hcju/isabelleps huggingface dataset:
equiintegrable_on_insert: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and F :: "('a \<Rightarrow> 'b) set" assumes "f integrable_on cbox a b"   and "F equiintegrable_on cbox a b" shows "insert f F equiintegrable_on cbox a b"
lookup_pair_sort_oalist_aux: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and ko :: "'o" assumes "oalist_inv xs" shows "lookup_pair ko (sort_oalist_aux ko xs) = lookup_raw xs"
Par_transC_invert: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "(c1 | c2, s) \<rightarrow>c (c', s')" obtains c1' :: "('test, 'atom) com" where "(c1, s) \<rightarrow>c (c1', s')" and "c' = c1' | c2"   | "(c1, s) \<rightarrow>t s'" and "c' = c2"   | c2' :: "('test, 'atom) com" where "(c2, s) \<rightarrow>c (c2', s')" and "c' = c1 | c2'"   | "(c2, s) \<rightarrow>t s'" and "c' = c1"
rep_iso_class: fixes f :: "('a, 'a set) Maps.arr" assumes "Maps.ide f" shows "Maps.isomorphic (Span.VVV.iso_class_rep (Maps.iso_class f)) f"
elements_matD: fixes a :: "'b"   and A :: "'b mat" assumes "a \<in> elements_mat A" shows "\<exists>i j. i < dim_row A \<and> j < dim_col A \<and> a = A $$ (i, j)"
cadjoint_univ_prop: fixes G :: "'c \<Rightarrow> 'd" assumes "bounded_clinear G" shows "\<forall>x y. (G\<^sup>\<dagger>) x \<bullet>\<^sub>C y = x \<bullet>\<^sub>C G y"
remove_comp_support_set: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and x :: "'c" shows "support_set (qty_rem_comp pf1 x) \<subseteq> support_set pf1 - {x}"
remaining_qty_adapt: fixes asset :: "'b"   and pf :: "'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and v :: "real"   and n :: "nat" assumes "borel_adapt_stoch_proc F (prices Mkt asset)"   and "trading_strategy pf"   and "support_adapt Mkt pf" shows "remaining_qty Mkt v pf asset n \<in> borel_measurable (F n)"
l_neg: fixes x :: "'c" assumes "x \<in> carrier V" shows "\<ominus>\<^bsub>V\<^esub> x \<oplus>\<^bsub>V\<^esub> x = \<zero>\<^bsub>V\<^esub>"
open_fbint_altdef: shows "open = generate_topology {arith_prog_fb a b |a b. 0 < b}"
simps_1: fixes ys :: "'g list" shows "splice [] ys = ys"
order_trans: fixes x :: "'c list"   and y :: "'c list"   and z :: "'c list" assumes "suffix x y"   and "suffix y z" shows "suffix x z"
valid_K_cfg: fixes cfg :: "'s cfg"   and cfg' :: "'s cfg" assumes "cfg \<in> valid_cfg"   and "cfg' \<in> set_pmf (K_cfg cfg)" shows "cfg' \<in> valid_cfg"
hyps: assumes "a \<in> set xs"   and "sorted (map f xs)"   and "hd (filter (\<lambda>x. f a = f x) xs) = a" shows "insort_key f a (remove1 a xs) = xs"
base_in_hom_2: shows "\<guillemotleft>r : r \<Rightarrow>\<^sub>C r\<guillemotright>"
step: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> X" shows "P \<leadsto>[(X \<union> bisim)] Q \<and> (Q, P) \<in> bisim \<union> X"
polyn_ring_nonzero: shows "1\<^sub>r \<noteq> \<zero>"
simps: fixes i :: "'a"   and j :: "nat"   and p :: "('a, 'b) event \<Rightarrow> bool" shows "event_pred (i, j) p = (is_valid_event_id (i, j) \<and> p (events i ! j))"
conv_compl_aux: fixes x :: "'a" shows "(- x)\<^sup>T = (- x)\<^sup>T + - x\<^sup>T"
Abs_mod_ring_cases: fixes x :: "'b mod_ring" obtains y :: "int" where "x = Abs_mod_ring y" and "y \<in> {0..<int CARD('b)}"
