hcju/isabelleps huggingface dataset:
seq: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and c\<^sub>1 :: "('a, 'c, 'd) Stmt"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c\<^sub>1' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b"   and c\<^sub>2 :: "('a, 'c, 'd) Stmt" assumes "sifum_lang eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "(((c\<^sub>1, mds), mem), (c\<^sub>1', mds'), mem') \<in> sifum_lang.eval\<^sub>w eval\<^sub>A eval\<^sub>B" shows "(((c\<^sub>1 ;; c\<^sub>2, mds), mem), (c\<^sub>1' ;; c\<^sub>2, mds'), mem') \<in> sifum_lang.eval\<^sub>w eval\<^sub>A eval\<^sub>B"
open_eq_continuous_inverse_map: fixes X :: "'c topology"   and f :: "'c \<Rightarrow> 'd"   and Y :: "'d topology"   and g :: "'d \<Rightarrow> 'c" assumes "\<And>x. x \<in> topspace X \<Longrightarrow> f x \<in> topspace Y \<and> g (f x) = x"   and "\<And>y. y \<in> topspace Y \<Longrightarrow> g y \<in> topspace X \<and> f (g y) = y" shows "open_map X Y f = continuous_map Y X g"
assms_1: shows "ccpo.admissible Sup (\<le>) P"
D_def: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "QR.D p q = {x \<in> {0<..(int p * int q - 1) div 2}. x mod int p \<in> {(int p - 1) div 2<..int p - 1} \<and> x mod int q \<in> {(int q - 1) div 2<..int q - 1}}"
vlrestriction_vdomain: shows "xs' \<restriction>\<^sup>l\<^sub>\<circ> \<D>\<^sub>\<circ> xs' = xs'"
SF_def: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a" shows "SF(A)_v \<equiv> \<box>\<diamond>Enabled (<A>_v) \<longrightarrow> \<box>\<diamond><A>_v"
assms: shows "\<forall>i. proj2_no_3_Col (range (($) (p $ i)))"
termdiff_converges_all: fixes c :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "\<And>x. summable (\<lambda>n. c n * x ^ n)" shows "summable (\<lambda>n. diffs c n * x ^ n)"
hyps_3: fixes \<P> :: "('a list \<times> nat) multiset \<Rightarrow> 'a" assumes "\<P> \<in> carrier M" shows "index_free \<P> (P, i)"
cfE_ss_invE: shows "cf.E \<subseteq> E \<union> E\<inverse>"
cong_3: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and P' :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<And>\<sigma>. P \<sigma> = P' \<sigma>" shows "\<diamond>P = \<diamond>P'"
AE_pair_measure: fixes P :: "'d \<times> 'c \<Rightarrow> bool" assumes "{x \<in> space (S \<Otimes>\<^sub>M T). P x} \<in> sets (S \<Otimes>\<^sub>M T)"   and "AE x in S. AE y in T. P (x, y)" shows "almost_everywhere (S \<Otimes>\<^sub>M T) P"
valid_program_history: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "causal_program_history is sb"
wf: shows "CFGExit_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"
continuous_map_alt: fixes T1 :: "'b topology"   and T2 :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "continuous_map T1 T2 f = ((\<forall>U. openin T2 U \<longrightarrow> openin T1 (f -` U \<inter> topspace T1)) \<and> f ` topspace T1 \<subseteq> topspace T2)"
chain_mono: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'b set" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x y"   and "pred_on.chain A P C" shows "pred_on.chain A Q C"
i_expand_i_drop_mod: fixes n :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'd message_af" assumes "n mod k = 0" shows "f \<odot> k \<Up> n = f \<Up> (n div k) \<odot> k"
IH: assumes "k' \<le> k"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n" shows "fw m n k i j \<le> fw m n k' i j"
Chinese_remTr3: fixes n :: "nat"   and J :: "nat \<Rightarrow> 'a set"   and B :: "nat \<Rightarrow> 'a set Ring"   and S :: "nat \<Rightarrow> 'a \<Rightarrow> 'a set" assumes "\<forall>k\<le>Suc n. ideal R (J k)"   and "\<forall>k\<le>Suc n. B k = R /\<^sub>r J k"   and "\<forall>k\<le>Suc n. S k = pj R (J k)"   and "\<forall>i\<le>Suc n. \<forall>j\<le>Suc n. i \<noteq> j \<longrightarrow> coprime_ideals R (J i) (J j)" shows "surjec\<^bsub>R,(r\<Pi>\<^bsub>{j. j \<le> Suc n}\<^esub> B)\<^esub> A_to_prodag R {j. j \<le> Suc n} S B"
