hcju/isabelleps huggingface dataset:
dim_col_mat_1: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'm" shows "dim_col (mat nr nc f) = nc"
split_asm: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'c"   and sum :: "'d + 'e" shows "P (case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = (\<not> ((\<exists>x1. sum = Inl x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x2. sum = Inr x2 \<and> \<not> P (f2 x2))))"
unfold_rtrancl2: fixes R :: "('b \<times> 'b) set" shows "R\<^sup>* = Id \<union> R O R\<^sup>*"
connected_continuous_image: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "continuous_on s f"   and "connected s" shows "connected (f ` s)"
awalk_induce: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and S :: "'a set" assumes "awalk u p v"   and "set (awalk_verts u p) \<subseteq> S" shows "pre_digraph.awalk (G \<restriction> S) u p v"
\<N>covers: fixes C :: "'a set"   and x :: "'a" assumes "C \<in> \<M> \<and> x \<in> C" shows "\<exists>D. D \<in> \<N> \<and> x \<in> D \<and> D \<subseteq> C"
eq: shows "wlp (While M S) P = adjoint (M 0) * P * M 0 + adjoint (M 1) * wlp S (wlp (While M S) P) * M 1"
assms_2: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> I" shows "v11 (f \<restriction>\<^sup>l\<^sub>\<circ> A i)"
inter_guards_execn_Fault: fixes c1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and s :: "('d, 'f) xstate"   and n :: "nat"   and f :: "'f" assumes "(c1 \<inter>\<^sub>g c2) = Some c"   and "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> Fault f" shows "\<Gamma>\<turnstile> \<langle>c1,s\<rangle> =n\<Rightarrow> Fault f \<or> \<Gamma>\<turnstile> \<langle>c2,s\<rangle> =n\<Rightarrow> Fault f"
functor_category_axioms: shows "functor_category (\<cdot>\<^sub>J) (\<cdot>)"
le_type_scheme_def: fixes sch' :: "type_scheme"   and sch :: "type_scheme" shows "(sch' \<le> sch) = (\<forall>t. t <| sch' \<longrightarrow> t <| sch)"
vsv_vinsert: fixes a :: "V"   and b :: "V" assumes "a \<notin>\<^sub>\<circ> \<D>\<^sub>\<circ> xs'" shows "vsv (vinsert \<langle>a, b\<rangle> xs')"
P2: shows "Col X Y P \<and> Col A B P \<and> P PerpAt X Y A B "
sset_streams: fixes s :: "'a stream"   and A :: "'a set" assumes "sset s \<subseteq> A" shows "s \<in> streams A"
cases: fixes r :: "'b set_arrow" obtains set_dom :: "'b set" and set_func :: "'b \<Rightarrow> 'b" and set_cod :: "'b set" where "r = \<lparr>set_dom = set_dom, set_func = set_func, set_cod = set_cod\<rparr>"
matching_Goto_1: shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[r], Undecided\<rangle> \<Rightarrow> t"
times_prod_def: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "x \<oplus> y = (fst x, snd x \<oplus> snd y)"
add_diff_inverse: fixes a :: "'d"   and b :: "'d" assumes "\<not> a < b" shows "b + (a - b) = a"
return_def: fixes x :: "'a" shows "return x = Heap_Monad.heap (Pair x)"
submoduleE_6: fixes H :: "(nat \<Rightarrow> 'a) set"   and x :: "nat \<Rightarrow> 'a" assumes "submodule H R P"   and "x \<in> H" shows "\<ominus>\<^bsub>P\<^esub> x \<in> H"