T_opt_def: fixes step :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'b"   and t :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> nat"   and s :: "'b"   and rs :: "'c list" shows "On_Off.T_opt step t s rs = Inf {On_Off.T step t s rs as |as. length as = length rs}"
g3mod1_def: shows "g3mod1 \<equiv> {(xa, x). g3mod1p xa x}"
intros_19: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and s' :: "'e"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> =n\<Rightarrow> Abrupt s'"   and "\<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal s'\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Catch c\<^sub>1 c\<^sub>2,Normal s\<rangle> =n\<Rightarrow> t"
v_1: shows "v\<^sub>1 \<in> carrier_vec n"
assms: shows "\<forall>\<^sub>F n in sequentially. fds_nth f n = fds_nth g n"
that: fixes a :: "real" assumes "A < a"   and "f \<in> O(\<lambda>x. real x powr p * (1 + integral (\<lambda>u. g' u / u powr (p + 1)) a (real x)))" shows "thesis"
assms_4: fixes s :: "complex Matrix.vec" shows "U * (|w\<rangle> \<Otimes> |s\<rangle>) = |w\<rangle> \<Otimes> |w\<rangle>"
cases: fixes a1 :: "'b"   and a2 :: "nat"   and a3 :: "'b fset" assumes "a1 |\<in>|\<^bsub>a2\<^esub> a3" obtains i :: "nat" and s :: "'b fset" where "a1 = list_of_fset s ! i" and "a2 = i" and "a3 = s" and "i < size s"
iwf_subst_freshen_outPort: fixes lc :: "('form, 'rule) graph_node \<Rightarrow> nat \<Rightarrow> 'subst \<Rightarrow> 'form fset \<times> 'form \<Rightarrow> bool"   and ts :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and ent :: "'form fset \<times> 'form" assumes "iwf lc ts ent" shows "snd ent = subst (iSubst ts) (freshen (iAnnot ts) (iOutPort ts))"
mprev: fixes I :: "\<I>"   and xs :: "'a option list set list"   and ts :: "nat list"   and ys :: "'a option list set list"   and xs' :: "'a option list set list"   and ts' :: "nat list"   and P :: "nat \<Rightarrow> 'a option list set \<Rightarrow> bool"   and i :: "nat"   and j' :: "nat"   and \<sigma> :: "'b trace"   and j :: "nat" assumes "mprev_next I xs ts = (ys, xs', ts')"   and "list_all2 P [i..<j'] xs"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [i..<j] ts"   and "i \<le> j'"   and "i < j" shows "list_all2 (\<lambda>i X. if mem (\<tau> \<sigma> (Suc i) - \<tau> \<sigma> i) I then P i X else X = empty_table) [i..<min j' (j - 1)] ys \<and> list_all2 P [min j' (j - 1)..<j'] xs' \<and> list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min j' (j - 1)..<j] ts'"
sumCases: fixes P :: "pi"   and Q :: "pi"   and Rs :: "residual" assumes "P \<oplus> Q \<longmapsto> Rs" obtains "P \<longmapsto> Rs"   | "Q \<longmapsto> Rs"
seq_compact_imp_countably_compact: fixes U :: "'b set" assumes "seq_compact U" shows "countably_compact U"
ae_filter_def: fixes M :: "'a measure" shows "ae_filter M = (\<Sqinter>N\<in>null_sets M. principal (space M - N))"
min_edge: shows "\<forall>(u', v')\<in>edges rg \<inter> C \<times> - C. w {u, v} \<le> w {u', v'}"
