hcju/isabelleps huggingface dataset:
red_def: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.red F p q = (\<exists>f\<in>F. \<exists>t. punit.red_single p q f t)"
assms_5: shows "\<forall>g\<in>set basis. (\<lambda>x. ln (g x)) \<in> o(\<lambda>x. ln (b' x))"
block_size_t: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and grouping :: "int"   and min_index :: "int" assumes "t_packing_design point_set block_collection u_block_size grouping min_index" shows "grouping \<le> u_block_size"
DD_subset_ds: fixes r :: "('a \<times> 'a) set"   and ars :: "('b \<times> 'a \<times> 'b) set"   and \<tau> :: "'b \<times> ('a \<times> 'b) list"   and \<sigma> :: "'b \<times> ('a \<times> 'b) list"   and \<sigma>' :: "'b \<times> ('a \<times> 'b) list"   and \<tau>' :: "'b \<times> ('a \<times> 'b) list"   and S :: "'a set" assumes "trans r"   and "DD ars r (\<tau>, \<sigma>, \<sigma>', \<tau>')"   and "set_mset (Decreasing_Diagrams.measure r (\<tau>, \<sigma>)) \<subseteq> r \<down>s S" shows "set_mset (Decreasing_Diagrams.measure r (\<sigma>', \<tau>')) \<subseteq> r \<down>s S"
F_D_part: fixes P :: "'c process" shows "F P = {(s, x). s \<in> D P} \<union> {(s, x). s \<notin> D P \<and> (s, x) \<in> F P}"
impl_composed: fixes implem :: "chan \<Rightarrow> msg"   and Z :: "chan" assumes "semivalid_implem implem" shows "composed (implem Z)"
augment_edge: fixes G1 :: "('v, 'w) graph"   and G2 :: "('v, 'w) graph"   and u :: "'v"   and v :: "'v" assumes "subgraph G1 G"   and "subgraph G2 G"   and "(u, v) \<in> {(a, b) |a b. nodes_connected G1 a b}"   and "(u, v) \<notin> {(a, b) |a b. nodes_connected G2 a b}" shows "\<exists>a b e. (a, b) \<notin> {(a, b) |a b. nodes_connected G2 a b} \<and> e \<notin> edges G2 \<and> e \<in> edges G1 \<and> (case e of (aa, w, ba) \<Rightarrow> a = aa \<and> b = ba \<or> a = ba \<and> b = aa)"
wf_atable_subset: fixes n :: "nat"   and V :: "nat set"   and X :: "'a option list set"   and Y :: "'a option list set" assumes "table n V X"   and "Y \<subseteq> X" shows "table n V Y"
cone_determines_arrow_2: fixes d :: "('b \<Rightarrow> 'a, 'b \<Rightarrow> 'a) concrete_category.arr"   and a :: "'a"   and \<chi> :: "'b \<Rightarrow> 'a" assumes "J_C.ide d"   and "Limit.cone J (\<cdot>) (J_C.Map d) a \<chi>" shows "J_C.Map (J_C.MkArr (constant_functor.map J (\<cdot>) a) (J_C.Map d) \<chi>) = \<chi>"
mergeFrames: 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 A\<^sub>F :: "name list"   and A\<^sub>G :: "name list"   and \<Psi>\<^sub>G :: "'d"   and \<Psi>\<^sub>F :: "'d" assumes "assertionAux SCompose SImp SBottom SChanEq"   and "A\<^sub>F \<sharp>* A\<^sub>G"   and "A\<^sub>F \<sharp>* \<Psi>\<^sub>G"   and "A\<^sub>G \<sharp>* \<Psi>\<^sub>F" shows "assertionAux.mergeFrame SCompose (\<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>) (\<langle>A\<^sub>G, \<Psi>\<^sub>G\<rangle>) = \<langle>(A\<^sub>F @ A\<^sub>G), SCompose \<Psi>\<^sub>F \<Psi>\<^sub>G\<rangle>"
safety_always_eventually: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "safety P"   and "\<forall>i. \<exists>j\<ge>i. \<exists>\<beta>. P (stake j \<sigma> @- \<beta>)" shows "P \<sigma>"
