hcju/isabelleps huggingface dataset:
curry_preserves_functors: fixes F :: "'a1 \<times> 'a2 \<Rightarrow> 'b" assumes "functor (\<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) F" shows "functor (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>[\<^sub>A\<^sub>2,\<^sub>B\<^sub>]) (local.curry F F F)"
dup_is_natural_transformation: shows "natural_transformation (\<cdot>) (\<cdot>) local.map \<Pi>o\<Delta>.map dup"
inf_def: shows "\<infinity>\<^sub>h \<equiv> abs_complex_homo \<infinity>\<^sub>h\<^sub>c"
dotsubstt_def: fixes \<theta> :: "trm" shows "dotsubstt \<theta> = (\<lambda>f. if f = dotid then Aterm \<theta> else undeft, Map.empty, Map.empty, Map.empty)"
pderiv_monom: fixes c :: "'a"   and n :: "nat" shows "pderiv (Polynomial.monom c n) = Polynomial.monom (of_nat n * c) (n - 1)"
permutation_insert_inj_on: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < Suc n"   and "j < Suc n" shows "inj_on (permutation_insert i j) {q. q permutes {0..<n}}"
finite_acyclic_wf: fixes r :: "('a \<times> 'a) set" assumes "finite r"   and "acyclic r" shows "wf r"
cring_simprules_15: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "cring R" shows "x \<ominus>\<^bsub>R\<^esub> y = x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> y"
assms_4: shows "(red F)\<^sup>*\<^sup>* p q"
rec1: fixes ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> nat \<times> 'd" assumes "ps \<noteq> []"   and "sps = sel gs bs ps (snd data)"   and "fst data \<le> count_const_lt_components (fst (compl gs bs (ps -- sps) sps (snd data)))" shows "P data bs ps (full_gb (gs @ bs))"
is_swap_def: fixes f :: "nat \<Rightarrow> nat" assumes "deutsch f" shows "deutsch.is_swap f = (\<forall>x\<in>{0, 1}. f x = 1 - x)"
integral_null: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "content (cbox a b) = 0" shows "integral (cbox a b) f = (0::'b)"
mset_subset_size: fixes A :: "'b multiset"   and B :: "'b multiset" assumes "A \<subset># B" shows "size A < size B"
stake_bottom_iff: fixes i :: "Integer"   and xs :: "[:'a:]" shows "(stake\<cdot>i\<cdot>xs = \<bottom>) = (i = \<bottom> \<or> xs = \<bottom>)"
parts_image_priEK: fixes i :: "nat"   and KK :: "nat set"   and evs :: "event list" assumes "Key (priEK (CA i)) \<in> parts (Key ` KK \<union> knows Spy evs)"   and "evs \<in> set_mr" shows "priEK (CA i) \<in> KK \<or> CA i \<in> bad"
HBL1: fixes var :: "'b set"   and trm :: "'c set"   and fmla :: "'d set"   and Var :: "'b \<Rightarrow> 'c"   and FvarsT :: "'c \<Rightarrow> 'b set"   and substT :: "'c \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'c"   and Fvars :: "'d \<Rightarrow> 'b set"   and subst :: "'d \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'd"   and num :: "'c set"   and eql :: "'c \<Rightarrow> 'c \<Rightarrow> 'd"   and cnj :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and imp :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and all :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and exi :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and prv :: "'d \<Rightarrow> bool"   and enc :: "'d \<Rightarrow> 'c"   and P :: "'d"   and \<phi> :: "'d" assumes "HBL1 var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv enc P"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "prv \<phi>" shows "prv (subst P (enc \<phi>) (Generic_Syntax.Variable var 10))"
distinct_2: fixes b' :: "'d"   and a :: "'c" shows "Inr b' \<noteq> Inl a"
add_mset_commute: fixes x :: "'b"   and y :: "'b"   and M :: "'b multiset" shows "add_mset x (add_mset y M) = add_mset y (add_mset x M)"
