hcju/isabelleps huggingface dataset:
scale_one: fixes x :: "('a, 'b) vec" shows "(1::'a) *s x = x"
ireturn_eq: fixes x :: "'b" shows "return x = SPEC ((=) x)"
folding_ex: fixes C :: "'a set" assumes "folding_g.chamber C"   and "C \<in> g \<turnstile> X" shows "\<exists>x. folding_g.chamber x \<and> x \<notin> g \<turnstile> X \<and> g ` x = C"
init_b_def: fixes r :: "'b rexp" shows "init_b r = (follow True (empty_mrexp r), nullable r)"
mult_mono: fixes plus :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and zero :: "'h"   and less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and times :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and a :: "'h"   and b :: "'h"   and c :: "'h"   and d :: "'h" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq c d"   and "less_eq zero b"   and "less_eq zero c" shows "less_eq (times a c) (times b d)"
mantissa_def: fixes f :: "Float.float" shows "mantissa f = fst (SOME p. f = 0 \<and> fst p = 0 \<and> snd p = 0 \<or> f \<noteq> 0 \<and> real_of_float f = real_of_int (fst p) * 2 powr real_of_int (snd p) \<and> odd (fst p))"
assms: shows "subst_domain \<delta> \<inter> set X = {}"
WN_onE: fixes r :: "('c \<times> 'c) set"   and A :: "'c set"   and a :: "'c" assumes "WN_on r A"   and "a \<in> A" obtains b :: "'c" where "(a, b) \<in> r\<^sup>!"
add_0_r: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and p :: "'c point" assumes "ell_field R" shows "ell_field.add R a p Infinity = p"
subset_code_2: fixes A :: "'a set"   and ys :: "'a list" shows "(A \<subseteq> List.coset ys) = (\<forall>y\<in>set ys. y \<notin> A)"
red_setI: fixes f :: "'t \<Rightarrow>\<^sub>0 'c"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "f \<in> F"   and "red_single p q f t" shows "red F p q"
sum_list_map_eq_sum_count: fixes f :: "'c \<Rightarrow> nat"   and xs :: "'c list" shows "sum_list (map f xs) = (\<Sum>x\<in>set xs. count_list xs x * f x)"
arith_prog_to_multi: fixes start :: "int"   and step :: "nat"   and c :: "nat" shows "arith_prog start step c = multi_arith_prog 1 start (\<lambda>_. step) (\<lambda>_. c)"
cont_true_cont_4: fixes v :: "i"   and p :: "\<o>" shows "[ContingentlyFalse p \<^bold>\<equiv> ContingentlyTrue (p\<^sup>-) in v]"
lea123456_lta__lta: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p" assumes "A B C LeA D E F"   and "D E F LtA G H I" shows "A B C LtA G H I"
dvdmE: fixes f :: "int poly"   and g :: "int poly" assumes "f dvdm g" obtains h :: "int poly" where "g =m f * h" and "Mp h = h"
Comp: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and c1 :: "stmt"   and c2 :: "stmt" assumes "E,dt\<Turnstile>c1\<Colon>\<surd>"   and "E,dt\<Turnstile>c2\<Colon>\<surd>" shows "E,dt\<Turnstile>c1;; c2\<Colon>\<surd>"
fset_strong_cases: fixes xs :: "'a fset" obtains "xs = {||}"   | ys :: "'a fset" and x :: "'a" where "x |\<notin>| ys" and "xs = finsert x ys"
smult_carrier_mat: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and k :: "'b" assumes "A \<in> carrier_mat nr nc" shows "k \<cdot>\<^sub>m A \<in> carrier_mat nr nc"
