hcju/isabelleps huggingface dataset:
IH_readable_1: shows "cnf F \<union> \<Union> (cnf ` \<Gamma>) \<turnstile> \<box>"
map_def: fixes f :: "'a" shows "local.map f = (if A.arr f then b else B.null)"
h_outside: fixes xy :: "'v \<times> 'v"   and n :: "nat" assumes "xy \<notin> \<^bold>V \<times> \<^bold>V" shows "h n xy = 0"
inf_img_fin_domE: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "finite (f ` A)"   and "infinite A" obtains y :: "'b" where "y \<in> f ` A" and "infinite (f -` {y})"
preserves_reflects_arr: fixes f :: "'a" shows "Right_a.arr (H\<^sub>R a f) = Right_a.arr f"
gso'_def: fixes i :: "nat" shows "fs.gs.gso' i = fs.gs.d i \<cdot>\<^sub>v fs.gs.gso i"
in_hom_char: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "S.in_hom f a b = (S.arr a \<and> S.arr b \<and> S.arr f \<and> \<guillemotleft>f : a \<Rightarrow>\<^sub>B b\<guillemotright>)"
unique_connecting_path_set_2: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and w :: "'a" assumes "Tree G"   and "v \<in> V\<^bsub>G\<^esub>"   and "w \<in> V\<^bsub>G\<^esub>" shows "w \<in> set (v \<leadsto>\<^bsub>G\<^esub> w)"
A_modifies_dma: shows "\<forall>x. dma mem\<^sub>1 [\<parallel>\<^sub>1 A] x \<noteq> dma mem\<^sub>1 x \<longrightarrow> \<not> var_asm_not_written mds x"
landau_function_family_axioms: shows "landau_function_family F (powr_closure g)"
c_7: fixes stateful_strand_step :: "('e, 'f) stateful_strand_step" assumes "is_NegChecks stateful_strand_step" shows "\<forall>bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p stateful_strand_step\<langle>\<or>\<noteq>: the_eqs stateful_strand_step \<or>\<notin>: the_ins stateful_strand_step\<rangle> = stateful_strand_step"
impl_N_def: fixes a :: "'a N"   and b :: "'a N" shows "a l\<rightarrow> b \<equiv> Abs_N ((Rep_N b - Rep_N a) \<sqinter> (0::'a))"
split_sel: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "('b, 'c) hd \<Rightarrow> 'a"   and f2 :: "('b, 'c) tm \<Rightarrow> ('b, 'c) tm \<Rightarrow> 'a"   and tm :: "('b, 'c) tm" shows "P (case tm of Hd x \<Rightarrow> f1 x | App x xa \<Rightarrow> f2 x xa) = ((tm = Hd (head tm) \<longrightarrow> P (f1 (head tm))) \<and> (tm = App (fun tm) (arg tm) \<longrightarrow> P (f2 (fun tm) (arg tm))))"
oth_class_taut_6_a: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>& \<psi> \<^bold>\<equiv> \<^bold>\<not>(\<^bold>\<not>\<phi> \<^bold>\<or> \<^bold>\<not>\<psi>) in v]"
start_of_nonzero_2: fixes ly :: "nat list"   and as :: "nat" shows "(start_of ly as = 0) = False"
eq: fixes x :: "'c"   and y :: "'c" shows "(c2 x y = Eq) = (x = y)"
U_subset_domain: fixes p :: "'a"   and x :: "'a" assumes "p \<in> carrier"   and "x \<in> U p" shows "x \<in> domain (\<psi> p)"
gauge_ball: fixes e :: "real" assumes "0 < e" shows "gauge (\<lambda>x. ball x e)"
mono: fixes a :: "'h"   and c :: "'h"   and b :: "'h"   and d :: "'h" assumes "a \<le> c"   and "b \<le> d" shows "min a b \<le> min c d"
par_beta_lc: fixes t :: "sterm"   and t' :: "sterm" assumes "t => t'" shows "lc t \<and> lc t'"
rel_eq_onp: fixes P :: "'a \<Rightarrow> bool" shows "list_all2 (eq_onp P) = eq_onp (list_all P)"
vars_subterm: fixes t :: "'b trm"   and p :: "indices list"   and s :: "'b trm" assumes "subterm t p s" shows "vars_of s \<subseteq> vars_of t"
