hcju/isabelleps huggingface dataset:
prems_2: fixes k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "(k, \<gamma>) \<in> set (a # l)" shows "valid_path \<gamma>"
exists_Union_lemma: fixes z :: "name"   and i :: "name"   and y :: "tm" assumes "atom z \<sharp> (i, y)"   and "atom i \<sharp> y" shows "{} \<turnstile> SyntaxN.Ex z (UnionP (Var i) y (Var z))"
deep_def: fixes pr :: "('e, 'f) Digit"   and m :: "('e, 'f) FingerTreeStruc"   and sf :: "('e, 'f) Digit" shows "deep pr m sf = Deep (gmd pr + gmft m + gmd sf) pr m sf"
sorted_wrt_append: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'g list" shows "sorted_wrt P (xs @ ys) = (sorted_wrt P xs \<and> sorted_wrt P ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. P x y))"
diagonal_admits_SNF_iff_bezout_ring: shows "class.bezout_ring (*) (1::'b) (+) (0::'b) (-) uminus = (\<forall>A. admits_SNF_JNF A)"
assms_3: shows "\<guillemotleft>\<omega> : w \<star> w \<Rightarrow> w\<guillemotright>"
sincreasing_shift: fixes s :: "'b stream"   and xs :: "'b list" assumes "sincreasing s" shows "sincreasing (xs @- s)"
F_def: shows "F \<equiv> {C \<in> components (U - S). C \<inter> K \<noteq> {}}"
hfref_cons: fixes f :: "'e \<Rightarrow> 'f Heap"   and g :: "'g \<Rightarrow> 'h nres"   and P :: "'g \<Rightarrow> bool"   and R :: "('g \<Rightarrow> 'e \<Rightarrow> assn) \<times> ('g \<Rightarrow> 'e \<Rightarrow> assn)"   and S :: "'h \<Rightarrow> 'f \<Rightarrow> assn"   and P' :: "'g \<Rightarrow> bool"   and R' :: "('g \<Rightarrow> 'e \<Rightarrow> assn) \<times> ('g \<Rightarrow> 'e \<Rightarrow> assn)"   and S' :: "'h \<Rightarrow> 'f \<Rightarrow> assn" assumes "(f, g) \<in> [P]\<^sub>a R \<rightarrow> S"   and "\<And>x. P' x \<Longrightarrow> P x"   and "\<And>x y. fst R' x y \<Longrightarrow>\<^sub>t fst R x y"   and "\<And>x y. snd R x y \<Longrightarrow>\<^sub>t snd R' x y"   and "\<And>x y. S x y \<Longrightarrow>\<^sub>t S' x y" shows "(f, g) \<in> [P']\<^sub>a R' \<rightarrow> S'"
unique_connecting_path_properties_4: fixes v :: "nat"   and w :: "nat" assumes "v \<in> V"   and "w \<in> V" shows "last (v \<leadsto> w) = w"
choiceFn_def: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "choiceFn A r \<equiv> \<forall>A'\<subseteq>A. A' \<noteq> {} \<longrightarrow> choiceSet A' r \<noteq> {}"
lcm_neg2: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ring_gcd one zero times gcd lcm divide plus minus unit_factor normalize uminus" shows "lcm a (uminus b) = lcm a b"
fls_inverse_of_int: fixes n :: "int" shows "inverse (of_int n) = fls_const (inverse (of_int n))"
seq_compact_imp_Heine_Borel: fixes S :: "'a set" assumes "seq_compact S" shows "compact S"
error_def: fixes e :: "char list" shows "Parser_Monad.error e = (\<lambda>_. Inl e)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs" shows "l_new_document_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs"
mcont_case_resumption: fixes lub :: "'e set \<Rightarrow> 'e"   and ord :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d option \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> ('g \<Rightarrow> ('d, 'f, 'g) Resumption.resumption) \<Rightarrow> ('d, 'f, 'g) Resumption.resumption \<Rightarrow> 'e" assumes "mcont (flat_lub None) option_ord lub ord f"   and "\<And>out. mcont resumption.lub_fun resumption.le_fun lub ord (\<lambda>c. g out c (Resumption.resumption.Pause out c))"   and "class.ccpo lub ord (mk_less ord)"   and "\<And>x. ord (f None) x" shows "mcont resumption_lub resumption_ord lub ord (\<lambda>r. case r of Resumption.resumption.Done x \<Rightarrow> f x | Resumption.resumption.Pause out c \<Rightarrow> g out c r)"
