hcju/isabelleps huggingface dataset:
stutter_extend_concat: fixes u :: "nat \<Rightarrow> 'b"   and v :: "nat \<Rightarrow> 'b"   and w :: "'b list" assumes "u \<approx> v" shows "w \<frown> u \<approx> w \<frown> v"
monofun_cfun_fun: fixes f :: "'b \<rightarrow> 'c"   and g :: "'b \<rightarrow> 'c"   and x :: "'b" assumes "f \<sqsubseteq> g" shows "f\<cdot>x \<sqsubseteq> g\<cdot>x"
ab_specD1: fixes ab :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and gs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and ns :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'h" assumes "ab_spec ab" shows "set (ab gs bs ns data) = set bs \<union> set ns"
dim_row: fixes n :: "nat"   and v :: "complex Matrix.mat" assumes "state n v" shows "dim_row v = 2 ^ n"
fds_ln_power: fixes f :: "'a fds"   and l :: "'a"   and n :: "nat"   and l' :: "'a" assumes "fds_nth f 1 \<noteq> (0::'a)"   and "l = of_nat n * l'" shows "fds_ln l (f ^ n) = of_nat n * fds_ln l' f"
has_integral_const: fixes c :: "'d"   and a :: "'c"   and b :: "'c" shows "((\<lambda>x. c) has_integral content (cbox a b) *\<^sub>R c) (cbox a b)"
distinct_step: fixes s :: "'a list"   and r :: "'a"   and as :: "nat \<times> nat list" shows "distinct (step s r as) = distinct s"
compute_keys_pp: fixes xs :: "('a, 'b) fmap" shows "keys (Pm_fmap xs) = fmdom' (clearjunk0 xs)"
simple_bochner_integral_bounded: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and s :: "'c \<Rightarrow> 'd"   and t :: "'c \<Rightarrow> 'd" assumes "f \<in> borel_measurable M"   and "simple_bochner_integrable M s"   and "simple_bochner_integrable M t" shows "ennreal (norm (simple_bochner_integral M s - simple_bochner_integral M t)) \<le> \<integral>\<^sup>+ x. ennreal (norm (f x - s x)) \<partial>M + \<integral>\<^sup>+ x. ennreal (norm (f x - t x)) \<partial>M"
FAssReds1: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>F{D} := e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>e'\<bullet>F{D} := e\<^sub>2,s'\<rangle>"
ID_ccompare': fixes ccompare :: "('c \<Rightarrow> 'c \<Rightarrow> order) option"   and c :: "'c \<Rightarrow> 'c \<Rightarrow> order" assumes "class.ccompare ccompare"   and "ID ccompare = Some c" shows "comparator c"
F: shows "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F"
enttD: fixes A :: "assn"   and B :: "assn" assumes "A \<Longrightarrow>\<^sub>t B" shows "A \<Longrightarrow>\<^sub>A B * true"
trans_ParCL: fixes c1 :: "('test, 'atom) com"   and s :: "'state"   and c1' :: "('test, 'atom) com"   and s' :: "'state"   and c2 :: "('test, 'atom) com" assumes "(c1, s) \<rightarrow>c (c1', s')" shows "(c1 | c2, s) \<rightarrow>c (c1' | c2, s')"
Fake_parts_sing: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> synth (analz H)" shows "parts {X} \<subseteq> synth (analz H) \<union> parts H"