chain_ocritE: fixes data :: "nat \<times> 'e"   and hs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and ps :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and p :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and q :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g" assumes "chain_ocrit data hs ps p q"   and "unique_idx (p # q # hs @ map (fst \<circ> snd) ps @ map (snd \<circ> snd) ps) data" obtains h :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g" where "h \<in> set hs" and "fst h \<noteq> 0" and "h \<noteq> p" and "h \<noteq> q" and "lt (fst h) adds\<^sub>t term_of_pair (lcs (lp (fst p)) (lp (fst q)), component_of_term (lt (fst p)))" and "(p, h) \<in>\<^sub>p snd ` set ps" and "(q, h) \<in>\<^sub>p snd ` set ps"
rsum_bound: fixes p :: "('a \<times> 'a set) set"   and a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b"   and e :: "real" assumes "p tagged_division_of cbox a b"   and "\<forall>x\<in>cbox a b. norm (f x) \<le> e" shows "norm (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) \<le> e * content (cbox a b)"
qtable_eval_agg: fixes b :: "nat"   and n :: "nat"   and \<phi> :: "Formula.formula"   and R :: "event_data list set"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and i :: "nat"   and rel :: "event_data option list set"   and y :: "nat"   and \<omega> :: "Formula.agg_type \<times> event_data"   and f :: "Formula.trm"   and g0 :: "bool" assumes "qtable (b + n) (fv \<phi>) (mem_restr (lift_envs' b R)) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>) rel"   and "\<forall>x\<in>fv (formula.Agg y \<omega> b f \<phi>). x < n"   and "y + b \<notin> fv \<phi>"   and "{0..<b} \<subseteq> fv \<phi>"   and "fv_trm f \<subseteq> fv \<phi>"   and "g0 = (fv \<phi> \<subseteq> {0..<b})" shows "qtable n (fv (formula.Agg y \<omega> b f \<phi>)) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i (formula.Agg y \<omega> b f \<phi>)) (eval_agg n g0 y \<omega> b f rel)"
smooth_on_cong: fixes k :: "enat"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "k-smooth_on S g"   and "open S"   and "\<And>x. x \<in> S \<Longrightarrow> f x = g x" shows "k-smooth_on S f"
Abs1_eq_iff_3: fixes a :: "'d"   and x :: "'a"   and b :: "'d"   and y :: "'a" shows "([[atom a]]lst. x = [[atom b]]lst. y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> x = (a \<leftrightarrow> b) \<bullet> y \<and> atom a \<sharp> y)"
sets_Ball: fixes I :: "'d set"   and A :: "'d \<Rightarrow> 'e set"   and M :: "'d \<Rightarrow> 'e measure"   and i :: "'d" assumes "\<forall>i\<in>I. A i \<in> sets (M i)"   and "i \<in> I" shows "A i \<in> sets (M i)"
runit_def: fixes f :: "'a" shows "Right.runit f \<equiv> THE \<mu>'. Right.in_hom \<mu>' (Right.R f) f \<and> Right.R \<mu>' = Right.comp (f \<star> \<i>[src \<mu>]) \<a>[f, src \<mu>, src \<mu>]"
add_less_cancel_left: fixes x :: "V"   and y :: "V"   and z :: "V" shows "(x + y < x + z) = (y < z)"