sum_self_financing: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and pf2 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('d, 'c) discrete_market" assumes "portfolio pf1"   and "portfolio pf2"   and "self_financing Mkt pf1"   and "self_financing Mkt pf2" shows "self_financing Mkt (qty_sum pf1 pf2)"
list_set_rel_range: fixes R :: "('c \<times> 'b) set" shows "Range (\<langle>R\<rangle>list_set_rel) = {S. finite S \<and> S \<subseteq> Range R}"
Retr_defs_1: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('c, 'd, 'a) cmd \<times> ('c, 'd, 'a) cmd) set" assumes "PL_Indis cval indis" shows "PL_Indis.Sretr aval tval cval indis theta \<equiv> {(c, d). PL_Indis.matchC_C aval tval cval indis theta c d}"
cltn2_transpose_id: shows "cltn2_transpose cltn2_id = cltn2_id"
subset_imp_LeadsETo: fixes A :: "'d set"   and B :: "'d set"   and F :: "'d program"   and CC :: "'d set set" assumes "A \<subseteq> B" shows "F \<in> A LeadsTo[CC] B"
eq_scons: fixes w :: "'d stream"   and a :: "'d"   and v :: "'d stream" shows "(w = a ## v) = (a = shd w \<and> v = stl w)"
mod_ring_mult32: fixes pp :: "uint32" assumes "p = int_of_uint32 pp"   and "p \<le> 65535" shows "(mod_ring_rel32 ===> mod_ring_rel32 ===> mod_ring_rel32) (mult_p32 pp) (*)"
num_params_tm_norm'_le: fixes t :: "taylor_model"   and X :: "nat"   and prec :: "nat"   and I :: "float interval list"   and a :: "float interval list" assumes "num_params (tm_poly t) \<le> X" shows "num_params (tm_poly (tm_round_floats_of_normed prec I a t)) \<le> X"
alice_step_2_state: fixes \<phi> :: "complex Matrix.mat" assumes "state 1 \<phi>" shows "state 3 ((CNOT \<Otimes> Quantum.Id 1) * (\<phi> \<Otimes> |\<beta>\<^sub>0\<^sub>0\<rangle>))"
IH: fixes S :: "'a set" assumes "S \<subseteq> T"   and "n = card S" shows "set (f S n) = S"
rb_aux_is_min_sig_GB: fixes z :: "nat" assumes "rword_strict = rw_rat_strict" shows "is_min_sig_GB dgrad (set (fst (fst (rb_aux (([], Koszul_syz_sigs fs, map Inr [0..<length fs]), z)))))"
simps_2: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat" shows "change_tv f (Indet n) = Indet (f n)"
cring_fieldI: assumes "Units (law_restrict R) = carrier (law_restrict R) - {\<zero>\<^bsub>law_restrict R\<^esub>}" shows "field (law_restrict R)"
prems_6: shows "edges_of_walk xs \<subseteq> \<Union> (edges_of_walk ` paths_with_new)"
expectation_gpv_mono: fixes fail :: "ennreal"   and fail' :: "ennreal"   and f :: "'h \<Rightarrow> ennreal"   and g :: "'h \<Rightarrow> ennreal"   and \<I> :: "('i, 'j) \<I>"   and gpv :: "('h, 'i, 'j) gpv" assumes "fail \<le> fail'"   and "f \<le> g" shows "expectation_gpv fail \<I> f gpv \<le> expectation_gpv fail' \<I> g gpv"
prime_elem_dvd_mult_iff: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and p :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.prime_elem times one zero p" shows "dvd.dvd times p (times a b) = (dvd.dvd times p a \<or> dvd.dvd times p b)"
prems_3: shows "length \<alpha>s = length (m1 # ms)"
ssort'_termination: fixes p :: "'b \<times> 'a list" assumes "inv (fst p)" shows "ssort'_dom p"
rt_strictly_fresher_trans: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt3 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "rt1 \<sqsubset>\<^bsub>dip\<^esub> rt2"   and "rt2 \<sqsubset>\<^bsub>dip\<^esub> rt3" shows "rt1 \<sqsubset>\<^bsub>dip\<^esub> rt3"