make_step_def: fixes selector :: "nat \<times> ('d \<times> nat \<times> nat) set \<Rightarrow> ((('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) \<times> ('e \<times> nat) set) option"   and S :: "nat \<times> ('d \<times> nat \<times> nat) set" shows "make_step selector S \<equiv> case selector S of None \<Rightarrow> S | Some (R, f) \<Rightarrow> let g = extend (fst S) R f in (max (fst S) (nextMax (Range g)), on_triple g `` edges (snd R) \<union> snd S)"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a"   and vertex :: "'b vertex" shows "P (case vertex of SOURCE \<Rightarrow> f1 | vertex.SINK \<Rightarrow> f2 | \<langle>x\<rangle> \<Rightarrow> f3 x) = (\<not> (vertex = SOURCE \<and> \<not> P f1 \<or> vertex = vertex.SINK \<and> \<not> P f2 \<or> (\<exists>x3. vertex = \<langle>x3\<rangle> \<and> \<not> P (f3 x3))))"
extremum_strict: fixes a :: "'a multiset" shows "\<not> a \<subset># {#}"
shadow_root_ptr_kinds_commutes: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap" shows "(cast shadow_root_ptr |\<in>| object_ptr_kinds h) = (shadow_root_ptr |\<in>| shadow_root_ptr_kinds h)"
max_ideal_shapeD: fixes X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" assumes "finite X"   and "F \<subseteq> P[X]"   and "generates_max_ideal X F" obtains a :: "'c \<Rightarrow> 'd" where "ideal F = ideal ((\<lambda>x. monomial (1::'d) (monomial 1 x) - monomial (a x) 0) ` X)"
ran_map_upd: fixes m :: "'d \<Rightarrow> 'c option"   and a :: "'d"   and b :: "'c" assumes "m a = None" shows "ran (m(a \<mapsto> b)) = {b} \<union> ran m"
map_of_zip_enum_is_Some: fixes ys :: "'a list"   and x :: "'b" assumes "length ys = length enum" shows "\<exists>y. map_of (zip enum ys) x = Some y"
NSconvergent_def: fixes X :: "nat \<Rightarrow> 'a" shows "NSconvergent X = (\<exists>L. X \<longlonglongrightarrow>\<^sub>N\<^sub>S L)"
simps_1: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and cfg1 :: "('a, 'c, 'b) configuration"   and p :: "'a"   and v :: "bool"   and cfg2 :: "('a, 'c, 'b) configuration" shows "asynchronousSystem.steps trans sends cfg1 <p, inM v> cfg2 = ((\<forall>s. (s = p \<longrightarrow> states cfg2 p = trans p (states cfg1 p) (Bool v)) \<and> (s \<noteq> p \<longrightarrow> states cfg2 s = states cfg1 s)) \<and> enabled cfg1 <p, inM v> \<and> msgs cfg2 = sends p (states cfg1 p) (Bool v) \<union># msgs cfg1 -# <p, inM v>)"
l1_obs_secrecy: shows "oreach l1 \<subseteq> l1_secrecy"
A: shows "is_Zp_cauchy ((\<bullet>) f \<circ> ns)"
distinct_1: fixes x1 :: "'b"   and x2 :: "'b" shows "Var x1 \<noteq> Const x2"
intros_9: fixes F :: "'b formula" shows "\<^bold>\<not> (F::'b formula) \<^bold>\<rightarrow> F \<^bold>\<rightarrow> \<bottom> \<in> AX10"
WhileTrue: fixes b :: "bexp"   and s\<^sub>1 :: "char list \<Rightarrow> int \<Rightarrow> int"   and \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and s\<^sub>2 :: "char list \<Rightarrow> int \<Rightarrow> int"   and s\<^sub>3 :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "bval b s\<^sub>1"   and "\<pi>: (c, s\<^sub>1) \<Rightarrow> s\<^sub>2"   and "\<pi>: (WHILE b DO c, s\<^sub>2) \<Rightarrow> s\<^sub>3" shows "\<pi>: (WHILE b DO c, s\<^sub>1) \<Rightarrow> s\<^sub>3"
right: fixes t1 :: "('c, 'd) tm"   and t2 :: "('c, 'd) tm" shows "App t1 t2 \<rightarrow>\<^sub>e\<^sub>m\<^sub>b t2"
paths_second_in_V: fixes xs :: "'a list" assumes "xs \<in> paths" shows "hd (tl xs) \<in> V"
