hcju/isabelleps huggingface dataset:
lookup_Some_in\<delta>: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and \<pi> :: "char list"   and op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "ast_problem.lookup_operator problem \<pi> = Some op" shows "op \<in> set (ast_problem.ast\<delta> problem)"
constant_rep_design_axioms: shows "constant_rep_design \<V> \<B>' \<r>'"
isZero_def: shows "isZero \<equiv> \<Lambda> (ValN\<cdot>n). if n = 0 then ValTT else ValFF"
hom_add: fixes x :: "'c"   and y :: "'c" shows "[:x + y:] = [:x:] + [:y:]"
ac_Zp_one: shows "ac_Zp \<one> = \<one>"
raw_induct: fixes P :: "bool Heap \<Rightarrow> 'b Heap \<Rightarrow> heap \<Rightarrow> heap \<Rightarrow> unit \<Rightarrow> bool"   and p :: "bool Heap"   and f :: "'b Heap"   and h :: "heap"   and h' :: "heap"   and r :: "unit" assumes "\<And>while p f pa fa r. \<lbrakk>\<And>a b h h' r. effect (while a b) h h' r \<Longrightarrow> P a b h h' r; effect (p \<bind> (\<lambda>b. if b then f \<bind> (\<lambda>_. while p f) else return ())) pa fa r\<rbrakk> \<Longrightarrow> P p f pa fa r"   and "effect (while p f) h h' r" shows "P p f h h' r"
upto_enum_def: fixes n :: "'b"   and m :: "'b" shows "[n .e. m] \<equiv> map toEnumAlt [fromEnumAlt n..<Suc (fromEnumAlt m)]"
borel_Ioi: shows "borel = sigma UNIV (range greaterThan)"
HEndPhase0_HInv5_q2: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase0 s s' p"   and "p \<noteq> q"   and "\<exists>D\<in>MajoritySet. \<exists>qq. \<forall>d\<in>D. bal (dblock s q) < mbal (disk s d qq) \<and> \<not> hasRead s q d qq" shows "\<exists>D\<in>MajoritySet. \<exists>qq. \<forall>d\<in>D. bal (dblock s' q) < mbal (disk s' d qq) \<and> \<not> hasRead s' q d qq"
set_to_map_dom: fixes S :: "('b \<times> 'c) set" shows "dom (set_to_map S) = fst ` S"
cfs_one: fixes R :: "('c, 'd) ring_scheme"   and n :: "nat" assumes "UP_ring R" shows "\<one>\<^bsub>UP R\<^esub> n = (if n = 0 then \<one>\<^bsub>R\<^esub> else \<zero>\<^bsub>R\<^esub>)"
lemma_order_pderiv: fixes n :: "nat"   and p :: "'b poly"   and a :: "'b"   and q :: "'b poly" assumes "0 < n"   and "pderiv p \<noteq> 0"   and "p = [:- a, 1::'b:] ^ n * q"   and "\<not> [:- a, 1::'b:] dvd q" shows "n = Suc (order a (pderiv p))"
homology_exactness_triple_3: fixes T :: "'a set"   and S :: "'a set"   and p :: "int"   and X :: "'a topology" assumes "T \<subseteq> S" shows "exact_seq ([relative_homology_group p X S, relative_homology_group p X T, relative_homology_group p (subtopology X S) T], [hom_induced p X T X S id, hom_induced p (subtopology X S) T X T id])"
llist_all2_LCons2: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and y :: "'d"   and ys :: "'d llist" shows "llist_all2 P xs (LCons y ys) = (\<exists>x xs'. xs = LCons x xs' \<and> P x y \<and> llist_all2 P xs' ys)"
in_\<R>n': shows "((cm\<^sub>An', mem\<^sub>A'), cm\<^sub>Cn', mem\<^sub>C') \<in> \<R>n"
bounded_iff: fixes a :: "'b upred"   and b :: "'b upred"   and c :: "'b upred" shows "((b \<and> c) \<sqsubseteq> a) = (b \<sqsubseteq> a \<and> c \<sqsubseteq> a)"
power2_commute: fixes x :: "'a"   and y :: "'a" shows "(x - y)\<^sup>2 = (y - x)\<^sup>2"
open_subset: fixes S :: "'c set"   and T :: "'c set" assumes "S \<subseteq> T"   and "open S" shows "openin (top_of_set T) S"
