hcju/isabelleps huggingface dataset:
nprv_ptrmE_uni: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and \<sigma> :: "'c"   and n :: "nat"   and F :: "'c set"   and t1 :: "'b"   and t2 :: "'b"   and \<psi> :: "'c" assumes "Deduct_with_False_Disj_Rename var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv"   and "\<sigma> \<in> Deduct_with_False_Disj_Rename.ptrm var fmla Var FvarsT Fvars eql cnj imp all exi prv n"   and "Deduct_with_False_Disj.nprv cnj imp fls prv F (subst \<sigma> t1 (Generic_Syntax.Variable var 0))"   and "Deduct_with_False_Disj.nprv cnj imp fls prv F (subst \<sigma> t2 (Generic_Syntax.Variable var 0))"   and "Deduct_with_False_Disj.nprv cnj imp fls prv (insert (eql t1 t2) F) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<psi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm" shows "Deduct_with_False_Disj.nprv cnj imp fls prv F \<psi>"
rotate_conv_mod': fixes n :: "nat"   and ls :: "'a list" shows "rotate (n mod |ls|) ls = rotate n ls"
assms_2: fixes f :: "rat poly"   and p :: "nat" assumes "(f, p) \<in> set fs" shows "f \<in> set ftrs"
IS_LEFT_UNIQUE_string_rel: shows "IS_LEFT_UNIQUE string_rel"
f_finite: fixes e :: "'v \<times> 'v" shows "f e \<noteq> \<top>"
bind_eq_Some_conv: fixes f :: "'c option"   and g :: "'c \<Rightarrow> 'b option"   and x :: "'b" shows "(Option.bind f g = Some x) = (\<exists>y. f = Some y \<and> g y = Some x)"
reachable_awalk: fixes u :: "'a"   and v :: "'a" shows "(u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v) = (\<exists>p. H.awalk u p v)"
F_def: shows "F = (\<lambda>k. if inverse (real (Suc k)) \<le> c then c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1) else 0)"
simps_2: fixes x :: "udom"   and ys :: "udom\<cdot>maybe" shows "plusU\<cdot>(Just\<cdot>x)\<cdot>ys = Just\<cdot>x"
Spy_see_shrK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> otway" shows "(Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
lnth_ltake: fixes m :: "nat"   and n :: "enat"   and xs :: "'b llist" assumes "enat m < n" shows "ltake n xs $ m = xs $ m"
simps_16: fixes a :: "floatarith" shows "max_Var_floatarith (Ln a) = max_Var_floatarith a"
quotients_length: shows "length comp\<GG>.quotients + 1 = length \<GG>"
sorted_add_spairs: fixes ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sorted_wrt pair_ord ps" shows "sorted_wrt pair_ord (add_spairs ps bs p)"
null_Boolean_def: shows "null = (\<lambda>\<tau>. \<lfloor>\<bottom>\<rfloor>)"
multiplicative_dirichlet_prodD1: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "multiplicative_function (dirichlet_prod f g)"   and "multiplicative_function f"   and "g 0 = (0::'b)" shows "multiplicative_function g"
fps_lr_inverse_eq0_imp_starting0_1: fixes f :: "'a fps"   and x :: "'a" assumes "fps_left_inverse f x = 0" shows "x = (0::'a)"
xy_2: shows "y /\<^sub>R (1 - k) \<in> S"
lsc_iff: fixes f :: "'b \<Rightarrow> ereal" shows "lsc f = (\<forall>y. closed {x. f x \<le> y}) \<and> lsc f = closed (Epigraph UNIV f)"
vconn_implies_hpar_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b"   and f :: "'b"   and g :: "'b" assumes "horizontal_homs V src trg"   and "partial_magma.in_hom V \<mu> f g" shows "src \<mu> = src f"
lim: fixes x :: "int" assumes "a \<le> real_of_int x" shows "((\<lambda>y. EM_remainder' n f (real_of_int x) (real_of_int y)) \<longlongrightarrow> EM_remainder n f x) at_top"