polynomial_split_on_var2: fixes x' :: "nat"   and s :: "int mpoly"   and p :: "int mpoly" assumes "x' \<notin> vars s" obtains q :: "int mpoly" and r :: "int mpoly" where "p = (MPoly_Type.monom (monomial (Suc 0) x') 1 - s) * q + r" and "x' \<notin> vars r"
assms: fixes a :: "'c" assumes "a \<in> A" shows "f a = \<one>"
action_rews_6: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and x :: "state \<Rightarrow> 'b"   and y :: "state \<Rightarrow> 'c" shows "f<x, y>$ \<equiv> f<x$, y$>"
refl_sreds: fixes ts :: "dB list" shows "ts [\<rightarrow>\<^sub>s] ts"
PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m_1_dir1: fixes N :: "nat"   and ps :: "nat list" assumes "DiningPhilosophers N"   and "0 < length ps" shows "DiningPhilosophers.PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m N 1 ps \<sqsubseteq> DiningPhilosophers.LPHIL0\<^sub>n\<^sub>o\<^sub>r\<^sub>m N (ps ! 0)"
dominates_trans: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and g :: "'a"   and n :: "'b"   and n' :: "'b"   and n'' :: "'b" assumes "graph_Entry \<alpha>e \<alpha>n invar inEdges' Entry"   and "invar g"   and "graph_Entry_base.dominates \<alpha>n invar inEdges' Entry g n n'"   and "graph_Entry_base.dominates \<alpha>n invar inEdges' Entry g n' n''" shows "graph_Entry_base.dominates \<alpha>n invar inEdges' Entry g n n''"
Tau: fixes P :: "pi" shows "\<tau>.P \<Longrightarrow>\<tau> \<prec> P"
unfolding_sound: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and s :: "state"   and t :: "state" assumes "wf_chain \<Gamma> rs"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>process_call \<Gamma> rs, s\<rangle> \<Rightarrow> t" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
has_field_derivative_at_within: fixes f :: "'c \<Rightarrow> 'c"   and f' :: "'c"   and x :: "'c"   and s :: "'c set" assumes "(f has_field_derivative f') (at x)" shows "(f has_field_derivative f') (at x within s)"
kat_2_equiv_opp: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "(y \<cdot> ar z \<le> ar x \<cdot> y) = (r x \<cdot> (y \<cdot> ar z) = \<bottom>)"
vone_ne: shows "1\<^sub>\<nat> \<noteq> []\<^sub>\<circ>"
is_rbt_def: fixes t :: "('a, 'd) rbt" shows "is_rbt t = (inv1 t \<and> inv2 t \<and> color_of t = B \<and> rbt_sorted t)"
addr_loc_type_def: fixes typeof_addr :: "'a \<Rightarrow> htype option"   and P :: "'c prog" shows "heap_base'.addr_loc_type TYPE('b) typeof_addr P = heap_base.addr_loc_type (\<lambda>_. typeof_addr) P undefined"
rb_aux_no_zero_red': fixes args :: "(('t \<Rightarrow>\<^sub>0 'b) list \<times> 't list \<times> (('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list) \<times> nat" assumes "hom_grading dgrad"   and "is_regular_sequence fs"   and "rb_aux_inv2 (fst args)" shows "snd (rb_aux args) = snd args"
independentD: fixes s :: "'b set"   and t :: "'b set"   and u :: "'b \<Rightarrow> 'a"   and v :: "'b" assumes "s \<subseteq> U\<^sub>M"   and "\<not> local.dependent s"   and "finite t"   and "t \<subseteq> s"   and "(\<Sum>v\<in>t. u v *s v) = (0::'b)"   and "v \<in> t" shows "u v = (0::'a)"