ladder_stepdown_i_0: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and L :: "(nat \<times> nat \<times> nat) list" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "1 < length L" shows "ladder_i (ladder_stepdown L) 0 = ladder_i L 1 + ladder_ix L 1"
simps_1: fixes T1 :: "'fsym trm"   and T2 :: "'fsym trm" shows "Ik.wtA (Eq T1 T2) = (Ik.wt T1 \<and> Ik.wt T2 \<and> Ik.tpOf T1 = Ik.tpOf T2)"
well_com_mat_incr: shows "well_com (Utrans_P vars2 (mat_incr K))"
set_faces_splitFace: fixes g :: "graph"   and f :: "face"   and v1 :: "nat"   and v2 :: "nat"   and vs :: "nat list"   and f1 :: "face"   and f2 :: "face"   and g' :: "graph" assumes "minGraphProps g"   and "f \<in> \<F> g"   and "pre_splitFace g v1 v2 f vs"   and "(f1, f2, g') = splitFace g v1 v2 f vs" shows "\<F> g' = {f1, f2} \<union> (\<F> g - {f})"
lookup_def: fixes destr :: "'b \<Rightarrow> 'a"   and n :: "'c"   and s :: "'c \<Rightarrow> 'b" shows "lookup destr n s = destr (s n)"
invalid_list_split: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and xs :: "'a list"   and y :: "'b"   and ys :: "'b list" shows "invalid_assn (list_assn A) (x # xs) (y # ys) \<Longrightarrow>\<^sub>t invalid_assn A x y * invalid_assn (list_assn A) xs ys"
rep_eq_card: shows "Rep a + 1 = int CARD('a)"
dim_solution_set_not_zero_imp_infinite_solutions_no_homogeneous: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" assumes "0 < vec.dim (solution_set A 0)"   and "consistent A b" shows "infinite (solution_set A b)"
fmdom'_restrict_set: fixes A :: "'d set"   and m :: "('d, 'e) fmap" shows "fmdom' (fmrestrict_set A m) \<subseteq> A"
at_left_to_top: shows "at_left 0 = filtermap (\<lambda>x. - inverse x) at_top"
root_set_finite: fixes les :: "(real \<times> real \<times> real) set" assumes "finite les"   and "\<not> (\<exists>(a, b, c)\<in>les. a = 0 \<and> b = 0 \<and> c = 0)" shows "finite (root_set les)"
assms: shows "x \<in>\<^sub>\<circ> fconst_on A c \<restriction>\<^sup>l\<^sub>\<bullet> B"
shiftl_t2n: fixes w :: "'b word"   and n :: "nat" shows "w << n = 2 ^ n * w"
tri_three_paths: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "MinkowskiPrimitive.kinematic_triangle \<E> \<P> a b c" shows "\<exists>ab bc ca. (ab \<in> \<P> \<and> a \<in> ab \<and> b \<in> ab \<and> a \<noteq> b) \<and> (bc \<in> \<P> \<and> b \<in> bc \<and> c \<in> bc \<and> b \<noteq> c) \<and> (ca \<in> \<P> \<and> c \<in> ca \<and> a \<in> ca \<and> c \<noteq> a) \<and> ab \<noteq> bc \<and> ab \<noteq> ca \<and> bc \<noteq> ca"
convex_hull_empty_2: fixes A :: "'b vec set"   and n :: "nat" assumes "A \<subseteq> carrier_vec n" shows "(gram_schmidt.convex_hull n A = {}) = (A = {})"
distinct_7: fixes x2 :: "real \<times> real \<times> real"   and x3 :: "real \<times> real \<times> real" shows "EqUni x2 \<noteq> LeqUni x3"
eventually_subst': fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "(\<forall>\<^sub>F x in F. P x (f x)) = (\<forall>\<^sub>F x in F. P x (g x))"
get_root_node_document: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and root :: "(_) object_ptr" assumes "l_get_owner_document_wf_get_root_node_wf heap_is_wellformed type_wf known_ptr known_ptrs get_root_node get_owner_document"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_root_node ptr \<rightarrow>\<^sub>r root"   and "is_document_ptr_kind root" shows "h \<turnstile> get_owner_document ptr \<rightarrow>\<^sub>r the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r root)"
E_ss: fixes G :: "('a, 'b) graph_rec_scheme" assumes "graph G" shows "g_E G \<subseteq> g_V G \<times> g_V G"