that: assumes "\<sigma>' = Some (snd (Array.alloc xs (the_state \<sigma>)))"   and "r = fst (Array.alloc xs (the_state \<sigma>))"   and "Array.get (the_state \<sigma>') r = xs" shows "thesis"
split_sel: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and option :: "'b option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (option = Some (the option) \<longrightarrow> P (f2 (the option))))"
igVarIPresIGWls_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igVarIPresIGWls MOD \<equiv> \<forall>xs x. igWls MOD (asSort xs) (igVar MOD xs x)"
eucl_rel_poly_iff: fixes x :: "'b poly"   and y :: "'b poly"   and q :: "'b poly"   and r :: "'b poly" shows "eucl_rel_poly x y (q, r) = (x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y))"
prod_inversef: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.field divide inverse times one plus zero minus uminus" shows "comm_monoid_mult.prod times one (inverse \<circ> f) A = inverse (comm_monoid_mult.prod times one f A)"
map_comp_map: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" shows "map f \<circ> map g = map (f \<circ> g)"
pre_id_unch: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "STUTINV F" shows "|~ F \<and> Unchanged id \<longrightarrow> \<circle>F"
B: shows "B = (if i_end = l then A else step_3_c_inner_loop (A $$ (i_end, k) / x) l i_end (Suc i_end - i_begin) A)"
combine_in_carrier: fixes Ks :: "'a list"   and Us :: "'a list" assumes "set Ks \<subseteq> carrier R"   and "set Us \<subseteq> carrier R" shows "combine Ks Us \<in> carrier R"
coplanar_linear_image: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "coplanar S"   and "linear f" shows "coplanar (f ` S)"
simple_function_restrict_space: fixes \<Omega> :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "\<Omega> \<inter> space M \<in> sets M" shows "simple_function (restrict_space M \<Omega>) f = simple_function M (\<lambda>x. indicat_real \<Omega> x *\<^sub>R f x)"
fpxs_nth_val_nonzero: fixes f :: "'b fpxs" assumes "f \<noteq> 0" shows "fpxs_nth f (fpxs_val f) \<noteq> (0::'b)"
pell_valuation_0_iff: fixes D :: "nat"   and z :: "int \<times> int" assumes "pell D" shows "(pell.pell_valuation D z = 0) = (z = (0, 0))"
tf_def: shows "x \<in>\<^sub>i range_tm e' tf"
FAcc_\<tau>red1t_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and F :: "String.literal"   and D :: "String.literal" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h (e\<bullet>F{D}, xs) (e'\<bullet>F{D}, xs')"
2: fixes z :: "complex" assumes "z \<in> ball p t" shows "cmod (deriv f z) \<le> 2 * cmod (deriv f p)"
var_pow_closed: fixes K :: "'a list set"   and n :: "nat" assumes "subring K (K [X])" shows "X\<^bsub>K [X]\<^esub> [^]\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> n \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"
monad_zero_minus_iff: fixes x :: "'a star" shows "(x \<in> monad 0) = (- x \<in> monad 0)"
path_in_edges: fixes v :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "is_trail v ps v'" shows "set ps \<subseteq> E"
sign_simps_44: fixes am :: "'a"   and bm :: "'a"   and ck :: "'a" shows "am * (bm - ck) = am * bm - am * ck"
n_mult_lb1: fixes x :: "'a"   and y :: "'a" shows "n x \<cdot> n y \<sqsubseteq> n x"
abs_eq: fixes xa :: "'b \<Rightarrow> nat"   and x :: "'b \<Rightarrow> nat" assumes "eq_onp (\<lambda>x. x \<in> multiset) xa xa"   and "eq_onp (\<lambda>x. x \<in> multiset) x x" shows "Abs_multiset xa + Abs_multiset x = Abs_multiset (\<lambda>a. xa a + x a)"