p_def: fixes x :: "nat"   and y :: "nat"   and u :: "nat" shows "p x y u = q (c_pair x y) u"
snoc_Cons: fixes xs :: "'a list"   and x :: "'a"   and y :: "'b"   and ys :: "'b list" assumes "as = xs @ [x]"   and "bs = y # ys" shows "P"
sel_1: fixes x1 :: "'l"   and x2 :: "'m" shows "fst (x1, x2) = x1"
oStrictLimit_ub: fixes f :: "nat \<Rightarrow> ordinal"   and n :: "nat" shows "f n < oStrictLimit f"
IH: shows "fsafe \<phi> \<longrightarrow> (\<forall>\<nu>. (\<nu> \<in> fml_sem I (FUrename x y \<phi>)) = (Radj x y \<nu> \<in> fml_sem I \<phi>))"
star_rtc1_eq: fixes x :: "'a" shows "1' + x + x\<^sup>\<star> ; x\<^sup>\<star> = x\<^sup>\<star>"
in_idem: fixes A :: "'a word set"   and x :: "'a word" assumes "finite A"   and "x \<in> A" shows "signed.max x (signed.Max A) = signed.Max A"
orientation_symmetric: fixes y :: "'a"   and x :: "'a" assumes "orientation x y" shows "symmetric x"
simps_1: fixes q :: "'b fm" shows "imply [] q = q"
init_InvGlobalRecordCount: fixes s :: "('c, 'd) configuration stream" assumes "holds init_config s" shows "holds InvGlobalRecordCount s"
bindU_bindU: fixes fmapU :: "(udom \<rightarrow> udom) \<rightarrow> udom\<cdot>'d \<rightarrow> udom\<cdot>'d"   and returnU :: "udom \<rightarrow> udom\<cdot>'d"   and bindU :: "udom\<cdot>'d \<rightarrow> (udom \<rightarrow> udom\<cdot>'d) \<rightarrow> udom\<cdot>'d"   and xs :: "udom\<cdot>'d"   and f :: "udom \<rightarrow> udom\<cdot>'d"   and g :: "udom \<rightarrow> udom\<cdot>'d" assumes "class.monad fmapU returnU bindU" shows "bindU\<cdot>(bindU\<cdot>xs\<cdot>f)\<cdot>g = bindU\<cdot>xs\<cdot>(\<Lambda> x. bindU\<cdot>(f\<cdot>x)\<cdot>g)"
last_snoc: fixes xs :: "'b list"   and x :: "'b" shows "last (xs \<cdot> [x]) = x"
N2: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<turnstile> F = G" shows "|~ \<circle>F = \<circle>G"
aact''_fst: shows "fst \<circ> aact'' = fst \<circ> aact"
sup_ge1: fixes x :: "'d multiset"   and y :: "'d multiset" shows "x \<subseteq># x \<union># y"
name_supp_abs: fixes a :: "name"   and x :: "'a" shows "supp ([a].x) = supp x - {a}"
read_tmps_distinct_axioms: shows "read_tmps_distinct ts\<^sub>s\<^sub>b"
unitLiteralIsEntailed: fixes uClause :: "Literal list"   and uLiteral :: "Literal"   and valuation :: "Literal list"   and formula :: "Literal list list" assumes "isUnitClause uClause uLiteral valuation"   and "formulaEntailsClause formula uClause" shows "formulaEntailsLiteral (formula @ val2form valuation) uLiteral"
suffix_to_prefix: fixes xs :: "'b list"   and ys :: "'b list" shows "xs \<le>s ys = (rev xs \<le>p rev ys)"
simps_2: shows "Dom \<^bold>\<I> = \<^bold>\<I>"
app_iso_inv: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" assumes "digraph_isomorphism hom" shows "app_iso (inv_iso hom) (app_iso hom G) = G"
l'_def: fixes F' :: "'a filter"   and g' :: "'a \<Rightarrow> 'b" shows "o[F'](g') = {f. \<forall>c>0. \<forall>\<^sub>F x in F'. norm (f x) \<le> c * norm (g' x)}"
antisym_conv: fixes less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and y :: "'b::type"   and x :: "'b::type" assumes "class.order (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (y::'b::type) (x::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (y::'b::type) = (x = y)"
select_result_eq: fixes h :: "'y"   and f :: "('y, 'z, 'x) prog"   and h' :: "'y" assumes "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x = h' \<turnstile> f \<rightarrow>\<^sub>r x" shows "|h \<turnstile> f|\<^sub>r = |h' \<turnstile> f|\<^sub>r"
