hcju/isabelleps huggingface dataset:
mset_set_Suc_Suc: fixes a :: "nat"   and b :: "nat" shows "mset_set {Suc a..<Suc b} = image_mset Suc (mset_set {a..<b})"
assms_1: shows "A \<subseteq>\<^sub>\<circ> B"
parallel_decomp: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<parallel> ys" shows "\<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs"
divide_less_eq_numeral_1: fixes b :: "'b"   and c :: "'b"   and w :: "num" shows "(b / c < numeral w) = (if (0::'b) < c then b < numeral w * c else if c < (0::'b) then numeral w * c < b else (0::'b) < numeral w)"
cond_action_oks_shr_change: fixes ls :: "('a, 'b) locks"   and ts :: "'b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option"   and m :: "'c"   and ws :: "'b \<Rightarrow> 'd wait_set_status option"   and "is" :: "'b set"   and t :: "'b"   and cts :: "'b conditional_action list"   and m' :: "'e" assumes "cond_action_oks (ls, (ts, m), ws, is) t cts" shows "cond_action_oks (ls, (ts, m'), ws, is) t cts"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_member: fixes l :: "'e strand_label"   and x :: "('f, 'g) stateful_strand_step"   and A :: "('e strand_label \<times> ('f, 'g) stateful_strand_step) list" assumes "(l, x) \<in> set A"   and "\<not> stateful_strand_step.is_Receive x"   and "\<not> stateful_strand_step.is_Send x" shows "(l, x) \<in> set (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A)"
distinct_names_map: fixes cld_dcl_list :: "(cld \<times> char list) list" assumes "(\<forall>(cld, y)\<in>set cld_dcl_list. class_name_f cld = y) \<and> distinct (map snd cld_dcl_list)" shows "distinct_names_f (map fst cld_dcl_list)"
dec_word_def: fixes \<Sigma> :: "'b list" assumes "formula \<Sigma>" shows "dec_word \<equiv> map fst"
notin_range_Some: fixes x :: "'b option" shows "(x \<notin> range Some) = (x = None)"
inv_safe: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('a, 'c) configuration" assumes "dataflow_topology summary results_in"   and "dataflow_topology.inv_imps_work_sum summary results_in c"   and "inv_implications_nonneg c" shows "dataflow_topology.inv_safe summary results_in c"
open_iff: fixes A :: "'b set"   and m :: "'b llist set" shows "(m open\<^bsub>topo (\<Union>s\<in>A\<^sup>\<star>. {infsuff A s})\<^esub>) = (m \<in> topo (\<Union>s\<in>A\<^sup>\<star>. {infsuff A s}))"
remove_child_removes_child: fixes h :: "(_) heap"   and ptr' :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> Shadow_DOM.remove_child ptr' child \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h" shows "child \<notin> set children"
chamber_in_subcomplex: fixes X :: "'b set set"   and Y :: "'b set set"   and C :: "'b set" assumes "ChamberComplex X"   and "ChamberComplex.ChamberSubcomplex X Y"   and "C \<in> Y"   and "SimplicialComplex.maxsimp X C" shows "SimplicialComplex.maxsimp Y C"
l_create_character_data_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_create_character_data_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr known_ptrs type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel set_disconnected_nodes set_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name get_tag_name_locs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs set_val set_val_locs create_character_data known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
assms_5: shows "\<forall>p\<in>set ps. real_of_int l - \<delta> \<le> real_of_int (fst p) \<and> real_of_int (fst p) \<le> real_of_int l + \<delta>"
C\<rho>_def: shows "C\<rho> = classical_operator \<rho>"
