hcju/isabelleps huggingface dataset:
assms_2: shows "\<parallel>c\<parallel>\<^bsub>t n'\<^esub>"
merge_sorted_wrt_partitions: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and lo :: "nat"   and hi :: "nat"   and p :: "nat" assumes "isPartition_wrt R xs lo hi p"   and "sorted_sublist_wrt R xs lo (p - Suc 0)"   and "sorted_sublist_wrt R xs (Suc p) hi"   and "lo \<le> hi"   and "lo \<le> p"   and "p \<le> hi"   and "hi < length xs"   and "\<forall>i j. lo \<le> i \<and> i < p \<and> p < j \<and> j \<le> hi \<longrightarrow> R (xs ! i) (xs ! j)" shows "sorted_sublist_wrt R xs lo hi"
chain_compr: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "Complete_Partial_Order.chain ord A" shows "Complete_Partial_Order.chain ord {x \<in> A. P x}"
eq_fold: fixes A :: "'b set" shows "Lcm\<^sub>f\<^sub>i\<^sub>n A = (if finite A then Finite_Set.fold lcm (1::'b) A else (0::'b))"
image_insert: fixes f :: "'h \<Rightarrow> 'g"   and a :: "'h"   and B :: "'h set" shows "f ` insert a B = insert (f a) (f ` B)"
rprodl_simps: fixes a :: "'c"   and b :: "'d"   and c :: "'e" shows "rprodl ((a, b), c) = (a, b, c)"
aval'_correct: fixes s :: "char list \<Rightarrow> int"   and S :: "char list \<Rightarrow> 'av"   and a :: "aexp" assumes "s \<in> \<gamma>\<^sub>s S" shows "aval a s \<in> \<gamma> (aval' a S)"
fix_F_stable_pair_on: fixes X :: "'a set \<times> 'a set dual"   and ds :: "'b set" assumes "X = F ds X" shows "stable_pair_on ds (map_prod id undual X)"
ec: shows "P,shp s2 \<turnstile>\<^sub>b (e2,b2) \<surd>"
bind_def: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'b list" shows "List.bind xs f = concat (map f xs)"
Snorm_def: shows "Snorm = map (\<lambda>s. s /\<^sub>R norm s) S"
rep_eq: fixes x :: "nat set" shows "Sequence.seq.Rep_seq (bit_seq_of_nat_set x) = (\<lambda>i. i \<in> x)"
neq0: shows "v \<noteq> 0\<^sub>v n"
tendsto_norm_zero_iff: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'c filter" shows "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> (0::'d)) F"
subset_Diff_insert: fixes A :: "'c set"   and B :: "'c set"   and x :: "'c"   and C :: "'c set" shows "(A \<subseteq> B - Set.insert x C) = (A \<subseteq> B - C \<and> x \<notin> A)"
assms_2: shows "(u, u') \<in> red\<^sup>*"
genideal_self: fixes S :: "'a set" assumes "S \<subseteq> carrier R" shows "S \<subseteq> Idl S"
set_add_def: fixes G :: "('c, 'd) ring_scheme" shows "(<+>\<^bsub>G\<^esub>) = (<#>\<^bsub>add_monoid G\<^esub>)"
spn_props_1: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "\<guillemotleft>spn \<mu> : src (tab\<^sub>0 (local.dom \<mu>)) \<rightarrow> src (tab\<^sub>0 (cod \<mu>))\<guillemotright>"
domInterMT: fixes a :: "'a \<Rightarrow> 'b option"   and b :: "'a \<Rightarrow> 'c option"   and x :: "'a" assumes "dom a \<inter> dom b = {}"   and "x \<in> dom a" shows "x \<notin> dom b"
phi_inf_principal: fixes x :: "'c regular"   and y :: "'c dense" shows "\<exists>z. up_filter z = stone_phi x \<sqinter> up_filter y"
pbij_lens_def: fixes x :: "'a \<Longrightarrow> 'b" shows "pbij_lens x \<equiv> weak_lens x \<and> pbij_lens_axioms x"
ring_tOp_commute: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c"   and y :: "'c" assumes "Ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> y = y \<cdot>\<^sub>r\<^bsub>R\<^esub> x"