regular_edge: fixes c :: "'form"   and "is" :: "nat list" assumes "c \<in> set conclusions"   and "is \<in> it_paths (it' c)" shows "edge_at c is \<in> edges"
resp: fixes x :: "'a" shows "(\<lambda>u. eq_app_right language `` {u @ [x]}) respects eq_app_right language"
decomps_3: shows "\<forall>(f, g)\<in>fundfoldpairs. s = Abs_induced_automorph f g \<longrightarrow> B \<in> g \<turnstile> \<C>"
D_def: shows "D = mat n n (\<lambda>(i, j). if i = j then csqrt (B $$ (i, i)) else 0)"
that_2: fixes args :: "term list"   and name :: "name" assumes "linears args"   and "t = name $$ args" shows "thesis"
B_2: fixes x :: "'a"   and y :: "'a" assumes "x < y" shows "\<exists>b\<in>B. x < b \<and> b \<le> y"
assms: shows "butlast (cs\<^bsup>\<pi>\<^esup> m) = butlast (cs\<^bsup>\<pi>\<^esup> n)"
InvariantEquivalentZL_def: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and F0 :: "Literal list list" shows "InvariantEquivalentZL F M F0 = equivalentFormulae (F @ val2form (elements (prefixToLevel 0 M))) F0"
replace_basis_one_step: fixes vs :: "'v list"   and us :: "'v list"   and w :: "'v" assumes "set vs \<subseteq> V"   and "set us \<subseteq> V"   and "R_lin_independent (us @ vs)"   and "w \<in> RSpan (us @ vs) - RSpan us" shows "\<exists>xs y ys. vs = xs @ y # ys \<and> set (w # us @ xs @ ys) \<subseteq> RSpan (us @ vs) \<and> RSpan (us @ vs) = RSpan (w # us @ xs @ ys) \<and> R_lin_independent (w # us @ xs @ ys)"
i_cut_defs_1: fixes I :: "'a set"   and t :: "'a" shows "I \<down>\<le> t \<equiv> {x \<in> I. x \<le> t}"
sums_mult_iff: fixes c :: "'a"   and f :: "nat \<Rightarrow> 'a"   and d :: "'a" assumes "c \<noteq> (0::'a)" shows "(\<lambda>n. c * f n) sums (c * d) = f sums d"
mimicking: fixes r :: "'r"   and s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and \<alpha> :: "'r \<Rightarrow> 'r"   and \<beta> :: "'l \<Rightarrow> 'l" assumes "revision_step r s s'"   and "bij \<alpha>"   and "bij \<beta>" shows "revision_step (\<alpha> r) (\<R>\<^sub>G \<alpha> \<beta> s) (\<R>\<^sub>G \<alpha> \<beta> s')"
map_le_override: fixes h :: "'c \<Rightarrow> 'd option"   and h' :: "'c \<Rightarrow> 'd option" assumes "h \<bottom> h'" shows "h \<subseteq>\<^sub>m h ++ h'"
intros_20: fixes s :: "s"   and i :: "nat"   and j :: "nat"   and m :: "mem"   and n :: "nat"   and vs :: "v list" assumes "smem_ind s i = Some j"   and "s.mem s ! j = m"   and "mem_size m = n" shows "\<lparr>s;vs;[$Current_memory]\<rparr> \<leadsto>_ i \<lparr>s;vs;[$C ConstInt32 (int_of_nat n)]\<rparr>"
npAdd: fixes x :: "'a"   and y :: "'a"   and m :: "nat"   and n :: "nat" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "x^\<^bsup>R m\<^esup> = \<zero>"   and "y^\<^bsup>R n\<^esup> = \<zero>" shows "(x \<plusminus> y)^\<^bsup>R (m + n)\<^esup> = \<zero>"
exp_times_scaleR_commute: fixes t :: "real"   and A :: "'a" shows "exp (t *\<^sub>R A) * A = A * exp (t *\<^sub>R A)"
simps_2: fixes c :: "char"   and cs :: "char list" shows "string_expose (c # cs) = Inr (c, cs)"
less_trans: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" assumes "x <s y"   and "y <s z" shows "x <s z"
map_def: fixes f :: "'c" shows "\<tau>'\<tau>.map f = (if \<tau>.A.arr f then B (\<tau>' (\<tau>.A.cod f)) (\<tau> f) else K.A.null)"
cancel_sym: fixes a :: "bool \<times> 'c"   and b :: "bool \<times> 'c" assumes "canceling a b" shows "canceling b a"