D: fixes z :: "'a" assumes "z \<in> X" shows "dist x z \<le> D"
linear_order_on_well_order_on: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "finite r" shows "linear_order_on A r = well_order_on A r"
h_2: shows "steps0 (st, l', r') tp stp = (a', b', c')"
interval_integral_Icc_approx_nonneg_1: fixes a :: "ereal"   and b :: "ereal"   and l :: "nat \<Rightarrow> real"   and u :: "nat \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and C :: "real" assumes "a < b"   and "einterval a b = (\<Union>i. {l i..u i})"   and "incseq u"   and "decseq l"   and "\<And>i. l i < u i"   and "\<And>i. a < ereal (l i)"   and "\<And>i. ereal (u i) < b"   and "(\<lambda>x. ereal (l x)) \<longlonglongrightarrow> a"   and "(\<lambda>x. ereal (u x)) \<longlonglongrightarrow> b"   and "\<And>i. set_integrable lborel {l i..u i} f"   and "AE x in lborel. a < ereal x \<longrightarrow> ereal x < b \<longrightarrow> 0 \<le> f x"   and "set_borel_measurable lborel (einterval a b) f"   and "(\<lambda>i. interval_lebesgue_integral lborel (ereal (l i)) (ereal (u i)) f) \<longlonglongrightarrow> C" shows "set_integrable lborel (einterval a b) f"
internalize_card_of_ordLess: fixes A :: "'a set"   and r :: "('b \<times> 'b) set" shows "(|A| <o r) = (\<exists>B\<subset>Field r. |A| =o |B| \<and> |B| <o r)"
type_equiv_entailment: fixes t :: "'BExp set"   and P :: "'BExp set"   and t' :: "'BExp set"   and P' :: "'BExp set" assumes "t =:\<^sub>P t'"   and "P' \<turnstile> P" shows "t =:\<^sub>P' t'"
less_eq_level: fixes l1 :: "level"   and l2 :: "level" shows "l1 \<le> l2 \<equiv> l1 = Lo \<or> l2 = Hi"
Nfunc_LeastTr2_1: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set" assumes "i \<le> n"   and "\<forall>l\<le>n. G \<guillemotright> f l "   and "\<forall>l\<le>n. G \<guillemotright> g l "   and "NfuncPair_neq_at f g i" shows "NfuncPair_neq_at f g (LEAST k. NfuncPair_neq_at f g k)"
abs_eq: shows "1 = Abs_fpxs (\<lambda>r. if r = 0 then 1::'b else (0::'b))"
a1: shows "s' = snd (fst (flush_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
mgu_None_is_subst_neq: fixes s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "mgu s t = None" shows "s \<cdot> \<delta> \<noteq> t \<cdot> \<delta>"
dom_locals_evaln_mono_elim: fixes G :: "prog"   and s0 :: "state"   and t :: "Term.term"   and n :: "nat"   and v :: "vals"   and s1 :: "state" assumes "G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v, s1)" obtains "dom (locals (snd s0)) \<subseteq> dom (locals (snd s1))" and "\<And>vv s val. \<lbrakk>v = In2 vv; normal s1\<rbrakk> \<Longrightarrow> dom (locals (snd s)) \<subseteq> dom (locals (snd (snd vv val s)))"
roots1_def: fixes p :: "'b poly" shows "roots1 p = - coeff p 0 / coeff p 1"
iD_def: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" shows "iD rt \<equiv> {dip. flag rt dip = Some Aodv_Basic.inv}"
add_nonneg_pos: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "{#} \<subseteq># a"   and "{#} \<subset># b" shows "{#} \<subset># a + b"
chart_last_inv_inverse: fixes x :: "'b" shows "chart_last (chart_last_inv x) = x"
induct: fixes x1 :: "'c"   and x2 :: "'c list"   and x3 :: "'c"   and P :: "'c \<Rightarrow> 'c list \<Rightarrow> 'c \<Rightarrow> bool" assumes "trace x1 x2 x3"   and "\<And>final. P final [] final"   and "\<And>conf' T final conf. \<lbrakk>trace conf' T final; P conf' T final; conf \<Rightarrow> conf'\<rbrakk> \<Longrightarrow> P conf (conf' # T) final" shows "P x1 x2 x3"