full_quicksort_ref_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and h :: "'a \<Rightarrow> 'b"   and xs :: "'a list" shows "full_quicksort_ref R h xs \<equiv> if List.null xs then RETURN xs else quicksort_ref R h (0, Polynomials.degree xs, xs)"
preserves_dom: fixes f :: "'a" assumes "S.arr f" shows "S.dom (R f) = R (S.dom f)"
da: shows "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd s0)) \<guillemotright>t\<guillemotright> A"
index_mat_addcol_2: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat"   and a :: "'a"   and l :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "addcol a j l A $$ (i, j) = a * A $$ (i, l) + A $$ (i, j)"
degree_eq_oneE: fixes p :: "'b poly" assumes "degree p = 1" obtains a :: "'b" and b :: "'b" where "p = [:a, b:]" and "b \<noteq> (0::'b)"
simps_1: fixes uu :: "'c" shows "mem_idx uu [] = None"
PPf: fixes t1 :: "'b"   and t2 :: "'b" assumes "t1 \<in> trm"   and "t2 \<in> trm"   and "xx \<notin> FvarsT t1" shows "PPf t1 t2 \<in> fmla"
dvdE: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and a :: "'d" assumes "dvd.dvd times b a" obtains k :: "'d" where "a = times b k"
word_oa_dist: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" shows "x && y || z = (x || z) && (y || z)"
matachAny_matchNone: fixes m :: "'b match_expr" assumes "\<not> has_primitive m" shows "matcheq_matchAny m = (\<not> matcheq_matchNone m)"
range_vars_subst_compose_subset: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<tau> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "range_vars (\<sigma> \<circ>\<^sub>s \<tau>) \<subseteq> range_vars \<sigma> - subst_domain \<tau> \<union> range_vars \<tau>"
rep_eq: fixes x :: "nat"   and xa :: "'a pdevs" shows "pdevs_apply (trunc_pdevs x xa) = trunc_pdevs_raw x (pdevs_apply xa)"
summatory_divisor_count_asymptotics'': shows "sum_upto (\<lambda>n. real (divisor_count n)) \<sim>[at_top] (\<lambda>x. x * ln x)"
ll: shows "\<forall>i<n. strictly_maximal_wrt (As ! i \<cdot>a \<sigma>) (Cs ! i \<cdot> \<sigma>)"
Compositionality_While: fixes d :: "'d"   and b :: "'exp"   and b' :: "'exp"   and c :: "('exp, 'id) MWLfCom"   and c' :: "('exp, 'id) MWLfCom" assumes "b \<equiv>\<^bsub>d\<^esub> b'"   and "[c] \<approx>\<^bsub>d\<^esub> [c']" shows "[while b do c od] \<approx>\<^bsub>d\<^esub> [while b' do c' od]"
homeomorphic_translation: fixes S :: "'a set"   and a :: "'a" shows "S homeomorphic (+) a ` S"
level_start_0: fixes dm :: "nat" shows "level (start dm) = 0"
measure_map_spmf: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d spmf"   and A :: "'c set" shows "Sigma_Algebra.measure (measure_spmf (map_spmf f p)) A = Sigma_Algebra.measure (measure_spmf p) (f -` A)"
aodv_ex_label: fixes p :: "(state, msg, pseqp, pseqp label) seqp" shows "\<exists>l. l \<in> labels \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V p"
A'_def: shows "A' = {x \<in> A. 0 \<noteq> (\<Sum>\<^sub>ay\<in>B x. norm (f (x, y)))}"
m_eq: fixes N :: "'a stream measure"   and M :: "'a stream measure" assumes "R N M" shows "M = measure_pmf (p N M) \<bind> (\<lambda>y. distr (m N M y) (stream_space (count_space UNIV)) ((##) y))"