p_all_zeroes: fixes D :: "real"   and t :: "real" assumes "D = D2" shows "(ego_other.p t = 0) = (a\<^sub>o - a\<^sub>e \<noteq> 0 \<and> 0 \<le> D \<and> (t = (- (v\<^sub>o - v\<^sub>e) + sqrt D) / (a\<^sub>o - a\<^sub>e) \<or> t = (- (v\<^sub>o - v\<^sub>e) - sqrt D) / (a\<^sub>o - a\<^sub>e)) \<or> a\<^sub>o - a\<^sub>e = 0 \<and> v\<^sub>o - v\<^sub>e = 0 \<and> s\<^sub>o - s\<^sub>e = 0 \<or> a\<^sub>o - a\<^sub>e = 0 \<and> v\<^sub>o - v\<^sub>e \<noteq> 0 \<and> t = - (s\<^sub>o - s\<^sub>e) / (v\<^sub>o - v\<^sub>e))"
assms_2: shows "I \<in>\<^sub>\<circ> Vset \<alpha>"
abs_eq: shows "1 = Interval (1::'b, 1::'b)"
simps_1: fixes n :: "'node" shows "valid_SDG_node (CFG_node n) = valid_node n"
jkbpC_tLength_inv: fixes t :: "'s Trace"   and n :: "nat" assumes "t \<in> SPR.jkbpC"   and "tLength t = n" shows "t \<in> SPR.jkbpCn n"
S5_successors: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $S5 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<longrightarrow> S5 rmhist p$ \<or> S6 rmhist p$"
min_pair: shows "\<forall>pair'\<in>set_of_candidate_values S v. (snd pair', snd pair) \<notin> trm_ord"
sym_joinable: fixes R :: "('c \<times> 'd) set" shows "sym (joinable R)"
Iff_cong: fixes H :: "fm set"   and A :: "fm"   and A' :: "fm"   and B :: "fm"   and B' :: "fm" assumes "H \<turnstile> A IFF A'"   and "H \<turnstile> B IFF B'" shows "H \<turnstile> (A IFF B) IFF A' IFF B'"
validt_def: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and F :: "'g set"   and P :: "'f set"   and c :: "('f, 'e, 'g) com"   and Q :: "'f set"   and A :: "'f set" shows "\<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A \<equiv> \<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A \<and> (\<forall>s\<in>Normal ` P. \<Gamma>\<turnstile>c \<down> s)"
cccFix_choose_def: fixes cccExp :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "CoCallArityAnalysis.cccFix_choose cccExp \<Gamma> = (if nonrec \<Gamma> then case hd \<Gamma> of (x, xa) \<Rightarrow> CoCallArityAnalysis.cccFix_nonrec cccExp x xa else CoCallArityAnalysis.cccFix cccExp \<Gamma>)"
bot_least: fixes bot :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d" assumes "class.order_bot bot less_eq less" shows "less_eq bot a"
astG_def: shows "astG \<equiv> case problem of (D, I, G, \<delta>) \<Rightarrow> G"
cval_add_def: fixes u :: "'b \<Rightarrow> 'd"   and d :: "'d" shows "u \<oplus> d = (\<lambda>x. u x + d)"
vars_subst_var_supset: fixes lp :: "linear_poly"   and x\<^sub>j :: "nat"   and lp' :: "linear_poly" shows "vars lp - {x\<^sub>j} - vars lp' \<subseteq> vars (subst_var x\<^sub>j lp' lp)"
trm_rep_is_lower_root_red: fixes S :: "'a eclause set"   and x :: "'a trm" assumes "\<not> subterm_reduction_applicable S x"   and "min_trms (set_of_candidate_values S x) \<noteq> {}" shows "(trm_rep x S, x) \<in> trm_ord"
sets_pair_measure: fixes A :: "'c measure"   and B :: "'d measure" shows "sets (A \<Otimes>\<^sub>M B) = sigma_sets (space A \<times> space B) {a \<times> b |a b. a \<in> sets A \<and> b \<in> sets B}"
imp_option_eq_def: fixes eq :: "'c \<Rightarrow> 'd \<Rightarrow> bool Heap"   and a :: "'c option"   and b :: "'d option" shows "imp_option_eq eq a b \<equiv> case (a, b) of (None, None) \<Rightarrow> return True | (None, Some aa) \<Rightarrow> return False | (Some a, None) \<Rightarrow> return False | (Some a, Some b) \<Rightarrow> eq a b"