mem_rel_interior_ball: fixes x :: "'a"   and S :: "'a set" shows "(x \<in> rel_interior S) = (x \<in> S \<and> (\<exists>e>0. ball x e \<inter> affine hull S \<subseteq> S))"
create_element_known_ptr: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and tag :: "char list"   and new_element_ptr :: "(_) element_ptr" assumes "h \<turnstile> create_element document_ptr tag \<rightarrow>\<^sub>r new_element_ptr" shows "known_ptr (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr)"
Un_upper1: fixes A :: "'d set"   and B :: "'d set" shows "A \<subseteq> A \<union> B"
IH_2: shows "\<forall>scc\<in>sccs s. is_scc E scc"
fact_fact_dvd_fact: fixes k :: "nat"   and n :: "nat" shows "fact k * fact n dvd fact (k + n)"
minus: fixes x :: "real"   and xa :: "'a" shows "- x *\<^sub>R xa = - (x *\<^sub>R xa)"
path2_forget_last: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m" shows "g \<turnstile> n-ns\<rightarrow>last ns"
cases: fixes x :: "tm \<times> tm \<times> tm" obtains x :: "name" and y :: "name" and f :: "tm" and k :: "tm" and g :: "tm" where "atom x \<sharp> (y, f, k, g)" and "atom y \<sharp> (f, k, g)" and "xa__ = (f, k, g)"
simps: fixes \<phi> :: "'b ltl"   and \<nu> :: "'b set" shows "af_letter_opt \<phi> \<nu> = Unf (step \<phi> \<nu>)"
no_end_point_char_converse: fixes p :: "'a"   and x :: "'a" assumes "point p" shows "(x ; p \<noteq> 0) = (p \<le> x\<^sup>T ; 1)"
top_right_normalize: fixes a :: "'a" shows "gcd a (0::'a) = normalize a"
StepConf_HAStates: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status" assumes "TS \<in> HPT ST" shows "StepConf (HA ST) (Conf ST) TS \<subseteq> HAStates (HA ST)"
Cons: fixes x :: "'c"   and xs :: "'c list" shows "prod_list (x # xs) = x * prod_list xs"
zero_in_omega: shows "[]\<^sub>\<circ> \<in>\<^sub>\<circ> \<omega>"
static_portfolio_self_financing: fixes p :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('d, 'c) discrete_market" assumes "\<forall>x\<in>support_set p. \<forall>w i. p x i w = p x (Suc i) w" shows "self_financing Mkt p"
induct: fixes x :: "trm"   and P :: "trm \<Rightarrow> bool" assumes "SN x"   and "\<And>t. \<lbrakk>\<And>t'. t \<mapsto> t' \<Longrightarrow> SN t'; \<And>t'. t \<mapsto> t' \<Longrightarrow> P t'\<rbrakk> \<Longrightarrow> P t" shows "P x"
lebesgue_integral_count_space_finite_support: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite {a \<in> A. f a \<noteq> (0::'b)}" shows "integral\<^sup>L (count_space A) f = (\<Sum>a | a \<in> A \<and> f a \<noteq> (0::'b). f a)"
eq_iff: fixes a :: "nat"   and b :: "nat" shows "((a::nat) = (b::nat)) = (a dvd b \<and> b dvd a)"
terminates_take_raw: fixes g :: "'e \<Rightarrow> ('f, 'e) step" assumes "terminates g" shows "terminates (take_raw g)"
CI: shows "C \<subseteq> \<int>\<^sub>v"
AGM_rel_sel: fixes K :: "'a set"   and \<phi> :: "'a" shows "\<gamma>\<^sub>A\<^sub>G\<^sub>M\<^sub>T\<^sub>R K \<phi> \<equiv> if K .\<bottom>. \<phi> = {} then {K} else {B \<in> K .\<bottom>. \<phi>. \<forall>C\<in>K .\<bottom>. \<phi>. AGM_relation C K B}"