l_span_sub_submodule: fixes A :: "'b set"   and N :: "'a set"   and H :: "'a set" assumes "ideal R A"   and "submodule R M N"   and "H \<subseteq> N" shows "linear_span R M A H \<subseteq> N"
v: fixes v :: "v" assumes "v \<in> set vs" shows "is_cupcake_value v"
subintegral: fixes H :: "'e set"   and R :: "('e, 'f) ring_scheme"   and h1 :: "'e"   and h2 :: "'e" assumes "subdomain H R"   and "h1 \<in> H"   and "h2 \<in> H"   and "h1 \<otimes>\<^bsub>R\<^esub> h2 = \<zero>\<^bsub>R\<^esub>" shows "h1 = \<zero>\<^bsub>R\<^esub> \<or> h2 = \<zero>\<^bsub>R\<^esub>"
rreq_rrep_fresh_def: fixes crt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and m :: "msg" shows "rreq_rrep_fresh crt m \<equiv> case m of Rreq hopsc x xa xb oipc osnc ipcc xc \<Rightarrow> ipcc \<noteq> oipc \<longrightarrow> oipc \<in> kD crt \<and> (osnc < sqn crt oipc \<or> sqn crt oipc = osnc \<and> the (dhops crt oipc) \<le> hopsc \<and> the (flag crt oipc) = val) | Rrep hopsc dipc dsnc x ipcc \<Rightarrow> ipcc \<noteq> dipc \<longrightarrow> dipc \<in> kD crt \<and> sqn crt dipc = dsnc \<and> the (dhops crt dipc) = hopsc \<and> the (flag crt dipc) = val | _ \<Rightarrow> True"
relcl_def: fixes L :: "'b set set" shows "relcl L \<equiv> {(x, y). y \<in> cl L {x}}"
dom_simp: fixes f :: "'c local.term set" assumes "\<F>\<^sub>SC.arr f" shows "\<F>\<^sub>SC.dom f = local.dom f"
rprodl_extend_state_oracle: fixes "oracle" :: "'c \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'c) spmf" shows "map_fun rprodl (map_fun id (map_spmf (map_prod id lprodr))) \<dagger>\<dagger>oracle = \<dagger>oracle"
optimal_paths: fixes paths' :: "'a list set"   and P_new' :: "'a list" assumes "ProofStepInduct G v0 v1 paths' P_new' sep_size" shows "Digraph.distance (remove_vertex v0) (last P_new) v1 \<le> Digraph.distance (remove_vertex v0) (last P_new') v1"
c1_on_open_def: fixes f :: "'d \<Rightarrow> 'd"   and f' :: "'d \<Rightarrow> 'd \<Rightarrow>\<^sub>L 'd"   and X :: "'d set" shows "c1_on_open f f' X \<equiv> open X \<and> (\<forall>x. x \<in> X \<longrightarrow> (f has_derivative blinfun_apply (f' x)) (at x)) \<and> continuous_on X f'"
simps_2: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" shows "list \<langle>l, x, r\<rangle> = x # splice (list l) (list r)"
insert_Collect: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "insert a (Collect P) = {u. u \<noteq> a \<longrightarrow> P u}"
lm052: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite A"   and "A \<noteq> {}" shows "Max (f ` A) \<in> f ` A"
image_dual: shows "(`) \<partial> \<circ> (`) \<partial> = id"
singleton_sub: fixes a :: "'e"   and A :: "'e set" assumes "a \<in> A" shows "{a} \<subseteq> A"
pq_order_subgrp_normal: fixes q :: "nat"   and p :: "nat" assumes "finite (carrier (G\<lparr>carrier := H\<rparr>))"   and "order (G\<lparr>carrier := H\<rparr>) = q * p"   and "prime p"   and "prime q"   and "p < q" shows "(THE Q. Q \<in> groupH.subgroups_of_size q) \<lhd> G\<lparr>carrier := H\<rparr>"
cl_in_def: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set"   and X :: "'b set" assumes "matroid carrier indep" shows "matroid.cl_in indep \<E> X = matroid.cl \<E> (indep_system.indep_in indep \<E>) X"