lookup_pair_inj: fixes x :: "'o"   and xs :: "('a \<times> 'c) list"   and ys :: "('a \<times> 'c) list" assumes "oalist_inv_raw x xs"   and "oalist_inv_raw x ys"   and "lookup_pair x xs = lookup_pair x ys" shows "xs = ys"
ofilter_Restr_under: fixes r :: "('c \<times> 'c) set"   and A :: "'c set"   and a :: "'c" assumes "Well_order r"   and "ofilter r A"   and "a \<in> A" shows "under (Restr r A) a = under r a"
finite_list_append: fixes xs :: "['b]"   and ys :: "['b]" shows "finite_list (xs ++ ys) = (finite_list xs \<and> finite_list ys)"
prime_factorization_empty_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "(factorial_semiring.prime_factorization zero times one normalize x = {#}) = (x = zero \<or> dvd.dvd times x one)"
act2: fixes i :: "nat"   and R :: "nat set" shows "\<langle>i : onp i : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). a \<noteq> \<tau> \<and> (\<forall>d. a \<noteq> i:deliver(d)) \<longrightarrow> S (\<sigma> i) (\<sigma>' i))"
while_termiE: fixes b :: "state \<Rightarrow> bool"   and c :: "com"   and s :: "state" assumes "WHILE b DO c \<down> s"   and "b s" shows "c \<down> s"
fmap_rel_def: fixes K :: "('a \<times> 'c) set"   and V :: "('b \<times> 'd) set" shows "\<langle>K, V\<rangle>fmap_rel \<equiv> {(m1, m2). (\<forall>i j. i |\<in>| fmdom m2 \<longrightarrow> (j, i) \<in> K \<longrightarrow> (the (fmlookup m1 j), the (fmlookup m2 i)) \<in> V) \<and> fset (fmdom m1) \<subseteq> Domain K \<and> fset (fmdom m2) \<subseteq> Range K \<and> (\<forall>i j. (i, j) \<in> K \<longrightarrow> (j |\<in>| fmdom m2) = (i |\<in>| fmdom m1))}"
Mon2Rel_def: fixes f :: "'a \<Rightarrow> (unit \<times> 'a) option" shows "Mon2Rel f = {(x, y). f x = Some ((), y)}"
sel_2: fixes x1 :: "'b list"   and x2 :: "nat" shows "arity (Fundef x1 x2) = x2"
finite_vars_imp_finite_adapt_a: fixes e :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd"   and s :: "'c \<Rightarrow> nat" assumes "finite (Aexp.vars e)" shows "finite (Aexp.vars (adapt_aexp e s))"
length_0_conv: fixes xs :: "'b list" shows "(length xs = 0) = (xs = [])"
assms: shows "low_equal s1 s2 \<and> get_S (cpu_reg_val PSR s1) = 0 \<and> get_S (cpu_reg_val PSR s2) = 0 \<and> \<not> snd (dispatch_instruction instr s1) \<and> \<not> snd (dispatch_instruction instr s2) \<and> t1 = snd (fst (dispatch_instruction instr s1)) \<and> t2 = snd (fst (dispatch_instruction instr s2))"
free_Abelian_group_def: fixes S :: "'b set" shows "free_Abelian_group S = \<lparr>carrier = {c. Poly_Mapping.keys c \<subseteq> S}, mult = (+), one = 0\<rparr>"
amult_1_both: fixes w :: "ant"   and x :: "ant" assumes "0 < w"   and "x * w = 1" shows "x = 1 \<and> w = 1"
poincare_mapsto_def: fixes P :: "'a set"   and X0 :: "('a \<times> 'b \<Rightarrow>\<^sub>L 'a) set"   and S :: "'a set"   and CX :: "'a set"   and Y :: "('a \<times> 'b \<Rightarrow>\<^sub>L 'a) set" shows "poincare_mapsto P X0 S CX Y = (\<forall>(x, d)\<in>X0. returns_to P x \<and> fst ` X0 \<subseteq> S \<and> return_time P differentiable at x within S \<and> (\<exists>D. (poincare_map P has_derivative blinfun_apply D) (at x within S) \<and> (poincare_map P x, D o\<^sub>L d) \<in> Y) \<and> (\<forall>t\<in>{0<..<return_time P x}. flow0 x t \<in> CX))"
seqE: fixes g :: "('o, 'a) arr"   and f :: "('o, 'a) arr" assumes "seq g f" obtains "arr f" and "arr g" and "local.dom g = cod f"