fine_single_boundaries: fixes \<delta> :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and D :: "(real \<times> real \<times> real) list"   and d :: "real"   and t :: "real"   and e :: "real" assumes "fine \<delta> (a, b) D"   and "D = [(d, t, e)]" shows "a = d \<and> b = e"
iteration_unfoldl_distl: fixes y :: "'a"   and x :: "'a" shows "y + y \<cdot> x \<cdot> x\<^sup>\<infinity> = y \<cdot> x\<^sup>\<infinity>"
lookup_env_cons: fixes b :: "'g"   and es :: "'h \<Rightarrow> ('f, 'g, 'h) env option"   and x :: "'h"   and xs :: "'h list" shows "lookup (Env b es) (x # xs) = (case es x of None \<Rightarrow> None | Some e \<Rightarrow> lookup e xs)"
inf_distribD_2: fixes f :: "'g \<Rightarrow> 'h" assumes "inf_distrib f" shows "cont f"
main_2: shows "{x. poly (root_unity n) x = 0} = {x. x ^ n = 1}"
quasi_isometry_onD_1: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "lambda C -quasi_isometry_on X f"   and "x \<in> X"   and "y \<in> X" shows "dist (f x) (f y) \<le> lambda * dist x y + C"
ParMerge_def: fixes tr1 :: "'c list"   and tr2 :: "'c list"   and cs :: "'c set" shows "ParMerge tr1 tr2 cs = set (ParMergel tr1 tr2 cs)"
inf_bot_left: fixes x :: "'f multiset" shows "{#} \<inter># x = {#}"
mutual_information_eq_entropy_conditional_entropy: fixes M :: "'a measure"   and b :: "real"   and X :: "'a \<Rightarrow> 'b"   and Y :: "'a \<Rightarrow> 'c" assumes "information_space M b"   and "simple_function M X"   and "simple_function M Y" shows "prob_space.mutual_information M b (count_space (X ` space M)) (count_space (Y ` space M)) X Y = prob_space.entropy M b (count_space (X ` space M)) X - prob_space.conditional_entropy M b (count_space (X ` space M)) (count_space (Y ` space M)) X Y"
2: assumes "u \<in> Fvars (rawpsubst \<phi> (zip (map Var us) (map snd txs)))" shows "False"
ideD_1: fixes a :: "'a \<times> 'a \<times> 'a" assumes "VVV.ide a" shows "VVV.arr a"
leM1: shows "adjoint M1 * M1 \<le>\<^sub>L 1\<^sub>m d"
elims_2: fixes x :: "bool list"   and xa :: "tree" assumes "path x xa" obtains T :: "tree" where "x = []" and "xa = T"   | d :: "bool" and ds :: "bool list" and T1 :: "tree" and T2 :: "tree" where "x = d # ds" and "xa = Branching T1 T2" and "if d then path ds T1 else path ds T2"
wqo_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "class.wqo less_eq less \<equiv> class.preorder less_eq less \<and> class.wqo_axioms less_eq"
diff_mult_distrib2: fixes V :: "'b set"   and x :: "'b"   and a :: "real"   and b :: "real" assumes "vectorspace V"   and "x \<in> V" shows "(a - b) \<cdot> x = a \<cdot> x - b \<cdot> x"
monom_a_inv: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and n :: "nat" assumes "UP_ring R"   and "a \<in> carrier R" shows "monom (UP R) (\<ominus>\<^bsub>R\<^esub> a) n = \<ominus>\<^bsub>UP R\<^esub> monom (UP R) a n"
v: shows "v \<in> V - attractor p (deadends p**) - attractor p** (deadends p****)"
poly_lift_hom_mult: fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e"   and p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "cring S"   and "\<phi> \<in> ring_hom R S"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)" shows "poly_lift_hom R S \<phi> (p \<otimes>\<^bsub>UP R\<^esub> q) = poly_lift_hom R S \<phi> p \<otimes>\<^bsub>UP S\<^esub> poly_lift_hom R S \<phi> q"