equiv_pivot_eq: fixes x\<^sub>j :: "nat"   and eq :: "nat \<times> linear_poly"   and v :: "nat \<Rightarrow> 'a" assumes "x\<^sub>j \<in> rvars_eq eq"   and "lhs eq \<notin> rvars_eq eq" shows "v \<Turnstile>\<^sub>e pivot_eq eq x\<^sub>j = v \<Turnstile>\<^sub>e eq"
assms_2: shows "x \<notin> nonzero Q\<^sub>p"
simps_1: fixes r :: "'b llist \<rightarrow> 'b llist \<rightarrow> 'b llist" shows "lrev_body2\<cdot>r\<cdot>lnil = ID"
lex_ord_raw_EqI: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and xs :: "('b \<times> 'c) list \<times> 'a"   and ys :: "('b \<times> 'd) list \<times> 'a"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> order option"   and ko :: "'a" assumes "oalist_raw.oalist_inv rep_key_order xs"   and "oalist_raw.oalist_inv rep_key_order ys"   and "\<And>k. k \<in> fst ` set (fst xs) \<union> fst ` set (fst ys) \<Longrightarrow> f k (oalist_raw.lookup_raw rep_key_order xs k) (oalist_raw.lookup_raw rep_key_order ys k) = Some Eq" shows "oalist_raw.lex_ord_raw rep_key_order ko f xs ys = Some Eq"
equality: fixes r :: "'a StoreImpl_scheme"   and r' :: "'a StoreImpl_scheme" assumes "newOID r = newOID r'"   and "newAID r = newAID r'"   and "vals r = vals r'"   and "more r = more r'" shows "r = r'"
is_sentence_def: fixes s :: "symbol list" shows "is_sentence s = list_all is_symbol s"
star_scaleR_def: fixes r :: "real" shows "(*\<^sub>R) r \<equiv> *f* (*\<^sub>R) r"
reds_hext_incr: fixes es :: "char list exp list"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and es' :: "char list exp list"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "P \<turnstile> \<langle>es,(h, l, sh),b\<rangle> [\<rightarrow>] \<langle>es',(h', l', sh'),b'\<rangle>" shows "h \<unlhd> h'"
id2: shows "IArray.of_fun (\<mu>' mm) (Suc j) = IArray (map (\<mu>' mm) [0..<Suc j])"
moebius_circline_comp_inv_left: fixes M :: "moebius"   and H :: "circline" shows "moebius_circline (moebius_inv M) (moebius_circline M H) = H"
that: fixes y :: "'c" assumes "y \<in> keys t"   and "x \<in> indets (f y)" shows "thesis"
ex_Length_0: fixes idx :: "'n" shows "\<exists>\<AA>. Length \<AA> = 0 \<and> #\<^sub>V \<AA> = idx"
r_step_Pr_prim: shows "prim_recfn 1 r_step_Pr"
Basic: fixes i :: "'b"   and l :: "'a FOL_Fitting.term list"   and G :: "('a, 'b) form list" shows "\<stileturn> Pred i l # Neg (Pred i l) # G"
times_divide_eq_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "times (divide b c) a = divide (times b a) c"
mod2_gr_0: fixes m :: "nat" shows "(0 < m mod 2) = (m mod 2 = 1)"
cases: fixes r :: "('a, 'd, 'b, 'e, 'c, 'f) Functor" obtains CatDom :: "('a, 'b, 'c) Category_scheme" and CatCod :: "('d, 'e, 'f) Category_scheme" and MapM :: "'b \<Rightarrow> 'e" where "r = \<lparr>CatDom = CatDom, CatCod = CatCod, MapM = MapM\<rparr>"
that: fixes g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "piecewise_continuous_on a i I g"   and "piecewise_continuous_on i b I h"   and "\<And>x. \<lbrakk>a \<le> x; x < i\<rbrakk> \<Longrightarrow> g x = f x"   and "\<And>x. \<lbrakk>i < x; x \<le> b\<rbrakk> \<Longrightarrow> h x = f x" shows "thesis"
gtrace_alt_def: fixes r :: "(nat \<times> 'c) stream"   and v :: "nat \<times> 'c" shows "gtrace r v = r"
pr_var_def: fixes x :: "'d \<Longrightarrow> 'e" shows "&x = x"
sup_bot_right: fixes x :: "'a" shows "x \<squnion> bot = x"
lstrict_prefix_code_3: fixes x :: "'c"   and xs :: "'c llist" shows "lstrict_prefix (LCons x xs) LNil = False"
sum_list_mono: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x" shows "sum_list (map f xs) \<le> sum_list (map g xs)"