HR_quot_fm: fixes e :: "(name, hf) finfun"   and A :: "fm" shows "HR \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e \<lbrakk>\<guillemotleft>\<guillemotleft>A\<guillemotright>\<guillemotright>\<rbrakk>e"
R27_R13: shows "pmf (sds R27) a = pmf (sds R13) a"
someI2: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and Q :: "'a \<Rightarrow> bool" assumes "P a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (SOME x. P x)"
cond_def: fixes s :: "('v, 'es) state_scheme" shows "cond s = ((V0 \<subseteq> {v. is_discovered v s} \<longrightarrow> \<not> is_empty_stack s) \<and> \<not> is_break param s)"
bind_split: fixes P :: "'a option \<Rightarrow> bool"   and m :: "'b option"   and f :: "'b \<Rightarrow> 'a option" shows "P (Option.bind m f) = ((m = None \<longrightarrow> P None) \<and> (\<forall>v. m = pure v \<longrightarrow> P (f v)))"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and T1 :: "('fsym, 'tp) efsym trm"   and T2 :: "('fsym, 'tp) efsym trm" shows "GE.satA \<xi> (Eq T1 T2) = (GE.int \<xi> T1 = GE.int \<xi> T2)"
upd_left_comm: fixes f :: "'a \<Rightarrow> 'b"   and g :: "('a, 'c) finfun" shows "comp_fun_commute (\<lambda>a c. finfun_update c a (f a, finfun_apply g a))"
integrable_EM_remainder': fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and n :: "nat" assumes "continuous_on {a..b} f" shows "(\<lambda>t. pbernpoly n t *\<^sub>R f t) integrable_on {a..b}"
vifunion_vintersection_index_left: fixes B :: "V"   and A :: "V" shows "\<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>B. a \<inter>\<^sub>\<circ> A) `\<^sub>\<circ> B) = \<Union>\<^sub>\<circ> B \<inter>\<^sub>\<circ> A"
add_mset_def: shows "add_mset \<equiv> map_fun id (map_fun count Abs_multiset) (\<lambda>a M b. if b = a then Suc (M b) else M b)"
boundOutputPar2Dest': fixes xvec :: "name list"   and M :: "'d"   and P :: "('d, 'e, 'f) psi"   and yvec :: "name list"   and N :: "'d"   and Q :: "('d, 'e, 'f) psi"   and R :: "('d, 'e, 'f) psi" assumes "\<lparr>\<nu>*xvec\<rparr>M \<prec>' P = \<lparr>\<nu>*yvec\<rparr>N \<prec>' (Q \<parallel> R)"   and "xvec \<sharp>* yvec" obtains T :: "('d, 'e, 'f) psi" and p :: "(name \<times> name) list" where "set p \<subseteq> set xvec \<times> set yvec" and "P = (p \<bullet> Q) \<parallel> T" and "\<lparr>\<nu>*xvec\<rparr>M \<prec>' T = \<lparr>\<nu>*yvec\<rparr>N \<prec>' R"
tab\<^sub>0_in_hom_1: fixes r :: "'a" assumes "ide r" shows "\<guillemotleft>tab\<^sub>0 r : src (tab\<^sub>0 r) \<rightarrow> src r\<guillemotright>"
f_def: shows "f = (\<lambda>r. {i \<in> I. r < Sigma_Algebra.measure M (A i)})"
geom_spick_Suc: fixes A :: "bool stream set"   and n :: "nat"   and x :: "bool" assumes "A \<in> {geom_proc (Suc n) -` B |B. B \<in> sets borel}" shows "(\<lambda>w. spick w n x) -` A \<in> {geom_proc n -` B |B. B \<in> sets borel}"
strict_implies_order: fixes a :: "'b::order"   and b :: "'b::order" assumes "(a::'b::order) < (b::'b::order)" shows "(a::'b::order) \<le> (b::'b::order)"
trivial_limit_at_left_real: fixes x :: "'b" shows "at_left x \<noteq> bot"
R'_def: shows "R' \<equiv> SOME R'. hd (tl R'_walk) \<leadsto>R'\<leadsto> z \<and> set R' \<subseteq> set (tl R'_walk)"