submoduleI: fixes H :: "'c set" assumes "H \<subseteq> carrier M"   and "\<zero>\<^bsub>M\<^esub> \<in> H"   and "\<And>a. a \<in> H \<Longrightarrow> \<ominus>\<^bsub>M\<^esub> a \<in> H"   and "\<And>a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<oplus>\<^bsub>M\<^esub> b \<in> H"   and "\<And>a x. \<lbrakk>a \<in> carrier R; x \<in> H\<rbrakk> \<Longrightarrow> a \<odot>\<^bsub>M\<^esub> x \<in> H" shows "submodule H R M"
case_4: fixes f1 :: "'a"   and f2 :: "bool \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> 'a"   and f4 :: "real \<Rightarrow> 'a"   and f5 :: "val \<Rightarrow> val \<Rightarrow> 'a"   and x4 :: "real" shows "(case RealVal x4 of UnitVal \<Rightarrow> f1 | BoolVal x \<Rightarrow> f2 x | IntVal x \<Rightarrow> f3 x | RealVal x \<Rightarrow> f4 x | <|x, xa|> \<Rightarrow> f5 x xa) = f4 x4"
iso_is_arr: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.iso f" shows "S.arr f"
all_valuate_zero: fixes p :: "linear_poly" assumes "\<And>v. p \<lbrace> v \<rbrace> = (0::'a)" shows "p = 0"
get_parent_is_l_get_parent: shows "l_get_parent DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs"
polynomial_product: fixes m :: "nat"   and a :: "nat \<Rightarrow> 'c"   and n :: "nat"   and b :: "nat \<Rightarrow> 'c"   and x :: "'c" assumes "\<And>i. m < i \<Longrightarrow> a i = (0::'c)"   and "\<And>j. n < j \<Longrightarrow> b j = (0::'c)" shows "(\<Sum>i\<le>m. a i * x ^ i) * (\<Sum>j\<le>n. b j * x ^ j) = (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. a k * b (r - k)) * x ^ r)"
is_zero_row_def: fixes i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec" shows "is_zero_row i A = is_zero_row_upt_k i (ncols A) A"
prj_simps_2: shows "C.dom \<mu>\<nu>.prj\<^sub>1 = \<mu>.leg0 \<down>\<down> \<nu>.leg1"
eq: shows "degree (prod f J) = (\<Sum>i\<in>J. degree (f i))"
assms: shows "((\<sigma>, k), \<sigma>', k') \<in> cp"
wordinterval_Union_def: fixes ws :: "'b wordinterval list" shows "wordinterval_Union ws = wordinterval_compress (foldr wordinterval_union ws Empty_WordInterval)"
restriction_mon2: fixes v :: "view"   and u :: "view"   and w :: "view"   and f :: "cars \<Rightarrow> nat_int"   and c :: "cars" shows "v=u--w \<longrightarrow> restrict w f c \<le> restrict v f c"
intro: fixes s\<^sub>0 :: "'a"   and lstep :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>s. system.reachable s\<^sub>0 lstep s \<Longrightarrow> \<exists>s' l. lstep l s s'" shows "df_system s\<^sub>0 lstep"
perm1: fixes p :: "perm" assumes "supp p \<sharp>* c" shows "p \<bullet> f a x c = f (p \<bullet> a) (p \<bullet> x) c"
prod_list_multf: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and xs :: "'c list" assumes "class.comm_monoid_mult times one" shows "monoid_mult.prod_list one times (map (\<lambda>x. times (f x) (g x)) xs) = times (monoid_mult.prod_list one times (map f xs)) (monoid_mult.prod_list one times (map g xs))"