adj_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" shows "f \<stileturn> g = (\<forall>x y. (f x \<le> y) = (x \<le> g y))"
axioms_1: fixes G :: "('d, 'e) monoid_scheme"   and H :: "('f, 'g) monoid_scheme"   and h :: "'d \<Rightarrow> 'f" assumes "group_hom G H h" shows "Group.group G"
supp_map_bset_Rep_formula: fixes xset :: "('a, 'b, 'c, 'd) formula set['e]" shows "supp (map_bset Rep_formula xset) = supp xset"
vimage2p_refl: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "\<And>x. R x x" shows "vimage2p f f R x x"
card_finite: fixes S :: "'b set" assumes "card S = CARD('c)" shows "finite S"
Vr_potent_eq: fixes v :: "'a \<Rightarrow> ant"   and n :: "ant"   and m :: "ant" assumes "valuation K v"   and "0 \<le> n"   and "0 \<le> m"   and "vp K v\<^bsup> Vr K v n\<^esup> = vp K v\<^bsup> Vr K v m\<^esup>" shows "n = m"
seqr_pre_out: fixes p :: "('c, 'b) urel"   and Q :: "('c, 'b) urel"   and R :: "'b hrel" assumes "out\<alpha> \<sharp> p" shows "(p \<and> Q) ;; R = (p \<and> Q ;; R)"
floor_divide_lower: fixes q :: "'c"   and p :: "'c" assumes "(0::'c) < q" shows "of_int \<lfloor>p / q\<rfloor> * q \<le> p"
laz_eq: fixes a :: "'a list"   and b :: "'a list" shows "list_all_zip (=) a b = (a = b)"
fold_once_fold_remdups: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and xs :: "'b list" assumes "rsp_fold f" shows "fold_once f xs = fold f (remdups xs)"
simps_5: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "restricted_subformulas_inner (\<phi> R\<^sub>n \<psi>) = restricted_subformulas_inner \<phi> \<union> restricted_subformulas_inner \<psi>"
mset_subset_eq_exists_conv: fixes A :: "'f multiset"   and B :: "'f multiset" shows "(A \<subseteq># B) = (\<exists>C. B = A + C)"
Max_le_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less_eq (linorder.Max less_eq A) x = (\<forall>a\<in>A. less_eq a x)"
parts_insert: fixes X :: "msg"   and H :: "msg set" shows "insert X (parts H) \<subseteq> parts (insert X H)"
\<omega>consistentStd1_def: shows "B.\<omega>consistentStd1 \<equiv> \<forall>\<phi>\<in>fmla. \<forall>x\<in>var. Fvars \<phi> = {x} \<longrightarrow> (\<forall>n\<in>num. bprv (neg (subst \<phi> n x))) \<longrightarrow> \<not> bprv (exi x \<phi>)"
conj_subset_def: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "(A \<subseteq> _Collect x (P x \<and> Q x)) = (A \<subseteq> _Collect x (P x) \<and> A \<subseteq> _Collect x (Q x))"
identity_arrow_of_tabulations_in_maps_axioms: shows "identity_arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg (r \<star> s) \<rho>\<sigma>.tab (tab\<^sub>0 s \<star> p\<^sub>0) (tab\<^sub>1 r \<star> p\<^sub>1) (r \<star> s) tab (tab\<^sub>0 (r \<star> s)) (tab\<^sub>1 (r \<star> s)) (r \<star> s)"
ub_def: shows "ub \<equiv> 2 / ln 2 * sqrt x + 8 * ln 2 * x / (ln x)\<^sup>2"
clock_simVal_def: shows "clock_simVal \<equiv> envVal \<circ> snd"
infsetsum_reindex_bij_betw: fixes g :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "bij_betw g A B" shows "(\<Sum>\<^sub>ax\<in>A. f (g x)) = infsetsum f B"
sup_bot_right: fixes x :: "'g multiset" shows "x \<union># {#} = x"
split_list_first_propE: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" obtains ys :: "'b list" and x :: "'b" and zs :: "'b list" where "xs = ys @ x # zs" and "P x" and "\<forall>y\<in>set ys. \<not> P y"
distinct_6: fixes trm1' :: "trm"   and trm2' :: "trm"   and name :: "name"   and trm :: "trm" shows "App trm1' trm2' \<noteq> Lam [name].trm"