rel_S_neg_distr_cond_def: fixes Co1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Co1' :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and Co2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Co2' :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and Contra1 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and Contra1' :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra2 :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and Contra2' :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and uu :: "('n \<times> 'o \<times> 'p \<times> 'q \<times> 'r \<times> 's \<times> 't) itself" shows "rel_S_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' uu = (\<forall>L1 L1' L2 L2'. rel_S (L1 OO L1') (L2 OO L2') (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') \<le> rel_S L1 L2 Co1 Co2 Contra1 Contra2 OO rel_S L1' L2' Co1' Co2' Contra1' Contra2')"
F_SKIP: shows "F SKIP = {(s, X). s = [] \<and> tick \<notin> X} \<union> {(s, X). s = [tick]}"
polar_identity_minus: fixes x :: "'b"   and y :: "'b" shows "(norm (x - y))\<^sup>2 = (norm x)\<^sup>2 + (norm y)\<^sup>2 - 2 * Re \<langle>x, y\<rangle>"
lex_scale2_zero: fixes v :: "real"   and u :: "real \<times> real" shows "Counterclockwise_2D_Arbitrary.lex 0 (v *\<^sub>R u) = (if 0 < v then Counterclockwise_2D_Arbitrary.lex 0 u else if v < 0 then Counterclockwise_2D_Arbitrary.lex u 0 else True)"
assms_3: shows "ipassmt (Iface (p_oiface p)) = Some p_ips \<and> p_dst p \<in> ipcidr_union_set (set p_ips)"
reachable_target: fixes q :: "'state"   and p :: "'state"   and r :: "'action list" assumes "q \<in> reachable p"   and "path r q" shows "target r q \<in> reachable p"
pgallery: fixes xs :: "'a set list" assumes "pgallery xs" shows "folding_g.gallery xs"
assms_1: shows "f \<in> carrier (function_ring (carrier R) R)"
replicate_MkI_plus_1_1: fixes n :: "int"   and x :: "'b" assumes "0 \<le> n" shows "replicate\<cdot>(MkI\<cdot>(n + 1))\<cdot>x = x : replicate\<cdot>(MkI\<cdot>n)\<cdot>x"
IH: fixes x :: "('b, 'a) Term.term" assumes "x \<in> set S"   and "distinct (fv_list x)"   and "Fun f T \<sqsubseteq> x" shows "fv (T ! i) \<inter> fv (T ! j) = {}"
widen_Class: fixes G :: "prog"   and C :: "qtname"   and T :: "ty" assumes "G\<turnstile>Class C\<preceq>T" shows "(\<exists>D. T = Class D) \<or> (\<exists>I. T = Iface I)"
assms_2: shows "(\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> eta"
set_tree_insert: fixes t :: "('a \<times> color) tree"   and x :: "'a" assumes "Tree2.bst t" shows "Tree2.set_tree (insert_rbt x t) = {x} \<union> Tree2.set_tree t"
on_tripleD_1: fixes l1 :: "'e"   and v1 :: "'f"   and v2 :: "'f"   and l2 :: "'e"   and v3 :: "'g"   and v4 :: "'g"   and R :: "('f \<times> 'g) set" assumes "((l1, v1, v2), l2, v3, v4) \<in> on_triple R" shows "l2 = l1"
path_components_of_def: fixes X :: "'c topology" shows "path_components_of X \<equiv> path_component_of_set X ` topspace X"
le_sup_iff: fixes x :: "'g multiset"   and y :: "'g multiset"   and z :: "'g multiset" shows "(x \<union># y \<subseteq># z) = (x \<subseteq># z \<and> y \<subseteq># z)"
gp_int: shows "integrable M (\<lambda>x. f x * enn2real (nn_transfer_operator gp x))"
dim_row_matrix_A: fixes signs :: "rat list list"   and subsets :: "nat list list" shows "dim_row (M_mat signs subsets) = length subsets"
Ints_0: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd" assumes "class.ring_1 one times plus zero minus uminus" shows "zero \<in> ring_1.Ints one plus zero minus"
