hcju/isabelleps huggingface dataset:
tree2_induct: fixes P :: "('b \<times> 'c) tree \<Rightarrow> bool"   and treea :: "('b \<times> 'c) tree" assumes "P \<langle>\<rangle>"   and "\<And>x1 a b x3. \<lbrakk>P x1; P x3\<rbrakk> \<Longrightarrow> P \<langle>x1, (a, b), x3\<rangle>" shows "P treea"
NextPropagate'_preserves_inv_init_imp_prop_safe: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b"   and c1 :: "('b, 't, 'loc) Combined.configuration" assumes "prop_invs (prop_config c0 p)"   and "inv_init_imp_prop_safe c0"   and "NextPropagate' c0 c1 p" shows "inv_init_imp_prop_safe c1"
m'_def: shows "m' = InsertMessage p new_id q \<sigma>"
Rep_scene: fixes x :: "'a scene" shows "Rep_scene x \<in> {F. overrider F}"
eq: shows "uint (2 ^ LENGTH(31)) = 2 ^ LENGTH(31)"
otherassms_3: shows "((\<sigma>, p), a, \<sigma>', q) \<in> automaton.trans A"
vm_path_def: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c" shows "vm_path G P v0 \<equiv> ParityGame G \<and> vm_path_axioms G P v0"
image_is_empty: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(f ` A = {}) = (A = {})"
inverse_positive_iff_positive: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less zero (inverse a) = less zero a"
not_eq_newpkt_pkt: fixes d :: "nat"   and dip :: "nat"   and sip :: "nat" shows "\<not> eq_newpkt (Pkt d dip sip)"
sorted_merge_wrt: fixes rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list" assumes "transp rel"   and "\<And>x y. x \<noteq> y \<Longrightarrow> rel x y \<or> rel y x"   and "sorted_wrt rel xs"   and "sorted_wrt rel ys" shows "sorted_wrt rel (merge_wrt rel xs ys)"
add_rel_class_of: fixes U :: "'a set"   and x :: "'b"   and V :: "'a set"   and y :: "'b"   and W :: "'a set" assumes "(U, x) \<in> Sigma I \<FF>"   and "(V, y) \<in> Sigma I \<FF>"   and "W \<in> I"   and "W \<subseteq> U"   and "W \<subseteq> V" shows "add_rel \<lfloor>U, x\<rfloor> \<lfloor>V, y\<rfloor> = \<lfloor>W, +\<^bsub>W\<^esub> (\<rho> U W x) (\<rho> V W y)\<rfloor>"
binrelchain_obtain_proper: fixes x :: "'b"   and y :: "'b"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "x \<noteq> y"   and "binrelchain P (x # xs @ [y])" shows "\<exists>zs. order.greater_eq (set xs) (set zs) \<and> order.greater_eq (length xs) (length zs) \<and> proper_binrelchain P (x # zs @ [y])"
le_sqrtI: fixes x :: "nat"   and y :: "nat" assumes "x\<^sup>2 \<le> y" shows "x \<le> local.sqrt y"
cert_ok_def: fixes cert :: "'c list"   and n :: "nat"   and T :: "'c"   and B :: "'c"   and A :: "'c set" shows "cert_ok cert n T B A = ((\<forall>i<n. cert ! i \<in> A \<and> cert ! i \<noteq> T) \<and> cert ! n = B)"
q1_def: shows "q1 \<equiv> [:\<i>, - 1:]"
fmfilter_add_distrib: fixes P :: "'d \<Rightarrow> bool"   and m :: "('d, 'e) fmap"   and n :: "('d, 'e) fmap" shows "fmfilter P (m ++\<^sub>f n) = fmfilter P m ++\<^sub>f fmfilter P n"
lower_float_interval: fixes prec :: "nat"   and X :: "float interval" shows "lower (sqrt_float_interval prec X) = lb_sqrt prec (lower X)"
grounding_of_cls_empty: shows "\<G>_F {#} = {{#}}"
iUntil_iEx_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "(True. t' \<U> t I. P t) = (\<diamond> t I. P t)"
U_le_F: fixes x :: "'s"   and y :: "'s" shows "U x y \<le> F x y"
Un_left_commute: fixes A :: "'x set"   and B :: "'x set"   and C :: "'x set" shows "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
ring_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "ring_hom hom \<equiv> semiring_hom hom"