ide_in_hom_1: fixes f :: "'a" assumes "C.ide f" shows "\<guillemotleft>f : src\<^sub>C f \<rightarrow>\<^sub>C trg\<^sub>C f\<guillemotright>"
some_eq_ex: fixes P :: "'g \<Rightarrow> bool" shows "P (SOME x. P x) = (\<exists>x. P x)"
count_filter_mset_less: fixes M :: "'a multiset"   and N :: "'a multiset" shows "(\<forall>a. count (filter_mset (Domainp R) M) a \<le> count (filter_mset (Domainp R) N) a) = (\<forall>a\<in>{x. Domainp R x}. count M a \<le> count N a)"
mulex1_union: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'b multiset"   and N :: "'b multiset"   and K :: "'b multiset" assumes "mulex1 P M N" shows "mulex1 P (K + M) (K + N)"
square_free_m_factor_1: fixes f :: "int poly"   and g :: "int poly" assumes "p.square_free_m (f * g)" shows "p.square_free_m f"
fa_nonzero: fixes p :: "int"   and f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "hensel p f a" shows "UP_cring.to_fun (padic_int p) f a \<noteq> \<zero>\<^bsub>padic_int p\<^esub>"
sqrt_int_ceiling_bound: fixes x :: "int" assumes "0 \<le> x" shows "x \<le> (sqrt_int_ceiling x)\<^sup>2"
is_projection_on_image: fixes \<pi> :: "'b \<Rightarrow> 'b"   and M :: "'b set" assumes "is_projection_on \<pi> M" shows "range \<pi> = M"
SourcesA3_L0: shows "Sources level0 sA3 = {sA1, sA2, sA3, sA4}"
bb: fixes u :: "'a event list"   and t :: "'a event list"   and f :: "nat \<Rightarrow> 'a event list"   and xa :: "nat"   and j :: "nat" assumes "front_tickFree u \<and> tickFree t \<and> x = trace_hide t (ev ` A) @ u \<and> isInfHiddenRun f (Y xa) A \<and> f j \<in> D (Y xa) \<and> t \<in> range f" shows "\<exists>t u. front_tickFree u \<and> tickFree t \<and> x = trace_hide t (ev ` A) @ u \<and> t \<in> D (Y xa)"
IS_LEFT_TOTAL_alt: fixes R :: "('c \<times> 'd) set" shows "IS_LEFT_TOTAL R = (\<forall>x. \<exists>y. (x, y) \<in> R)"
Q_axioms_def: shows "Q_axioms \<equiv> {uu_. \<exists>A X1 X2. uu_ = A \<and> X1 \<noteq> X2 \<and> (A = neg (zer EQ suc (Var X1)) \<or> A = suc (Var X1) EQ suc (Var X2) IMP Var X1 EQ Var X2 \<or> A = Var X2 EQ zer OR exi X1 (Var X2 EQ suc (Var X1)) \<or> A = pls (Var X1) zer EQ Var X1 \<or> A = pls (Var X1) (suc (Var X2)) EQ suc (pls (Var X1) (Var X2)) \<or> A = tms (Var X1) zer EQ zer \<or> A = tms (Var X1) (suc (Var X2)) EQ pls (tms (Var X1) (Var X2)) (Var X1))}"
subspace_0: fixes S :: "('d \<Rightarrow>\<^sub>0 'c) set" assumes "phull.subspace S" shows "0 \<in> S"
simps_2: fixes p :: "('b \<times> 'a \<times> 'b) list" shows "path_weight' (Some p) = Num (path_weight p)"
applicableBacktrackCharacterization: fixes stateA :: "State"   and F0 :: "Literal list list" shows "applicableBacktrack stateA F0 = (formulaFalse F0 (elements (getM stateA)) \<and> decisions (getM stateA) \<noteq> [])"
Max_ge_iff: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "(x \<le> Max A) = (\<exists>a\<in>A. x \<le> a)"
limited_dom_neq_restricted_neq: fixes f1 :: "('c, 'd) fmap"   and vs :: "'c set"   and f2 :: "('c, 'd) fmap" assumes "fmdom' f1 \<subseteq> vs"   and "f1 ++ f2 \<noteq> f2" shows "fmrestrict_set vs (f1 ++ f2) \<noteq> fmrestrict_set vs f2"
inext_iprev: fixes I :: "nat set"   and n :: "nat" assumes "iMin I \<noteq> n" shows "inext (iprev n I) I = n"