domintros: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and C :: "acom" assumes "bval b s \<Longrightarrow> times_dom (postQ C s, b, C)" shows "times_dom (s, b, C)"
strict_coboundedI2: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "b < c" shows "inf a b < c"
to_nat_eq: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "(mod_type.to_nat Rep x = mod_type.to_nat Rep y) = (x = y)"
inj_on_subset: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "inj_on f A"   and "B \<subseteq> A" shows "inj_on f B"
initial\<^sub>v_axioms_def: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> 'd \<times> nat set \<times> 'e) list"   and valid :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'f list \<times> 'd \<times> nat set \<times> ('g \<Rightarrow> 'h option)) list \<Rightarrow> bool" shows "initial\<^sub>v_axioms ts valid \<equiv> (\<forall>i p is xs sb \<D> \<O> \<R>. i < length ts \<longrightarrow> ts ! i = (p, is, xs, sb, \<D>, \<O>, \<R>) \<longrightarrow> is = []) \<and> valid (map (\<lambda>(p, is, \<theta>, sb, \<D>, \<O>, \<R>). (p, is, \<theta>, [], \<D>, \<O>, Map.empty)) ts)"
vars\<^sub>s\<^sub>s\<^sub>t_append: fixes A :: "('f, 'e) stateful_strand_step list"   and B :: "('f, 'e) stateful_strand_step list" shows "vars\<^sub>s\<^sub>s\<^sub>t (A @ B) = vars\<^sub>s\<^sub>s\<^sub>t A \<union> vars\<^sub>s\<^sub>s\<^sub>t B"
represents_mult: fixes p :: "int poly"   and x :: "'a"   and q :: "int poly"   and y :: "'a" assumes "p represents x"   and "q represents y"   and "poly q 0 \<noteq> 0" shows "poly_mult p q represents x * y"
fps_of_poly_normalize_field: fixes p :: "'b poly" shows "fps_of_poly (normalize p) = fps_of_poly p * fps_const (inverse (lead_coeff p))"
last: shows "last (last Cs # rev Ds') = last (Cs @ tl (last Cs # rev Ds'))"
ide_in_hom: fixes a :: "'a" shows "\<phi>.A.ide a = \<guillemotleft>a : a \<rightarrow>\<^sub>A a\<guillemotright>"
lens_comp_lb: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'e \<Longrightarrow> 'f" assumes "vwb_lens X" shows "X ;\<^sub>L Y \<subseteq>\<^sub>L Y"
rel_induct: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a list"   and y :: "'b list"   and Q :: "'a list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "list_all2 R x y"   and "Q [] []"   and "\<And>a21 a22 b21 b22. \<lbrakk>R a21 b21; Q a22 b22\<rbrakk> \<Longrightarrow> Q (a21 # a22) (b21 # b22)" shows "Q x y"
conj_idem: fixes P :: "'d upred" shows "(P \<and> P) = P"
zero_not_infinity: fixes x :: "('a, 'b) IEEE.float" assumes "IEEE.is_zero x" shows "\<not> is_infinity x"
Tinv_quasi_measure_preserving: assumes "invertible_qmpt" shows "Tinv \<in> quasi_measure_preserving M M"
c1: assumes "ps \<noteq> []"   and "ls = []" shows "thesis"
same_suffix: shows "drop (j + 1) (t[j - 1 := t ! j, j := t ! (j - 1)]) = drop (j + 1) t"
ordinal_mono_oLog: fixes b :: "ordinal" shows "mono (oLog b)"
assms_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> (y \<cdot> z) = x \<cdot> y \<cdot> z"
liftAll2_lift_ext: fixes inp :: "'a \<Rightarrow> 'b option"   and inp' :: "'a \<Rightarrow> 'b option"   and f :: "'b \<Rightarrow> 'c" shows "(sameDom inp inp' \<and> liftAll2 (\<lambda>v v'. f v = f v') inp inp') = (lift f inp = lift f inp')"
injective_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<sqinter> id x \<le> top" shows "x \<sqinter> y \<sqinter> id (x \<sqinter> y) \<le> top"
Abinds_reorder: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and \<Delta> :: "(Vars.var \<times> exp) list" assumes "map_of \<Gamma> = map_of \<Delta>" shows "ABinds \<Gamma> = ABinds \<Delta>"
