> dist p (last candidates) \<le> dist v (p $ k) then candidates else nearest_nbors n candidates p l)"
conn2_3: shows "distinct (map of_int_hom.vec_hom fs')"
has_fields_is_class: fixes P :: "'a prog"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list" assumes "P \<turnstile> C has_fields FDTs" shows "is_class P C"
filter_eqI: fixes vs :: "'c list"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "\<And>v. v \<in> set vs \<Longrightarrow> P v = Q v" shows "filter P vs = filter Q vs"
invar: shows "2 * g * x = 2 * g * h + v * v"
swapEnv_defs_3: fixes xs :: "'f"   and x :: "'g"   and y :: "'g"   and rho :: "'f \<Rightarrow> 'g \<Rightarrow> 'h" shows "swapEnvDom xs x y rho \<equiv> \<lambda>zs z. rho zs (z @zs[x \<and> y]_xs)"
CI_def: fixes J :: "nat set" shows "CI J = distr (C 0 (up_to J) (\<lambda>x. undefined)) (Pi\<^sub>M J (\<lambda>_. M)) (\<lambda>f. restrict f J)"
preserves_dom: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>\<^sub>SC.dom (\<F>C.D f) = \<F>C.D (\<F>C.dom f)"
fps_unit_dvd_right: fixes f :: "'a fps" assumes "fps_nth f 0 \<noteq> (0::'a)" shows "\<exists>g. 1 = g * f"
sumCP_V: fixes v :: "'v" assumes "v \<in> V" shows "(\<Sum>g\<in>G. CP g v) \<in> V"
one_dim_loewner_order: fixes B :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a"   and A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a" shows "(B \<le> A) = (one_dim_iso B \<le> one_dim_iso A)"
simps_3: fixes f1 :: "'b"   and f2 :: "'b" shows "(case True of True \<Rightarrow> f1 | False \<Rightarrow> f2) = f1"
tendsto_at_top_iff: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c" shows "(f \<longlongrightarrow> l) at_top = (\<forall>e>0. \<exists>x0. \<forall>x\<ge>x0. dist (f x) l < e)"
filter_neg_atm_of_S: fixes C :: "'a literal multiset" shows "{#Neg (atm_of L). L \<in># S C#} = S C"
proper_interval_Compl_set_aux_fusion_def: fixes g1 :: "('a, 'd) generator"   and g2 :: "('a, 'e) generator"   and ao :: "'a option"   and s1 :: "'d"   and s2 :: "'e" shows "proper_interval_Compl_set_aux_fusion g1 g2 ao s1 s2 = proper_interval_Compl_set_aux ao (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
Borsuk_map_essential_bounded_component: fixes S :: "'a set"   and a :: "'a" assumes "compact S"   and "a \<notin> S" shows "bounded (connected_component_set (- S) a) = (\<nexists>c. homotopic_with_canon (\<lambda>x. True) S (sphere (0::'a) 1) (\<lambda>x. (x - a) /\<^sub>R norm (x - a)) (\<lambda>x. c))"
insort_not_Nil: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and xs :: "'b list" shows "insort_key f a xs \<noteq> []"
tensor_mat_id: shows "ps_P.tensor_mat (1\<^sub>m ps_P.d1) (1\<^sub>m ps_P.d2) = 1\<^sub>m ps_P.d"
PP_def: shows "wrepr.PP \<equiv> \<lambda>t. subst P t xx"
bet2_out_out: fixes B :: "'p"   and A :: "'p"   and B' :: "'p"   and C :: "'p"   and C' :: "'p" assumes "B \<noteq> A"   and "B' \<noteq> A"   and "A Out C C'"   and "Bet A B C"   and "Bet A B' C'" shows "A Out B B'"
dg_tiny_digraph_if_ge_Limit: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "tiny_digraph \<beta> \<AA>"
set_less_eq_aux''_refl: fixes A :: "'a set" shows "(A \<sqsubseteq>''' A) = (A \<in> infinite_complement_partition)"
neutral_const: fixes A :: "'c set" shows "(\<Sum>_\<in>A. (0::'b)) = (0::'b)"
fail_nondet_def: shows "fail_nondet = NondetT (return empty)"
healthy_wp_Embed: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "healthy t" shows "healthy (wp (Embed t))"