remove_vertex_walk_add: fixes x :: "'a"   and xs :: "'a list" assumes "Digraph.walk (remove_vertex x) xs" shows "walk xs"
abcd_dcba_only_14: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "[[a b c d]]" shows "\<not> [[c b a d]]"
Rep_0: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "Rep zero = 0"
assms_4: shows "b = insertion (nth_default 0 (xs' @ y # xs)) B"
subdensity: fixes T :: "'d \<Rightarrow> 'e"   and P :: "'d measure"   and Q :: "'e measure"   and M :: "'f measure"   and X :: "'f \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> ennreal"   and Y :: "'f \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> ennreal" assumes "T \<in> P \<rightarrow>\<^sub>M Q"   and "distributed M P X f"   and "distributed M Q Y g"   and "Y = T \<circ> X" shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
nfoldli_simps_1: fixes c :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a nres"   and s :: "'a" shows "nfoldli [] c f s = RETURN s"
insert_iff: fixes a :: "'k"   and b :: "'k"   and A :: "'k set" shows "(a \<in> insert b A) = (a = b \<or> a \<in> A)"
density_1: fixes M :: "'b measure" shows "density M (\<lambda>_. 1) = M"
sDef: shows "Sources i S = DSources i S \<union> \<Union> (Sources i ` DSources i S)"
intros_1: fixes a :: "'d"   and r :: "('d \<times> 'd) set" shows "(a, a) \<in> r\<^sup>*"
less_Suc_numeral: fixes n :: "nat"   and k :: "num" shows "(Suc n < numeral k) = (n < pred_numeral k)"
D'_p: shows "D' \<in> Q_of_state (Liminf_state Sts) \<and> D' \<cdot> \<sigma>' = C \<and> is_ground_subst \<sigma>'"
to_stream_def: fixes w :: "nat \<Rightarrow> 'b" shows "to_stream w \<equiv> smap w nats"
is_homeomorphism: shows "homeomorphism X is_open Spec is_zariski_open f"
simps_4: fixes c :: "int"   and a :: "MIR.num"   and b :: "MIR.num" shows "zsplit0 (CF c a b) = zsplit0 (Add (Mul c (Floor a)) b)"
Rep_inject: fixes x :: "'a"   and y :: "'a" shows "(Rep x = Rep y) = (x = y)"
set_set_swap: fixes r :: "'a array"   and r' :: "'b array"   and x :: "'a list"   and x' :: "'b list"   and h :: "heap" assumes "r =!!= r'" shows "Array.set r x (Array.set r' x' h) = Array.set r' x' (Array.set r x h)"
Branch_not_Terminal: fixes c :: "'h"   and rs :: "'i \<Rightarrow> ('f, 'g, 'h, 'i) resumption"   and a :: "'f" shows "Branch c rs \<noteq> Terminal a"
hom_0_iff: fixes x :: "'b" shows "(to_fract x = 0) = (x = (0::'b))"
simple_integral_def: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" shows "integral\<^sup>S M f = (\<Sum>x\<in>f ` space M. x * emeasure M (f -` {x} \<inter> space M))"
simps: fixes n :: "nat"   and x :: "nat" shows "loc_upb n x = (let n1 = c_fst n mod 7 in if n1 = 0 then c_pair (c_pair n x) 0 + 1 else if n1 = 1 then c_pair (c_pair n x) 0 + 1 else if n1 = 2 then c_pair (c_pair n x) 0 + 1 else if n1 = 3 then c_pair (c_pair n x) 0 + 1 else if n1 = 4 then let m = c_snd n; m1 = c_fst m; m2 = c_snd m; y = c_assoc_value (pr_gr (loc_upb m2 x)) (c_pair m2 x) in c_pair (c_pair n x) (loc_upb m2 x + loc_upb m1 y) + 1 else if n1 = 5 then let m = c_snd n; m1 = c_fst m; m2 = c_snd m in c_pair (c_pair n x) (loc_upb m1 x + loc_upb m2 x) + 1 else if n1 = 6 then let m = c_snd n; m1 = c_fst m; m2 = c_snd m; y1 = c_fst x; x1 = c_snd x in if y1 = 0 then c_pair (c_pair n x) (loc_upb m1 x1) + 1 else let y2 = y1 - 1; t1 = c_assoc_value (pr_gr (loc_upb n (c_pair y2 x1))) (c_pair n (c_pair y2 x1)); t2 = c_pair (c_pair y2 t1) x1 in c_pair (c_pair n x) (loc_upb n (c_pair y2 x1) + loc_upb m2 t2) + 1 else 0)"