rell_F_def: fixes L1 :: "'n \<Rightarrow> 'x \<Rightarrow> bool"   and L2 :: "'o \<Rightarrow> 'y \<Rightarrow> bool"   and L3 :: "'p \<Rightarrow> 'z \<Rightarrow> bool" shows "rell_F L1 L2 L3 = rel_F L1 L2 L3 (=) (=) (=) (=) (=) (=)"
freshBinp_def: fixes xs :: "'d"   and x :: "'e"   and binp :: "'f \<Rightarrow> ('g, 'h, 'd, 'e, 'i) qAbs set option" shows "freshBinp xs x binp \<equiv> liftAll (freshAbs xs x) binp"
bdd_below_insert: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and a :: "'b::type"   and A :: "'b::type set" assumes "class.lattice (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)" shows "preorder.bdd_below (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (insert (a::'b::type) (A::'b::type set)) = preorder.bdd_below less_eq A"
loose_bvar_Suc: fixes t :: "Core.term"   and k :: "nat" assumes "loose_bvar t (Suc k)" shows "loose_bvar t k"
bij_betw_imp_funcset: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" assumes "bij_betw f A B" shows "f \<in> A \<rightarrow> B"
from_list_correct_4: fixes l :: "(('v \<times> 'v) \<times> nat) list"   and u :: "'v"   and v :: "'v"   and d :: "nat" assumes "valid_wgraph_repr l"   and "((u, v), d) \<in> set l" shows "\<alpha>w (from_list l) {u, v} = d"
length_isort: fixes xs :: "'c list" shows "length (isort xs) = length xs"
pj_def: shows "pj < length ys \<and> ys ! pj = j"
w_prop: shows "w \<in> root_list [:c, b, a:] \<and> x \<le> w \<and> w \<le> y"
pochhammer_minus': fixes b :: "'a"   and k :: "nat" shows "pochhammer (b - of_nat k + (1::'a)) k = (- (1::'a)) ^ k * pochhammer (- b) k"
idx_inj: fixes bound :: "'b bound"   and x :: "'b"   and y :: "'b" assumes "in_bound less_eq1 less1 bound x"   and "in_bound less_eq1 less1 bound y"   and "idx1 bound x = idx1 bound y" shows "x = y"
finite_imageI'': fixes A :: "'d set"   and B :: "'e set"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "finite (A \<times> B)" shows "finite {f x y |x y. x \<in> A \<and> y \<in> B \<and> R x y}"
WT_parallel_wiring: fixes \<I>1 :: "('i, 'm) \<I>"   and \<I>2 :: "('j, 'n) \<I>"   and \<I>3 :: "('k, 'o) \<I>"   and \<I>4 :: "('l, 'p) \<I>" shows "(\<I>1 \<oplus>\<^sub>\<I> \<I>2) \<oplus>\<^sub>\<I> (\<I>3 \<oplus>\<^sub>\<I> \<I>4), (\<I>1 \<oplus>\<^sub>\<I> \<I>3) \<oplus>\<^sub>\<I> (\<I>2 \<oplus>\<^sub>\<I> \<I>4) \<turnstile>\<^sub>C parallel_wiring \<surd>"
not_singleton_def: shows "class.not_singleton TYPE('b) \<equiv> \<exists>x y. x \<noteq> y"
lossless_spmf_spmf_of_spmf: fixes p :: "'a pmf" shows "lossless_spmf (spmf_of_pmf p)"
P2_case_l_neq_1_gt_x0_rewrite: fixes t :: "nat"   and u0 :: "nat"   and s :: "nat" assumes "t < order \<G>"   and "t \<noteq> 0" shows "\<^bold>g [^] (t * (u0 + s * nat (fst (bezw t (order \<G>)) mod int (order \<G>)))) = \<^bold>g [^] (t * u0) \<otimes> \<^bold>g [^] s"
Inf_closed_set_def: fixes X :: "'b set" shows "Inf_closed_set X = (\<forall>Y\<subseteq>X. \<Sqinter> Y \<in> X)"
simps_1: fixes x :: "char list" shows "fromB (B x) = Some x"
gallery_CConsI: fixes w :: "'a set"   and x :: "'a set"   and xs :: "'a set list" assumes "folding.chamber w"   and "folding.gallery (x # xs)"   and "w \<sim> x" shows "folding.gallery (w # x # xs)"