filter_replace2: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and ys :: "'a list"   and xs :: "'a list" assumes "\<not> P x"   and "\<forall>y\<in>set ys. \<not> P y" shows "filter P (replace x ys xs) = filter P xs"
\<tau>_eq: shows "lst \<tau>3' = fst \<tau>3''"
uminus_def_2: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.linear_aggregation_algebra one times uminus inf less_eq less sup bot top plus conv" shows "uminus x = (if x = bot then top else bot)"
h4c: shows "aEval (Eq (A + B * Var var)) (L[var := sqrt Cv]) = eval (And (fm.Atom (Leq (A * B))) (fm.Atom (Eq (A\<^sup>2 - B\<^sup>2 * c)))) (L[var := sqrt Cv])"
simps_2: fixes l :: "('b \<times> 'c) tree"   and a :: "'b"   and uu :: "'c"   and r :: "('b \<times> 'c) tree"   and x :: "'b" shows "isin \<langle>l, (a, uu), r\<rangle> x = (case cmp x a of LT \<Rightarrow> isin l x | EQ \<Rightarrow> True | GT \<Rightarrow> isin r x)"
x2: shows "flow0 x t2 \<in> {a<--<flow0 x t1}"
power2_diff: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "power.power one times (minus x y) 2 = minus (plus (power.power one times x 2) (power.power one times y 2)) (times (times (numeral.numeral one plus (num.Bit0 num.One)) x) y)"
next_trans: fixes c :: "('p, 's, 'm) configuration"   and p :: "'p"   and s :: "'s"   and s' :: "'s"   and c' :: "('p, 's, 'm) configuration" shows "c \<turnstile> Trans p s s' \<mapsto> c' = (can_occur (Trans p s s') c \<and> states c' p = s' \<and> (\<forall>r. r \<noteq> p \<longrightarrow> states c' r = states c r) \<and> (\<forall>i. msgs c' i = msgs c i) \<and> no_snapshot_change c c')"
min_max_set_dual: shows "(`) \<partial> \<circ> min_set = max_set \<circ> (`) \<partial>"
axioms_2: fixes V\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>B :: "'c \<Rightarrow> 'c"   and src\<^sub>B :: "'c \<Rightarrow> 'c"   and trg\<^sub>B :: "'c \<Rightarrow> 'c"   and V\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>C :: "'d \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and V\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<a>\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<i>\<^sub>D :: "'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and H :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>H :: "'d \<times> 'd \<Rightarrow> 'e"   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" assumes "pseudonatural_transformation_whisker_left V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<tau>\<^sub>0 \<tau>\<^sub>1" shows "bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"
vrrestriction_vrange: fixes A :: "V" shows "(r \<restriction>\<^sup>l\<^sub>\<circ> A) \<restriction>\<^sup>r\<^sub>\<circ> \<R>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A) = r \<restriction>\<^sup>l\<^sub>\<circ> A"
simps_26: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> 'c"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'c"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'c"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'c"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'c"   and x11 :: "nat"   and x12 :: "nat"   and x13 :: "nat"   and x14 :: "k"   and x15 :: "nat"   and x16 :: "nat"   and x17 :: "nat"   and x18 :: "bool" shows "(case Rreq x11 x12 x13 x14 x15 x16 x17 x18 of Rreq x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f1 x11 x12 x13 x14 x15 x16 x17 x18"