assms_5: shows "\<not> (\<exists>op\<in>set ((\<Phi>\<inverse> \<Pi> \<A> t) ! k). v \<in> set (add_effects_of op) \<or> v \<in> set (delete_effects_of op))"
unfoldr_simps_1: fixes g :: "('b, 'c) generator"   and s :: "'c" assumes "list.has_next g s" shows "list.unfoldr g s = fst (list.next g s) # list.unfoldr g (snd (list.next g s))"
iB: fixes i :: "nat"   and j :: "nat"   and s :: "'s" shows "iterates body G i (M j) s \<le> b"
full_abstraction_and_enc_respects_pred_impl_SRel_iff_TRel_preserve: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set"   and Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)"   and "enc_respects_pred Pred" shows "rel_preserves_pred (_Collect (P, Q) (\<exists>SP SQ. SP \<in>S P \<and> SQ \<in>S Q \<and> (SP, SQ) \<in> SRel)) Pred = rel_preserves_pred (_Collect (P, Q) (\<exists>SP SQ. \<lbrakk>SP\<rbrakk> \<in>T P \<and> \<lbrakk>SQ\<rbrakk> \<in>T Q \<and> (\<lbrakk>SP\<rbrakk>, \<lbrakk>SQ\<rbrakk>) \<in> TRel)) Pred"
dual_filtered: shows "ord.filtered (\<lambda>x y. y \<le> x) = directed"
cases: fixes a1 :: "('a, 'b) ta_rule set"   and a2 :: "'b tree"   and a3 :: "'a" assumes "accs a1 a2 a3" obtains q :: "'a" and f :: "'b" and qs :: "'a list" and \<delta> :: "('a, 'b) ta_rule set" and ts :: "'b tree list" where "a1 = \<delta>" and "a2 = NODE f ts" and "a3 = q" and "q \<rightarrow> f qs \<in> \<delta>" and "length ts = length qs" and "\<And>i. i < length qs \<Longrightarrow> accs \<delta> (ts ! i) (qs ! i)"
llist_all2_reflI: fixes xs :: "'c llist"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. x \<in> lset xs \<Longrightarrow> P x x" shows "llist_all2 P xs xs"
before_def: fixes ev :: "event"   and evs :: "event list" shows "before ev on evs = takeWhile (\<lambda>z. z \<noteq> ev) (rev evs)"
anb: shows "fmset G as \<noteq> fmset G bs"
rbt_is_empty_def: fixes t :: "'c tree" shows "rbt_is_empty t \<equiv> t = \<langle>\<rangle>"
cases: fixes a :: "'a list" assumes "steps a" obtains x :: "'a" where "a = [x__]"   | x :: "'a" and y :: "'a" and xs :: "'a list" where "a = x # y # xs" and "E x y" and "steps (y # xs)"
assms: shows "f maps\<^bsub>SET\<^esub> X to Y"
unit_imp_det_non_zero: fixes A :: "'b mat"   and n :: "nat"   and b :: "'c" assumes "A \<in> Units (ring_mat TYPE('b) n b)" shows "det A \<noteq> (0::'b)"
rep_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C" shows "free_monoidal_category.rep C f \<equiv> free_monoidal_category.norm C (SOME t. t \<in> f)"
of_nat_mult: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and m :: "nat"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (m * n) = times (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
pos_emb_step_at_disjunct: fixes q :: "dir list"   and p :: "dir list"   and t :: "('a, 'b) tm"   and d1 :: "dir"   and d2 :: "dir" assumes "take (length q) p \<noteq> q"   and "take (length p) q \<noteq> p" shows "position_of t (p @ [d1]) = position_of (emb_step_at q d2 t) (p @ [d1])"
normalized_min_not_None: fixes xs :: "('c, 'd) bintree option list" assumes "normalized xs"   and "xs \<noteq> []" shows "Binomial_Queue.min xs \<noteq> None"
vrange_vrrestriction: fixes r :: "V"   and A :: "V" shows "\<R>\<^sub>\<circ> (r \<restriction>\<^sup>r\<^sub>\<circ> A) = \<R>\<^sub>\<circ> r \<inter>\<^sub>\<circ> A"
assms_4: shows "\<forall>x. S1 x \<triangle> S2 x \<le> S1 x"