Mul_PBInv_def: shows "Mul_PBInv \<equiv> \<lambda>s. Safe s \<or> obc s \<subset> Blacks (M s) \<or> l s < Queue s \<or> (\<forall>i<ind s. \<not> BtoW (E s ! i, M s)) \<and> l s \<le> Queue s"
le_zero_iff_pprt_id: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "less_eq zero a = (lattice_ab_group_add.pprt zero sup a = a)"
pq: shows "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>"
distinct_5: fixes x1 :: "bool"   and x41 :: "'b"   and x42 :: "value list" shows "Bc x1 \<noteq> In x41 x42"
that: fixes b :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" shows "thesis"
linorder_lattice_sup_inf_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "sup x (inf y z) = inf (sup x y) (sup x z)"
sum_constant: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and y :: "'c"   and A :: "'d set" assumes "class.semiring_1 one times plus zero" shows "comm_monoid_add.sum plus zero (\<lambda>x. y) A = times (semiring_1.of_nat one plus zero (card A)) y"
lincomb_indexed_Ints_vec: fixes C :: "'a vec set"   and c :: "'a vec \<Rightarrow> 'a"   and I :: "nat set" assumes "\<And>x. x \<in> C \<Longrightarrow> c x \<in> \<int>"   and "C \<subseteq> carrier_vec n"   and "C \<subseteq> indexed_Ints_vec I" shows "lincomb c C \<in> indexed_Ints_vec I"
inject_2: fixes x21 :: "'b aexp"   and x22 :: "'b aexp"   and y21 :: "'b aexp"   and y22 :: "'b aexp" shows "(Eq x21 x22 = Eq y21 y22) = (x21 = y21 \<and> x22 = y22)"
numsubst0_numbound0: fixes t :: "Cooper.num"   and a :: "Cooper.num" assumes "numbound0 t" shows "numbound0 (numsubst0 t a)"
"2_4": fixes x :: "real \<times> real" assumes "x \<in> set (z # zs)" shows "x \<noteq> 0"
fmfilter_comm: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and m :: "('a, 'b) fmap" shows "fmfilter P (fmfilter Q m) = fmfilter Q (fmfilter P m)"
relation_of_state_update_after: fixes sc :: "'d \<times> 'd \<Rightarrow> bool"   and Ac :: "('c, 'd) action" shows "action.relation_of (state_update_after sc Ac) = (action.relation_of Ac ;; R (true \<turnstile> \<lambda>(A, A'). sc (alpha_rp.more A, alpha_rp.more A') \<and> \<not> wait A' \<and> tr A = tr A'))"
pth_5: fixes c :: "real"   and d :: "real"   and x :: "'b" shows "c *\<^sub>R d *\<^sub>R x \<equiv> (c * d) *\<^sub>R x"
rbt_greater_prop: fixes k :: "'a"   and t :: "('a, 'c) rbt" shows "(k \<guillemotleft>| t) = (\<forall>x\<in>set (keys t). k < x)"
affine_dependent_def: fixes s :: "'a set" shows "affine_dependent s = (\<exists>x\<in>s. x \<in> affine hull (s - {x}))"
exhaust: fixes y :: "'b btree" obtains "y = Leaf"   | x21 :: "('b btree \<times> 'b) list" and x22 :: "'b btree" where "y = Node x21 x22"
sem_5d: shows "\<forall>X Y Z. Y \<sqsubseteq> X \<and> ob X Y \<and> X \<sqsubseteq> Z \<longrightarrow> ob Z ((Z \<sqinter> (\<sim>X)) \<squnion> Y)"
has_integral_cbox: fixes f :: "'c \<Rightarrow> 'd"   and I :: "'d"   and a :: "'c"   and b :: "'c" shows "(f has_integral I) (cbox a b) = (sum (\<lambda>(x, k). content k *\<^sub>R f x) \<longlongrightarrow> I) (division_filter (cbox a b))"
IH: shows "nths (replicate n x) A = replicate (card {i. i < n \<and> i \<in> A}) x"