set: fixes x1 :: "16 word"   and x2 :: "'c set"   and x3 :: "'b" shows "set_flow_entry_match (OFEntry x1 x2 x3) = {x3}"
aux4lose1: fixes d :: "nat"   and k :: "nat"   and lose :: "nat \<Rightarrow> bool list"   and t :: "nat"   and j :: "nat" assumes "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = [False]"   and "j \<le> k" shows "lose (t + 2 + j) = [False]"
elims: fixes x :: "trilean"   and y :: "trilean" assumes "\<not>? x = y" obtains "x = true" and "y = false"   | "x = false" and "y = true"   | "x = invalid" and "y = invalid"
closed_halfspace_Re_eq: fixes b :: "real" shows "closed {z. Re z = b}"
full_abstraction_and_trans_relation_contains_TRel_impl_SRel: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set"   and Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)"   and "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel"   and "\<forall>S1 S2. ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel) = ((TargetTerm (\<lbrakk>S1\<rbrakk>), TargetTerm (\<lbrakk>S2\<rbrakk>)) \<in> Rel)"   and "trans (Rel \<union> _Collect (P, Q) (\<exists>S. \<lbrakk>S\<rbrakk> \<in>T P \<and> S \<in>S Q))" shows "SRel = _Collect (S1, S2) ((SourceTerm S1, SourceTerm S2) \<in> Rel)"
simps_2: fixes uu :: "'b"   and x :: "'b"   and xs :: "['b]" shows "last\<cdot>(uu : x : xs) = last\<cdot>(x : xs)"
elims_3: fixes x :: "('b ivl \<times> 'b) tree" assumes "\<not> invar x" obtains t :: "('b ivl \<times> 'b) tree" where "x = t" and "\<not> (inv_max_hi t \<and> Sorted_Less.sorted (Tree2.inorder t))"
return_time_le: fixes P :: "'a set"   and x :: "'a"   and t :: "real" assumes "returns_to P x"   and "closed P"   and "flow0 x t \<in> P"   and "0 < t" shows "return_time P x \<le> t"
trg_dom: fixes \<mu> :: "'d" assumes "D.arr \<mu>" shows "trg\<^sub>D (D.dom \<mu>) = trg\<^sub>D \<mu>"
radical_idealI: fixes X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and f :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and x :: "'c" assumes "finite X"   and "F \<subseteq> P[X]"   and "f \<in> P[X]"   and "x \<notin> X"   and "\<V> (insert (1 - punit.monom_mult (1::'d) (monomial 1 x) f) F) = {}" shows "f \<in> \<surd>ideal F"
zcount_union: fixes M :: "'b zmultiset"   and N :: "'b zmultiset"   and a :: "'b" shows "zcount (M + N) a = zcount M a + zcount N a"
intros_4: shows "IK_nf (I \<circ> K)"
correctCompositionKeys_subcomp2: fixes C :: "specID"   and xb :: "specID"   and xc :: "Keys" assumes "correctCompositionKeys C"   and "xb \<in> subcomponents C"   and "xc \<in> specKeys xb" shows "xc \<in> specKeys C"
lagr: shows "card (carrier G) = card (rcosets\<^bsub>G\<^esub> kernel G H f) * card (kernel G H f)"
select_convs_2: fixes xc :: "int"   and yc :: "int"   and zc :: "int"   and more :: "'a" shows "point3d.more \<lparr>xc = xc, yc = yc, zc = zc, \<dots> = more\<rparr> = more"
Bex_set: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" shows "Bex (set xs) P = list_ex P xs"
char_poly_hom: fixes hom :: "'a \<Rightarrow> 'b"   and A :: "'a mat"   and n :: "nat" assumes "comm_ring_hom hom"   and "A \<in> carrier_mat n n" shows "char_poly (map_mat hom A) = map_poly hom (char_poly A)"
assms_3: shows "\<Theta>(g) \<equiv> BIGTHETA_CONST c2 \<Theta>(eval_primfuns gs)"
hyps_2: shows "Q'' \<longmapsto> \<tau> \<prec> Q'''"