splitAtRec_distinct: fixes vs :: "'c list"   and s :: "'c list"   and ram :: "'c" assumes "distinct vs"   and "distinct s"   and "set s \<inter> set vs = {}" shows "set (fst (splitAtRec ram s vs)) \<inter> set (snd (splitAtRec ram s vs)) = {}"
OclANY_singleton_exec: fixes a :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "Set{a}->any\<^sub>S\<^sub>e\<^sub>t() = a"
tableauproof_def: fixes ps :: "('d, 'e) form list"   and p :: "('d, 'e) form" shows "tableauproof ps p \<equiv> (\<stileturn> Neg p # ps)"
field_intro2: fixes R :: "('c, 'd) ring_scheme" assumes "cring R"   and "\<zero>\<^bsub>R\<^esub> \<noteq> \<one>\<^bsub>R\<^esub>"   and "\<And>x. x \<in> carrier R - {\<zero>\<^bsub>R\<^esub>} \<Longrightarrow> x \<in> Units R" shows "field R"
face_cycle_succ_inI: fixes x :: "'b"   and y :: "'b" assumes "x \<in> G.face_cycle_set y" shows "G.face_cycle_succ x \<in> G.face_cycle_set y"
rep_eq: fixes x :: "'c dense"   and xa :: "'c dense" shows "Rep_dense (x \<squnion> xa) = Rep_dense x \<squnion> Rep_dense xa"
NEGn_decreasing: fixes X :: "ctrm set"   and Y :: "ctrm set"   and B :: "ty" assumes "X \<subseteq> Y" shows "NEGn B Y \<subseteq> NEGn B X"
carrier_matD_2: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "dim_col A = nc"
more_rejective_def: fixes p :: "'c \<Rightarrow> 'd decision option"   and q :: "'c \<Rightarrow> 'd decision option" shows "p \<sqsubseteq>\<^sub>D q = (\<forall>x. case q x of \<bottom> \<Rightarrow> True | \<lfloor>allow y\<rfloor> \<Rightarrow> True | \<lfloor>deny y\<rfloor> \<Rightarrow> \<exists>z. p x = \<lfloor>deny z\<rfloor>)"
run_shift: fixes execute :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'a list"   and p :: "'b"   and s :: "'a stream" assumes "transition_system.path execute enabled r p"   and "transition_system.run execute enabled s (fold execute r p)" shows "transition_system.run execute enabled (r @- s) p"
outstanding_non_volatile_refs_owned_or_read_only_tl: fixes \<S> :: "nat \<Rightarrow> bool option"   and t :: "'b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "outstanding_non_volatile_refs_owned_or_read_only \<S> (t # ts)" shows "outstanding_non_volatile_refs_owned_or_read_only \<S> ts"
jump_linear_comp: fixes c :: "real"   and f :: "real \<Rightarrow> real"   and b :: "real"   and x :: "real" assumes "c \<noteq> 0" shows "jump (f \<circ> (\<lambda>x. c * x + b)) x = (if 0 < c then jump f (c * x + b) else - jump f (c * x + b))"
set_keys: fixes t :: "('b, unit) RBT.rbt" shows "Set t = set (RBT.keys t)"
VA2: shows "Vagree \<nu> \<nu>' (FVT \<theta>\<^sub>2)"
approx_from_below_dense_linorder: fixes y :: "'b"   and x :: "'b" assumes "y < x" shows "\<exists>u. (\<forall>n. u n < x) \<and> u \<longlonglongrightarrow> x"
le2: shows "matrix_sum d f n \<le>\<^sub>L matrix_sum d g n"
zero_in_bigomega_iff: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c" shows "((\<lambda>_. 0::'c) \<in> \<Omega>[F](f)) = (\<forall>\<^sub>F x in F. f x = (0::'c))"
tseytin_tran1_atoms: fixes H :: "'a formula"   and G :: "'a formula" assumes "H \<in> set (tseytin_tran1 (tseytin_toatom F) G)"   and "G \<in> set (subformulae F)" shows "atoms H \<subseteq> atoms F \<union> (\<Union>I\<in>set (subformulae F). atoms (tseytin_toatom F I))"