vsv_is_VLambda_on_vlrestriction: fixes A :: "V"   and Aa :: "V" assumes "A \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> Aa)" shows "VLambda A (app (r \<restriction>\<^sup>l\<^sub>\<circ> Aa)) = (r \<restriction>\<^sup>l\<^sub>\<circ> Aa) \<restriction>\<^sup>l\<^sub>\<circ> A"
Kcknows_3: shows "spr_simAbs ec = SPRdet.sim_equiv_class a t"
B_2: shows "prefix (B @ [(l, s')]) (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>)"
well_def_subsets1: shows "all_list_constr subsets1 (length qs1)"
not_mem_frontier_diff: fixes t :: "'t"   and M :: "'t zmultiset"   and N :: "'t zmultiset" assumes "t \<notin>\<^sub>A frontier M - frontier N"   and "t \<notin>\<^sub>A frontier N - frontier M" shows "zcount (frontier_changes M N) t = 0"
chain_map_diff: fixes p :: "nat"   and g :: "'c \<Rightarrow> 'b"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "chain_map p g (a - b) = chain_map p g a - chain_map p g b"
list_all2_append1: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list"   and zs :: "'d list" shows "list_all2 P (xs @ ys) zs = (\<exists>us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and> list_all2 P xs us \<and> list_all2 P ys vs)"
hyps: shows "list_all (\<lambda>a. TE.int \<xi> (tNN a) = Ik.int \<xi> a) Tl"
minus_div_mult_eq_mod: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "minus a (times (divide a b) b) = modulo a b"
BijGroup_def: fixes S :: "'d set" shows "BijGroup S = \<lparr>carrier = Bij S, monoid.mult = \<lambda>g\<in>Bij S. restrict (compose S g) (Bij S), one = \<lambda>x\<in>S. x\<rparr>"
bounded_linear_Blinfun_apply: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_linear f" shows "(*\<^sub>v) (Blinfun f) = f"
Sup_apply: fixes A :: "('c \<Rightarrow> 'b) set"   and x :: "'c" shows "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
dom_llp1: fixes x :: "'a"   and y :: "'a" assumes "x \<le> x \<cdot> ar (ar y)" shows "ar (ar x) \<le> ar (ar y)"
rcos_sum: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "H \<lhd> G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "H #>\<^bsub>G\<^esub> x <#>\<^bsub>G\<^esub> (H #>\<^bsub>G\<^esub> y) = H #>\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> y"
continuous_at_Sup_mono: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "mono f"   and "continuous (at_left (Sup S)) f"   and "S \<noteq> {}"   and "bdd_above S" shows "f (Sup S) = Sup (f ` S)"
many_strongly_connected_implies_12: fixes x :: "'a" assumes "many_strongly_connected x" shows "x\<^sup>\<star> ; x ; x\<^sup>T \<le> x\<^sup>+"
encode12_inv: fixes k :: "nat"   and dims :: "nat list"   and vars :: "nat set" assumes "k < state_sig.d dims" shows "partial_state.encode12 dims vars (partial_state.encode1 dims vars k, partial_state.encode2 dims vars k) = k"
intro: fixes set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>ptr h h'. \<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_host_locs. r h h')" shows "l_set_disconnected_nodes_get_host set_disconnected_nodes_locs get_host_locs"
fast_bv_to_nat_Cons0: fixes bs :: "bit list"   and bin :: "num" shows "fast_bv_to_nat_helper (\<zero> # bs) bin = fast_bv_to_nat_helper bs (num.Bit0 bin)"