image_mset_const_eq: fixes c :: "'k"   and M :: "'l multiset" shows "{#c. a \<in># M#} = replicate_mset (size M) c"
vunion_union: fixes A :: "V"   and B :: "V" shows "A \<union>\<^sub>\<circ> B = set (elts A \<union> elts B)"
setdist_compact_closed: fixes A :: "'b set"   and B :: "'b set" assumes "compact A"   and "closed B"   and "A \<noteq> {}"   and "B \<noteq> {}" shows "\<exists>x\<in>A. \<exists>y\<in>B. dist x y = setdist A B"
assms_20: shows "InvariantClCurrentLevel (getCl state) (getM state)"
a_pos_m: fixes m :: "int" assumes "0 < m" shows "ant m * - \<infinity> = - \<infinity>"
gvut2: fixes s :: "('d, 'c, 'e) configuration_scheme stream"   and q :: "'d"   and t :: "'c" assumes "GlobVacantUpto (shd s) q t"   and "next_sendupd s" shows "GlobVacantUpto (shd (stl s)) q t"
get_set_neq: fixes r :: "'a array"   and s :: "'b array"   and x :: "'b list"   and h :: "heap" assumes "r =!!= s" shows "Array.get (Array.set s x h) r = Array.get h r"
monotone_parametric: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "bi_total A" shows "rel_fun (rel_fun A (rel_fun A (=))) (rel_fun (rel_fun B (rel_fun B (=))) (rel_fun (rel_fun A B) (=))) monotone monotone"
order_tendsto_iff: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'a"   and F :: "'c filter" shows "(f \<longlongrightarrow> x) F = ((\<forall>l<x. \<forall>\<^sub>F x in F. l < f x) \<and> (\<forall>u>x. \<forall>\<^sub>F x in F. f x < u))"
Collect_mono: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>x. P x \<longrightarrow> Q x" shows "Collect P \<subseteq> Collect Q"
a: shows "i \<in> {a * b..<(a + 1) * b}"
write_def: fixes c :: "'b \<Rightarrow> 'a"   and a :: "'b"   and P :: "'a process" shows "c`!`a \<rightarrow> P \<equiv> Mprefix {c a} (\<lambda>x. P)"
assms: shows "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"
simps_7: fixes xs :: "'a list"   and a :: "fexpr"   and b :: "fexpr" shows "feval xs (FDiv a b) = feval xs a \<oslash> feval xs b"
functor_FF: shows "functor VV.comp Span.VV.comp SPN.FF"
Gromov_completion_rel_to_const: fixes u :: "nat \<Rightarrow> 'a"   and x :: "'a"   and n :: "nat" assumes "Gromov_completion_rel u (\<lambda>n. x)" shows "u n = x"
Polys_closed_plus: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and X :: "'a set"   and q :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p \<in> P[X]"   and "q \<in> P[X]" shows "p + q \<in> P[X]"
dist_bs: fixes a :: "'a" assumes "a \<in> set as" shows "distinct (bs a)"
g_newcond: shows "\<forall>g\<in>set (take (Suc n) gs). terminate g xs \<and> (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_exec g xs # 0 \<up> (xb - Suc xa) @ xc}))"
strict_implies_order: fixes a :: "'c word"   and b :: "'c word" assumes "a <s b" shows "a \<le>s b"
map_index_map: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "map_index f (map g xs) = map_index (\<lambda>n x. f n (g x)) xs"
log_le_one_cancel_iff: fixes a :: "real"   and x :: "real" assumes "1 < a"   and "0 < x" shows "(log a x \<le> 1) = (x \<le> a)"
inf_or_of_complex: fixes z :: "complex_homo" shows "z = \<infinity>\<^sub>h \<or> (\<exists>x. z = of_complex x)"
concat_xss': shows "lnull (lconcat (llist_of xss'))"
sum_roots_unity: fixes n :: "nat" assumes "1 < n" shows "\<Sum> {z. z ^ n = 1} = 0"
power_commutes: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and n :: "nat" assumes "class.monoid_mult one times" shows "times (power.power one times a n) a = times a (power.power one times a n)"