poly: fixes C :: "'a set" assumes "C \<in> \<M>" shows "polytope C"
UNIV_set_less_eq: fixes A :: "'a set" shows "(UNIV \<sqsubseteq> A) = (A = UNIV)"
ax_MethdN: fixes G :: "prog"   and P :: "'b assn"   and C :: "qtname"   and sig :: "sig"   and Q :: "'b assn"   and A :: "'b triple set" assumes "G,insert ({Normal P} Methd C sig-\<succ> {Q}) A\<turnstile>{Normal P} body G C sig-\<succ> {Q}" shows "G,A\<turnstile>{Normal P} Methd C sig-\<succ> {Q}"
folding: fixes C :: "'a set" assumes "folding_g.chamber C"   and "C \<in> f \<turnstile> X" shows "\<exists>!D. folding_g.chamber D \<and> D \<notin> f \<turnstile> X \<and> f ` D = C"
rel_map_1: fixes Sb :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'d \<Rightarrow> 'a"   and x :: "('c, 'd) fmap"   and y :: "('c, 'b) fmap" shows "fmrel Sb (fmmap i x) y = fmrel (\<lambda>x. Sb (i x)) x y"
le_Derives1_take: fixes i :: "nat"   and j :: "nat"   and a :: "symbol list"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "i \<le> j"   and "Derives1 a j r b" shows "take i a = take i b"
aux0_5: fixes x :: "('a, 'b) Combinators list" assumes "\<nexists>a b. a \<oplus> b \<in> set x" shows "singleCombinators x"
modelAppendTautology: fixes valuation :: "Literal list"   and F :: "Literal list list"   and c :: "Literal list" assumes "model valuation F"   and "clauseTautology c"   and "vars F \<union> vars c \<subseteq> vars valuation" shows "model valuation (F @ [c])"
sets: fixes i :: "'a" assumes "i \<in> I" shows "E i \<in> sets (M i)"
omega_closed_add: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "\<alpha> \<in>\<^sub>\<circ> \<omega>"   and "\<beta> \<in>\<^sub>\<circ> \<omega>" shows "\<alpha> + \<beta> \<in>\<^sub>\<circ> \<omega>"
symmetric_top_closed: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and conv :: "'c \<Rightarrow> 'c" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv" shows "conv top = top"
summable_exp: fixes x :: "'a" shows "summable (\<lambda>n. inverse (fact n) * x ^ n)"
simps_3: fixes e :: "(name, nat) finfun"   and t :: "trm" shows "eval_trm e (suc t) = Suc (eval_trm e t)"
zero_closed: fixes R :: "('h, 'i) ring_scheme"   and N :: "'j set"   and M :: "('h, 'j, 'k) module_scheme" assumes "LinearCombinations.submodule R N M" shows "\<zero>\<^bsub>M\<^esub> \<in> N"
Cons_eq_filter_iff: fixes x :: "'g"   and xs :: "'g list"   and P :: "'g \<Rightarrow> bool"   and ys :: "'g list" shows "(x # xs = filter P ys) = (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
filter_mset_eq: fixes L :: "'a"   and A :: "'a multiset" shows "filter_mset ((=) L) A = replicate_mset (count A L) L"
fds_converges_imp_abs_converges: fixes f :: "'a fds"   and s :: "'a"   and s' :: "'a" assumes "fds_converges f s"   and "s \<bullet> (1::'a) + 1 < s' \<bullet> (1::'a)" shows "fds_abs_converges f s'"
bin_eqI: fixes x :: "int"   and y :: "int" assumes "\<And>n. bit x n = bit y n" shows "x = y"
node_ptr_kinds_eq_h: shows "node_ptr_kinds h = node_ptr_kinds h2"
irreflD: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "RPRs.irrefl A r"   and "x \<^bsub>r\<^esub>\<preceq> y" shows "hasw [x, y] A"