induct: fixes x1 :: "'node SDG_node"   and x2 :: "'node SDG_node list"   and x3 :: "'node SDG_node"   and P :: "'node SDG_node \<Rightarrow> 'node SDG_node list \<Rightarrow> 'node SDG_node \<Rightarrow> bool" assumes "x1 i-x2\<rightarrow>\<^sub>d* x3"   and "\<And>n. valid_SDG_node n \<Longrightarrow> P n [] n"   and "\<And>n ns n'' n'. \<lbrakk>n i-ns\<rightarrow>\<^sub>d* n''; P n ns n''; n'' \<longrightarrow>\<^bsub>cd\<^esub> n'\<rbrakk> \<Longrightarrow> P n (ns @ [n'']) n'"   and "\<And>n ns n'' V n'. \<lbrakk>n i-ns\<rightarrow>\<^sub>d* n''; P n ns n''; n'' -V\<rightarrow>\<^sub>d\<^sub>d n'; n'' \<noteq> n'\<rbrakk> \<Longrightarrow> P n (ns @ [n'']) n'" shows "P x1 x2 x3"
assms_1: shows "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L"
cap_simps_1: fixes x :: "'v"   and y :: "'v" shows "cap (\<langle>x\<rangle>, \<langle>y\<rangle>) = (if edge \<Gamma> x y then h 0 (x, y) else if edge \<Gamma> y x then max (weight \<Gamma> x) (weight \<Gamma> y) else 0)"
Functional: fixes \<phi> :: "('t, 'f) Language"   and i1 :: "('o, 'm) IType"   and i2 :: "('o, 'm) IType" assumes "L\<lbrakk>\<phi>\<rbrakk> \<rightarrow> i1"   and "L\<lbrakk>\<phi>\<rbrakk> \<rightarrow> i2" shows "i1 = i2"
hyperpow_two: fixes r :: "'b star" shows "r pow 2 = r * r"
Rep_BinomialHeap_inverse: fixes x :: "('a, 'b) BinomialHeap" shows "Abs_BinomialHeap (Rep_BinomialHeap x) = x"
conv_radius_conv_Sup': fixes f :: "nat \<Rightarrow> 'a" shows "conv_radius f = Sup {r. \<forall>z. ereal (norm z) < r \<longrightarrow> summable (\<lambda>n. norm (f n * z ^ n))}"
rref_upt_condition4_explicit: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat"   and i :: "'c"   and j :: "'c" assumes "reduced_row_echelon_form_upt_k A k"   and "\<not> is_zero_row_upt_k i k A"   and "i \<noteq> j" shows "A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'a)) = (0::'a)"
finite_measure_Diff: fixes A :: "'b set"   and M :: "'b pmf"   and B :: "'b set" assumes "A \<in> measure_pmf.events M"   and "B \<in> measure_pmf.events M"   and "B \<subseteq> A" shows "measure_pmf.prob M (A - B) = measure_pmf.prob M A - measure_pmf.prob M B"
a1: shows "E \<subseteq> backflows (filternew_flows_state \<T>)"
order_trans: fixes x :: "'a list"   and y :: "'a list"   and z :: "'a list" assumes "subseq x y"   and "subseq y z" shows "subseq x z"
iMODb_iprev: fixes t :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" assumes "t \<in> [ r, mod m, c ]"   and "r < t" shows "iprev t [ r, mod m, c ] = t - m"
schroeder_5_p: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y \<le> - z) = (y \<sqinter> id z \<le> - id x)"
not: fixes bbs :: "bool list"   and bs :: "int list"   and p :: "fm" shows "Ifm bbs bs (not p) = Ifm bbs bs (NOT p)"
safe: fixes c :: "('loc, 't) Propagate.configuration" assumes "inv_imps_work_sum c"   and "inv_implications_nonneg c"   and "\<And>loc. c_work c loc = {#}\<^sub>z" shows "safe c"
SPAWN: fixes ss :: "'s"   and ce :: "'s multiset"   and ss' :: "'s"   and csp :: "'s multiset" assumes "c = add_mset ss ce"   and "ce' = add_mset ss' (csp + ce)"   and "((ss, ce), e, ss', add_mset s' (csp + ce)) \<in> gtrs" shows "P"
simps_1: fixes t1 :: "('a, 'c) trm"   and t2 :: "('a, 'c) trm" shows "SIGF (Geq t1 t2) = {Inl x |x. x \<in> SIGT t1 \<union> SIGT t2}"
social_decision_scheme_axioms: shows "social_decision_scheme agents alts RD"