t_n_closed: fixes x :: "'a" shows "ad (ad (ad x)) = ad x"
abs_eq: fixes x :: "nat \<Rightarrow> rat" assumes "realrel x x" shows "Real.positive (Real.Real x) = (\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < x n)"
transIrreflexiveOrderingIsAcyclic: fixes r :: "('b \<times> 'b) set" assumes "trans r"   and "\<forall>x. (x, x) \<notin> r" shows "acyclic r"
integrable_ln_Gamma_aux: fixes s :: "complex"   and n :: "nat"   and N :: "nat" assumes "s \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "(\<lambda>x. complex_of_real (pbernpoly n x) / (complex_of_real x + s) ^ n) integrable_on {0..real N}"
merge_poly_merge_poly: shows "(merge_poly, merge_poly) \<in> poly_rel \<rightarrow> poly_rel \<rightarrow> poly_rel"
index_transpose_mat_2: fixes A :: "'a mat" shows "dim_row A\<^sup>T = dim_col A"
initial\<^sub>d_axioms_def: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> 'd \<times> nat set \<times> ('e \<Rightarrow> 'f option)) list"   and valid :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'g list \<times> 'd \<times> nat set \<times> ('e \<Rightarrow> 'f option)) list \<Rightarrow> bool" shows "initial\<^sub>d_axioms ts valid \<equiv> (\<forall>i p is xs sb \<D> \<O> \<R>. i < length ts \<longrightarrow> ts ! i = (p, is, xs, sb, \<D>, \<O>, \<R>) \<longrightarrow> is = []) \<and> (\<forall>i p is xs sb \<D> \<O> \<R>. i < length ts \<longrightarrow> ts ! i = (p, is, xs, sb, \<D>, \<O>, \<R>) \<longrightarrow> \<R> = Map.empty) \<and> valid (map (\<lambda>(p, is, \<theta>, sb, \<D>, \<O>, \<R>). (p, is, \<theta>, [], \<D>, \<O>, \<R>)) ts)"
LI_preserves_tfr_stp_all_single: fixes S :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S' :: "('fun, 'var) strand_step list"   and \<theta>' :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "(S, \<theta>) \<leadsto> (S', \<theta>')"   and "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S \<theta>"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"   and "list_all tfr\<^sub>s\<^sub>t\<^sub>p S"   and "tfr\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>t S)"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>t S)" shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p S'"
not_below2not_eq: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.po below"   and "\<not> below x y" shows "x \<noteq> y"
cT_ne: shows "cbox (- c) c \<inter> T \<noteq> {}"
hop3: shows "rel_spmf (\<lambda>x xa. snd (snd x) = snd xa \<and> (\<not> snd xa \<longrightarrow> fst x = fst xa)) game_semi_forgetful game_A"
P: shows "prime (p\<^sup>2 + N * q\<^sup>2)"
subdigraph_axioms: shows "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<AA>"
le_minus_plus_same_hmset: fixes m :: "hmultiset"   and n :: "hmultiset" shows "m \<le> m - n + n"
is_cadjoint_unique: fixes F1 :: "'a \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> 'a"   and F2 :: "'a \<Rightarrow> 'b" assumes "is_cadjoint F1 G"   and "is_cadjoint F2 G" shows "F1 = F2"
ws_prog_ideclD: fixes G :: "prog"   and I :: "qtname"   and i :: "iface"   and J :: "qtname" assumes "iface G I = Some i"   and "J \<in> set (isuperIfs i)"   and "ws_prog G" shows "is_iface G J \<and> (J, I) \<notin> (subint1 G)\<^sup>+"
cobounded1: fixes a :: "'a upred"   and b :: "'a upred" shows "a \<sqsubseteq> (a \<and> b)"
H'_def: shows "H' = G\<lparr>carrier := H\<rparr>"
del_point_def: fixes pa :: "'a"   and p :: "'a" shows "add_point_sys.del_point pa p \<equiv> del_block_sys.add_point pa - {p}"