interleavings_distinct: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "interleavings xs ys zs" shows "(distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {}) = distinct zs"
set_14: fixes x61 :: "('b, 'a, 'c) expr"   and x62 :: "('b, 'a, 'c) expr" shows "LID\<^sub>E (Assign x61 x62) = LID\<^sub>E x61 \<union> LID\<^sub>E x62"
in_listspI: fixes xs :: "'a list"   and A :: "'a \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. A x" shows "listsp A xs"
uminus_of_uminus_id: fixes x :: "'a literal" shows "- (- x) = x"
permutesI: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "bij_betw f A A"   and "\<forall>x. x \<notin> A \<longrightarrow> f x = x" shows "f permutes A"
multiplicity_one: fixes p :: "'a" shows "multiplicity p (1::'a) = 0"
dist_float_eq_0_iff: fixes x :: "float"   and y :: "float" shows "(dist x y = 0) = (x = y)"
circline_equation_quadratic_equation: fixes A :: "complex"   and z :: "complex"   and B :: "complex"   and D :: "complex"   and x :: "real"   and y :: "real"   and bx :: "real"   and "by" :: "real" assumes "circline_equation A B (cnj B) D z 1"   and "Re z = x"   and "Im z = y"   and "Re B = bx"   and "Im B = by" shows "A * cor (x\<^sup>2) + A * cor (y\<^sup>2) + cor (2 * bx * x) + cor (2 * by * y) + D = 0"
cases: fixes x :: "('e, 'f, 'g, 'h) SLin_state_scheme \<times> ('i, 'f, 'j, 'k) SLin_state_scheme" obtains s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme" and s2 :: "('i, 'f, 'j, 'k) SLin_state_scheme" where "x = (s1, s2)"
assms_2: shows "f\<lparr>ArrCod\<rparr> = A"
diff_mod_le: fixes t :: "nat"   and r :: "nat"   and m :: "nat" shows "(t - r) mod m \<le> t"
elims_2: fixes x :: "'a option list"   and xa :: "nat set"   and xb :: "'a option list" assumes "isSameIntersection x xa xb" obtains t1 :: "'a option list" and s :: "nat set" and t2 :: "'a option list" where "x = t1" and "xa = s" and "xb = t2" and "\<forall>i\<in>s. t1 ! i = t2 ! i"
fstar_Inf_pres: fixes f :: "'b \<Rightarrow> 'b" assumes "Inf_pres f" shows "Inf_pres (fstar f)"
invertible_inverse_closed: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c" assumes "Group_Theory.monoid M composition unit"   and "monoid.invertible M composition unit u"   and "u \<in> M" shows "monoid.inverse M composition unit u \<in> M"
lift_pref_profile_Pareto_strict_iff: fixes x :: "'b"   and y :: "'b" shows "x \<prec>[Pareto R'] y = (x \<in> alts' \<and> y \<in> alts' \<and> (x \<notin> alts \<and> y \<in> alts \<or> x \<prec>[Pareto R] y))"
intros_8: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" assumes "s \<in> b"   and "\<Gamma>\<turnstile>c\<^sub>1 \<down> Normal s" shows "\<Gamma>\<turnstile>Cond b c\<^sub>1 c\<^sub>2 \<down> Normal s"
relcomp3_I: fixes t :: "'a"   and u :: "'b"   and A :: "('a \<times> 'b) set"   and s :: "'b"   and B :: "('b \<times> 'a) set"   and v :: "'a" assumes "(t, u) \<in> A"   and "(s, t) \<in> B"   and "(u, v) \<in> B" shows "(s, v) \<in> B O A O B"
hd_rev: fixes xs :: "'f list" assumes "xs \<noteq> []" shows "hd (rev xs) = last xs"
nat_pow_is_Zp_continuous: fixes f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and n :: "nat" assumes "is_Zp_continuous f" shows "is_Zp_continuous (f [^]\<^bsub>function_ring (carrier Zp) Zp\<^esub> n)"
eq_elapsed: shows "PTA_Reachability.pairwise eq_elapsed xs"