cons: shows "\<forall>s\<in>P. \<forall>t. \<exists>P' Q' A'. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P' c Q',A' \<and> ((s \<in> P' \<longrightarrow> t \<in> Normal ` Q' \<union> Abrupt ` A') \<longrightarrow> t \<in> Normal ` Q \<union> Abrupt ` A)"
h7: shows "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set [[1], [0], [- 1]]"
assms: shows "x \<in> N \<times> (Pow \<Omega> - {{}})"
fsubset_pfsubset_trans: fixes A :: "'b fset"   and B :: "'b fset"   and C :: "'b fset" assumes "A |\<subseteq>| B"   and "B |\<subset>| C" shows "A |\<subset>| C"
decode_state_at_def: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and k :: "nat" shows "\<Phi>\<^sub>S\<inverse> \<Pi> \<A> k \<equiv> let vs = \<Pi>\<^sub>\<V>; state_encoding_to_assignment = \<lambda>v. (v, \<A> (State k (index vs v))) in map_of (map state_encoding_to_assignment vs)"
path2_cases: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m" obtains "ns = [n]" and "m = n"   | "g \<turnstile> hd (tl ns)-tl ns\<rightarrow>m" and "n \<in> set (predecessors g (hd (tl ns)))"
assms_1: shows "H \<turnstile> HaddP x' y z'"
closed_segment_image_interval: fixes a :: "'a"   and b :: "'a" shows "closed_segment a b = (\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) ` {0..1}"
asym_inv_image: fixes R :: "('g \<times> 'g) set"   and f :: "'h \<Rightarrow> 'g" assumes "asym R" shows "asym (inv_image R f)"
error_free_raise_if: fixes s :: "state"   and p :: "bool"   and x :: "xname" assumes "error_free s" shows "error_free (abupd (raise_if p x) s)"
f2_def: fixes x1 :: "'a"   and x2 :: "'b" shows "f2 x1 x2 = snd (f x1 x2)"
finite_measure_distr: fixes f :: "val \<Rightarrow> 'a"   and M' :: "'a measure" assumes "f \<in> M \<rightarrow>\<^sub>M M'" shows "finite_measure (distr M M' f)"
app_ins2: fixes x' :: "hf"   and x :: "hf"   and y :: "hf"   and f :: "hf" assumes "x' \<noteq> x" shows "app (f \<triangleleft> \<langle>x, y\<rangle>) x' = app f x'"
set: fixes y :: "'a" assumes "y \<in> set L" shows "P y"
index_Cons: fixes x :: "'b"   and xs :: "'b list"   and a :: "'b" shows "index (x # xs) a = (if x = a then 0 else index xs a + 1)"
split_sels_1: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "nat \<Rightarrow> 'b"   and nat :: "nat" shows "P (case nat of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = ((nat = 0 \<longrightarrow> P f1) \<and> (nat = Suc (nat.pred nat) \<longrightarrow> P (f2 (nat.pred nat))))"
axioms_2: fixes ccw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set" assumes "ccw_vector_space ccw S" shows "ccw_vector_space_axioms ccw"
max_li_is_basis: fixes A :: "'c set" assumes "maximal A (\<lambda>S. S \<subseteq> carrier (V.vs kerT) \<and> ker.lin_indpt S)" shows "ker.basis A"
intros: fixes base :: "nat"   and M :: "hmultiset"   and e :: "nat" assumes "2 \<le> base"   and "\<forall>m\<in>#hmsetmset M. encode base 0 e \<le> m" shows "aligned\<^sub>e base e M"
coboundedI: fixes A :: "'b word set"   and a :: "'b word" assumes "finite A"   and "a \<in> A" shows "a \<le>s signed.Max A"
assms_2: fixes s :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<paragraph>''invar-pres''"   and "modifies modset s s\<^sub>0"   and "I s"   and "bval b s" shows "wlp \<pi> c I s"