fls_compose_power_add: fixes f :: "'b fls"   and g :: "'b fls"   and d :: "nat" shows "fls_compose_power (f + g) d = fls_compose_power f d + fls_compose_power g d"
env_restr_UNIV: fixes m :: "'c \<Rightarrow> 'd" shows "m f|` UNIV = m"
strict_def: fixes f :: "'b \<Rightarrow> 'c" shows "strict f \<equiv> f bot = bot"
list_all2_nthD: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'h list"   and p :: "nat" assumes "list_all2 P xs ys"   and "p < length xs" shows "P (xs ! p) (ys ! p)"
dvd_nat_abs_iff: fixes n :: "nat"   and k :: "int" shows "(n dvd nat \<bar>k\<bar>) = (int n dvd k)"
mono: fixes x :: "real"   and y :: "real" assumes "a \<le> x"   and "x \<le> y"   and "y \<le> b" shows "g x \<le> g y"
transpose_uminus: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "(- A)\<^sup>T = - A\<^sup>T"
base: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and xs :: "'c stream" assumes "\<phi> xs" shows "ev \<phi> xs"
compare_double_antisym: fixes x :: "double"   and y :: "double" assumes "compare_double x y \<le> 0"   and "compare_double y x \<le> 0"   and "\<not> Code_Double.is_zero x \<or> \<not> Code_Double.is_zero y" shows "x = y"
no_better_subset_pref: fixes x :: "'a"   and y :: "'a" assumes "x \<succeq> y" shows "no_better_than y carrier relation \<subseteq> no_better_than x carrier relation"
ground_heads_arity\<^sub>h: fixes f :: "'s"   and \<zeta> :: "('s, 'v) hd" assumes "f \<in> local.ground_heads \<zeta>" shows "arity_hd\<^sub>h \<zeta> \<le> arity_sym\<^sub>h f"
len: shows "length (cs\<^bsup>\<pi>\<^esup> k) < length (cs\<^bsup>\<pi>\<^esup> l)"
saturated_Extend: fixes S :: "('a, 'b) fm set"   and f :: "nat \<Rightarrow> ('a, 'b) fm" assumes "consistent (Extend S f)"   and "surj f" shows "saturated (Extend S f)"
least_dual: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and A :: "'c set" shows "least (inv_gorder L) x A = greatest L x A"
cInf_restrict_nat: fixes x :: "nat"   and A :: "nat set" assumes "x \<in> A" shows "\<Sqinter> A = \<Sqinter> {y \<in> A. y \<le> x}"
semiring_normalization_rules_26: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "'a"   and p :: "nat"   and q :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "times (power.power one times x p) (power.power one times x q) = power.power one times x (p + q)"
homotopic_paths_refl: fixes s :: "'c set"   and p :: "real \<Rightarrow> 'c" shows "homotopic_paths s p p = (path p \<and> path_image p \<subseteq> s)"
pointwise_convergent_to_def: fixes x :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and l :: "'b \<Rightarrow> 'c" shows "x \<midarrow>pointwise\<rightarrow> l = (\<forall>t. (\<lambda>n. x n t) \<longlonglongrightarrow> l t)"
assms: shows "\<Gamma> \<turnstile> e1 \<guillemotleft>bop\<guillemotright> e2 : T"
inext_eq_infin_fin: fixes a :: "nat"   and I :: "nat set"   and b :: "nat" assumes "inext a I = inext b I"   and "a \<noteq> Max I \<and> b \<noteq> Max I \<or> infinite I" shows "a = b"
Beta: fixes r :: "lam"   and x :: "name"   and s :: "lam"   and ss :: "lam list" assumes "WN (r[x::=s] \<degree>\<degree> ss)" shows "WN ((Lam [x].r) \<degree> s \<degree>\<degree> ss)"
dist_geq_iff_midpoint_in_between: fixes A :: "'b"   and B :: "'b"   and X :: "'b" assumes "between (A, B) X" shows "(dist A X \<le> dist B X) = between (X, B) (midpoint A B)"