prime_elem_dvd_power: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and p :: "'b"   and x :: "'b"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.prime_elem times one zero p"   and "dvd.dvd times p (power.power one times x n)" shows "dvd.dvd times p x"
not_numeral_le_zero: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less_eq (numeral.numeral one plus n) zero"
ord_iso_alt: fixes f :: "'b \<Rightarrow> 'c" shows "ord_iso f = (ord_embed f \<and> top_pres ((`) f))"
integrable_on_powr_from_0: fixes a :: "real"   and c :: "real" assumes "- 1 < a"   and "0 \<le> c" shows "(\<lambda>x. x powr a) integrable_on {0..c}"
catch_gpv_def: fixes gpv :: "('g, 'h, 'i) gpv" shows "catch_gpv gpv = TRY map_gpv Some id gpv ELSE Generative_Probabilistic_Value.Done None"
lateEarlyCong: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim>\<^sup>s\<^sub>l Q" shows "P \<sim>\<^sup>s\<^sub>e Q"
matcheq_matchAny: fixes m :: "'c match_expr"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d" assumes "\<not> has_primitive m" shows "matcheq_matchAny m = matches \<gamma> m p"
fmapU_maybe_simps_1: fixes f :: "udom \<rightarrow> udom" shows "fmapU\<cdot>f\<cdot>\<bottom> = \<bottom>"
adj_mk_symmetric_eq: assumes "symmetric (with_proj (list_digraph G))" shows "parcs (mk_symmetric (with_proj (list_digraph G))) = parcs (list_digraph G)"
simps_1: fixes S :: "'b"   and S' :: "'b" shows "(SKIP {S} \<sqsubseteq> SKIP {S'}) = (S \<sqsubseteq> S')"
simps_6: fixes ux :: "'a" shows "(Top \<le> Val ux) = False"
star_sum_unfold: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<star> + x\<^sup>\<star> \<cdot> y \<cdot> (x + y)\<^sup>\<star> = (x + y)\<^sup>\<star>"
assms: fixes Gxy :: "'a set"   and x :: "'a"   and y :: "'a"   and Gxz :: "'a set"   and z :: "'a"   and Gyz :: "'a set"   and w :: "'a" assumes "geodesic_segment_between Gxy x y"   and "geodesic_segment_between Gxz x z"   and "geodesic_segment_between Gyz y z"   and "w \<in> Gxy" shows "infdist w (Gxz \<union> Gyz) \<le> delta"
currentD_weight_OUT: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "current \<Gamma> f" shows "d_OUT f x \<le> weight \<Gamma> x"
fls_right_inverse': fixes x :: "'a"   and f :: "'a fls"   and y :: "'a" assumes "x * fls_nth f (fls_subdegree f) = (1::'a)"   and "fls_nth f (fls_subdegree f) * y = (1::'a)" shows "f * fls_left_inverse f x = 1"
DisjAssoc1: fixes H :: "fm set"   and A :: "fm"   and B :: "fm"   and C :: "fm" assumes "H \<turnstile> A OR B OR C" shows "H \<turnstile> (A OR B) OR C"
pow_ge_lgth: fixes tid :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "honest tid"   and "\<parallel>tid\<parallel>\<^bsub>t n\<^esub>" shows "length (bc (\<sigma>\<^bsub>tid\<^esub>t n)) \<le> PoW t n"
sigma_algebra_trivial: fixes \<Omega> :: "'c set" shows "sigma_algebra \<Omega> {{}, \<Omega>}"
tr_par_deduct_iff: fixes B :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and D :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and I :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term" assumes "B \<in> set (tr\<^sub>p\<^sub>c A D)" shows "(ik\<^sub>s\<^sub>t (unlabel B) \<cdot>\<^sub>s\<^sub>e\<^sub>t I \<turnstile> t) = (ik\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t I \<turnstile> t)"
ccBindsExtra_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "ccBindsExtra \<Gamma> = (\<Lambda> i. snd i \<squnion> ccBinds \<Gamma>\<cdot>i \<squnion> (\<Squnion>x\<mapsto>e\<in>map_of \<Gamma>. fv e G\<times> ccNeighbors x (snd i)))"
less_eq_multiset\<^sub>H\<^sub>O: fixes M :: "'a multiset"   and N :: "'a multiset" shows "(M \<le> N) = (\<forall>y. count N y < count M y \<longrightarrow> (\<exists>x>y. count M x < count N x))"
append_butlast_last_id: fixes xs :: "'d list" assumes "xs \<noteq> []" shows "butlast xs @ [last xs] = xs"