maximal_ideal_ideal: fixes R :: "('c, 'd) Ring_scheme"   and mx :: "'c set" assumes "Ring R"   and "maximal_ideal R mx" shows "ideal R mx"
norm: fixes ui :: "int poly" assumes "ui \<in> set (berlekamp_hensel p n f)" shows "poly_mod.Mp (p ^ n) ui = ui"
coprime_bezout_strong: fixes a :: "nat"   and b :: "nat" assumes "coprime a b"   and "b \<noteq> 1" shows "\<exists>x y. a * x = b * y + 1"
mult_cong: fixes x :: "int"   and y :: "int" shows "x mod int p \<otimes>\<^bsub>R\<^esub> (y mod int p) = x * y mod int p"
case_1: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Static of Static \<Rightarrow> f1 | NonStatic \<Rightarrow> f2) = f1"
extended_predist_ereal: fixes x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "ereal (extended_predist x y) = min (esqrt (extended_Gromov_distance x y)) (eexp (ereal (- epsilonG TYPE('a)) * extended_Gromov_product_at basepoint x y))"
locate2_def: fixes f :: "'b \<Rightarrow> nat"   and al :: "'b list"   and i :: "nat" shows "locate2 f al i \<equiv> snd (locate f al i)"
Vset_mono: fixes m :: "hf"   and n :: "hf" assumes "Ord m"   and "n \<le> m" shows "Vset n \<le> Vset m"
integral_nonneg_AE: fixes f :: "'e \<Rightarrow> real"   and M :: "'e measure" assumes "AE x in M. 0 \<le> f x" shows "0 \<le> integral\<^sup>L M f"
subs_2: shows "cbox (u, w) (v, z) \<subseteq> s"
fminus_triv: fixes A :: "'a fset"   and B :: "'a fset" assumes "A |\<inter>| B = {||}" shows "A |-| B = A"
upper_3_def: fixes x :: "real_alg_3"   and i :: "nat" shows "upper_3 x i \<equiv> interval.upper (get_itvl_3 ((tighten_bounds_3 ^^ i) x))"
ncoplanar_perm_8: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet B C A D"
bound: shows "2 ^ LENGTH('f) \<le> nat \<bar>mantissa x\<bar> * 2 ^ (Suc LENGTH('f) - nat (bitlen \<bar>mantissa x\<bar>))"
eventually_at_topological: fixes "open" :: "'c set \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and a :: "'c"   and s :: "'c set" assumes "class.topological_space open" shows "eventually P (topological_space.at_within open a s) = (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x))"
matrix_to_lpolies_coeff_access: fixes i :: "nat"   and A :: "rat mat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "Abstract_Linear_Poly.coeff (matrix_to_lpolies A ! i) j = A $$ (i, j)"
CONSTRAINT_def: fixes P :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "CONSTRAINT P x \<equiv> P x"
trans_less_equiv: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "comparator cmp"   and "cmp a b = Less"   and "cmp b c = Equiv" shows "cmp a c = Less"
rep_eq: fixes x :: "('a, 'b) pfun" shows "pfun_graph x = map_graph (Rep_pfun x)"
simps_6: fixes a :: "'c"   and b :: "'c" shows "dbm_add (Lt a) (Le b) = Lt (a + b)"
bigtheta_mult_eq: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "\<Theta>[F](\<lambda>x. f x * g x) = \<Theta>[F](f) * \<Theta>[F](g)"