top_higher: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c" assumes "weak_partial_order_top L"   and "x \<in> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> \<top>\<^bsub>L\<^esub>"
fac_acc_work_final_fac_acc_work3_eq: shows "fac_acc_body_final = fac_acc_body3"
foldl_conv_fold: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and s :: "'c"   and xs :: "'d list" shows "foldl f s xs = fold (\<lambda>x s. f s x) xs s"
coprime_0_left_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one zero a = dvd.dvd times a one"
continuous_at_within_inverse: fixes a :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous (at a within s) f"   and "f a \<noteq> (0::'b)" shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
ComplD: fixes c :: "'d"   and A :: "'d set" assumes "c \<in> - A" shows "c \<notin> A"
set_sel: fixes a :: "'a option" assumes "a \<noteq> None" shows "the a \<in> set_option a"
eventually_always_imp_always_eventually: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "(\<diamond>\<box>P) \<sigma>" shows "(\<box>\<diamond>P) \<sigma>"
suf_ext: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "xs \<le>s ys" shows "xs \<le>s zs \<cdot> ys"
lens_indep_def: fixes X :: "'c \<Longrightarrow> 'd"   and Y :: "'e \<Longrightarrow> 'd" 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>)"
not_know_s_not_ine: fixes A :: "specID"   and m :: "Secrets" assumes "\<not> know A (sKS m)" shows "\<not> ine A (sE m)"
composition_closed: fixes a :: "('a set \<times> 'b) set"   and b :: "('a set \<times> 'b) set" assumes "a \<in> stk2.carrier_stalk"   and "b \<in> stk2.carrier_stalk" shows "stk2.add_stalk a b \<in> stk2.carrier_stalk"
rev_map: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "rev (map f xs) = map f (rev xs)"
assms_2: shows "fold (\<lambda>(T, U) subs. Option.bind subs (raw_match' T U)) (zip (xs @ [x]) (ys @ [y])) (Some subs) = Some subs'"
intros: fixes \<S> :: "s_context"   and \<C>s :: "t_context list"   and tfs :: "tf list"   and ns :: "nat list"   and ms :: "nat list"   and tgs :: "tg list"   and insts :: "inst list"   and fs :: "cl list"   and tclss :: "cl option list list"   and bss :: "mem list"   and gs :: "global list" assumes "\<S> = \<lparr>s_inst = \<C>s, s_funcs = tfs, s_tab = ns, s_mem = ms, s_globs = tgs\<rparr>"   and "list_all2 (inst_typing \<S>) insts \<C>s"   and "list_all2 (cl_typing \<S>) fs tfs"   and "list_all (tab_agree \<S>) (concat tclss)"   and "list_all2 (\<lambda>tcls n. n \<le> length tcls) tclss ns"   and "list_all2 mem_agree bss ms"   and "list_all2 glob_agree gs tgs" shows "store_typing \<lparr>inst = insts, funcs = fs, tab = tclss, mem = bss, globs = gs\<rparr> \<S>"
reduce_range_destination_def: fixes l :: "('a \<times> 'b wordinterval) list" shows "reduce_range_destination l \<equiv> let ps = remdups (map fst l); c = \<lambda>s. (wordinterval_Union \<circ> map snd \<circ> filter ((=) s \<circ> fst)) l in map (\<lambda>p. (p, c p)) ps"
linorder_neqE_linordered_idom: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and x :: "'e"   and y :: "'e" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "x \<noteq> y" obtains "less x y"   | "less y x"
of_bits_int_simps_4: fixes bs :: "bool list"   and b :: "bool" shows "of_bits (bs @ [b]) = horner_sum of_bool 2 bs - 2 ^ length bs * of_bool b"
degree0_coeffs: fixes p :: "'b poly" assumes "degree p = 0" shows "\<exists>a. p = [:a:]"