source: fixes S1 :: "'procS"   and S2 :: "'procS"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "(S1, S2) \<in> SRel" shows "SourceTerm S1 \<sim>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> SourceTerm S2"
end_smult_comp_comm_right: fixes S :: "'v \<Rightarrow> 'v"   and a :: "'f" assumes "range S \<subseteq> V" shows "T \<circ> a \<cdot>\<cdot> S = a \<cdot>\<cdot> (T \<circ> S)"
sum_lt: fixes x :: "'b"   and t :: "'b" shows "x + t < (0::'b) \<equiv> x < - t"
inject_2: fixes x2 :: "nat"   and y2 :: "nat" shows "(Intv x2 = Intv y2) = (x2 = y2)"
defval_conf: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and P :: "'d prog"   and h :: "'b"   and T :: "ty" shows "heap_base.conf typeof_addr P h (default_val T) T"
2: shows "is_map_to_sorted_list ordR Rk Rv Rm tsl"
colossless_converterD: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter"   and a :: "'a" assumes "colossless_converter \<I> \<I>' conv"   and "a \<in> outs_\<I> \<I>" shows "colossless_gpv \<I>' (run_converter conv a) \<and> (\<forall>(b, conv')\<in>results_gpv \<I>' (run_converter conv a). colossless_converter \<I> \<I>' conv')"
"1_1": fixes xa :: "nat \<times> nat"   and xb :: "nat"   and ya :: "nat" assumes "\<not> n \<le> 1"   and "xa = divmod_nat n 2"   and "(xb, ya) = xa"   and "\<And>i. i \<le> xb \<Longrightarrow> to_fract x ^ i / to_fract y ^ (i - 1) \<in> range to_fract" shows "to_fract (dichotomous_Lazard x y xb) = to_fract x ^ xb / to_fract y ^ (xb - 1)"
S5ClerkUnch: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> [MClkNext memCh crCh cst p]_(c p) \<and> $S5 rmhist p \<longrightarrow> unchanged c p"
netglobalsimp: fixes np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'e) \<times> 'f, 'd seq_action) automaton"   and sr :: "'c \<Rightarrow> 'e \<times> 'f"   and P :: "(nat \<Rightarrow> 'e) \<Rightarrow> bool"   and s :: "'c net_state" assumes "openproc np onp sr" shows "openproc.netglobal np sr P s = P (fst (openproc.initmissing np sr (netgmap sr s)))"
fds_abs_converges_comparison_test: fixes f :: "'b fds"   and g :: "real fds"   and s :: "'b" assumes "\<forall>\<^sub>F n in sequentially. norm (fds_nth f n) \<le> fds_nth g n"   and "fds_converges g (s \<bullet> (1::'b))" shows "fds_abs_converges f s"
FP'_2: shows "F I0 \<in> F ` P"
replicate_mset_eq_empty_iff: fixes n :: "nat"   and a :: "'b" shows "(replicate_mset n a = {#}) = (n = 0)"
set_intros_1: fixes x21 :: "'g"   and x22 :: "'g list" shows "x21 \<in> set (x21 # x22)"
cdf_is_right_cont: fixes a :: "real" shows "continuous (at_right a) (cdf M)"
pow_carrier_mat: fixes A :: "'b mat"   and n :: "nat"   and k :: "nat" assumes "A \<in> carrier_mat n n" shows "A ^\<^sub>m k \<in> carrier_mat n n"
"1_1": shows "\<forall>m. Suc m \<le> length \<GG> \<longrightarrow> (\<forall>x. m = length x \<longrightarrow> (\<forall>xa xb. Group.group xb \<longrightarrow> finite (carrier xb) \<longrightarrow> composition_series xb x \<longrightarrow> composition_series xb xa \<longrightarrow> mset (map group.iso_class (normal_series.quotients xb x)) = mset (map group.iso_class (normal_series.quotients xb xa))))"
list_max_nonempty: fixes xs :: "'t list" assumes "xs \<noteq> []" shows "list_max xs = ord_term_lin.Max (set xs)"