is_processT7_S: fixes s :: "'a event list"   and t :: "'a event list"   and P :: "'a process" assumes "tickFree s"   and "front_tickFree t"   and "s \<in> D P" shows "s @ t \<in> D P"
composition_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> G"   and "b \<in> G" shows "a \<cdot> b \<in> G"
2: fixes x :: "'a" assumes "x \<in> P"   and "p \<sqsubseteq> x" shows "False"
exhaust: fixes y :: "'e + 'f" obtains a :: "'e" where "y = pure a"   | b :: "'f" where "y = Inr b"
hyps_3: shows "Subobjs\<^sub>R P D Cs"
vsv_vimage_eqI: fixes a :: "V"   and b :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> f"   and "f\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> A" shows "b \<in>\<^sub>\<circ> f `\<^sub>\<circ> A"
eventually_at_topological: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c"   and s :: "'c set" shows "eventually P (at a within s) = (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x))"
trancl_into_trancl2: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set"   and c :: "'c" assumes "(a, b) \<in> r"   and "(b, c) \<in> r\<^sup>+" shows "(a, c) \<in> r\<^sup>+"
insert_not_elem: fixes x :: "'a"   and A :: "'a fset" assumes "x |\<notin>| A"   and "A \<noteq> {||}" shows "F (finsert x A) = x \<^bold>* F A"
trans_tm_abs: fixes e :: "name list"   and name :: "name"   and t :: "tm" shows "trans_tm (e @ [name]) t = abst_dbtm name (length e) (trans_tm e t)"
indicator_times_eq_if_2: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and S :: "'b set" shows "f x * indicator S x = (if x \<in> S then f x else (0::'a))"
inject_1: fixes x1 :: "('a, 'b) Term.term"   and y1 :: "('a, 'b) Term.term" shows "(send\<langle>x1\<rangle>\<^sub>s\<^sub>t = send\<langle>y1\<rangle>\<^sub>s\<^sub>t) = (x1 = y1)"
irreducible_primitive_part_fract: fixes p :: "'b fract poly" assumes "irreducible p" shows "irreducible (primitive_part_fract p)"
asIMOD_igFreshCls: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "sWlsDisj SEM" shows "igFreshCls (asIMOD SEM)"
1: shows "Inf (PLS s as) \<le> problem_plan_bound_s PROB s"
run_tell_state: fixes tell :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and w :: "'c"   and m :: "('b, 'a) stateT"   and s :: "'b" shows "run_state (tell_state tell w m) s = tell w (run_state m s)"
integral: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a"   and b :: "'a" assumes "field R"   and "a \<otimes>\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub>"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "a = \<zero>\<^bsub>R\<^esub> \<or> b = \<zero>\<^bsub>R\<^esub>"
delta_hedging_replicating: fixes der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and N :: "bool stream measure" assumes "der \<in> borel_measurable (G matur)"   and "N = bernoulli_stream ((1 + r - d) / (u - d))" shows "replicating_portfolio (delta_hedging N der matur) der matur"
AssignIdxE: fixes \<pi> :: "char list \<Rightarrow> com option"   and x :: "char list"   and i :: "aexp"   and a :: "aexp"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<pi>: (x[i] ::= a, s) \<Rightarrow> t" obtains "t = s(x := (s x)(aval i s := aval a s))"
proper_design_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" shows "proper_design point_set block_collection \<equiv> design point_set block_collection \<and> proper_design_axioms block_collection"