grun_subset: fixes A :: "('a, 'b) nba"   and w :: "'a stream"   and V :: "(nat \<times> 'b) set"   and r :: "(nat \<times> 'b) stream"   and v :: "nat \<times> 'b"   and U :: "(nat \<times> 'b) set" assumes "grun A w V r v"   and "sset (gtrace r v) \<subseteq> U" shows "grun A w U r v"
subspace_span: fixes S :: "'c set" shows "vs2.subspace (vs2.span S)"
simps: fixes a1 :: "char list"   and a2 :: "char list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "P \<turnstile> a1 \<prec>\<^sup>1 a2 = (\<exists>C Bs rest D. a1 = C \<and> a2 = D \<and> class P C = \<lfloor>(Bs, rest)\<rfloor> \<and> D \<in> baseClasses Bs)"
stake_shift: fixes i :: "nat"   and w :: "'b list"   and s :: "'b stream" shows "stake i (w @- s) = take i w @ stake (i - length w) s"
returns_heap_eq: fixes h :: "'d"   and f :: "('d, 'e, 'f) prog"   and h' :: "'d"   and h'' :: "'d" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "h \<turnstile> f \<rightarrow>\<^sub>h h''" shows "h' = h''"
mult_row_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and a :: "'f"   and q :: "'d" shows "mult_row A a q = (\<chi>i j. if i = a then q * A $ a $ j else A $ i $ j)"
sup_idem: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.semilattice_sup sup less_eq less" shows "sup x x = x"
vimage2p_def: fixes f :: "'c \<Rightarrow> 'f"   and g :: "'d \<Rightarrow> 'g"   and R :: "'f \<Rightarrow> 'g \<Rightarrow> 'e" shows "BNF_Def.vimage2p f g R = (\<lambda>x y. R (f x) (g y))"
ce_OB_eq_ce_t: shows "\<I>( fst ; OB ) = \<I>( fst ; t \<circ> OB )"
lemma1: fixes r :: "nat \<Rightarrow> nat \<Rightarrow> b"   and p :: "nat \<times> nat list \<times> nat"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "is_path r p i j k" shows "is_path r p (Suc i) j k"
NextPropagateD_3: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b" assumes "NextPropagate' c0 c1 p" shows "\<forall>p'. Some (prop_config c1 p') = (if p' = p then propagate_all (prop_config c0 p') else Some (prop_config c0 p'))"
vlrestriction_atD: fixes Aa :: "V"   and A :: "V"   and a :: "V"   and b :: "V" assumes "((r \<restriction>\<^sup>r\<^sub>\<circ> Aa) \<restriction>\<^sup>l\<^sub>\<circ> A)\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>r\<^sub>\<circ> Aa)"   and "a \<in>\<^sub>\<circ> A" shows "(r \<restriction>\<^sup>r\<^sub>\<circ> Aa)\<lparr>a\<rparr> = b"
Retr_defs_2: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set" shows "ZOretr theta \<equiv> {(c, d). matchC_LC theta c d}"
M_2: fixes S' :: "'a set" assumes "S' \<in> S"   and "M \<subseteq> S'" shows "S' = M"
real_sqrt_ge_zero: fixes x :: "real" assumes "0 \<le> x" shows "0 \<le> sqrt x"
ffmember_filter: fixes x :: "'m"   and P :: "'m \<Rightarrow> bool"   and A :: "'m fset" shows "(x |\<in>| ffilter P A) = (x |\<in>| A \<and> P x)"
BinOpRedsVal: fixes e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp"   and v\<^sub>2 :: "val"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>2 :: "bool"   and bop :: "bop"   and v :: "val" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v\<^sub>1,s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1,b\<^sub>1\<rangle> \<rightarrow>* \<langle>Val v\<^sub>2,s\<^sub>2,b\<^sub>2\<rangle>"   and "binop (bop, v\<^sub>1, v\<^sub>2) = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,s\<^sub>2,b\<^sub>2\<rangle>"
add_left_mono: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subseteq># b" shows "c + a \<subseteq># c + b"
const: fixes name :: "name"   and rhs :: "value"   and rs :: "(name \<times> value) list"   and C_info :: "(name, dt_def) fmap"   and \<Gamma> :: "(name, value) fmap" assumes "(name, rhs) \<in> set rs" shows "special_constants.veval C_info rs \<Gamma> (Sconst name) rhs"