assms_2: fixes f :: "nat \<Rightarrow> nat option" assumes "f \<in> U" shows "\<exists>i. \<psi> i = f"
assms_3: fixes n :: "nat" assumes "0 < n"   and "P n" shows "P (Suc n)"
cut_le_not_empty_iff: fixes I :: "'a set"   and t :: "'a" shows "(I \<down>\<le> t \<noteq> {}) = (\<exists>x\<in>I. x \<le> t)"
Polys_def: fixes X :: "'b set" shows "P[X] = {p. keys p \<subseteq> .[X]}"
C1: fixes \<Psi>' :: "'b"   and \<Psi>\<^sub>U :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and U :: "('a, 'b, 'c) psi"   and A\<^sub>U :: "name list" assumes "(\<Psi>' \<otimes> \<Psi>\<^sub>U, S, T) \<in> Rel"   and "extractFrame U = \<langle>A\<^sub>U, \<Psi>\<^sub>U\<rangle>"   and "A\<^sub>U \<sharp>* \<Psi>'"   and "A\<^sub>U \<sharp>* S"   and "A\<^sub>U \<sharp>* T" shows "(\<Psi>', S \<parallel> U, T \<parallel> U) \<in> Rel'"
rel_pmfI: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'b pmf"   and q :: "'c pmf" assumes "rel_set R (set_pmf p) (set_pmf q)"   and "\<And>x y. \<lbrakk>x \<in> set_pmf p; y \<in> set_pmf q; R x y\<rbrakk> \<Longrightarrow> measure_pmf.prob p {x. R x y} = measure_pmf.prob q {y. R x y}" shows "rel_pmf R p q"
obtain_frontier_elem: fixes M :: "'a zmultiset"   and t :: "'a" assumes "0 < zcount M t" obtains u :: "'a" where "u \<in>\<^sub>A frontier M" and "u \<le> t"
wellorder_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "class.wellorder less_eq less \<equiv> class.linorder less_eq less \<and> class.wellorder_axioms less"
add_le_add_imp_diff_le: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and i :: "'c"   and k :: "'c"   and n :: "'c"   and j :: "'c" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq (plus i k) n"   and "less_eq n (plus j k)"   and "less_eq (plus i k) n"   and "less_eq n (plus j k)" shows "less_eq (minus n k) j"
bot_pres_multr: fixes y :: "'c" shows "\<bottom> \<cdot> y = \<bottom>"
arg: fixes rs :: "(name \<times> sterm) list"   and u :: "sterm"   and u' :: "sterm"   and t :: "sterm" assumes "rs \<turnstile>\<^sub>s u \<longrightarrow> u'" shows "rs \<turnstile>\<^sub>s t $\<^sub>s u \<longrightarrow> t $\<^sub>s u'"
boundedE: fixes b :: "'c \<times> 'c \<Rightarrow> 'd"   and c :: "'c \<times> 'c \<Rightarrow> 'd"   and a :: "'c \<times> 'c \<Rightarrow> 'd" assumes "b \<oplus> c \<preceq> a" obtains "b \<preceq> a" and "c \<preceq> a"
w_in_hom_2: shows "\<guillemotleft>w : w \<Rightarrow> w\<guillemotright>"
ua: shows "ua \<in> insert v E \<rightarrow> carrier K"
less_extended_simps_4: fixes h :: "'a extended" shows "(\<infinity> < h) = False"
Abs_reg_lan_inverse: fixes y :: "'a list set" assumes "y \<in> range lang" shows "Rep_reg_lan (Abs_reg_lan y) = y"
fds_abs_converges: fixes f :: "'a fds"   and s :: "'a" assumes "abs_conv_abscissa f < ereal (s \<bullet> (1::'a))" shows "fds_abs_converges f s"
valid_history_axioms: fixes program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'a \<Rightarrow> 'a \<times> instr list \<Rightarrow> bool" shows "valid_history program_step ts\<^sub>s\<^sub>b"
all_total_totalize: fixes F :: "'b program" shows "all_total (totalize F)"
simps_7: fixes cl :: "('a, 'b, 'c) cmd list" shows "finished (ParT cl) = Ball (set cl) finished"