SN_order_pair_axioms: shows "SN_order_pair S NS"
H2: shows "ss [\<rightarrow>\<^sub>s] ss'"
add_le_cancel_left: fixes c :: "'b zmultiset"   and a :: "'b zmultiset"   and b :: "'b zmultiset" shows "(c + a \<subseteq>#\<^sub>z c + b) = (a \<subseteq>#\<^sub>z b)"
remove_heap_is_wellformed_preserved_2: fixes h :: "(_) heap"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> remove child \<rightarrow>\<^sub>h h'"   and "known_ptrs h"   and "type_wf h" shows "known_ptrs h'"
assms_2: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<otimes> c = b \<otimes> c"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "a = b"
Not_Domain_rtrancl: fixes x :: "'b"   and R :: "('b \<times> 'b) set"   and y :: "'b" assumes "x \<notin> Domain R" shows "((x, y) \<in> R\<^sup>*) = (x = y)"
exhaust: fixes y :: "'h \<times> 'i" obtains a :: "'h" and b :: "'i" where "y = (a, b)"
assms: fixes c' :: "real" assumes "c < ereal c'" shows "\<exists>s. s \<bullet> (1::'a) = c' \<and> fds_abs_converges f s"
minit_safe_def: fixes \<phi> :: "'a MFOTL.formula" shows "minit_safe \<phi> = (if mmonitorable_exec \<phi> then minit \<phi> else undefined)"
reflcl_trancl: fixes r :: "('c \<times> 'c) set" shows "(r\<^sup>+)\<^sup>= = r\<^sup>*"
arc_mono_def: fixes P :: "('c, 'd) pre_digraph \<Rightarrow> bool" shows "arc_mono P \<equiv> \<forall>H1 H2. P H1 \<and> subgraph H1 H2 \<and> verts H1 = verts H2 \<longrightarrow> P H2"
line_in: fixes t :: "real" assumes "t \<in> {0..1}" shows "x + t *\<^sub>R y \<in> S"
overrider_def: fixes F :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "overrider F \<equiv> (\<forall>x y z. F (F x y) z = F x z) \<and> (\<forall>x y z. F x (F y z) = F x z)"
sets_Collect_imp: fixes M :: "'c measure"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "{x \<in> space M. P x} \<in> sets M"   and "{x \<in> space M. Q x} \<in> sets M" shows "{x \<in> space M. Q x \<longrightarrow> P x} \<in> sets M"
power_one: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times one n = one"
zero_sub_one: shows "0\<^sub>h -\<^sub>h 1\<^sub>h = of_complex (- 1)"
simps_1: fixes x :: "('b, 'c) Combinators" shows "list2policyR [x] = x"
kab_3: shows "a div gcd a b = gcd (a div gcd a b) (b div gcd a b) * ka'"
rel_of_enex_def: fixes enex :: "('d \<Rightarrow> 'f set) \<times> ('f \<Rightarrow> 'd \<Rightarrow> 'e)" shows "rel_of_enex enex \<equiv> let (en, ex) = enex in {(s, ex a s) |s a. a \<in> en s}"
comp_ipresIGSubst: fixes h :: "'u \<Rightarrow> 'v"   and MOD :: "('w, 'x, 'varSort, 'sort, 'y, 'z, 'u, 'aa, 'ab) model_scheme"   and MOD' :: "('ac, 'ad, 'varSort, 'sort, 'ae, 'z, 'v, 'af, 'ag) model_scheme"   and h' :: "'v \<Rightarrow> 'ah"   and MOD'' :: "('ai, 'aj, 'varSort, 'ak, 'al, 'z, 'ah, 'am, 'an) model_scheme" assumes "ipresIGWls h MOD MOD'"   and "ipresIGSubst h MOD MOD'"   and "ipresIGSubst h' MOD' MOD''" shows "ipresIGSubst (h' \<circ> h) MOD MOD''"
not_eq_order_implies_strict: fixes a :: "'e"   and b :: "'e" assumes "a \<noteq> b"   and "a \<le> b" shows "a < b"
equivalence_of_bicategories_axioms: shows "equivalence_of_bicategories (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<eta>\<^sub>0 \<eta>\<^sub>1 \<epsilon>\<^sub>0 \<epsilon>\<^sub>1"
real_cond_exp_def: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> real" shows "real_cond_exp M F f = (\<lambda>x. enn2real (nn_cond_exp M F (\<lambda>x. ennreal (f x)) x) - enn2real (nn_cond_exp M F (\<lambda>x. ennreal (- f x)) x))"