CFG_axioms_def: fixes \<alpha>n :: "'d \<Rightarrow> 'e list"   and invar :: "'d \<Rightarrow> bool"   and defs :: "'d \<Rightarrow> 'e \<Rightarrow> 'f set"   and "uses" :: "'d \<Rightarrow> 'e \<Rightarrow> 'f set" shows "CFG_axioms \<alpha>n invar defs uses \<equiv> ((\<forall>n g. n \<in> set (\<alpha>n g) \<longrightarrow> defs g n \<inter> uses g n = {}) \<and> (\<forall>g n. finite (defs g n))) \<and> (\<forall>v g n. v \<in> uses g n \<longrightarrow> n \<in> set (\<alpha>n g)) \<and> (\<forall>g n. finite (uses g n)) \<and> (\<forall>g. invar g)"
hyps: fixes x :: "real" shows "bernpoly n (x + 1) - bernpoly n x = real n * x ^ (n - 1)"
simps_1: fixes rv :: "nat option" shows "step ([], rv) = ([], rv)"
cases: fixes a :: "'a nmultiset" assumes "no_elem a" obtains M :: "'a nmultiset multiset" where "a = MSet M" and "\<And>X. X \<in># M \<Longrightarrow> no_elem X"
Smith_normal_form_upt_k_condition2: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat"   and a :: "'c"   and b :: "'b" assumes "Smith_normal_form_upt_k A k"   and "to_nat a \<noteq> to_nat b"   and "to_nat a < k \<or> to_nat b < k" shows "A $ a $ b = (0::'a)"
comp: fixes r :: "Vertex \<Rightarrow> Vertex"   and s :: "Vertex \<Rightarrow> Vertex" assumes "r \<in> simple_rotations"   and "s \<in> complex_rotations" shows "r \<circ> s \<in> complex_rotations"
adverse_in_V01: shows "\<exists>m n. local.adverse 0 m n \<in> V\<^sub>0\<^sub>1 \<and> local.adverse 1 m n \<in> V\<^sub>0\<^sub>1"
bind_def: fixes xs :: "'d list"   and f :: "'d \<Rightarrow> 'c list" shows "List.bind xs f = concat (map f xs)"
Confidentiality_B_temporal: fixes B :: "agent"   and Tk :: "nat"   and A :: "agent"   and K :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace> \<in> parts (knows Spy evs)"   and "\<not> expiredK Tk evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerb_gets" shows "Key K \<notin> analz (knows Spy evs)"
drop_Suc: fixes n :: "nat"   and xs :: "'k list" shows "drop (Suc n) xs = drop n (tl xs)"
comm_monoid_add_ow: shows "comm_monoid_add_ow UNIV (+) (0::'b)"
set_of_insrt: fixes x :: "'a"   and t :: "'a tree\<^sub>0" shows "set_of (insrt\<^sub>0 x t) = insert x (set_of t)"
i_Exec_Comp_Stream_Acc_Output__eq_iAll_iSince_conv: fixes k :: "nat"   and s :: "nat \<Rightarrow> 'a message_af"   and output_fun :: "'b \<Rightarrow> 'a message_af"   and trans_fun :: "'c message_af \<Rightarrow> 'b \<Rightarrow> 'b"   and input :: "nat \<Rightarrow> 'c message_af"   and c :: "'b"   and t :: "nat"   and m :: "'a message_af" assumes "0 < k"   and "s = output_fun \<circ> i_Exec_Comp_Stream trans_fun (input \<odot> k) c" shows "(i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = m) = ((m = NoMsg \<longrightarrow> (\<box> t1 [t * k\<dots>,k - Suc 0]. s t1 = NoMsg)) \<and> (m \<noteq> NoMsg \<longrightarrow> (s t2 = NoMsg. t2 \<S> t1 [t * k\<dots>,k - Suc 0]. s t1 = m)))"
I_lift_dnf_qe_anormal: fixes qe :: "atom list \<Rightarrow> atom fm"   and f :: "atom fm"   and xs :: "int list" assumes "\<And>as. \<forall>a\<in>set as. hd_coeff a \<noteq> 0 \<Longrightarrow> qfree (qe as)"   and "\<And>as. \<forall>a\<in>set as. hd_coeff a \<noteq> 0 \<and> divisor a \<noteq> 0 \<Longrightarrow> Z.normal (qe as)"   and "\<And>xs as. \<forall>a\<in>set as. hd_coeff a \<noteq> 0 \<and> divisor a \<noteq> 0 \<Longrightarrow> Z.is_dnf_qe qe as"   and "Z.normal f" shows "Z.I (Z.lift_dnf_qe qe f) xs = Z.I f xs"