fps_unit_factor_decompose: fixes f :: "'a fps" shows "f = unit_factor f * fps_X ^ subdegree f"
orderI: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "a = a \<union># b" shows "b \<subseteq># a"
inject_4: fixes x51 :: "'b fm"   and x52 :: "'b fm"   and y51 :: "'b fm"   and y52 :: "'b fm" shows "((x51 \<^bold>\<longrightarrow> x52) = (y51 \<^bold>\<longrightarrow> y52)) = (x51 = y51 \<and> x52 = y52)"
moebius_mu_not_squarefree: fixes n :: "nat" assumes "\<not> squarefree n" shows "moebius_mu n = (0::'b)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> char list option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_attribute type_wf set_attribute set_attribute_locs"   and "l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"   and "l_set_attribute_get_child_nodes_axioms set_attribute_locs get_child_nodes_locs" shows "l_set_attribute_get_child_nodes type_wf set_attribute set_attribute_locs known_ptr get_child_nodes get_child_nodes_locs"
DF_unfold: fixes A :: "'b set" shows "DF A = (\<sqinter>z\<in>A \<rightarrow>  DF A)"
ring_distribs_1: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "a * (b + c) = a * b + a * c"
intros_5: fixes BMap :: "'a \<Rightarrow> bool"   and E :: "'b \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and b :: "'b"   and m :: "'c \<Rightarrow> 'a"   and c1 :: "('b, 'c) MWLfCom"   and c2 :: "('b, 'c) MWLfCom" assumes "BMap (E b m) = True" shows "((if b then c1 else c2 fi, m), [c1], m) \<in> MWLf_semantics.MWLfSteps_det E BMap"
ext: fixes g :: "'d"   and f :: "'d" assumes "Dop.comp g f \<noteq> Dop.null" shows "Dop.seq g f"
k_def_1: shows "[[a\<^sub>1 b f k]]"
simps_1: fixes lens_get :: "'c \<Rightarrow> 'd"   and lens_put :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and more :: "'e" shows "get\<^bsub>\<lparr>lens_get = lens_get, lens_put = lens_put, \<dots> = more\<rparr>\<^esub> = lens_get"
cyc_free_diag_dest': fixes n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'c"   and i :: "nat" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len m i i xs"   and "i \<le> n" shows "\<one> \<le> m i i"
phis_wf: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and phis :: "'a \<Rightarrow> 'b \<times> 'd \<Rightarrow> 'd list option"   and g :: "'a"   and n :: "'b"   and v :: "'d"   and args :: "'d list" assumes "CFG_SSA \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis"   and "phis g (n, v) = Some args" shows "length (graph_path_base.predecessors inEdges' g n) = length args"
before_between2: fixes f :: "face"   and u :: "nat"   and v :: "nat"   and w :: "nat" assumes "before (verticesFrom f u) v w"   and "distinct (vertices f)"   and "u \<in> \<V> f" shows "u = v \<or> u \<in> set (between (vertices f) w v)"
B: shows "msgs cJ1 <pJ, inM True> = 1"
homeomorphism_sym: fixes S :: "'b set"   and t :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" shows "homeomorphism S t f g = homeomorphism t S g f"
is_nth_powerE: fixes n :: "nat"   and x :: "'a" assumes "is_nth_power n x" obtains y :: "'a" where "x = y ^ n"
reachable_target_elim: fixes q :: "'c"   and A :: "('d, 'c) ngba"   and p :: "'c" assumes "q \<in> NGBA.reachable A p" obtains r :: "('d \<times> 'c) list" where "NGBA.path A r p" and "q = target r p"