star_subid: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and star :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.left_near_kleene_algebra plus times one less_eq less star"   and "less_eq x one" shows "star x = one"
map_is_empty: fixes empty :: "'a"   and update :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and delete :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and invar :: "'a \<Rightarrow> bool"   and lookup :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and is_empty :: "'a \<Rightarrow> bool"   and getmin :: "'a \<Rightarrow> 'b \<times> 'c"   and m :: "'a" assumes "PrioMap empty update delete invar lookup is_empty getmin"   and "invar m" shows "is_empty m = (lookup m = Map.empty)"
one_notin_unit_dic: shows "1\<^sub>h \<notin> unit_disc"
cases: fixes a :: "'a llist" assumes "lsorted a" obtains "a = LNil"   | x :: "'a" where "a = LCons x__ LNil"   | x :: "'a" and y :: "'a" and xs :: "'a llist" where "a = LCons x (LCons y xs)" and "x \<le> y" and "lsorted (LCons y xs)"
1: shows "v\<^sup>T * w\<^sup>T\<^sup>\<star> \<le> v\<^sup>T"
hyps_4: shows "sp_fundef F1 gd (take n (body gd)) = Ok (map typeof \<Sigma>2)"
big_mono: fixes f :: "'m \<Rightarrow> 'n"   and g :: "'m \<Rightarrow> 'n"   and F :: "'m filter" assumes "\<forall>\<^sub>F x in F. norm (f x) \<le> norm (g x)" shows "f \<in> O[F](g)"
cases: fixes x :: "int iarray iarray \<times> nat" obtains dmus :: "int iarray iarray" and i :: "nat" where "x = (dmus, i)"
simps_2: fixes b :: "bexp"   and c1 :: "'a acom"   and c2 :: "'a acom"   and S :: "'a" shows "sub\<^sub>1 (IF b THEN c1 ELSE c2 {S}) = c1"
flin_independentD_all_scalars: fixes rs :: "'f list"   and ms :: "'v list" assumes "set rs \<subseteq> UNIV"   and "set ms \<subseteq> V"   and "length rs \<le> length ms"   and "R_scalar_mult.R_lin_independent UNIV (\<sharp>\<cdot>) ms"   and "rs \<bullet>\<sharp>\<cdot> ms = (0::'v)" shows "set rs \<subseteq> 0"
intro: fixes carrier :: "'c set"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "total_preorder_on carrier le"   and "finite_total_preorder_on_axioms carrier" shows "finite_total_preorder_on carrier le"
elims_2: fixes x :: "const"   and xa :: "const" assumes "x \<le> xa" obtains x :: "const" and y :: "const" where "xa__ = x" and "xa = y" and "y = Any \<or> x = y"
var_def: fixes lit :: "lit" shows "var lit = (case lit of Lit x1 x2 \<Rightarrow> x2)"
Int_empty_left: fixes B :: "'d set" shows "{} \<inter> B = {}"
L_infinity_AE_bound: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "f \<in> space\<^sub>N (\<LL> \<infinity> M)" shows "AE x in M. \<bar>f x\<bar> \<le> Norm (\<LL> \<infinity> M) f"
abs_eq: fixes x :: "('a \<times> 'a) set" shows "converse (rel_of_set x) = rel_of_set (x\<inverse>)"
inf_right_idem: fixes x :: "'k multiset"   and y :: "'k multiset" shows "x \<inter># y \<inter># y = x \<inter># y"
simps_2: fixes lens_get :: "'a \<Rightarrow> 'b"   and lens_put :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and more :: "'c" shows "put\<^bsub>\<lparr>lens_get = lens_get, lens_put = lens_put, \<dots> = more\<rparr>\<^esub> = lens_put"