fps_one_nth: fixes n :: "nat" shows "1 $ n = (if n = 0 then 1::'a else (0::'a))"
of_nat_eq_1_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero n = one) = (n = 1)"
lex_comp_pp_def: shows "lex_comp = lex_comp'"
closure_sequential: fixes l :: "'c"   and S :: "'c set" shows "(l \<in> closure S) = (\<exists>x. (\<forall>n. x n \<in> S) \<and> x \<longlonglongrightarrow> l)"
card_2_iff': fixes S :: "'a set" shows "(card S = 2) = (\<exists>x\<in>S. \<exists>y\<in>S. x \<noteq> y \<and> (\<forall>z\<in>S. z = x \<or> z = y))"
that: fixes n :: "nat" assumes "\<And>l. n \<le> l \<Longrightarrow> card (level A w (2 * k) l) \<le> card (nodes A) - k" shows "thesis"
smap_comm: fixes x :: "'a"   and Y :: "'a set" shows "smap (insert x (\<eta> (smap Y))) = smap (insert (smap Y) (\<eta> x))"
case_pl_inf: fixes w1 :: "32 Word.word"   and P :: "32 Word.word \<Rightarrow> 32 Word.word \<Rightarrow> bool"   and w2 :: "32 Word.word" assumes "w1 = NEG_INF \<Longrightarrow> P w1 w2"   and "w2 = NEG_INF \<Longrightarrow> P w1 w2"   and "\<lbrakk>w1 = POS_INF; w2 = POS_INF\<rbrakk> \<Longrightarrow> P w1 w2"   and "\<lbrakk>w1 = POS_INF; w2 \<noteq> POS_INF; w2 \<noteq> NEG_INF\<rbrakk> \<Longrightarrow> P w1 w2"   and "\<lbrakk>w1 \<noteq> POS_INF; w1 \<noteq> NEG_INF; w2 = POS_INF\<rbrakk> \<Longrightarrow> P w1 w2"   and "\<lbrakk>w1 \<noteq> POS_INF; w1 \<noteq> NEG_INF; w2 \<noteq> POS_INF; w2 \<noteq> NEG_INF\<rbrakk> \<Longrightarrow> P w1 w2" shows "P w1 w2"
1: shows "x * x\<^sup>T\<^sup>\<star>\<^sup>T \<le> x\<^sup>T\<^sup>\<star>\<^sup>T"
nres_rel_def: fixes R :: "('b \<times> 'c) set" shows "\<langle>R\<rangle>nres_rel \<equiv> {(c, a). c \<le> \<Down> R a}"
conjugation_is_inj: fixes g :: "'a"   and h1 :: "'a"   and h2 :: "'a" assumes "g \<in> carrier G"   and "h1 \<in> carrier G"   and "h2 \<in> carrier G"   and "g \<otimes> h1 \<otimes> inv g = g \<otimes> h2 \<otimes> inv g" shows "h1 = h2"
tok_id_pubk_prik_msg: fixes n :: "nat"   and m :: "nat" assumes "n \<in> bad_tok_prik" shows "{ID n (Sec_PubKey m), Hash (ID n (Sec_PubKey m)), \<langle>n, ID n (Sec_PubKey m)\<rangle>} \<subseteq> spied (tok_id_pubk_prik n m)"
\<Delta>'''_sel_2: fixes \<Delta> :: "('a, 'b) network_scheme"   and x :: "'a"   and y :: "'a" assumes "countable_network \<Delta>" shows "capacity (countable_network.\<Delta>''' \<Delta>) (x, y) = (if x = sink \<Delta> \<or> y = source \<Delta> then 0 else capacity \<Delta> (x, y))"
assert_bound_tableau: fixes s :: "('i, 'a) state"   and a :: "'i \<times> 'a atom"   and s' :: "('i, 'a) state" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "assert_bound a s = s'" shows "\<T> s' = \<T> s \<and> \<nabla> s'"
bind_def: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'd set" shows "Set.bind A f = {x. \<exists>B\<in>f ` A. x \<in> B}"
rk: assumes "r = 1" shows "k \<noteq> 0"
impl_ded: fixes a :: "'a N"   and b :: "'a N"   and c :: "'a N" shows "a * b l\<rightarrow> c = a l\<rightarrow> b l\<rightarrow> c"
continuous_limit: fixes f :: "'b set \<Rightarrow> 'b set" assumes "continuous f" shows "continuous (limit f)"
rtranclp_unfold: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" shows "r\<^sup>*\<^sup>* a b \<equiv> a = b \<or> r\<^sup>+\<^sup>+ a b"