assms_5: fixes A :: "'a trm"   and B :: "'a trm" assumes "P A"   and "P B" shows "P (SimplyTyped.Pair A B)"
handle2b: fixes env :: "v sem_env"   and s1 :: "'c state"   and e :: "exp0"   and s2 :: "'c state"   and v1 :: "v"   and pes :: "(pat \<times> exp0) list"   and err :: "v error_result" assumes "Big_Step_Unclocked.evaluate env s1 e (s2, Rerr (Rraise v1))"   and "match_result env s2 v1 pes v1 = Rerr err" shows "Big_Step_Unclocked.evaluate env s1 (Handle e pes) (s2, Rerr err)"
preservation: shows "rel_weakly_preserves_barbs (indRelTEQ TRel) (STCalWB SWB TWB)"
assms: fixes a :: "'a" assumes "a \<in> ds \<pi> \<sigma>" shows "a \<notin> ptrm_fvs M"
drop_append: fixes n :: "nat"   and xs :: "'b list"   and ys :: "'b list" shows "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
uniqDistinct: fixes l :: "'a list" shows "uniq l = distinct l"
finiteInsert: fixes a :: "'b"   and A :: "'b set" shows "finite (insert a A) = finite A"
def: shows "usubsta \<sigma> \<phi> \<noteq> undeff"
simps_5: fixes v :: "'a rexp"   and va :: "'a rexp" shows "local.le_rexp (Times v va) Zero = False"
faithful_functor_axioms: shows "faithful_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G"
not_neutral_contains_not_neutral: fixes g :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "prod g A \<noteq> (1::'b)" obtains a :: "'c" where "a \<in> A" and "g a \<noteq> (1::'b)"
qfree_lift_dnfeq_qe: fixes qe :: "atom list \<Rightarrow> atom fm"   and \<phi> :: "atom fm" assumes "\<And>as. \<forall>a\<in>set as. depends\<^sub>R a \<Longrightarrow> qfree (qe as)" shows "qfree (R\<^sub>e.lift_dnfeq_qe qe \<phi>)"
case_eq_if: fixes f1 :: "'a"   and f2 :: "enat \<Rightarrow> 'a"   and enat :: "enat" shows "(case enat of 0 \<Rightarrow> f1 | eSuc x \<Rightarrow> f2 x) = (if enat = 0 then f1 else f2 (epred enat))"
iii: shows "properdiv_set 75 = {1, 3, 5, 15, 25}"
26564: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - (- x \<sqinter> y) = top"
fold_invariant: fixes xs :: "'e list"   and Q :: "'e \<Rightarrow> bool"   and P :: "'f \<Rightarrow> bool"   and s :: "'f"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" assumes "\<And>x. x \<in> set xs \<Longrightarrow> Q x"   and "P s"   and "\<And>x s. \<lbrakk>Q x; P s\<rbrakk> \<Longrightarrow> P (f x s)" shows "P (fold f xs s)"
not_disjoint_implies_equal: fixes a :: "'a"   and b :: "'a" assumes "Class a \<inter> Class b \<noteq> {}"   and "a \<in> S"   and "b \<in> S" shows "Class a = Class b"
measure_pmf_in_subprob_algebra: fixes x :: "'c pmf" shows "measure_pmf x \<in> space (subprob_algebra (count_space UNIV))"
mk_v_agree: fixes I :: "('e, 'f, 'd) interp"   and ODE :: "('e, 'd) ODE"   and \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and sol :: "(real, 'd) vec" shows "Vagree (mk_v I ODE \<nu> sol) \<nu> (- semBV I ODE) \<and> Vagree (mk_v I ODE \<nu> sol) (mk_xode I ODE sol) (semBV I ODE)"
wprepare_invs_2: fixes m :: "nat"   and lm :: "nat list"   and l :: "cell list"   and r :: "cell list" shows "wprepare_goto_first_end m lm (l, r) = (\<exists>ml mr rn. l = Oc \<up> ml \<and> r = Oc \<up> mr @ Bk # <lm> @ Bk \<up> rn \<and> ml + mr = Suc (Suc m))"
subset_imp_less_mset: fixes A :: "'a multiset"   and B :: "'a multiset" assumes "A \<subset># B" shows "A < B"
elims: fixes x :: "'a trm"   and y :: "Preliminaries.var set" assumes "nvT x = y" obtains x :: "Preliminaries.var" where "xa__ = Var x" and "y = {x}"   | f :: "'a" and Tl :: "'a trm list" where "xa__ = Fn f Tl" and "y = {}"
o_apply: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "(f \<circ> g) x = f (g x)"