simps_1: fixes x21 :: "dag"   and x22 :: "ref"   and x23 :: "dag"   and y21 :: "dag"   and y22 :: "ref"   and y23 :: "dag" shows "(Node x21 x22 x23 = Node y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
power_commutes: fixes a :: "'a"   and n :: "nat" shows "power.power 1' (;) a n ; a = a ; power.power 1' (;) a n"
f_def: shows "f = (\<lambda>(x, y) b. if A = {} then undefined else if b then x else y)"
compat: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "snd (g x y)"   and "fst (g y z)" shows "fst (g x z)"
sg2: shows "if hd (y t) < 700 then z t = [] \<and> l t = One else z t = [Zero] \<and> l t = Zero"
index_add_mat_2: fixes A :: "'b mat"   and B :: "'b mat" shows "dim_row (A + B) = dim_row B"
ideduct_restricted_mono: fixes M :: "('fun, 'var) Term.term set"   and P :: "('fun, 'var) Term.term \<Rightarrow> bool"   and t :: "('fun, 'var) Term.term"   and M' :: "('fun, 'var) Term.term set" assumes "\<langle>M;P\<rangle> \<turnstile>\<^sub>r t"   and "M \<subseteq> M'" shows "\<langle>M';P\<rangle> \<turnstile>\<^sub>r t"
list_graph_to_graph_def: fixes G :: "'a list_graph" shows "list_graph_to_graph G = \<lparr>nodes = set (nodesL G), edges = set (edgesL G)\<rparr>"
cong': fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "sum' g A = sum' h B"
mono: fixes c :: "'c"   and a :: "'c"   and d :: "'c"   and b :: "'c" assumes "c \<sqsubseteq> a"   and "d \<sqsubseteq> b" shows "c \<squnion> d \<sqsubseteq> a \<squnion> b"
set_rw: shows "insert (k - 1) ({0..<dim_row (mat_delete (make_mat n k B) 0 0)} - {k - 1}) = {0..<dim_row (mat_delete (make_mat n k B) 0 0)}"
rtrancl_path_distinct: fixes r :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and xs :: "'k list"   and y :: "'k" assumes "rtrancl_path r x xs y" obtains xs' :: "'k list" where "rtrancl_path r x xs' y" and "distinct (x # xs')" and "set xs' \<subseteq> set xs"
nhds_discrete_open: fixes x :: "'a" assumes "open {x}" shows "nhds x = principal {x}"
cos_angle: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "cos (angle a b c) = (a - b) \<bullet> (c - b) / (dist a b * dist c b)"
lfp_dual_var: fixes f :: "'b \<Rightarrow> 'b" shows "\<partial> (lfp f) = gfp (\<partial>\<^sub>F f)"
bind_apply_cong: fixes f :: "'d \<Rightarrow> ('c \<times> 'd) set \<times> bool"   and s :: "'d"   and f' :: "'d \<Rightarrow> ('c \<times> 'd) set \<times> bool"   and s' :: "'d"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'd) set \<times> bool"   and g' :: "'c \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'd) set \<times> bool" assumes "f s = f' s'"   and "\<And>rv st. (rv, st) \<in> fst (f' s') \<Longrightarrow> g rv st = g' rv st" shows "(f >>= g) s = (f' >>= g') s'"
scaleR_compose: fixes c :: "real"   and x :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e" shows "c *\<^sub>R x \<circ> f = c *\<^sub>R (x \<circ> f)"
bre_algo_def: fixes Qi :: "'d set"   and \<delta> :: "('d, 'e) ta_rule set" shows "bre_algo Qi \<delta> \<equiv> \<lparr>wa_cond = bre_cond Qi, wa_step = br_step \<delta>, wa_initial = {br_initial \<delta>}, wa_invar = br_invar \<delta>\<rparr>"
Id_onE: fixes c :: "'b \<times> 'b"   and A :: "'b set" assumes "c \<in> s2r A" obtains x :: "'b" where "x \<in> A" and "c = (x, x)"
face_cycle_set_self: fixes a :: "'b" shows "a \<in> GM.face_cycle_set a"