assms: fixes x :: "'x"   and s :: "'a" shows "f x s \<noteq> dSUCCEED"
ik_append: fixes S :: "('a, 'b) strand_step list"   and S' :: "('a, 'b) strand_step list" shows "ik\<^sub>s\<^sub>t (S @ S') = ik\<^sub>s\<^sub>t S \<union> ik\<^sub>s\<^sub>t S'"
rep_eq: fixes x :: "('c, 'd) \<I>" shows "Rep_\<I> (stop_\<I> x) = (\<lambda>x. if Rep_\<I> x x = {} then {} else insert None (Some ` Rep_\<I> x x))"
AbstFormP_dbfm: fixes i :: "name"   and n :: "nat"   and db :: "dbfm" shows "{} \<turnstile> AbstFormP \<guillemotleft>Var i\<guillemotright> (ORD_OF n) (quot_dbfm db) (quot_dbfm (abst_dbfm i n db))"
equals_dom_if_value_is_ide: assumes "C.ide a'" shows "A'.map = A'.F.map"
dickson_grading_varnum_empty: shows "dickson_grading (varnum {})"
length_greater_0_conv: fixes xs :: "'b list" shows "(0 < \<^bold>|xs\<^bold>|) = (xs \<noteq> \<epsilon>)"
poly_map_poly_eval_poly: fixes p :: "'c poly" shows "poly (map_poly (\<lambda>a. [:a:]) p) = eval_poly (\<lambda>a. [:a:]) p"
norm_sgn: fixes x :: "'b" shows "\<parallel>sgn x\<parallel> = (if x = (0::'b) then 0 else 1)"
fcomp_apply: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'a"   and x :: "'b" shows "fcomp f g x = g (f x)"
sum_strict_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "class.strict_ordered_comm_monoid_add plus zero less_eq less"   and "finite A"   and "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> less (f x) (g x)" shows "less (comm_monoid_add.sum plus zero f A) (comm_monoid_add.sum plus zero g A)"
refl_sred: fixes t :: "lam" shows "t \<rightarrow>\<^sub>s t"
step: fixes A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and l :: "'d"   and Z :: "('e \<Rightarrow> 'f) set"   and l' :: "'d"   and Z' :: "('e \<Rightarrow> 'f) set"   and l'' :: "'d"   and Z'' :: "('e \<Rightarrow> 'f) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', Z'\<rangle>"   and "A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto>* \<langle>l'', Z''\<rangle>" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>* \<langle>l'', Z''\<rangle>"
sinl_def: shows "sinl = (\<Lambda> a. Abs_ssum (seq\<cdot>a\<cdot>TT, a, \<bottom>))"
le_oLimitE: fixes x :: "ordinal"   and f :: "nat \<Rightarrow> ordinal" assumes "x \<le> oLimit f" obtains n :: "nat" where "x \<le> f n"   | "x = oLimit f"
npMulDistr: fixes R :: "('e, 'f) Ring_scheme"   and x :: "'e"   and n :: "nat"   and m :: "nat" assumes "Ring R"   and "x \<in> carrier R" shows "x^\<^bsup>R n\<^esup> \<cdot>\<^sub>r\<^bsub>R\<^esub> x^\<^bsup>R m\<^esup> = x^\<^bsup>R (n + m)\<^esup>"
diff1_eq_iff: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "(n - Suc 0 = m) = (n = m + 1)"
reflcl_trancl: fixes r :: "('a \<times> 'a) set" shows "(r\<^sup>+)\<^sup>= = r\<^sup>*"
os_reverse_rule: fixes xs :: "'a list"   and p :: "'a node ref option" shows "<os_list xs p> os_reverse p <os_list (rev xs)>"
ent_pure_post_iff: fixes P :: "assn"   and Q :: "assn"   and b :: "bool" shows "(P \<Longrightarrow>\<^sub>A Q * \<up> b) = ((\<forall>h. h \<Turnstile> P \<longrightarrow> b) \<and> (P \<Longrightarrow>\<^sub>A Q))"
inv_letter_pair: fixes a :: "accent"   and x :: "'a" shows "inv_letter (a, x) = (case a of Acute \<Rightarrow> Grave | Grave \<Rightarrow> Acute | Macron \<Rightarrow> Macron, x)"