s_top_induced: fixes L :: "'a set"   and H :: "'a set"   and K :: "'a set" assumes "G \<guillemotright> L "   and "H \<subseteq> L"   and "K \<subseteq> L" shows "H \<diamondop>\<^bsub>\<natural>L\<^esub> K = H \<diamondop> K"
cone_in_polyhedral_cone: fixes C :: "'a vec set"   and A :: "'a mat"   and nr :: "nat" assumes "C \<subseteq> polyhedral_cone A"   and "A \<in> carrier_mat nr n" shows "cone C \<subseteq> polyhedral_cone A"
inf_le2: fixes inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.semilattice_inf inf less_eq less" shows "less_eq (inf x y) y"
map\<^sub>1_simps_1: fixes f :: "'c" assumes "C.ide f" shows "D.arr (\<sigma>\<^sub>1 f)"
keys_update: fixes k :: "'b"   and v :: "'c"   and m :: "('b, 'c) mapping" shows "keys (update k v m) = insert k (keys m)"
is_mgu_def: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" shows "Unifiers.is_mgu \<sigma> E = (\<sigma> \<in> unifiers E \<and> (\<forall>\<tau>\<in>unifiers E. \<exists>\<gamma>. \<tau> = comp_subst_abbrev \<sigma> \<gamma>))"
set_renaming_perm2: fixes bs :: "atom set"   and p :: "perm" shows "\<exists>q. (\<forall>b\<in>bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> p \<bullet> bs"
subst_term_perm: fixes x' :: "Syntax.var"   and x :: "Syntax.var"   and e :: "Syntax.term"   and v :: "Syntax.term" assumes "atom x' \<sharp> (x, e)" shows "e[v / x] = ((x \<leftrightarrow> x') \<bullet> e)[v / x']"
lappend_ltl: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "\<not> lnull xs" shows "lappend (ltl xs) ys = ltl (lappend xs ys)"
basis_wf_def: fixes basis :: "(real \<Rightarrow> real) list" shows "basis_wf basis = ((\<forall>f\<in>set basis. filterlim f at_top at_top) \<and> sorted_wrt (\<lambda>f g. (\<lambda>x. ln (g x)) \<in> o(\<lambda>x. ln (f x))) basis)"
nn_contains_Inf: fixes S :: "nat set" assumes "S \<noteq> {}" shows "\<Sqinter> S \<in> S"
rec_transfer: fixes R2 :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and R1 :: "'h \<Rightarrow> 'i \<Rightarrow> bool" shows "((R2 ===> S) ===> rel_nondetT' R1 R2 ===> S) rec_nondetT rec_nondetT"
except_decomp: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and U :: "'a set" shows "p = except p U + except p (- U)"
self_finance_trading_strat: fixes pf :: "'a \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real"   and asset :: "'a"   and v :: "real" assumes "trading_strategy pf"   and "portfolio pf"   and "borel_adapt_stoch_proc G (prices Mkt asset)"   and "support_adapt Mkt pf" shows "trading_strategy (self_finance Mkt v pf asset)"
get_child_nodes_impl: shows "get_child_nodes = Shadow_DOM.get_child_nodes"
test_def2: fixes xaa :: "'a option"   and x2ba :: "'b list" shows "test xaa x2ba = (case xaa of None \<Rightarrow> case x2ba of [] \<Rightarrow> 1::'a | x # xa \<Rightarrow> 2::'a | Some x1ba \<Rightarrow> x1ba)"
freshBoundOutputTransition: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> P'"   and "c \<sharp> P"   and "c \<noteq> x" shows "c \<sharp> P'"
bigo_const_mult2: fixes c :: "'b"   and f :: "'a \<Rightarrow> 'b" shows "O(\<lambda>x. c * f x) \<subseteq> O(f)"
triangular_to_jnf_steps_dims_5: fixes A :: "'b mat" shows "dim_row (??.Jordan_Normal_Form_Existence.step_3 A) = dim_row A"
list2FWpolicy2list: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" shows "Cp (list2FWpolicy (policy2list p)) = Cp p"
distinct_298: fixes x17 :: "nat"   and x151 :: "floatarith"   and x152 :: "nat" shows "Var x17 \<noteq> x151 ^\<^sub>e x152"