step: fixes f :: "'c"   and x :: "'b"   and h :: "heap"   and h' :: "heap"   and r :: "'a" assumes "\<And>x h h' r. effect (U f x) h h' r \<Longrightarrow> P x h h' r"   and "effect (U (F f) x) h h' r" shows "P x h h' r"
Posix_One: shows "[] \<in> One \<rightarrow> Void"
padic_add_comm: fixes p :: "int"   and x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "prime p"   and "x \<in> carrier (padic_int p)"   and "y \<in> carrier (padic_int p)" shows "x \<oplus>\<^bsub>padic_int p\<^esub> y = y \<oplus>\<^bsub>padic_int p\<^esub> x"
projecting_constrains: fixes C :: "'c program \<Rightarrow> 'c set"   and F :: "'a program"   and A :: "'a set"   and B :: "'a set" shows "projecting C h F (extend_set h A co extend_set h B) (A co B)"
merges_same_conv: fixes xs :: "'a list"   and n :: "nat"   and ps :: "(nat \<times> 'a) list" assumes "xs \<in> list n A"   and "\<And>x. x \<in> set ps \<Longrightarrow> case x of (p, x) \<Rightarrow> p < length xs \<and> x \<in> A" shows "(merges f ps xs = xs) = (\<forall>(p, x)\<in>set ps. x \<sqsubseteq>\<^bsub>r\<^esub> xs ! p)"
sym_digraphI: assumes "symmetric (with_proj G)" shows "sym_digraph (with_proj G)"
sub: shows "\<Union> (SFV \<sigma> ` (SDom \<sigma> \<inter> SIGP a)) \<subseteq> - U"
mult_left_cancel: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_no_zero_divisors_cancel plus zero times"   and "c \<noteq> zero" shows "(times c a = times c b) = (a = b)"
single_non_fresh_ordLess_var: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort" assumes "good X" shows "|{x. \<not> fresh xs x X}| <o |UNIV|"
set_zip_rightD: fixes x :: "'c"   and y :: "'d"   and xs :: "'c list"   and ys :: "'d list" assumes "(x, y) \<in> set (zip xs ys)" shows "y \<in> set ys"
wf_induct: fixes r :: "('k \<times> 'k) set"   and P :: "'k \<Rightarrow> bool"   and a :: "'k" assumes "wf r"   and "\<And>x. \<forall>y. (y, x) \<in> r \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
filterlim_at_top_dense: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "filterlim f at_top F = (\<forall>Z. \<forall>\<^sub>F x in F. Z < f x)"
is_process5_sing: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set"   and s :: "'a event list"   and x :: "'a event" assumes "is_process P"   and "(s, {x}) \<notin> FAILURES P"   and "(s, {}) \<in> FAILURES P" shows "(s @ [x], {}) \<in> FAILURES P"
fold_congs_4: fixes r :: "('e, 'f, 'g, 'h) SLin_state_scheme"   and r' :: "('e, 'f, 'g, 'h) SLin_state_scheme"   and v' :: "'f \<Rightarrow> SLin_status"   and f :: "('f \<Rightarrow> SLin_status) \<Rightarrow> 'f \<Rightarrow> SLin_status"   and f' :: "('f \<Rightarrow> SLin_status) \<Rightarrow> 'f \<Rightarrow> SLin_status" assumes "r = r'"   and "status r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "status_update f r = status_update f' r'"
fps_mult_code: fixes x :: "'b ratfps"   and y :: "'b ratfps" shows "fps_of_ratfps x * fps_of_ratfps y = fps_of_ratfps (x * y)"
edges_of_walk_in_E: fixes xs :: "'a list" assumes "H.walk xs" shows "H.edges_of_walk xs \<subseteq> E\<^bsub>H\<^esub>"
assms_3: shows "rk {A, B, \<gamma>} = 2"
k_mkAlgAuto_implements: shows "SPR.implements SPRsingle.k_mkAlgAuto"