induct: fixes P :: "\<I> \<Rightarrow> nat \<Rightarrow> (nat \<times> event_data option list set \<times> event_data option list set) list \<Rightarrow> bool"   and a0 :: "\<I>"   and a1 :: "nat"   and a2 :: "(nat \<times> event_data option list set \<times> event_data option list set) list" assumes "\<And>I nt. P I nt []"   and "\<And>I nt t a1 a2 aux. (enat t + right I < enat nt \<Longrightarrow> P I nt aux) \<Longrightarrow> P I nt ((t, a1, a2) # aux)" shows "P a0 a1 a2"
ts_ord_mult_isor: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "n x \<sqsubseteq> n y" shows "n x \<cdot> n z \<sqsubseteq> n y \<cdot> n z"
dirichlet_inverse_induct: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P 0"   and "P (Suc 0)"   and "\<And>n. \<lbrakk>1 < n; \<And>k. k < n \<Longrightarrow> P k\<rbrakk> \<Longrightarrow> P n" shows "P n"
extend_1_eval: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e'' :: "char list exp"   and s'' :: "heap \<times> sfields \<times> sheap"   and b'' :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e'',s'',b''\<rangle>"   and "P,shp s \<turnstile>\<^sub>b (e,b) \<surd>"   and "P \<turnstile> \<langle>e'',s''\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
the_apartment_isoD_2: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and A' :: "'b set set"   and C :: "'b set" assumes "ChamberComplexWithApartmentSystem X \<A>"   and "A \<in> \<A>"   and "A' \<in> \<A>"   and "SimplicialComplex.maxsimp X C"   and "C \<in> A \<inter> A'" shows "fixespointwise (ChamberComplexWithApartmentSystem.the_apartment_iso A A') (\<Union> (A \<inter> A'))"
fst_not_snd: fixes P :: "'b trm"   and Q :: "'b trm" shows "Fst P \<noteq> Snd Q"
1: shows "xs !! (LEAST n. xs !! n = x) = x"
is_weakly_dom_component_safe_impl: shows "is_weakly_dom_component_safe = local.a_is_weakly_dom_component_safe"
axioms_1: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "group_hom G H h" shows "Group.group G"
gallery_least_length: fixes X :: "'b set set"   and C :: "'b set"   and D :: "'b set" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D"   and "C \<noteq> D" shows "SimplicialComplex.maxsimpchain X (C # (ARG_MIN length Cs. SimplicialComplex.maxsimpchain X (C # Cs @ [D])) @ [D])"
abs_eq: fixes x :: "'a set" shows "fds x = Abs_scene (\<lambda>f g. override_on f g x)"
not_is_pole_holomorphic: fixes A :: "complex set"   and x :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "open A"   and "x \<in> A"   and "f holomorphic_on A" shows "\<not> is_pole f x"
bind_gpv_Pause: fixes out :: "'b"   and rpv :: "('d, 'b, 'c) rpv"   and f :: "'d \<Rightarrow> ('a, 'b, 'c) gpv" shows "Generative_Probabilistic_Value.Pause out rpv \<bind> f = Generative_Probabilistic_Value.Pause out (rpv \<bind> f)"
run_put_nondet: fixes put :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and s :: "'f"   and m :: "('e, 'd) nondetT" shows "run_nondet (put_nondet put s m) = put s (run_nondet m)"
functor_axioms: shows "functor (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>A\<^sub>1) A1.map"
dom_map_option_comp: fixes g :: "'d \<Rightarrow> 'c"   and m :: "'b \<Rightarrow> 'd option" shows "dom (map_option g \<circ> m) = dom m"
is_gcd_def: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and gcd' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.GCD_ring times one plus zero minus uminus" shows "GCD_ring.is_gcd times gcd' = (\<forall>a b. dvd.dvd times (gcd' a b) a \<and> dvd.dvd times (gcd' a b) b \<and> (\<forall>d'. dvd.dvd times d' a \<and> dvd.dvd times d' b \<longrightarrow> dvd.dvd times d' (gcd' a b)))"