reds_fresh: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and L :: "Vars.var list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and z :: "exp"   and x :: "Vars.var" assumes "\<Gamma> : e \<Down>\<^bsub>L\<^esub> \<Delta> : z"   and "atom x \<sharp> (\<Gamma>, e)" shows "atom x \<sharp> (\<Delta>, z) \<or> x \<in> domA \<Delta> - set L"
sec_id_identified_3: fixes n :: "nat"   and m :: "nat" shows "{Tok_PriK n, Rev_PriK} = {Tok_PriK n, Sec_PriK m, Rev_PriK} - {Sec_PriK m}"
proots_count_emtpy: fixes p :: "'b poly" shows "proots_count p {} = 0"
DBM_reset_complete_empty': fixes n :: "nat"   and v :: "'c \<Rightarrow> nat"   and k :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and d :: "'d"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and u :: "'c \<Rightarrow> 'd" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   and "\<forall>c. 0 < v c"   and "k \<le> n"   and "DBM_reset M n k d M'"   and "\<forall>u. \<not> u \<turnstile>\<^bsub>v,n\<^esub> M" shows "\<not> u \<turnstile>\<^bsub>v,n\<^esub> M'"
cod_inv: fixes f :: "'a" assumes "B.iso f" shows "B.cod (B.inv f) = B.dom f"
slg_rel_def: fixes R :: "('a \<times> 'b) set" shows "\<langle>R\<rangle>slg_rel = (R \<rightarrow> \<langle>R\<rangle>list_set_rel) O br (\<lambda>succs. {(u, v). v \<in> succs u}) (\<lambda>_. True)"
cons_interleave_split: fixes a :: "'b"   and l :: "'b list"   and l1 :: "'b list"   and l2 :: "'b list" assumes "a # l \<in> l1 \<otimes> l2" shows "\<exists>lh. l1 = a # lh \<and> l \<in> lh \<otimes> l2 \<or> l2 = a # lh \<and> l \<in> l1 \<otimes> lh"
one_mono_ordered_semiring_1_axioms: shows "one_mono_ordered_semiring_1 default (\<succ>)"
inject: fixes x21 :: "interval_tree"   and x22 :: "nat idx_interval"   and x23 :: "nat"   and x24 :: "interval_tree"   and y21 :: "interval_tree"   and y22 :: "nat idx_interval"   and y23 :: "nat"   and y24 :: "interval_tree" shows "(Node x21 x22 x23 x24 = Node y21 y22 y23 y24) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24)"
mult_assoc: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd"   and c :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<odot> b \<odot> c = a \<odot> (b \<odot> c)"
exec_bind_SE_success''': fixes \<sigma> :: "bool"   and A :: "bool \<Rightarrow> ('d \<times> bool) option"   and M :: "'d \<Rightarrow> bool \<Rightarrow> (bool \<times> bool) option" assumes "\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E A M" shows "\<exists>a. A \<sigma> = Some a \<and> snd a \<Turnstile> M (fst a)"
res_def: shows "S_CHAN.res \<equiv> RES (S_CHAN.sec_oracle \<oplus>\<^sub>O S_CHAN.send_oracle \<oplus>\<^sub>O S_CHAN.recv_oracle) Void"
elims: fixes x :: "'b list"   and xa :: "'b list"   and y :: "'b poly" assumes "horner_composition x xa = y" obtains cn :: "'b" and xis :: "'b list" where "x = [cn]" and "xa = xis__" and "y = [:cn:]"   | ci :: "'b" and v :: "'b" and va :: "'b list" and xi :: "'b" and xis :: "'b list" where "x = ci # vc__ # va" and "xa = xi # xis" and "y = horner_composition (vc__ # va) xis * [:- xi, 1::'b:] + [:ci:]"   | uv_ :: "'b list" where "x = []" and "xa = uv_" and "y = 0"   | v :: "'b" and vb :: "'b" and vc :: "'b list" where "x = v # vb # vc" and "xa = []" and "y = 0"
jordan_block_constant_bound: fixes n :: "nat" assumes "norm a < 1" shows "\<exists>p. \<forall>i j k. i < n \<longrightarrow> j < n \<longrightarrow> norm ((jordan_block n a ^\<^sub>m k) $$ (i, j)) \<le> p"