assms_4: shows "(state0, state) \<in> transitionRelation decisionVars"
"2_2": fixes xa :: "('a btree \<times> 'a) list"   and y :: "('a btree \<times> 'a) list"   and x21 :: "'a btree \<times> 'a"   and x22 :: "('a btree \<times> 'a) list"   and xb :: "'a btree"   and ya :: "'a"   and ti :: "'a btnode ref option" assumes "(xa, y) = split ts x"   and "y = x21 # x22"   and "(xb, ya) = x21"   and "ya \<noteq> x"   and "sorted_less (inorder xb)" shows "<btree_assn k xb ti> ins k x ti <btupi_assn k (abs_split.ins k x xb)>\<^sub>t"
normB: fixes b :: "'b" assumes "b \<in> basisB" shows "norm b = 1"
triangle_diff_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 "\<not> (\<exists>Q\<in>\<P>. a \<in> Q \<and> b \<in> Q \<and> c \<in> Q)"
prev: fixes f :: "complex \<Rightarrow> complex" assumes "continuous_on S f"   and "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0" shows "\<exists>g. continuous_on S g \<and> (\<forall>z\<in>S. f z = exp (g z))"
successively_iff_sorted_wrt_strong: fixes xs :: "'g list"   and P :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>x \<in> set xs; y \<in> set xs; z \<in> set xs; P x y; P y z\<rbrakk> \<Longrightarrow> P x z" shows "successively P xs = sorted_wrt P xs"
step: shows "\<Gamma>\<turnstile> (cs, cssxs, s) \<rightarrow> (cs', css'xs, t)"
nonzero_mult_div_cancel_left: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)" shows "a * b div a = b"
c_assoc_value_df: fixes y :: "nat"   and x :: "nat" shows "c_assoc_value y x = (if y = 0 then 0 else if c_fst (c_hd y) = x then c_snd (c_hd y) else c_assoc_value (c_tl y) x)"
no_subst: fixes k :: "'b"   and F :: "'b formula"   and G :: "'b formula" assumes "k \<notin> atoms F" shows "F[G/k] = F"
cod_simps_1: shows "cod.apex = \<mu>\<nu>.apex"
mdijkstra_def: shows "mdijkstra \<equiv> mdinit \<bind> (\<lambda>\<sigma>0. WHILE\<^sub>T\<^bsup>dinvarm\<^esup> (\<lambda>(wl, uu_). dom wl \<noteq> {}) (\<lambda>\<sigma>. mpop_min \<sigma> \<bind> (\<lambda>(v, wv, \<sigma>'). mupdate v wv \<sigma>')) \<sigma>0 \<bind> (\<lambda>(uu_, res). RETURN res))"
SN_defs: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "SN_on r A = (\<nexists>f. f 0 \<in> A \<and> (\<forall>i. (f i, f (Suc i)) \<in> r))"
dickson_grading_adds_imp_le: fixes d :: "'c \<Rightarrow> nat"   and s :: "'c"   and t :: "'c" assumes "dickson_grading d"   and "s adds t" shows "d s \<le> d t"
that: fixes us :: "'a list list" assumes "us \<in> lists G\<^sub>+"   and "1 < \<^bold>|us\<^bold>|"   and "concat us = w" shows "thesis"
of_nat_1_eq_iff: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(one = semiring_1.of_nat one plus zero n) = (n = 1)"
fls_inverse_def: fixes f :: "'a fls" shows "inverse f = fls_shift (fls_subdegree f) (fps_to_fls (inverse (fls_base_factor_to_fps f)))"
simps_2: fixes l :: "'b tree23"   and a :: "'b"   and r :: "'b tree23" shows "hI (OF l a r) = height l"
set_induct_2: fixes x :: "'a"   and a :: "('b, 'a) tllist"   and P :: "'a \<Rightarrow> ('b, 'a) tllist \<Rightarrow> bool" assumes "x \<in> set2_tllist a"   and "\<And>z. P z (TNil z)"   and "\<And>z1 z2 xb. \<lbrakk>xb \<in> set2_tllist z2; P xb z2\<rbrakk> \<Longrightarrow> P xb (TCons z1 z2)" shows "P x a"