prime_factorization_eqI_strong: 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 P :: "'a multiset"   and n :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "\<And>p. p \<in># P \<Longrightarrow> normalization_semidom.prime zero times one normalize p"   and "comm_monoid_mult.prod_mset times one P = n" shows "factorial_semiring.prime_factorization zero times one normalize n = P"
prems: shows "mat (row_length (a # M1)) (length (a # M1)) (a # M1) \<and> mat (row_length M2) (length M2) M2"
K_def: shows "K = (\<lambda>N. A2 \<inter> (\<Union>i<N. (T ^^ i) --` B))"
assms: shows "\<forall>i\<le>n. s (x + i) = []"
legal_rules_def: fixes Q :: "'b set" shows "legal_rules Q \<equiv> {q \<rightarrow> f qs |q f qs. q \<in> Q \<and> qs \<in> lists Q \<and> A f = Some (length qs)}"
sum_intro: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "CEM"   and "\<forall>w. O w z = (O w x \<or> O w y)" shows "x \<oplus> y = z"
simps_1: fixes ts :: "'a wordinterval list" shows "partitioningIps [] ts = ts"
consis_m: shows "reads_consistent False \<O> m (r # sb)"
vdomainI: fixes r :: "V"   and A :: "V" assumes "r \<in>\<^sub>\<circ> A" shows "\<langle>r, \<D>\<^sub>\<circ> r\<rangle> \<in>\<^sub>\<circ> vdomain A"
Units_nat_pow_Units: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and c :: "nat" assumes "Group.monoid G"   and "a \<in> Units G" shows "a [^]\<^bsub>G\<^esub> c \<in> Units G"
ideal_2: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> I" shows "multA b a \<in> I"
tail_monom_mult: fixes c :: "'c"   and t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.tail (local.punit.monom_mult c t p) = local.punit.monom_mult c t (punit.tail p)"
wp1ID: fixes p :: "('b, 'c) Combinators list" shows "wellformed_policy1_strong (insertDeny (removeShadowRules1 p))"
final_conc: fixes L :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and p :: "bool list" shows "fin (RegExp2NA.conc L R) p = (fin R (start R) \<and> (\<exists>s. p = True # s \<and> fin L s) \<or> (\<exists>s. p = False # s \<and> fin R s))"
apply_cltn2_abs: fixes x :: "(real, 3) vec"   and A :: "((real, 3) vec, 3) vec" assumes "x \<noteq> 0"   and "invertible A" shows "apply_cltn2 (proj2_abs x) (cltn2_abs A) = proj2_abs (x v* A)"
monot: shows "monotone (\<lambda>x y. y < x) (\<lambda>x y. y < x) f"
FSwSbImorph_def: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "'g \<Rightarrow> 'h"   and hA :: "'i \<Rightarrow> 'j"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'k, 'g, 'i, 'l) model_scheme"   and MOD' :: "('b, 'c, 'd, 'e, 'f, 'k, 'h, 'j, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.FSwSbImorph Delta h hA MOD MOD' \<equiv> FixSyn.FSwImorph Delta h hA MOD MOD' \<and> FixSyn.ipresIGSubstAll Delta h hA MOD MOD'"
word_smod_def: fixes x :: "'b word"   and y :: "'b word" shows "x smod y = (let x' = sint x; y' = sint y; negative = x' < 0; result = \<bar>x'\<bar> mod \<bar>y'\<bar> in word_of_int (if negative then - result else result))"
payload_parts: fixes X :: "msg"   and S :: "msg set" assumes "X \<in> parts S"   and "S \<subseteq> payload" shows "X \<in> payload"
ParamUses_in_Use: fixes n :: "'b"   and V :: "'c" assumes "valid_node n"   and "V \<in> \<Union> (set (ParamUses n))" shows "V \<in> Use n"
map_poly_pCons_hom: fixes a :: "rat"   and p :: "rat poly" shows "map_poly of_rat (pCons a p) = pCons (of_rat a) (map_poly of_rat p)"