in_keys_timesE: fixes v :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'b"   and q :: "'a \<Rightarrow>\<^sub>0 'b" assumes "v \<in> keys (p * q)" obtains t :: "'a" and u :: "'a" where "t \<in> keys p" and "u \<in> keys q" and "v = t + u"
antisym: fixes x :: "'c list"   and y :: "'c list" assumes "suffix x y"   and "suffix y x" shows "x = y"
dualization_rew_4: fixes A :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<not> \<diamond>\<langle>A\<rangle>_v \<equiv> \<box>[\<not> A]_v"
norm_eq_sqrt_cinner: fixes x :: "'a" shows "norm x = sqrt (cmod (cinner x x))"
arr_cod_iff_arr: fixes f :: "('o, 'a) arr" shows "arr (cod f) = arr f"
r_sv01_recfn: shows "recfn 1 r_sv01"
are_all_operator_effects_consistent_def: fixes ops :: "('a, 'b) strips_operator_scheme list" shows "are_all_operator_effects_consistent ops \<equiv> list_all (\<lambda>op. list_all (are_operator_effects_consistent op) ops) ops"
mem_val_w32_8_unchanged: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and a :: "word36"   and addr :: "word36"   and val :: "word8" assumes "mem_equal s1 s2 a" shows "mem_val_w32 8 a (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) = mem_val_w32 8 a (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>)"
simps: fixes a1 :: "'b list"   and a2 :: "'c sequent" shows "((a1, a2) \<in> upRules) = (\<exists>c R Fs ps. a1 = ps \<and> a2 = c \<and> SRCTransforms.mset c = \<LM> Compound R Fs  \<RM> \<and> ps \<noteq> [])"
CAND_ImpL_elim: fixes x :: "name"   and a :: "coname"   and M :: "trm"   and z :: "name"   and N :: "trm"   and B :: "ty" assumes "x:ImpL <a>.M z.N x \<in> \<parallel>(B)\<parallel>"   and "x:ImpL <a>.M z.N x \<notin> BINDINGn B (\<parallel><B>\<parallel>)" shows "\<exists>B1 B2. B = B1 IMP B2 \<and> <a>:M \<in> \<parallel><B1>\<parallel> \<and> z:N \<in> \<parallel>(B2)\<parallel>"
exhaust: fixes y :: "'b tree" obtains x1 :: "nat" and x2 :: "'b" and x3 :: "'b tree list" where "y = Node x1 x2 x3"
rep_eq: shows "\<lbrakk>\<bottom>\<rbrakk>\<^sub>e = (\<lambda>A. \<bottom>)"
alpha_Tree_induct': fixes P :: "('e, 'f, 'g) Tree \<Rightarrow> ('e, 'f, 'g) Tree \<Rightarrow> bool"   and a0 :: "('e, 'f, 'g) Tree"   and a1 :: "('e, 'f, 'g) Tree" assumes "\<And>tset1 tset2. (\<And>a b. \<lbrakk>a \<in> set_bset tset1; b \<in> set_bset tset2\<rbrakk> \<Longrightarrow> P a b) \<Longrightarrow> P (tConj tset1) (tConj tset2)"   and "\<And>t1 t2. P t1 t2 \<Longrightarrow> P (tNot t1) (tNot t2)"   and "\<And>\<phi>1 \<phi>2. P (tPred \<phi>1) (tPred \<phi>2)"   and "\<And>\<alpha>1 t1 \<alpha>2 t2. \<lbrakk>\<And>x. P (x \<bullet> t1) t2; \<And>x a b. P ((a \<rightleftharpoons> b) \<bullet> t1) t1; \<And>x a b. P ((a \<rightleftharpoons> b) \<bullet> t2) t2\<rbrakk> \<Longrightarrow> P (tAct \<alpha>1 t1) (tAct \<alpha>2 t2)"   and "\<And>v va. P (tNot v) (tConj va)"   and "\<And>v va. P (tNot v) (tPred va)"   and "\<And>v va vb. P (tNot v) (tAct va vb)"   and "\<And>v va. P (tPred v) (tConj va)"   and "\<And>v va. P (tPred v) (tNot va)"   and "\<And>v va vb. P (tPred v) (tAct va vb)"   and "\<And>v va vb. P (tAct v va) (tConj vb)"   and "\<And>v va vb. P (tAct v va) (tNot vb)"   and "\<And>v va vb. P (tAct v va) (tPred vb)"   and "\<And>va v. P (tConj va) (tNot v)"   and "\<And>va v. P (tPred va) (tNot v)"   and "\<And>va vb v. P (tAct va vb) (tNot v)"   and "\<And>va v. P (tConj va) (tPred v)"   and "\<And>va v. P (tNot va) (tPred v)"   and "\<And>va vb v. P (tAct va vb) (tPred v)"   and "\<And>vb v va. P (tConj vb) (tAct v va)"   and "\<And>vb v va. P (tNot vb) (tAct v va)"   and "\<And>vb v va. P (tPred vb) (tAct v va)" shows "P a0 a1"