DERIV_pos_imp_increasing_at_bot: fixes b :: "real"   and f :: "real \<Rightarrow> real"   and flim :: "real" assumes "\<And>x. x \<le> b \<Longrightarrow> \<exists>y. (f has_real_derivative y) (at x) \<and> 0 < y"   and "(f \<longlongrightarrow> flim) at_bot" shows "flim < f b"
Inf_dual_def: shows "Inf = \<partial> \<circ> Sup \<circ> (`) \<partial>\<^sup>-"
a_host_shadow_root_rel_code: fixes h :: "(_) heap" shows "local.a_host_shadow_root_rel h = set (concat (map (\<lambda>host. case |h \<turnstile> get_shadow_root host|\<^sub>r of None \<Rightarrow> [] | Some shadow_root \<Rightarrow> [(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)]) (sorted_list_of_fset (element_ptr_kinds h))))"
simps_5: fixes \<phi> :: "atomUni fmUni"   and \<psi> :: "atomUni fmUni" shows "negUni (OrUni \<phi> \<psi>) = AndUni (negUni \<phi>) (negUni \<psi>)"
resCasesF: fixes x :: "name"   and P :: "pi"   and \<alpha> :: "freeRes"   and RP' :: "pi"   and F :: "pi \<Rightarrow> bool" assumes "<\<nu>x>P \<longmapsto> \<alpha> \<prec> RP'"   and "x \<sharp> \<alpha>"   and "\<And>P'. P \<longmapsto> \<alpha> \<prec> P' \<Longrightarrow> F (<\<nu>x>P')" shows "F RP'"
cases_scheme: fixes r :: "'a State_scheme" obtains getF :: "Literal list list" and getM :: "(Literal \<times> bool) list" and getConflictFlag :: "bool" and getC :: "Literal list" and more :: "'a" where "r = \<lparr>getF = getF, getM = getM, getConflictFlag = getConflictFlag, getC = getC, \<dots> = more\<rparr>"
neqE: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.linorder less_eq less"   and "x \<noteq> y" obtains "less x y"   | "less y x"
zhmsetmset_less: fixes x :: "zhmultiset"   and xa :: "zhmultiset" shows "(zhmsetmset x < zhmsetmset xa) = (x < xa)"
is_greatest_fixpoint_def: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "is_greatest_fixpoint f x \<equiv> f x = x \<and> (\<forall>y. f y = y \<longrightarrow> y \<le> x)"
that: fixes k :: "nat" assumes "sdrop k w = sconst (w !! k)" shows "thesis"
inject: fixes x1 :: "'a \<Rightarrow> ('a, 'b) tree option"   and x2 :: "'b \<Rightarrow> ('a, 'b) tree option"   and y1 :: "'a \<Rightarrow> ('a, 'b) tree option"   and y2 :: "'b \<Rightarrow> ('a, 'b) tree option" shows "(Branch x1 x2 = Branch y1 y2) = (x1 = y1 \<and> x2 = y2)"
Q_Succ_cong: fixes H :: "fm set"   and x :: "tm"   and x' :: "tm" assumes "H \<turnstile> x EQ x'" shows "H \<turnstile> Q_Succ x EQ Q_Succ x'"
that: assumes "P is CSP1 healthy"   and "P is CSP2 healthy"   and "P is R healthy" shows "thesis"
bound_aux_C: fixes C :: "nat set"   and f :: "nat \<Rightarrow> real"   and x :: "real"   and p :: "nat"   and q :: "nat"   and r :: "nat" assumes "\<forall>l\<in>C. \<forall>m\<in>C. \<bar>f l - f m\<bar> \<le> x"   and "p \<in> C"   and "q \<in> C"   and "r \<in> C" shows "\<bar>fiX f p r - f q\<bar> \<le> x"
R: shows "graph_defs.is_run (degeneralize_ext m) r"
quot_set_def: fixes A :: "'a set"   and x :: "'a" shows "A \<div> x = (*) x -` A"
m_assoc: fixes x :: "rat vec"   and y :: "rat vec"   and z :: "rat vec" assumes "x \<in> Rn"   and "y \<in> Rn"   and "z \<in> Rn" shows "x + y + z = x + (y + z)"
diamond_sem: fixes I :: "('b, 'c, 'a) interp"   and \<alpha> :: "('b, 'c, 'a) hp"   and \<phi> :: "('b, 'c, 'a) formula" shows "fml_sem I (\<langle> \<alpha> \<rangle> \<phi>) = {\<nu>. \<exists>\<omega>. (\<nu>, \<omega>) \<in> prog_sem I \<alpha> \<and> \<omega> \<in> fml_sem I \<phi>}"
Icc_Suc2: fixes a :: "nat"   and b :: "nat" assumes "a \<le> b" shows "{a..Suc b} = insert (Suc b) {a..b}"