Abs_BinomialHeap_inverse: fixes y :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "y \<in> {q. BinomialHeapStruc.invar q}" shows "Rep_BinomialHeap (Abs_BinomialHeap y) = y"
assoc'_simps_5: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "C.cod \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, h] = (C.cod f \<star>\<^sub>C C.cod g) \<star>\<^sub>C C.cod h"
simps: fixes a :: "'b" shows "the_Inl (Inl a) = a"
t1_space: fixes "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.t1_space open"   and "x \<noteq> y" shows "\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
null_space_eq_left_null_space_transpose: fixes A :: "(('b, 'c) vec, 'd) vec" shows "null_space A = left_null_space (Finite_Cartesian_Product.transpose A)"
n_add_ub1: fixes x :: "'a"   and y :: "'a" shows "n x \<sqsubseteq> n x \<oplus> n y"
qe_eq_repeat_helper_eval_case1: fixes xs :: "real list"   and var :: "nat"   and A :: "real mpoly list"   and L :: "atom list"   and F :: "atom fm list"   and x :: "real"   and \<Gamma> :: "real list" assumes "length xs = var"   and "\<forall>p\<in>set A. MPoly_Type.degree p var = 1 \<or> MPoly_Type.degree p var = 2"   and "eval (list_conj (map (fm.Atom \<circ> Eq) A @ map fm.Atom L @ F)) (xs @ x # \<Gamma>)" shows "eval (qe_eq_repeat_helper var A L F) (xs @ x # \<Gamma>)"
that: fixes c1 :: "('a, 'e) chart"   and c2 :: "('b, 'f) chart" assumes "c1 \<in> src.atlas"   and "c2 \<in> dest.atlas"   and "x \<in> domain c1"   and "f ` domain c1 \<subseteq> domain c2"   and "k-smooth_on (codomain c1) (apply_chart c2 \<circ> f \<circ> inv_chart c1)" shows "thesis"
assms_2: shows "C \<notin> f \<turnstile> X"
take_bit_of_nat: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and divide :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and modulo :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and bit :: "'k \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and drop_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and take_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and n :: "nat"   and m :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit n (semiring_1.of_nat one plus zero m) = semiring_1.of_nat one plus zero (take_bit n m)"
pred_funI: fixes A :: "'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x. A x \<Longrightarrow> B (f x)" shows "pred_fun A B f"
simplt_islin: fixes t :: "tm" shows "islin (simplt t)"
induct: fixes x1 :: "nat"   and x2 :: "com"   and P :: "nat \<Rightarrow> com \<Rightarrow> bool" assumes "x1 \<turnstile>' x2"   and "\<And>l. P l SKIP"   and "\<And>x a l. \<lbrakk>sec a \<le> sec x; l \<le> sec x\<rbrakk> \<Longrightarrow> P l (x ::= a)"   and "\<And>l c\<^sub>1 c\<^sub>2. \<lbrakk>l \<turnstile>' c\<^sub>1; P l c\<^sub>1; l \<turnstile>' c\<^sub>2; P l c\<^sub>2\<rbrakk> \<Longrightarrow> P l (c\<^sub>1;; c\<^sub>2)"   and "\<And>b l c\<^sub>1 c\<^sub>2. \<lbrakk>sec b \<le> l; l \<turnstile>' c\<^sub>1; P l c\<^sub>1; l \<turnstile>' c\<^sub>2; P l c\<^sub>2\<rbrakk> \<Longrightarrow> P l (IF b THEN c\<^sub>1 ELSE c\<^sub>2)"   and "\<And>b l c. \<lbrakk>sec b \<le> l; l \<turnstile>' c; P l c\<rbrakk> \<Longrightarrow> P l (WHILE b DO c)"   and "\<And>l c l'. \<lbrakk>l \<turnstile>' c; P l c; l' \<le> l\<rbrakk> \<Longrightarrow> P l' c" shows "P x1 x2"
takeWhile_def: shows "takeWhile \<equiv> \<lambda>uu uua. rec_list (\<lambda>P. []) (\<lambda>x xs xsa P. if P x then x # xsa P else []) uua uu"