l_set_child_nodes_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_child_nodes_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf ShadowRootClass.known_ptr DocumentClass.type_wf DocumentClass.known_ptr Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs Core_DOM_Functions.set_child_nodes Core_DOM_Functions.set_child_nodes_locs get_shadow_root get_shadow_root_locs"
IH_2: shows "evalUni F2 x = evalUni (list_disj_Uni (map (list_conj_Uni \<circ> map AtomUni) (DNFUni F2))) x"
rev_image_eqI: fixes x :: "'b"   and A :: "'b set"   and b :: "'c"   and f :: "'b \<Rightarrow> 'c" assumes "x \<in> A"   and "b = f x" shows "b \<in> f ` A"
Y_def: fixes f :: "'c" shows "Y f \<equiv> Cop_S.Map (local.map f)"
exp_eq_zero_iff: fixes n :: "nat" shows "(2 ^ n = 0) = (LENGTH('b) \<le> n)"
gt_sym_total: fixes gt_sym :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a"   and g :: "'a" assumes "gt_sym gt_sym" shows "gt_sym f g \<or> gt_sym g f \<or> g = f"
build_index_loc_def: fixes m_ops :: "('a, 'b, 'c, 'd) map_ops_scheme"   and s_ops :: "('e, 'b, 'f) set_ops_scheme"   and t_ops :: "('e, 'g, 'h) set_ops_scheme" shows "build_index_loc m_ops s_ops t_ops \<equiv> index_loc m_ops s_ops \<and> StdSet t_ops"
r_inv_ex: fixes x :: "'a" assumes "x \<in> carrier G" shows "\<exists>y\<in>carrier G. x \<otimes> y = \<one>"
Union_mono: fixes A :: "'b set set"   and B :: "'b set set" assumes "order.greater_eq B A" shows "order.greater_eq (\<Union> B) (\<Union> A)"
generate_pow_card: fixes a :: "'a" assumes "a \<in> carrier G" shows "ord a = card (generate G {a})"
mon_c_mono: fixes c :: "'f list multiset"   and c' :: "'f list multiset"   and fg :: "('f, 'h, 'i, 'g, 'j) flowgraph_rec_scheme" assumes "c \<subseteq># c'" shows "mon_c fg c \<subseteq> mon_c fg c'"
simps: fixes a :: "'c"   and r :: "('c \<times> 'c) set" shows "(a \<in> Wellfounded.acc r) = (\<exists>x. a = x \<and> (\<forall>xa. (xa, x) \<in> r \<longrightarrow> xa \<in> Wellfounded.acc r))"
cod_simp: fixes f :: "'a" assumes "arr f" shows "cod f = C.cod f"
ChiPlus_HAStates_Right: fixes S :: "'a"   and T :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "(S, T) \<in> ChiPlus A" shows "T \<in> HAStates A"
finite_compl: fixes A :: "'a set" assumes "finite A" shows "finite (- A) = finite UNIV"
scale_eq_0_iff: fixes a :: "rat"   and x :: "'a" shows "(a *R x = (0::'a)) = (a = 0 \<or> x = (0::'a))"
ra_2: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "x ; (z \<cdot> y ; 1) = (x \<cdot> (y ; 1)\<^sup>\<smile>) ; z"
flatten_def: fixes p :: "'c \<Rightarrow>\<^sub>0 'c \<Rightarrow>\<^sub>0 'd" shows "flatten p = (\<Sum>t\<in>keys p. punit.monom_mult (1::'d) t (lookup p t))"
assms: fixes \<Psi>' :: "'b" shows "(\<Psi> \<otimes> \<Psi>', P, Q) \<in> Rel"
subst_subst_domain: fixes x :: "'c"   and t :: "('d, 'c) Term.term" shows "subst_domain (subst x t) = (if t = Var x then {} else {x})"
assms_1: shows "boruvka_inner_invariant j f h g d"
subset_fset: fixes xs :: "'b fset"   and ys :: "'b fset" shows "(xs |\<subseteq>| ys) = (fset xs \<subseteq> fset ys)"