RECT_rule_arb: fixes body :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c"   and V :: "('b \<times> 'b) set"   and pre :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and arb :: "'d"   and x :: "'b"   and M :: "'d \<Rightarrow> 'b \<Rightarrow> 'c" assumes "trimono body"   and "wf V"   and "pre arb x"   and "\<And>f arb x. \<lbrakk>\<And>arb' x'. \<lbrakk>pre arb' x'; (x', x) \<in> V\<rbrakk> \<Longrightarrow> f x' \<le> M arb' x'; pre arb x; REC\<^sub>T body = f\<rbrakk> \<Longrightarrow> body f x \<le> M arb x" shows "REC\<^sub>T body x \<le> M arb x"
FUpdate: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and v :: "'a"   and t :: "('b, 'a) Term.term" assumes "fsubstp \<theta>"   and "v \<notin> subst_domain \<theta>"   and "t \<noteq> Var v" shows "fsubstp (\<theta>(v := t))"
nSteps_add: fixes Cal :: "'a processCalculus"   and n1 :: "nat"   and n2 :: "nat" shows "\<forall>P Q R. P \<longmapsto>Cal\<^bsup>n1\<^esup> Q \<and> Q \<longmapsto>Cal\<^bsup>n2\<^esup> R \<longrightarrow> P \<longmapsto>Cal\<^bsup>(n1 + n2)\<^esup> R"
dim_def: shows "A.dim = (LEAST n. \<exists>Aa. finite Aa \<and> card Aa = n \<and> Aa \<subseteq> mat_kernel A \<and> A.span Aa = mat_kernel A)"
moebius_similarity_id: shows "moebius_similarity 1 0 = id_moebius"
nn_integral_embed_measure: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> ennreal"   and M :: "'c measure" assumes "inj f"   and "g \<in> borel_measurable (embed_measure M f)" shows "integral\<^sup>N (embed_measure M f) g = \<integral>\<^sup>+ x. g (f x) \<partial>M"
k: fixes i :: "'c"   and x :: "real \<times> 'c" assumes "i \<in> U"   and "x \<in> {0..1} \<times> U \<inter> {0..1} \<times> V i" shows "k x = fs i x"
not_le_imp_less: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "class.linorder less_eq less"   and "\<not> less_eq y x" shows "less x y"
ifbd_inj: fixes \<phi> :: "'c set \<Rightarrow> 'd set"   and \<psi> :: "'c set \<Rightarrow> 'd set" assumes "Sup_pres \<phi>"   and "Sup_pres \<psi>"   and "bd\<^sup>-\<^sub>\<F> \<phi> = bd\<^sup>-\<^sub>\<F> \<psi>" shows "\<phi> = \<psi>"
rel_pmf_imp_rel_set: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c pmf"   and q :: "'d pmf" assumes "rel_pmf R p q" shows "rel_set R (set_pmf p) (set_pmf q)"
power_Suc_less_one: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less zero a"   and "less a one" shows "less (power.power one times a (Suc n)) one"
last_message_Cons_NoMsg: fixes xs :: "'b message_af list" shows "last_message (NoMsg # xs) = last_message xs"
suc_above_AboveS: fixes B :: "'a set" assumes "B \<subseteq> Field r"   and "local.AboveS B \<noteq> {}" shows "local.AboveS B = local.above (suc B)"
inject_1: fixes x1 :: "real \<times> real \<times> real"   and y1 :: "real \<times> real \<times> real" shows "(LessUni x1 = LessUni y1) = (x1 = y1)"
prm_apply_id: fixes x :: "'b" shows "\<epsilon> $ x = x"
uminus_rel_frac: fixes r :: "'a"   and s :: "'a" assumes "(r, s) \<in> R \<times> S" shows "uminus_rel (r / s) = comm.additive.inverse r / s"
total_ES1_C1_inter_Upsilon1: shows "EventSystems.total ES1 (C\<^bsub>\<V>1\<^esub> \<inter> \<Upsilon>\<^bsub>\<Gamma>1\<^esub>)"
t: shows "t \<in> pair ` setops\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
simps_2: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and x :: "name"   and G :: "'d frame"   and F :: "'d frame" assumes "assertionAux SCompose SImp SBottom SChanEq"   and "x \<sharp> G" shows "assertionAux.mergeFrame SCompose (\<lparr>\<nu>x\<rparr>F) G = \<lparr>\<nu>x\<rparr>assertionAux.mergeFrame SCompose F G"
ideal_inc_unit: fixes I :: "'a set"   and a :: "'a" assumes "ideal R I"   and "a \<in> I"   and "Unit R a" shows "1\<^sub>r \<in> I"