to_fract_quot_of_fract: fixes x :: "'a fract" assumes "snd (quot_of_fract x) = (1::'a)" shows "to_fract (fst (quot_of_fract x)) = x"
nonregular_event_induces_snapshot: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and c :: "('a, 'b, 'c) configuration"   and p :: "'a"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration" assumes "distributed_system channel"   and "\<not> ps c p \<noteq> None"   and "distributed_system.next channel trans send recv c ev c'"   and "occurs_on ev = p"   and "\<not> regular_event ev" shows "\<not> regular_event ev \<longrightarrow> ps c' p \<noteq> None"
closed_rel_frontier: fixes S :: "'a set" shows "closed (rel_frontier S)"
wary_Hd: fixes \<zeta> :: "('s, 'v) hd" shows "wary (Hd \<zeta>)"
flatmap_def: fixes l :: "'c list"   and f :: "'c \<Rightarrow> 'b list" shows "flatmap l f = List.maps f l"
induct: fixes x1 :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and x2 :: "com"   and x3 :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<Rightarrow> com \<Rightarrow> ((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<Rightarrow> bool" assumes "\<turnstile>\<^sub>t {x1} x2 {x3}"   and "\<And>P. P P SKIP P"   and "\<And>P a x. P (\<lambda>l s. P l (s[a/x])) (x ::= a) P"   and "\<And>P\<^sub>1 c\<^sub>1 P\<^sub>2 c\<^sub>2 P\<^sub>3. \<lbrakk>\<turnstile>\<^sub>t {P\<^sub>1} c\<^sub>1 {P\<^sub>2}; P P\<^sub>1 c\<^sub>1 P\<^sub>2; \<turnstile>\<^sub>t {P\<^sub>2} c\<^sub>2 {P\<^sub>3}; P P\<^sub>2 c\<^sub>2 P\<^sub>3\<rbrakk> \<Longrightarrow> P P\<^sub>1 (c\<^sub>1;; c\<^sub>2) P\<^sub>3"   and "\<And>P b c\<^sub>1 Q c\<^sub>2. \<lbrakk>\<turnstile>\<^sub>t {\<lambda>l s. P l s \<and> bval b s} c\<^sub>1 {Q}; P (\<lambda>l s. P l s \<and> bval b s) c\<^sub>1 Q; \<turnstile>\<^sub>t {\<lambda>l s. P l s \<and> \<not> bval b s} c\<^sub>2 {Q}; P (\<lambda>l s. P l s \<and> \<not> bval b s) c\<^sub>2 Q\<rbrakk> \<Longrightarrow> P P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) Q"   and "\<And>P x c b. \<lbrakk>\<turnstile>\<^sub>t {\<lambda>l. P (l(x := Suc (l x)))} c {P}; P (\<lambda>l. P (l(x := Suc (l x)))) c P; \<forall>l s. 0 < l x \<and> P l s \<longrightarrow> bval b s; \<forall>l s. l x = 0 \<and> P l s \<longrightarrow> \<not> bval b s\<rbrakk> \<Longrightarrow> P (\<lambda>l s. \<exists>n. P (l(x := n)) s) (WHILE b DO c) (\<lambda>l. P (l(x := 0)))"   and "\<And>P' P c Q Q'. \<lbrakk>\<forall>l s. P' l s \<longrightarrow> P l s; \<turnstile>\<^sub>t {P} c {Q}; P P c Q; \<forall>l s. Q l s \<longrightarrow> Q' l s\<rbrakk> \<Longrightarrow> P P' c Q'" shows "P x1 x2 x3"
M_Says_AuthReq: fixes AuthReqData :: "msg"   and LID_M :: "nat"   and XID :: "nat"   and HOIData :: "msg"   and HOD :: "msg"   and M :: "agent"   and P_I :: "msg"   and evs :: "event list" assumes "AuthReqData = \<lbrace>Number LID_M, Number XID, HOIData, HOD\<rbrace>"   and "sign (priSK M) \<lbrace>AuthReqData, Hash P_I\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> set_pur"   and "M \<notin> bad" shows "\<exists>j trans KM. Notes M \<lbrace>Number LID_M, Agent (PG j), trans\<rbrace> \<in> set evs \<and> Says M (PG j) (EncB (priSK M) KM (pubEK (PG j)) AuthReqData P_I) \<in> set evs"
map_hd_lem: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "0 < n" shows "f 0 # map f [1..<n] = map f [0..<n]"
snil_def: shows "[::] \<equiv> slist_abs\<cdot>(sinl\<cdot>ONE)"