apartment_chamber_distance: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and C :: "'b set"   and D :: "'b set" assumes "ChamberComplexWithApartmentSystem X \<A>"   and "A \<in> \<A>"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D"   and "C \<in> A"   and "D \<in> A" shows "ChamberComplex.chamber_distance A C D = ChamberComplex.chamber_distance X C D"
exists_least_iff: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and P :: "'e \<Rightarrow> bool" assumes "class.wellorder less_eq less" shows "(\<exists>n. P n) = (\<exists>n. P n \<and> (\<forall>m. less m n \<longrightarrow> \<not> P m))"
combine_singleton: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and j :: "nat"   and i :: "nat"   and a :: "'b"   and b :: "'c" assumes "f (0::'b) (0::'c) = (0::'a)" shows "combine_matrix f (singleton_matrix j i a) (singleton_matrix j i b) = singleton_matrix j i (f a b)"
map_poly_map_poly: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and p :: "'d poly" assumes "f (0::'c) = (0::'b)"   and "g (0::'d) = (0::'c)" shows "map_poly f (map_poly g p) = map_poly (f \<circ> g) p"
real_divr_nonpos_pos_upper_bound: fixes x :: "real"   and y :: "real"   and prec :: "nat" assumes "x \<le> 0"   and "0 \<le> y" shows "real_divr prec x y \<le> 0"
lrev_wrap_def: shows "lrev_wrap \<equiv> wrapH\<cdot>lrev_work"
coboundedI1: fixes c :: "'b"   and a :: "'b"   and b :: "'b" assumes "c \<le> a" shows "c \<le> a \<squnion> b"
orbit_stabilizer_theorem: fixes x :: "'c" assumes "x \<in> E" shows "card (orbit G \<phi> x) * card (stabilizer G \<phi> x) = order G"
Inr: shows "min_rvar_incdec dir s x\<^sub>i = Inr x\<^sub>j"
AE_2_1: shows "AE x in M. \<bar>subcocycle_lim_ereal u x\<bar> \<noteq> \<infinity>"
\<beta>_inner: shows "inner_prod \<beta> \<beta> = 1"
Methd_sound: fixes G :: "prog"   and A :: "'b triple set"   and P :: "qtname \<Rightarrow> sig \<Rightarrow> 'b assn"   and Q :: "qtname \<Rightarrow> sig \<Rightarrow> 'b assn"   and ms :: "(qtname \<times> sig) set" assumes "G,A \<union> {{P} Methd-\<succ> {Q} | ms}|\<Turnstile>\<Colon>{{P} body G-\<succ> {Q} | ms}" shows "G,A|\<Turnstile>\<Colon>{{P} Methd-\<succ> {Q} | ms}"
D_preservation: fixes c :: "com"   and s :: "char list \<Rightarrow> int option"   and c' :: "com"   and s' :: "char list \<Rightarrow> int option"   and A :: "char list set" assumes "(c, s) \<rightarrow> (c', s')"   and "D (dom s) c A" shows "\<exists>A'. D (dom s') c' A' \<and> A \<subseteq> A'"
f_approx_bounded_below: obtains c :: "real" where "\<And>x. \<lbrakk>x\<^sub>0 \<le> x; x \<le> x\<^sub>1\<rbrakk> \<Longrightarrow> c \<le> f_approx x" and "0 < c"
perm_struct_induct: fixes p :: "perm"   and S :: "atom set"   and P :: "perm \<Rightarrow> bool" assumes "supp p \<subseteq> S"   and "P 0"   and "\<And>p a b. \<lbrakk>P p; supp p \<subseteq> S; a \<in> S; b \<in> S; a \<noteq> b; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> P ((a \<rightleftharpoons> b) + p)" shows "P p"
lbl_lbl: fixes xa :: "'b word"   and x :: "'b word" shows "length (to_bl xa) = length (to_bl x)"
intro: fixes ops :: "'a approximate_set_ops"   and set_of_appr :: "'a list \<Rightarrow> real list set"   and appr_rell :: "('a list \<times> real list set) set"   and optns :: "'a numeric_options" assumes "approximate_sets ops set_of_appr appr_rell optns" shows "approximate_sets_ode set_of_appr appr_rell optns ops"