add_cancel_right_right: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and minus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and a :: "'f"   and b :: "'f" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(a = plus a b) = (b = zero)"
no_vs_change_obtain_snapshot_bound_1st_step: fixes PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set"   and as :: "(('a, bool) fmap \<times> ('a, bool) fmap) list"   and vs :: "'a set"   and s :: "('a, bool) fmap" assumes "finite PROB"   and "vars_change as vs s = []"   and "sat_precond_as s as"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "\<exists>as'. exec_plan (fmrestrict_set (prob_dom (snapshot PROB (fmrestrict_set vs s))) s) as = exec_plan (fmrestrict_set (prob_dom (snapshot PROB (fmrestrict_set vs s))) s) as' \<and> subseq as' as \<and> length as' \<le> problem_plan_bound (snapshot PROB (fmrestrict_set vs s))"
assms_3: shows "eint (int k) < val_Zp_dist a b"
fglb_rep_eq: fixes A :: "'b fset set"   and t :: "'b fset" shows "\<langle>fglb A t\<rangle>\<^sub>f = (if A = {} then \<langle>t\<rangle>\<^sub>f else \<Inter> (fset ` A))"
distinct_card: fixes xs :: "'a list" assumes "distinct xs" shows "card (set xs) = length xs"
finsum_vec_closed: fixes f :: "'b \<Rightarrow> 'c vec"   and A :: "'b set"   and n :: "nat" assumes "f \<in> A \<rightarrow> carrier_vec n" shows "finsum_vec TYPE('c) n f A \<in> carrier_vec n"
consecutiveRes: fixes ts :: "traffic"   and c :: "cars" shows "|res ts c| = 2 \<longrightarrow> (\<exists>n. Rep_nat_int (res ts c) = {n, n + 1})"
ccApprox_mono': fixes t :: "Vars.var ttree"   and t' :: "Vars.var ttree" assumes "t \<sqsubseteq> t'" shows "ccApprox t \<sqsubseteq> ccApprox t'"
field_one_plus_frac1: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier K"   and "b \<in> carrier K"   and "b \<noteq> \<zero>" shows "1\<^sub>r \<plusminus> a \<cdot>\<^sub>r b\<^bsup>\<hyphen> K\<^esup> = (b \<plusminus> a) \<cdot>\<^sub>r b\<^bsup>\<hyphen> K\<^esup>"
topspace_nsphere_minus1: fixes x :: "nat \<Rightarrow> real"   and n :: "nat" assumes "x \<in> topspace (nsphere n)"   and "x n = 0" shows "x \<in> topspace (nsphere (n - Suc 0))"
assms: shows "pyf \<in> borel_measurable (G matur)"
cupcake_single_preserve0: fixes static_cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and env :: "v sem_env"   and e :: "exp0"   and res :: "(v, v) result" assumes "cakeml_static_env static_cenv"   and "cupcake_evaluate_single env e res"   and "cakeml_static_env.is_cupcake_all_env static_cenv env"   and "is_cupcake_exp e" shows "if_rval (cakeml_static_env.is_cupcake_value static_cenv) res"
order_refl: fixes x :: "'f word" shows "x \<le>s x"
ACI_norm_idem: fixes r :: "'c rexp" shows "\<guillemotleft>\<guillemotleft>r\<guillemotright>\<guillemotright> = \<guillemotleft>r\<guillemotright>"
select_convs_7: fixes counter :: "nat"   and discovered :: "'c \<Rightarrow> nat option"   and finished :: "'c \<Rightarrow> nat option"   and pending :: "('c \<times> 'c) set"   and stack :: "'c list"   and tree_edges :: "('c \<times> 'c) set"   and back_edges :: "('c \<times> 'c) set"   and cross_edges :: "('c \<times> 'c) set"   and more :: "'d" shows "back_edges \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, \<dots> = more\<rparr> = back_edges"
prime_factorization_1: 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" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.prime_factorization zero times one normalize one = {#}"
bisimScopeExt: fixes x :: "name"   and P :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi" assumes "x \<sharp> P" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(P \<parallel> Q) \<sim> P \<parallel> \<lparr>\<nu>x\<rparr>Q"