nneg_def: fixes P :: "'a \<Rightarrow> 'b" shows "nneg P = (\<forall>x. (0::'b) \<le> P x)"
modifiesclause: shows "\<tau> \<Turnstile> X->excluding\<^sub>S\<^sub>e\<^sub>t(x)->oclIsModifiedOnly()"
trans: fixes b :: "'a multiset"   and a :: "'a multiset"   and c :: "'a multiset" assumes "b \<subseteq># a"   and "c \<subseteq># b" shows "c \<subseteq># a"
Limsup_def: fixes F :: "'d filter"   and f :: "'d \<Rightarrow> 'c" shows "Limsup F f = (INF P\<in>{P. eventually P F}. Sup (f ` {x. P x}))"
leadsTo_induct_pre: fixes F :: "'b program"   and za :: "'b set"   and zb :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "F \<in> za \<longmapsto> zb"   and "P zb"   and "\<And>A B. \<lbrakk>F \<in> A ensures B; F \<in> B \<longmapsto> zb; P B\<rbrakk> \<Longrightarrow> P A"   and "\<And>S. \<forall>A\<in>S. F \<in> A \<longmapsto> zb \<and> P A \<Longrightarrow> P (\<Union> S)" shows "P za"
in_set_\<alpha>nI1: fixes g :: "'g"   and x :: "'node" assumes "invar g"   and "x \<in> getFrom ` \<alpha>e g" shows "x \<in> set (\<alpha>n g)"
intro: fixes h :: "'f \<Rightarrow> 'g"   and G :: "('f, 'h) monoid_scheme"   and H :: "('g, 'i) monoid_scheme" assumes "h \<in> hom G H" shows "group_hom_axioms G H h"
eval\<^sub>1_eval: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and Vs :: "char list list"   and ls :: "val list" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',(h', l', sh')\<rangle>"   and "fv e \<subseteq> set Vs"   and "l \<subseteq>\<^sub>m [Vs [\<mapsto>] ls]"   and "length Vs + max_vars e \<le> length ls" shows "\<exists>ls'. compP\<^sub>1 P \<turnstile>\<^sub>1 \<langle>compE\<^sub>1 Vs e,(h, ls, sh)\<rangle> \<Rightarrow> \<langle>fin\<^sub>1 e',(h', ls', sh')\<rangle> \<and> l' \<subseteq>\<^sub>m [Vs [\<mapsto>] ls']"
assms_8: fixes A :: "'a"   and B :: "'a" assumes "rewrite A B" shows "reflexive_rewrite A B"
has_derivative_plus_fun: fixes x :: "'c \<Rightarrow> 'd"   and x' :: "'c \<Rightarrow> 'd"   and a :: "'c"   and A :: "'c set"   and y :: "'c \<Rightarrow> 'd"   and y' :: "'c \<Rightarrow> 'd" assumes "(x has_derivative x') (at a within A)"   and "(y has_derivative y') (at a within A)" shows "(x + y has_derivative x' + y') (at a within A)"
n: shows "\<forall>x. eval r_phi [n, x] = eval psi [n, x]"
domain_def: fixes f :: "'c \<Rightarrow> 'd" shows "domain f \<equiv> {x. f x \<noteq> none}"
filterlim_at_infinity_imp_not_convergent: fixes f :: "nat \<Rightarrow> 'b" assumes "filterlim f at_infinity sequentially" shows "\<not> convergent f"
host_shadow_root_rel_host: fixes h :: "(_) heap"   and shadow_root :: "(_) shadow_root_ptr"   and host :: "(_) element_ptr" assumes "heap_is_wellformed h"   and "h \<turnstile> get_host shadow_root \<rightarrow>\<^sub>r host" shows "(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host, cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root) \<in> local.a_host_shadow_root_rel h"
of_int_not_eq: 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 bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and k :: "int" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_1.of_int one plus zero minus (NOT k) = not (ring_1.of_int one plus zero minus k)"
dupe_monic_def: fixes D :: "'a poly"   and H :: "'a poly"   and S :: "'a poly"   and T :: "'a poly"   and U :: "'a poly" shows "dupe_monic D H S T U = (case pdivmod_monic (T * U) D of (q, r) \<Rightarrow> (S * U + H * q, r))"