insert_mset: fixes q :: "('c, 'd) SkewBinomialTree list"   and e :: "'c"   and a :: "'d" assumes "queue_invar q" shows "queue_to_multiset (local.insert e a q) = queue_to_multiset q + {#(e, a)#}"
abelian_FactGroup: fixes G :: "('b, 'c) monoid_scheme"   and N :: "'b set" assumes "comm_group G"   and "subgroup N G" shows "comm_group (G Mod N)"
strip_lemma: fixes r :: "comb \<Rightarrow> comb \<Rightarrow> bool"   and x :: "comb"   and y :: "comb"   and y' :: "comb" assumes "diamond r"   and "r\<^sup>*\<^sup>* x y"   and "r x y'" shows "\<exists>z. r\<^sup>*\<^sup>* y' z \<and> r y z"
supp_Act: fixes \<alpha> :: "'d"   and x :: "('b, 'c, 'd) formula" shows "supp (Act \<alpha> x) = supp \<alpha> \<union> supp x - bn \<alpha>"
standard_rules_def: fixes C :: "'b set"   and L :: "'b Standard_Constant set" shows "standard_rules C L \<equiv> constant_rules C \<union> identity_rules L \<union> {top_rule S_Top, nonempty_rule}"
FR\<^sub>1_def: fixes \<phi> :: "atom fm" shows "FR\<^sub>1 \<phi> = (let as = atoms\<^sub>0 \<phi>; lbs = lbounds as; ubs = ubounds as; ebs = ebounds as; intrs = concat (map (\<lambda>l. map (\<lambda>u. subst \<phi> (between l u)) ubs) lbs) in list_disj (inf\<^sub>- \<phi> # inf\<^sub>+ \<phi> # intrs @ map (subst \<phi>) ebs))"
take_bit_xor: fixes n :: "nat"   and a :: "'a"   and b :: "'a" shows "take_bit n (a XOR b) = take_bit n a XOR take_bit n b"
opp_mult_def: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" shows "times.opp_mult times x y \<equiv> times y x"
K: fixes j :: "'d" assumes "j \<in> L" shows "K j \<subseteq> I"
distinct_10: fixes x31 :: "'b aexp"   and x32 :: "'b aexp"   and x21 :: "'b aexp"   and x22 :: "'b aexp" shows "Gt x31 x32 \<noteq> Eq x21 x22"
weakBisimResPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and x :: "name" assumes "\<Psi> \<rhd> P \<approx> Q"   and "x \<sharp> \<Psi>" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>P \<approx> \<lparr>\<nu>x\<rparr>Q"
ordsetmap: fixes a :: "'a"   and b :: "'a" assumes "a \<in> P"   and "b \<in> P"   and "b \<succeq> a" shows "codomain.greater_eq (f b) (f a)"
word_rotr_eq: fixes n :: "nat"   and w :: "'b word" shows "word_rotr n w = of_bl (rotater n (to_bl w))"
distinct_3: fixes x1 :: "('e, 'f, 'g) Tree set['e]"   and x3 :: "'f" shows "tConj x1 \<noteq> tPred x3"
X: fixes i :: "'n" shows "Cauchy (\<lambda>n. X n $ i)"
DBM_zone_repr_def: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'f DBMEntry"   and v :: "'e \<Rightarrow> nat"   and n :: "nat" shows "[M]\<^bsub>v,n\<^esub> = {u. u \<turnstile>\<^bsub>v,n\<^esub> M}"
is_sup_prod: fixes p :: "'c \<times> 'd"   and q :: "'c \<times> 'd" shows "is_sup p q (fst p \<squnion> fst q, snd p \<squnion> snd q)"
convex_halfspace_Re_gt: fixes b :: "real" shows "convex {x. b < Re x}"
n_val_t2p: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and y :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "y \<in> carrier K" shows "n_val K v (x \<cdot>\<^sub>r y) = n_val K v x + n_val K v y"
L: fixes ib :: "nat"   and jb :: "nat" assumes "ib < jb"   and "jb < length (Some l # xs)" shows "(fst (mk_eqcl (Some l # xs) zs ii T) ! ib = fst (mk_eqcl (Some l # xs) zs ii T) ! jb) = (\<not> tr_lookup T (ib + ii) (jb + ii))"