AndL_sim: fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula set"   and H :: "'b formula" assumes "F \<triangleright> G \<triangleright> \<Gamma> \<turnstile> H" shows "F \<^bold>\<and> G \<triangleright> \<Gamma> \<turnstile> H"
closed_Un'': fixes S :: "'a set"   and U :: "'a set"   and T :: "'a set"   and \<tau> :: "'a set \<Rightarrow> \<bool>" assumes "S \<subseteq> U"   and "T \<subseteq> U"   and "topological_space_ow U \<tau>"   and "closed_ow U \<tau> S"   and "closed_ow U \<tau> T" shows "closed_ow U \<tau> (S \<union> T)"
snd_SUP: fixes f :: "'e \<Rightarrow> 'd \<times> 'c"   and A :: "'e set" shows "snd (Sup (f ` A)) = (SUP x\<in>A. snd (f x))"
soundI: fixes P :: "'a \<Rightarrow> real" assumes "Expectations.bounded P"   and "nneg P" shows "sound P"
flow_solves_ode: fixes t0 :: "real"   and x0 :: "'a" assumes "t0 \<in> T"   and "x0 \<in> S" shows "(flow t0 x0 solves_ode f) (existence_ivl t0 x0) S"
v: shows "\<forall>a\<in>{i. i < n}. [v = if a = i then [:s_i:] else if a = j then [:s_j:] else [:0:]] (mod m a)"
refl_NS_point: fixes s :: "('f, 'v) Term.term" shows "(s, s) \<in> NS"
simps_2: fixes name :: "name" shows "quot_dbtm (DBVar name) = ORD_OF (Suc (nat_of_name name))"
all: shows "\<forall>Ds. Subobjs P C Ds \<and> last Ds = D \<longrightarrow> Ds = Cs @ tl Cs'"
code: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "iterates f x = LCons x (iterates f (f x))"
angle_inverse: fixes A :: "'b"   and C :: "'b"   and B :: "'b"   and D :: "'b" assumes "between (A, C) B"   and "A \<noteq> B"   and "B \<noteq> C" shows "angle A B D = pi - angle C B D"
progressing_appendD1: fixes p :: "('b \<times> 'b) list"   and q :: "('b \<times> 'b) list" assumes "progressing (p @ q)" shows "progressing p"
holds_Stream: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and s :: "'c stream" shows "holds P (x ## s) = P x"
continuous_map_nsphere_reflection: fixes n :: "nat"   and k :: "nat" shows "continuous_map (nsphere n) (nsphere n) (\<lambda>x i. if i = k then - x i else x i)"
comp_ide_self: fixes a :: "'j" assumes "J.ide a" shows "a \<cdot>\<^sub>J a = a"
bvars\<^sub>s\<^sub>t\<^sub>p_def: fixes strand_step :: "('d, 'c) strand_step" shows "bvars\<^sub>s\<^sub>t\<^sub>p strand_step = (case strand_step of \<forall>x41\<langle>\<or>\<noteq>: x42\<rangle>\<^sub>s\<^sub>t \<Rightarrow> x41 | _ \<Rightarrow> [])"
nearly_healthy_wlp_repeat: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat" assumes "nearly_healthy (wlp a)" shows "nearly_healthy (wlp (repeat n a))"
powr_nat_def: fixes x :: "real"   and y :: "real" shows "powr_nat x y = (if y = 0 then 1 else if x < 0 then cos (pi * y) * (- x) powr y else x powr y)"
T_insert3_d_def: fixes x :: "'a"   and t :: "'a tree \<times> nat \<times> nat" shows "T_insert3_d x t = time (insert3_d_tm x t)"
set_mset_replicate_mset_subset: fixes n :: "nat"   and x :: "'c" shows "set_mset (replicate_mset n x) = (if n = 0 then {} else {x})"
representation_add: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and basis :: "'d set"   and v :: "'d"   and u :: "'d" assumes "module scale"   and "\<not> module.dependent scale basis"   and "v \<in> module.span scale basis"   and "u \<in> module.span scale basis" shows "module.representation scale basis (u + v) = (\<lambda>b. module.representation scale basis u b + module.representation scale basis v b)"
lcp_mismatch_shorter: fixes u :: "'b list"   and v :: "'b list"   and x :: "'b"   and y :: "'b" assumes "\<^bold>|u\<^bold>| = \<^bold>|v\<^bold>|"   and "x \<noteq> y" shows "u \<cdot> [x] \<and>\<^sub>p v \<cdot> [y] = u \<and>\<^sub>p v"