prems_3: fixes f :: "'a match_expr \<Rightarrow> 'a match_expr option"   and m :: "'a match_expr"   and m' :: "'a match_expr" assumes "f \<in> set (f__ # fs)"   and "normalized_nnf_match m"   and "f m = Some m'" shows "normalized_nnf_match m'"
known_ptrs_implies: fixes h :: "(_) heap" assumes "DocumentClass.known_ptrs h" shows "ShadowRootClass.known_ptrs h"
vec_simps_33: fixes ah :: "'a"   and bh :: "'a"   and n :: "nat" shows "(ah * bh) ^ n = ah ^ n * bh ^ n"
that: fixes r :: "'a list"   and s :: "'a list" assumes "r \<cdot> s = u"   and "s \<cdot> r = v" shows "thesis"
1: shows "v \<in> (g_E G')\<^sup>* `` g_V0 G'"
face_distance_eq_0: fixes C :: "'a set"   and F :: "'a set" assumes "chamber C"   and "order.greater_eq C F" shows "face_distance F C = 0"
used_takeWhile_used: fixes x :: "msg"   and P :: "event \<Rightarrow> bool"   and X :: "event list" assumes "x \<in> used (takeWhile P X)" shows "x \<in> used X"
Iagree_refl: fixes I :: "('a, 'b, 'c) interp"   and A :: "('a + 'b + 'c) set" shows "Iagree I I A"
inv_closed: fixes x :: "'a vec" assumes "x \<in> carrier_vec n" shows "\<ominus>\<^bsub>V\<^esub> x \<in> carrier_vec n"
bind_gpv_Pause: fixes out :: "'c"   and rpv :: "('e, 'c, 'd) rpv"   and f :: "'e \<Rightarrow> ('b, 'c, 'd) gpv" shows "Pause out rpv \<bind> f = Pause out (rpv \<bind> f)"
l_neg: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<ominus> x \<oplus> x = \<zero>"
r_cancel_one: fixes x :: "'a"   and a :: "'a" assumes "x \<in> carrier G"   and "a \<in> carrier G" shows "(a \<otimes> x = x) = (a = \<one>)"
compact_fip: fixes "open" :: "'b set \<Rightarrow> bool"   and U :: "'b set" assumes "class.topological_space open" shows "topological_space.compact open U = (\<forall>A. (\<forall>a\<in>A. topological_space.closed open a) \<longrightarrow> (\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter> B \<noteq> {}) \<longrightarrow> U \<inter> \<Inter> A \<noteq> {})"
yun_rel_div: fixes F :: "int poly"   and c :: "rat"   and f :: "rat poly"   and G :: "int poly"   and d :: "rat"   and g :: "rat poly" assumes "yun_rel F c f"   and "yun_rel G d g"   and "G dvd F"   and "g dvd f" shows "yun_rel (F div G) (c / d) (f div g)"
eval_real_def: fixes c :: "real" shows "eval c = (\<lambda>_. c)"
subset_Zorn: fixes A :: "'c set set" assumes "\<And>C. subset.chain A C \<Longrightarrow> \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U" shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
rec_newstat_def: shows "rec_newstat = Cn 3 rec_add [Cn 3 rec_mult [Cn 3 rec_entry [recf.id 3 0, Cn 3 rec_add [Cn 3 rec_mult [Cn 3 (constn 4) [recf.id 3 0], Cn 3 rec_minus [recf.id 3 1, Cn 3 (constn 1) [recf.id 3 0]]], Cn 3 rec_add [Cn 3 rec_mult [Cn 3 (constn 2) [recf.id 3 0], Cn 3 rec_scan [recf.id 3 2]], Cn 3 (constn 1) [recf.id 3 0]]]], Cn 3 rec_noteq [recf.id 3 1, Cn 3 (constn 0) [recf.id 3 0]]], Cn 3 rec_mult [Cn 3 (constn 0) [recf.id 3 0], Cn 3 rec_eq [recf.id 3 1, Cn 3 (constn 0) [recf.id 3 0]]]]"
feq: fixes w :: "complex" assumes "w \<in> ball \<xi> r" shows "f w - f \<xi> = (w - \<xi>) ^ n * g w"
simps_2: fixes S :: "'b formula \<Rightarrow> 'b formula" shows "tseytin_tran1 S \<bottom> = [\<bottom> \<^bold>\<leftrightarrow> S \<bottom>]"
deriv: shows "resolution_deriv Cs Cs' \<and> {} \<in> Cs'"
lcm_unique_nat: fixes a :: "nat"   and d :: "nat"   and b :: "nat" shows "(a dvd d \<and> b dvd d \<and> (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e)) = (d = lcm a b)"
ThickChamberComplex_axioms: shows "ThickChamberComplex X"
d\<^sub>0_def: shows "d\<^sub>0 \<equiv> map_fun eval\<o> id Some"