assms_2: shows "ps' \<equiv> sturm_squarefree' p"
permutes_UNIV_permutation: fixes p :: "'c \<Rightarrow> 'c" shows "permutation p = (p permutes UNIV)"
\<G>\<F>_semantics: fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" shows "\<G>\<F> \<phi> w = {\<psi> \<in> subformulas\<^sub>\<mu> \<phi>. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>)}"
assms: shows "R \<subseteq> 1\<^sub>\<pi>"
geodesic_segment_commute: fixes G :: "'b set"   and x :: "'b"   and y :: "'b" assumes "geodesic_segment_between G x y" shows "geodesic_segment_between G y x"
smooth_on_norm: fixes k :: "enat"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "k-smooth_on S f"   and "(0::'b) \<notin> f ` S"   and "open S" shows "k-smooth_on S (\<lambda>x. norm (f x))"
closed_levelset_within: fixes S :: "'b set"   and s :: "'b \<Rightarrow> 'c"   and c :: "'c" assumes "continuous_on S s"   and "closed S" shows "closed {x \<in> S. s x = c}"
REL_IS_ID_def: fixes R :: "('a \<times> 'a) set" shows "REL_IS_ID R \<equiv> R = Id"
isTrivialPhi_def: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'c set"   and phis :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'c list option"   and g :: "'a"   and v :: "'c"   and v' :: "'c" shows "CFG_SSA_wf_base.isTrivialPhi \<alpha>n defs phis g v v' = (v' \<noteq> v \<and> (case CFG_SSA_wf_base.phi \<alpha>n defs phis g v of None \<Rightarrow> False | Some vs \<Rightarrow> set vs = {v, v'} \<or> set vs = {v'}))"
steps_reaches: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "Graph_Defs.steps E xs" shows "E\<^sup>*\<^sup>* (hd xs) (last xs)"
r: shows "s \<union> connected_component_set (- s) a \<subseteq> ball a r"
hyps_2: shows "valof x = real_of_int (denormal_mantissa x) * 2 powr real_of_int (denormal_exponent TYPE(('e, 'f) IEEE.float))"
narrow_type: fixes \<Delta> :: "binding list"   and X :: "tyvrs"   and Q :: "ty"   and \<Gamma> :: "binding list"   and t :: "trm"   and T :: "ty"   and P :: "ty" assumes "\<Delta> @ TVarB X Q # \<Gamma> \<turnstile> t : T"   and "\<Gamma>\<turnstile>P<:Q" shows "\<Delta> @ TVarB X P # \<Gamma> \<turnstile> t : T"
product_atMost_eq_Un: fixes A :: "nat set"   and m :: "nat" shows "A \<times> {..m} = (SIGMA i:A. {..m - i}) \<union> (SIGMA i:A. {m - i<..m})"
irreducible_dvd_prod_list: fixes p :: "'d poly"   and as :: "'d poly list" assumes "irreducible p"   and "p dvd prod_list as" shows "Bex (set as) ((dvd) p)"
euclidean_eqI: fixes x :: "'a"   and y :: "'a" assumes "\<And>b. b \<in> Basis \<Longrightarrow> x \<bullet> b = y \<bullet> b" shows "x = y"
rtrancl_trancl_reflcl: fixes r :: "('d \<times> 'd) set" shows "r\<^sup>* = (r\<^sup>+)\<^sup>="
intro: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'b" assumes "functor A A F" shows "endofunctor A F"
inorder_insert_d: fixes x :: "'a"   and t :: "'a tree"   and dl :: "nat"   and t' :: "'a tree"   and dl' :: "nat" assumes "insert_d x (t, dl) = (t', dl')"   and "Sorted_Less.sorted (inorder t)" shows "inorder t' = ins_list x (inorder t)"
map_2: fixes f :: "'d \<Rightarrow> 'c"   and x21 :: "'d"   and x22 :: "'d llist" shows "lmap f (LCons x21 x22) = LCons (f x21) (lmap f x22)"