NE: shows "{l v |v. (u, v) \<in> cf.E} \<noteq> {}"
rsquarefree_def: fixes p :: "'b poly" shows "rsquarefree p = (p \<noteq> 0 \<and> (\<forall>a. Polynomial.order a p = 0 \<or> Polynomial.order a p = 1))"
vsum_def: fixes A :: "V"   and B :: "V" shows "A \<Uplus> B \<equiv> vtimes (set {0}) A \<squnion> vtimes (set {1}) B"
H2_J: fixes P :: "'c alpha_d_scheme \<times> 'c alpha_d_scheme \<Rightarrow> bool" shows "(\<forall>A. case A of (A, A') \<Rightarrow> P (A, A'\<lparr>ok := False\<rparr>) \<longrightarrow> P (A, A'\<lparr>ok := True\<rparr>)) = P is H2 healthy"
butlast_tl: fixes xs :: "'d list" shows "butlast (tl xs) = tl (butlast xs)"
rb_aux_inv_D5: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list" assumes "rb_aux_inv (bs, ss, ps)" shows "sorted_wrt pair_ord ps"
mguPaaPax_def: shows "mguPaaPax = (\<lambda>x. if x = ''x'' then Fun ''a'' [] else \<epsilon> x)"
HashMap_inject: fixes x :: "('a, 'b) ArrayHashMap_Impl.hashmap"   and y :: "('a, 'b) ArrayHashMap_Impl.hashmap" assumes "x \<in> {hm. ArrayHashMap_Impl.ahm_invar hm}"   and "y \<in> {hm. ArrayHashMap_Impl.ahm_invar hm}" shows "(HashMap x = HashMap y) = (x = y)"
measure: fixes lrs :: "('a \<times> 'a \<times> 'a) set"   and ars :: "('a \<times> 'a) set"   and s :: "'a"   and \<alpha> :: "'a"   and t :: "'a"   and ss :: "('a \<times> 'a) list" assumes "lrs = {(a, c, b). c = a \<and> (a, b) \<in> ars}"   and "(s, (\<alpha>, t) # ss) \<in> seq lrs" shows "set (labels (t, ss)) \<subseteq> (ars\<^sup>+)\<inverse> \<down>s {\<alpha>}"
abs_plus: fixes m :: "nat"   and n :: "nat" shows "abs_nat m + abs_nat n = abs_nat (m + n)"
minus_dvd_iff: fixes 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 x :: "'a"   and y :: "'a" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "dvd.dvd times (uminus x) y = dvd.dvd times x y"
if_set_elim: fixes x :: "'a"   and b :: "bool"   and A :: "'a set"   and B :: "'a set" shows "(x \<in> (if b then A else B)) = (b \<and> x \<in> A \<or> \<not> b \<and> x \<in> B)"
cart_product_singleton_left: fixes a :: "'b"   and N :: "'b set multiset" shows "{#m \<in># {#a#} \<times># N. fst m \<in> snd m#} = {#a#} \<times># filter_mset ((\<in>) a) N"
id_func_preserves_comp: shows "preserves_comp (id_func AA)"
a: shows "f \<approx>>\<^bsub>MakeCat SET'\<^esub> g"
Vr_exp_f_exp: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and n :: "nat" assumes "valuation K v"   and "x \<in> carrier (Vr K v)" shows "x^\<^bsup>Vr K v n\<^esup> = x^\<^bsup>K n\<^esup>"
filter_eq_nths: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "filter P xs = nths xs {i. i < length xs \<and> P (xs ! i)}"
lookup_single_eq: fixes k :: "'c"   and v :: "'b" shows "lookup (monomial v k) k = v"
nprv_suc_congI: fixes F :: "'fmla set"   and t1 :: "'trm"   and t2 :: "'trm" assumes "nprv F (eql t1 t2)"   and "t1 \<in> atrm"   and "t2 \<in> atrm"   and "F \<subseteq> fmla"   and "finite F" shows "nprv F (eql (suc t1) (suc t2))"
satisfies_properties_sys: shows "satisfies_properties p qs subsets signs matrix"
subtopology_topspace: fixes U :: "'b topology" shows "subtopology U (topspace U) = U"