compress_pos_interfaces_Some: fixes ifces :: "iface list"   and ifce :: "iface"   and p_i :: "char list" assumes "compress_pos_interfaces ifces = Some ifce" shows "match_iface ifce p_i = (\<forall>i\<in>set ifces. match_iface i p_i)"
less_le_trans: fixes D :: "('e, 'f) Order_scheme"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "c \<in> carrier D"   and "a \<prec>\<^bsub>D\<^esub> b"   and "b \<preceq>\<^bsub>D\<^esub> c" shows "a \<prec>\<^bsub>D\<^esub> c"
assms_2: fixes f :: "'a" assumes "f \<in> space\<^sub>N N1" shows "Norm N2 f \<le> C * Norm N1 f"
r_pdec222_def: shows "r_pdec222 \<equiv> Cn 1 r_pdec2 [r_pdec22]"
hoare2o_valid_def: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "(\<Turnstile>\<^sub>2\<^sub>' {P} c {Q}) = (\<exists>k>0. \<forall>s. P s < \<infinity> \<longrightarrow> (\<exists>t p. (c, s) \<Rightarrow> p \<Down> t \<and> enat p + enat k * Q t \<le> enat k * P s))"
product_topology_homeomorphic_component: fixes i :: "'c"   and I :: "'c set"   and X :: "'c \<Rightarrow> 'd topology" assumes "i \<in> I"   and "\<And>j. \<lbrakk>j \<in> I; j \<noteq> i\<rbrakk> \<Longrightarrow> \<exists>a. topspace (X j) = {a}" shows "product_topology X I homeomorphic_space X i"
continuous_on_ln: fixes s :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "continuous_on s f"   and "\<forall>x\<in>s. f x \<noteq> 0" shows "continuous_on s (\<lambda>x. ln (f x))"
harry_avg_def: fixes n :: "nat" shows "harry_avg n = real (harry_sum n) / real (card (seqs n))"
is_strongly_dom_component_safe_def: shows "is_strongly_dom_component_safe \<equiv> l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_strongly_dom_component_safe get_root_node to_tree_order"
assms_2: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c) Tree\<^sub>\<alpha>" assumes "P t\<^sub>\<alpha>" shows "P (Not\<^sub>\<alpha> t\<^sub>\<alpha>)"
mult_mv_inv: fixes x :: "complex \<times> complex"   and A :: "complex \<times> complex \<times> complex \<times> complex"   and y :: "complex \<times> complex" assumes "x = A *\<^sub>m\<^sub>v y"   and "mat_det A \<noteq> 0" shows "y = mat_inv A *\<^sub>m\<^sub>v x"
lunit_simps_5: fixes f :: "'a" assumes "B.ide f" shows "B.cod (B.lunit f) = f"
mthr_delay_bisimulation_diverge_final: shows "delay_bisimulation_diverge_final r1.redT r2.redT mbisim mta_bisim m\<tau>move1 m\<tau>move2 r1.mfinal r2.mfinal"
changes_poly_pos_inf_def: fixes ps :: "'a poly list" shows "changes_poly_pos_inf ps = changes (map sgn_pos_inf ps)"
lift_Suc_mono_le: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and n' :: "nat" assumes "\<And>n. f n \<le> f (Suc n)"   and "n \<le> n'" shows "f n \<le> f n'"
ws': shows "ws' = xs @ remdups' (map_prod post post \<circ> snd) (filter (\<lambda>(uu_, p). map_prod post post p \<notin> N) (map (\<lambda>a. (a # w, delta a r', delta a s')) (\<sigma> n)))"
poincare_collinear_singleton: fixes u :: "complex_homo" assumes "u \<in> unit_disc" shows "poincare_collinear {u}"
simple_valid_cfg_iff: fixes ct :: "'s \<Rightarrow> 's pmf"   and s :: "'s" assumes "ct \<in> Pi S K" shows "(simple ct s \<in> valid_cfg) = (s \<in> S)"
leadsTo_Basis: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "F \<in> A ensures B" shows "F \<in> A \<longmapsto> B"