prj_simps_11: shows "C.seq src_src.prj\<^sub>1 \<p>\<^sub>0[f.leg0, src.leg1 \<cdot> src_src.prj\<^sub>1]"
no_finished_root: fixes r :: "'a"   and scc :: "'a set"   and x :: "'a" assumes "scc_root s r scc"   and "r \<in> dom (finished s)"   and "x \<in> scc" shows "x \<notin> set (tj_stack s)"
linfinite_llength: fixes w :: "'a llist"   and k :: "nat" assumes "linfinite w" shows "enat k < llength w"
set_to_rank_invar: fixes bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list"   and t :: "('a, 'b) BinomialHeapStruc.BinomialTree" assumes "\<forall>t'\<in>set bq. BinomialHeapStruc.rank t < BinomialHeapStruc.rank t'"   and "BinomialHeapStruc.rank_invar bq" shows "BinomialHeapStruc.rank_invar (t # bq)"
antimono_def: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f" assumes "class.order less_eq less" shows "order.antimono less_eq f = (\<forall>x y. less_eq x y \<longrightarrow> f y \<le> f x)"
fC_1: shows "f w \<notin> f ` (C \<inter> D)"
SFAssNone\<^sub>1: fixes P :: "nat exp prog"   and e\<^sub>2 :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and v :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "val list"   and sh\<^sub>2 :: "sheap"   and C :: "char list"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"   and "\<nexists>b t. P \<turnstile> C has F,b:t in D" shows "P \<turnstile>\<^sub>1 \<langle>C\<bullet>\<^sub>sF{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NoSuchFieldError,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"
F: shows "\<forall>u\<in>sinks_aux (con_comp_pol I) (con_comp_map D E p q) U xs. (u, con_comp_map D E p q x) \<notin> con_comp_pol I"
continuous_closed_vimage: fixes S :: "'d set"   and f :: "'e \<Rightarrow> 'd" assumes "closed S"   and "\<And>x. isCont f x" shows "closed (f -` S)"
lookup_OAlist_tc_map2_val_rneutr: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'c"   and xs :: "('b, 'c) oalist_tc"   and ys :: "('b, 'd) oalist_tc"   and k :: "'b" assumes "\<And>k x. f k x (0::'d) = x" shows "OAlist_tc_lookup (OAlist_tc_map2_val_rneutr f xs ys) k = f k (OAlist_tc_lookup xs k) (OAlist_tc_lookup ys k)"
elims: fixes x :: "arctic"   and xa :: "arctic"   and y :: "arctic" assumes "x * xa = y" obtains y :: "arctic" where "xa__ = MinInfty" and "xa = yb__" and "ya__ = MinInfty"   | v :: "int" where "xa__ = Num_arc v" and "xa = MinInfty" and "ya__ = MinInfty"   | x :: "int" and y :: "int" where "xa__ = Num_arc x" and "xa = Num_arc y" and "ya__ = Num_arc (x + y)"
last_cons_def: fixes g :: "('e, 'f) generator"   and x :: "'e option"   and s :: "'f" shows "last_cons g x s = (if unstream g s = [] then the x else last (unstream g s))"
ground_heads_axioms: shows "Lambda_Free_Term.ground_heads (>\<^sub>s) arity_sym arity_var ground_heads_var"
mult_left_mono: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and a :: "'k"   and b :: "'k"   and c :: "'k" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq zero c" shows "less_eq (times c a) (times c b)"
homeomorphism_symD: fixes S :: "'c set"   and t :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" assumes "homeomorphism S t f g" shows "homeomorphism t S g f"
simps_1: fixes x :: "'c" shows "Missing_List.min_list [x] = x"
rtranclp_r_rtranclp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "r\<^sup>*\<^sup>* x y"   and "r y z" shows "r\<^sup>*\<^sup>* x z"