init_throw_PD: fixes P :: "(char list list \<times> char list exp) prog"   and C' :: "char list"   and Cs :: "char list list"   and b :: "bool"   and s :: "heap \<times> sfields \<times> sheap"   and a :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>INIT C' (Cs,b) \<leftarrow> unit,s\<rangle> \<Rightarrow> \<langle>throw a,s'\<rangle>"   and "iconf (shp s) (INIT C' (Cs,b) \<leftarrow> unit)" shows "\<exists>sfs i. shp s' C' = \<lfloor>(sfs, Error)\<rfloor>"
cancel1_into_pcancel1: fixes xs :: "'d list"   and ys :: "'d list" assumes "cancel1 xs ys" shows "pcancel1 xs ys"
utp_theory_left_unital_axioms: shows "utp_theory_left_unital \<H> \<I>\<I>"
push_exn: fixes \<sigma> :: "heap option"   and c :: "'b Heap"   and r :: "'b" assumes "is_exn \<sigma>" shows "run c \<sigma> \<sigma> r"
simps_5: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and x :: "variable"   and \<phi> :: "fml" shows "usubstappf \<sigma> U (Exists x \<phi>) = Existso x (usubstappf \<sigma> (U \<union> {x}) \<phi>)"
adjacent_V_1: fixes G :: "('a, 'b) graph"   and v :: "'a"   and v' :: "'a" assumes "valid_unMultigraph G"   and "valid_unMultigraph.adjacent G v v'" shows "v \<in> nodes G"
exhaust: fixes y :: "'b tree" obtains "y = Leaf"   | x21 :: "nat" and x22 :: "nat" and x23 :: "'b option" and x24 :: "'b tree" and x25 :: "'b tree" where "y = Node x21 x22 x23 x24 x25"
measurable_compose_rev: fixes f :: "'b \<Rightarrow> 'c"   and L :: "'b measure"   and N :: "'c measure"   and g :: "'d \<Rightarrow> 'b"   and M :: "'d measure" assumes "f \<in> L \<rightarrow>\<^sub>M N"   and "g \<in> M \<rightarrow>\<^sub>M L" shows "(\<lambda>x. f (g x)) \<in> M \<rightarrow>\<^sub>M N"
simps_2: fixes \<xi> :: "nat \<Rightarrow> 'b set" shows "\<xi> \<Turnstile>\<^sub>c false\<^sub>c = False"
fls_of_int: fixes i :: "int" shows "of_int i = fls_const (of_int i)"
star_of_Infinitesimal_iff_0: fixes x :: "'a" shows "(star_of x \<in> Infinitesimal) = (x = (0::'a))"
commute: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "AC f" shows "f x y = f y x"
ESem_subst: fixes Fn :: "('a \<rightarrow> 'a) \<rightarrow> 'a"   and Fn_project :: "'a \<rightarrow> 'a \<rightarrow> 'a"   and B :: "bool discr \<rightarrow> 'a"   and B_project :: "'a \<rightarrow> 'a \<rightarrow> 'a \<rightarrow> 'a"   and tick :: "'a \<rightarrow> 'a"   and e :: "exp"   and \<sigma> :: "Vars.var \<Rightarrow> 'a"   and x :: "Vars.var"   and y :: "Vars.var" shows "semantic_domain.ESem Fn Fn_project B B_project tick e\<cdot>(\<sigma>(x := \<sigma> y)) = semantic_domain.ESem Fn Fn_project B B_project tick e[x::=y]\<cdot>\<sigma>"
elims: fixes x :: "'edge list"   and xa :: "'edge list"   and y :: "'edge list" assumes "upd_cs x xa = y" obtains cs :: "'edge list" where "x = csb__" and "xa = []" and "y = csb__"   | cs :: "'edge list" and a :: "'edge" and as :: "'edge list" where "x = cs" and "xa = a # as" and "y = (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> upd_cs (a # cs) as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> case cs of [] \<Rightarrow> upd_cs cs as | c' # cs' \<Rightarrow> upd_cs cs' as | _ \<Rightarrow> upd_cs cs as)"
b1: shows "inv_map (Some \<circ> \<pi>) \<circ>\<^sub>m (Some \<circ> \<pi>) = Some"
inducts: fixes x1 :: "lam"   and x2 :: "lam"   and P :: "lam \<Rightarrow> lam \<Rightarrow> bool" assumes "x1 \<longrightarrow>x x2"   and "\<And>E x t t'. P (E\<lbrakk>App (Lam [x].t) t'\<rbrakk>) (E\<lbrakk>t[x::=t']\<rbrakk>)" shows "P x1 x2"
par_beta_refl: fixes t :: "sterm" assumes "lc t" shows "t => t"
transaction_views_for_same_hash: fixes t\<^sub>m :: "transaction\<^sub>m"   and t :: "transaction" assumes "t\<^sub>m \<in> set (local.transaction_views_for t)" shows "hash_transaction t\<^sub>m = hash_transaction (embed_transaction t)"