Lim_bounded2: fixes f :: "nat \<Rightarrow> 'a"   and l :: "'a"   and N :: "nat"   and C :: "'a" assumes "f \<longlonglongrightarrow> l"   and "\<forall>n\<ge>N. C \<le> f n" shows "C \<le> l"
col_vec_mat_Tensor_prelim: fixes M :: "'a list list"   and v :: "'a list" shows "\<forall>j<length M. col (vec_mat_Tensor v M) j = vec_vec_Tensor v (col M j)"
simps_10: fixes f :: "atom \<Rightarrow> nat \<Rightarrow> atom"   and \<phi> :: "atom fm"   and n :: "nat" shows "map_fm_binders f (Neg \<phi>) n = Neg (map_fm_binders f \<phi> n)"
prems_1: shows "list.set (\<alpha> # \<alpha>s) \<subseteq> ON"
union_mset_add_mset_right: fixes A :: "'f multiset"   and a :: "'f"   and B :: "'f multiset" shows "A + add_mset a B = add_mset a (A + B)"
param_insert: fixes R :: "('a \<times> 'b) set" shows "(insert, insert) \<in> R \<rightarrow> \<langle>R\<rangle>set_rel \<rightarrow> \<langle>R\<rangle>set_rel"
top_implies: fixes x :: "'c" shows "- top \<squnion> x = x"
cong: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d" shows "comm_monoid_add.Sum_any plus zero \<equiv> comm_monoid_add.Sum_any plus zero"
Range_mono: fixes r :: "('a \<times> 'b) set"   and s :: "('a \<times> 'b) set" assumes "r \<subseteq> s" shows "Range r \<subseteq> Range s"
simps: fixes S :: "Vars.var set"   and env :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and G :: "CoCalls" shows "combined_restrict S (env, G) = (env f|` S, G G|` S)"
assms_5: fixes ui :: "int poly" assumes "ui \<in> set us" shows "pl.Mp ui = ui"
CleanRepr_Proofs_axioms: shows "CleanRepr_Proofs var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc fls dsj M.proof M.prfOf M.encPf Pf"
on_triple: fixes l1 :: "'e"   and v1 :: "'f"   and v2 :: "'f"   and l2 :: "'e"   and v3 :: "'g"   and v4 :: "'g"   and R :: "('f \<times> 'g) set" shows "(((l1, v1, v2), l2, v3, v4) \<in> on_triple R) = ((v1, v3) \<in> R \<and> (v2, v4) \<in> R \<and> l1 = l2)"
index_transpose_mat_1: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat" assumes "i < dim_col A"   and "j < dim_row A" shows "A\<^sup>T $$ (i, j) = A $$ (j, i)"
arity_def: fixes arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and s :: "('a, 'b) tm" shows "arity.arity arity_sym arity_var s = arity.arity_hd arity_sym arity_var (head s) - enat (num_args s)"
assms_1: shows "Domainp A = (\<lambda>x. x \<in> \<UU>A)"
src_inv: fixes \<mu> :: "'d" assumes "D.iso \<mu>" shows "src\<^sub>D (D.inv \<mu>) = src\<^sub>D \<mu>"
hyps: fixes ys :: "'a list" assumes "set xs \<subseteq> carrier G"   and "set ys \<subseteq> carrier G" shows "sumlist (xs @ ys) = sumlist xs \<oplus> sumlist ys"
oreachable_local: fixes s :: "'d \<times> 'e"   and A :: "('d \<times> 'e, 'f) automaton"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'f"   and s' :: "'d \<times> 'e" assumes "s \<in> oreachable A S U"   and "(s, a, s') \<in> automaton.trans A"   and "S (fst s) (fst s') a" shows "s' \<in> oreachable A S U"
Sup_downset_id: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup \<circ> ord.downset less_eq = id"
simps: fixes xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real" shows "- MS xs f = MS (uminus_ms_aux xs) (\<lambda>x. - f x)"
cinner_complex_def: fixes x :: "complex"   and y :: "complex" shows "x \<bullet>\<^sub>C y = cnj x * y"
rangeI: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "f x \<in> Set.range f"
abs_ok_2: shows "term_ok \<Theta> (Abs \<tau>' t)"
intros_1: fixes F :: "'b formula" assumes "F \<in> AX0" shows "F \<in> AX1"