ab: shows "a + b = zerosign (if is_zero a \<and> is_zero b \<and> sign a = sign b then sign a else 0) (IEEE.round To_nearest (valof a + valof b))"
exhaust: fixes y :: "binding_Rep" obtains x11 :: "vrs" and x12 :: "ty" where "y = VarB_Rep x11 x12"   | x21 :: "tyvrs" and x22 :: "ty" where "y = TVarB_Rep x21 x22"
conjugate_multr: fixes y :: "'a" shows "conjugate ((\<cdot>) y) = (\<rhd>) y"
ideI: fixes a :: "'a" assumes "\<guillemotleft>a : src (src f) \<rightarrow> src e\<guillemotright>"   and "ide a" shows "hom.ide a"
triple_valid2_def: fixes G :: "prog"   and n :: "nat"   and t :: "'b triple" shows "G\<Turnstile>n\<Colon>t = (case t of {P} t\<succ> {Q} \<Rightarrow> \<forall>Y s Z. P Y s Z \<longrightarrow> (\<forall>L. s\<Colon>\<preceq>(G, L) \<longrightarrow> (\<forall>T C A. (normal s \<longrightarrow> \<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile>t\<Colon>T \<and> \<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile> dom (locals (snd s)) \<guillemotright>t\<guillemotright> A) \<longrightarrow> (\<forall>Y' s'. G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (Y', s') \<longrightarrow> Q Y' s' Z \<and> s'\<Colon>\<preceq>(G, L)))))"
rep_eq: shows "Rep_lifted_distrib_lattice_top top = (\<lambda>f. top)"
real_is_int_add_int_of_real: fixes a :: "real"   and b :: "real" assumes "real_is_int a"   and "real_is_int b" shows "int_of_real (a + b) = int_of_real a + int_of_real b"
s'_s''_3: shows "\<R>\<^sub>G \<alpha>' \<beta>' s' = s''"
assms_5: shows "A1 A2 Perp C1 C2"
not_eq_order_implies_strict: fixes a :: "'b list"   and b :: "'b list" assumes "a \<noteq> b"   and "b \<le>s a" shows "b <s a"
stream_pred_coinduct: fixes R :: "'b stream \<Rightarrow> bool"   and w :: "'b stream"   and P :: "'b \<Rightarrow> bool" assumes "R w"   and "\<And>a w. R (a ## w) \<Longrightarrow> P a \<and> R w" shows "pred_stream P w"
genset_sub_free: fixes G :: "'b list set" shows "G \<subseteq> \<langle>G\<rangle>\<^sub>F"
simps_2: fixes uu :: "real" shows "evalUni TrueFUni uu = True"
has_subprob_densityD_3: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "has_subprob_density M N f" shows "M = density N f"
InfModel_intTI_intFI_intPI: fixes intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" assumes "MModel intT intF intP" shows "IInfModel intTI intFI intPI"
simps: fixes n :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_loop1_loop n (l, r) = (\<exists>i j. i + j + 1 = n \<and> (l, r) = (Oc \<up> i, Oc # Oc # Bk \<up> j @ Oc \<up> j) \<and> 0 < j)"
normal_lift_nnf_qe: fixes qe :: "'a fm \<Rightarrow> 'a fm"   and \<phi> :: "'a fm" assumes "\<And>\<phi>. nqfree \<phi> \<Longrightarrow> qfree (qe \<phi>)"   and "\<And>\<phi>. \<lbrakk>nqfree \<phi>; normal \<phi>\<rbrakk> \<Longrightarrow> normal (qe \<phi>)"   and "normal \<phi>" shows "normal (lift_nnf_qe qe \<phi>)"
map_gpv_conv_bind: fixes f :: "'d \<Rightarrow> 'a"   and x :: "('d, 'b, 'c) gpv" shows "map_gpv f (\<lambda>x. x) x = x \<bind> (\<lambda>x. Generative_Probabilistic_Value.Done (f x))"