expand: shows "f' = (\<Lambda> x. g\<cdot>x \<union> \<Union> (Rep_cfun f' ` R\<cdot>x))"
same_hyp_for_two_not_Lim: fixes f\<^sub>1 :: "nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and f\<^sub>2 :: "nat \<Rightarrow> nat option"   and n\<^sub>1 :: "nat"   and s :: "nat \<Rightarrow> nat option"   and h :: "nat option"   and n\<^sub>2 :: "nat"   and \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "f\<^sub>1 \<in> U"   and "f\<^sub>2 \<in> U"   and "f\<^sub>1 \<noteq> f\<^sub>2"   and "\<forall>n\<ge>n\<^sub>1. s (f\<^sub>1 \<triangleright> n) = h"   and "\<forall>n\<ge>n\<^sub>2. s (f\<^sub>2 \<triangleright> n) = h" shows "\<not> learn_lim \<psi> U s"
A: shows "Sigma (space M) A \<in> sets (M \<Otimes>\<^sub>M N)"
prems: shows "p \<noteq> 0 \<and> poly p a \<noteq> 0 \<and> count_roots_above p a = 0"
delete_edge_def: fixes v :: "'b"   and v' :: "'b"   and G :: "('b, 'c) graph_scheme" shows "FiniteGraph.delete_edge v v' G \<equiv> \<lparr>nodes = nodes G, edges = {(e1, e2). (e1, e2) \<in> edges G \<and> (e1, e2) \<noteq> (v, v')}\<rparr>"
qSkelInp_def: fixes qinp :: "'e \<Rightarrow> ('f, 'g, 'h, 'i, 'j) qTerm option" shows "qSkelInp qinp = lift qSkel qinp"
assms_1: shows "P \<parallel> !P \<Longrightarrow>\<^sub>l\<^sup>^Rs"
hyps_2: fixes Z :: "'b" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s = Z \<and> \<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F) \<and> \<Gamma>\<turnstile>the (\<Gamma> p) \<down> Normal \<sigma> \<and> (\<exists>cs css. \<Gamma>\<turnstile> ([the (\<Gamma> p)], [], Normal \<sigma>) \<rightarrow>\<^sup>* (c\<^sub>2 # cs, css, Normal s))} c\<^sub>2 {t. \<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal Z\<rangle> \<Rightarrow> Abrupt t}"
subset_Collect_conv: fixes S :: "'e set"   and P :: "'e \<Rightarrow> bool" shows "(S \<subseteq> Collect P) = (\<forall>x\<in>S. P x)"
singleton_inject: fixes a :: "'k"   and b :: "'k" assumes "{a} = {b}" shows "a = b"
invertible: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "invertible_lf s1 s2 f" shows "\<exists>g. g \<circ> f = id \<and> f \<circ> g = id"
Src_Cod: fixes t :: "'c local.term" assumes "Arr t" shows "Src (Cod t) = Src t"
distrib: fixes g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "finite {a. g a \<noteq> (1::'c)}"   and "finite {a. h a \<noteq> (1::'c)}" shows "(\<Prod>a. g a * h a) = Prod_any g * Prod_any h"
lm146: fixes A :: "'a set" assumes "card (Pow A) \<noteq> 0" shows "card A = Discrete.log (card (Pow A))"
assms_1: shows "insert C H \<turnstile> A IMP B"
assms_4: shows "openin (top_of_set (S \<union> T)) T"
sym_Int: fixes r :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "sym r"   and "sym s" shows "sym (r \<inter> s)"
subst_atm_list_length: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and As :: "'b list"   and \<sigma> :: "'c" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "length (substitution_ops.subst_atm_list subst_atm As \<sigma>) = length As"
of_bool_eq_iff: fixes p :: "bool"   and q :: "bool" shows "(of_bool p = of_bool q) = (p = q)"
monoid_congruence_axioms: shows "monoid_congruence G (\<cdot>) \<one> E"
c: shows "h \<turnstile> to_tree_order root_ptr \<rightarrow>\<^sub>r c"