inv_mult_group: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier S"   and "y \<in> carrier S" shows "\<ominus>\<^bsub>S\<^esub> (x \<oplus>\<^bsub>S\<^esub> y) = \<ominus>\<^bsub>S\<^esub> y \<oplus>\<^bsub>S\<^esub> \<ominus>\<^bsub>S\<^esub> x"
ord: shows "i \<le> order x ([:- y, 1:] ^ Suc j) + order x (\<Prod>(x, i)\<leftarrow>yjs. [:- x, 1:] ^ Suc i)"
gen_without_effect: fixes t :: "typ"   and sch :: "type_scheme list" assumes "free_tv t \<subseteq> free_tv sch" shows "gen sch t = mk_scheme t"
while_rule: fixes s :: "state"   and b :: "state \<Rightarrow> bool"   and c :: "com"   and n :: "nat"   and t :: "state"   and P :: "state \<Rightarrow> bool" assumes "s -WHILE b DO c-n\<rightarrow> t"   and "P s"   and "\<And>s s'. \<lbrakk>P s; b s; s -c-n\<rightarrow> s'\<rbrakk> \<Longrightarrow> P s'" shows "P t \<and> \<not> b t"
closure_of_subset_eq: fixes S :: "'a set"   and X :: "'a topology" shows "(S \<subseteq> topspace X \<and> X closure_of S \<subseteq> S) = closedin X S"
weak_transition_not_tau_iff: fixes \<alpha> :: "('act, 'pred) S_action"   and P :: "'state"   and P' :: "'state" assumes "\<alpha> \<noteq> S_action.Act \<tau>" shows "P \<Rightarrow>\<langle>\<alpha>\<rangle>\<^sub>S P' = P \<Rightarrow>{\<alpha>}\<^sub>S P'"
bc: shows "b - c = (1 - k) *\<^sub>R (b - a)"
"1_1": shows "(S ===> A ===> rel_gpv'' (rel_prod B (rel_sum (rel_converter A B C R) S)) C R) f g"
tpart1_def: fixes V :: "Preliminaries.var set" shows "tpart1 V \<equiv> \<Union>\<sigma>. part1 (V \<inter> tpOfV -` {\<sigma>})"
lens_indep_def: fixes X :: "'e \<Longrightarrow> 'f"   and Y :: "'g \<Longrightarrow> 'f" shows "X \<bowtie> Y \<equiv> (\<forall>\<sigma> v u. put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u = put\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) v) \<and> (\<forall>\<sigma> v. get\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) = get\<^bsub>X\<^esub> \<sigma>) \<and> (\<forall>\<sigma> u. get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) = get\<^bsub>Y\<^esub> \<sigma>)"
assms_5: shows "g \<in> B \<rightarrow> carrier G"
exhaust: fixes y :: "tree" obtains "y = Leaf"   | x21 :: "tree" and x22 :: "tree" where "y = Node x21 x22"
L10: fixes js :: "nat list"   and k :: "nat"   and i :: "nat" assumes "\<And>f h. associative f \<Longrightarrow> foldl1 f (map h js) = foldl1 f (map h [0..<k + 1])"   and "i < k" shows "foldl1 f2 (map (h2 i) js) = Two"
emeasure_bind_density: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd measure"   and N :: "'d measure"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> ennreal"   and X :: "'d set" assumes "space M \<noteq> {}"   and "\<And>x. x \<in> space M \<Longrightarrow> has_density (f x) N (g x)"   and "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "X \<in> sets N" shows "emeasure (M \<bind> f) X = \<integral>\<^sup>+ x. set_nn_integral N X (g x) \<partial>M"
simps_5: fixes va :: "('a, 'b) rbt"   and vb :: "'a"   and vc :: "'b"   and vd :: "('a, 'b) rbt"   and b :: "('a, 'b) rbt"   and k :: "'a"   and x :: "'b"   and c :: "('a, 'b) rbt" shows "combine (Branch B va vb vc vd) (Branch R b k x c) = Branch R (combine (Branch B va vb vc vd) b) k x c"
in_\<Omega>: fixes N :: "'a stream measure"   and M :: "'a stream measure"   and y :: "'a" assumes "(N, M) \<in> F"   and "y \<in> set_pmf (p N M)" shows "y \<in> \<Omega>"
log_powr: fixes x :: "real"   and b :: "real"   and y :: "real" assumes "x \<noteq> 0" shows "log b (x powr y) = y * log b x"
open_carrier: fixes charts :: "('c, 'd) chart set" shows "open (manifold.carrier charts)"