zm: shows "zm = sorted_list_of_fset (fmdom xm |\<inter>| fmdom ym)"
widen_trans: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and S :: "ty"   and U :: "ty"   and T :: "ty" assumes "P \<turnstile> S \<le> U"   and "P \<turnstile> U \<le> T" shows "P \<turnstile> S \<le> T"
hyps: fixes u :: "'a"   and S :: "'a set" assumes "cas u as v"   and "set (awalk_verts u as) \<subseteq> S" shows "pre_digraph.cas (G \<restriction> S) u as v"
assms_3: fixes x :: "'a" assumes "x \<in> X" shows "0 < f x"
VectorSpaceEnd: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and T :: "'c \<Rightarrow> 'c" assumes "FGModuleEnd G smult V T" shows "VectorSpaceEnd (aezfun_scalar_mult.fsmult smult) V T"
max_ip_addr_max_word: shows "max_ip_addr = - 1"
AVar_lemma1: fixes s :: "st"   and a :: "loc"   and obj :: "obj"   and ty :: "ty"   and i :: "int"   and i' :: "val"   and G :: "prog"   and Tb :: "ty"   and L :: "lenv" assumes "globs s (Inl a) = Some obj"   and "tag obj = Arr ty i"   and "the_Intg i' in_bounds i"   and "wf_prog G"   and "G\<turnstile>ty.[]\<preceq>Tb.[]"   and "Norm s\<Colon>\<preceq>(G, L)" shows "G,s\<turnstile>the (values obj (Inr (the_Intg i')))\<Colon>\<preceq>Tb"
simps_3: fixes m1 :: "(nat \<times> nat) convnet"   and m2 :: "(nat \<times> nat) convnet" shows "output_size (Pool m1 m2) = output_size m1"
bird_simps_2: shows "left bird = pure recip \<diamondop> (pure succ \<diamondop> bird)"
nofail_RES_conv: fixes m :: "'c nres" shows "nofail m = (\<exists>M. m = RES M)"
axioms_2: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" assumes "simple_incidence_system point_set block_collection" shows "simple_incidence_system_axioms block_collection"
simps_3: fixes x1 :: "nat" shows "pnat.PInfty \<noteq> pnat x1"
almost_full_on_imp_homogeneous_subseq: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and f :: "nat \<Rightarrow> 'a" assumes "almost_full_on P A"   and "\<forall>i. f i \<in> A" shows "\<exists>\<phi>. \<forall>i j. i < j \<longrightarrow> \<phi> i < \<phi> j \<and> P (f (\<phi> i)) (f (\<phi> j))"
advantage\<^sub>1_def: fixes \<A> :: "nat \<Rightarrow> (bool, unit + 'cipher + 'plain \<times> 'plain, 'ekey + 'plain option + 'cipher) gpv"   and \<eta> :: "nat" shows "advantage\<^sub>1 \<A> \<eta> = \<bar>spmf (ind_cca2\<^sub>1 \<A> \<eta>) True - 1 / 2\<bar>"
normal_imp_not_zero: fixes f :: "('a, 'b) IEEE.float" assumes "is_normal f" shows "\<not> is_zero f"
freeultrafilter_Ex: assumes "infinite UNIV" shows "\<exists>U. freeultrafilter U"
Sup1_I: fixes P :: "'b \<Rightarrow> bool"   and A :: "('b \<Rightarrow> bool) set"   and a :: "'b" assumes "P \<in> A"   and "P a" shows "Sup A a"
pt_swap_eq_aux: fixes y :: "'a"   and pi :: "('b \<times> 'b) list" assumes "pt TYPE('a) TYPE('b)"   and "\<forall>a b. [(a, b)] \<bullet> y = y" shows "pi \<bullet> y = y"
distinct_30: fixes x5 :: "nat"   and x3 :: "nat" shows "Key x5 \<noteq> Nonce x3"
PI: fixes x :: "'a set" assumes "\<forall>j\<in>J. pr_ideal R x (+) (\<cdot>) \<zero> \<one> \<and> \<aa> j \<subseteq> x" shows "pr_ideal R x (+) (\<cdot>) \<zero> \<one>"
Dube: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "finite F"   and "F \<subseteq> P[X]"   and "\<And>f. f \<in> F \<Longrightarrow> homogeneous f"   and "g \<in> punit.reduced_GB F" shows "poly_deg g \<le> Dube (card X) (maxdeg F)"
assms_2: shows "\<langle>c #\<^bsub>enat n\<^esub>t\<rangle> < \<langle>c #\<^bsub>n'\<^esub>t\<rangle>"