subseqs_of_length_of_indices: fixes n :: "nat"   and xs :: "'c list" shows "{ys. subseq_of_length n xs ys} = {map ((!) xs) is |is. subseq_of_length n [0..<length xs] is}"
arev_o_arev: shows "rev_H \<circ> rev_H = id"
P2_ideal_real_eq: fixes \<G> :: "'d cyclic_group"   and x1 :: "'d"   and x0 :: "'d"   and \<sigma> :: "bool"   and z :: "'f"   and \<A> :: "(bool \<Rightarrow> 'f \<Rightarrow> (('d \<times> 'd \<times> 'd \<times> 'd \<times> 'd) \<times> 'g) spmf) \<times> ('d \<times> 'd \<times> 'd \<times> 'd \<times> 'd \<Rightarrow> 'g \<Rightarrow> ((('d \<times> 'd \<times> 'd) \<times> nat) \<times> 'g) spmf) \<times> ('d \<times> 'd \<Rightarrow> 'd \<times> 'd \<Rightarrow> 'g \<Rightarrow> 'e spmf)" assumes "ot \<G>"   and "x1 \<in> carrier \<G>"   and "x0 \<in> carrier \<G>" shows "ot_base.P2_real_model \<G> (x0, x1) \<sigma> z \<A> = ot_base.P2_ideal_model \<G> (x0, x1) \<sigma> z \<A>"
proper_it'_def: fixes it :: "'f \<Rightarrow> ('g \<Rightarrow> bool) \<Rightarrow> ('h \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g"   and it' :: "'f \<Rightarrow> ('i \<Rightarrow> bool) \<Rightarrow> ('h \<Rightarrow> 'i \<Rightarrow> 'i) \<Rightarrow> 'i \<Rightarrow> 'i" shows "proper_it' it it' \<equiv> \<forall>s. proper_it (it s) (it' s)"
inf_I_intT: fixes \<sigma> :: "'tp" shows "infinite {a. I_intT \<sigma> a}"
simps_2: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and ys :: "'b list" shows "linorder_list0.selsort le (y # ys) = (let xm = fold (linorder_list0.min_for le) ys y; xs' = remove1 xm (y # ys) in xm # linorder_list0.selsort le xs')"
unl_lift2: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and x :: "'f \<Rightarrow> 'd"   and y :: "'f \<Rightarrow> 'e"   and w :: "'f" shows "(f<x, y>) w \<equiv> f (x w) (y w)"
r_2: shows "n = smn 1 (encode r_univuniv) [encode r]"
P5: shows "\<lfloor>\<lambda>w. \<nexists>x xa. (Contingent xa \<^bold>\<and> Necessary x \<^bold>\<and> xa explains x) w\<rfloor>"
plus_closed: fixes a :: "'ag"   and b :: "'ag" assumes "a \<in> U"   and "b \<in> U" shows "a +\<^sub>o\<^sub>w b \<in> U"
nonzero_of_complex_inverse: fixes x :: "complex" assumes "x \<noteq> 0" shows "of_complex (inverse x) = inverse (of_complex x)"
space_completion: fixes M :: "'b measure" shows "space (completion M) = space M"
safe_roomk_currk: fixes s :: "state"   and r :: "room" assumes "s \<in> reach"   and "safe s r" shows "roomk s r = currk s r"
p_def: fixes t :: "real" shows "ego.p t = s\<^sub>e + v\<^sub>e * t + 1 / 2 * a\<^sub>e * t\<^sup>2"
FORWEAK_elementwise_rule: fixes d :: "'c nres"   and XX :: "'d set"   and Inf_spec :: "'d \<Rightarrow> 'c nres"   and Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and comb :: "'c \<Rightarrow> 'c \<Rightarrow> 'c nres" assumes "nofail d"   and "\<And>X. X \<in> XX \<Longrightarrow> Inf_spec X \<le> SPEC (Q X)"   and "\<And>a b X Y. Q X a \<Longrightarrow> comb a b \<le> SPEC (Q X)"   and "\<And>a b X Y. Q X b \<Longrightarrow> comb a b \<le> SPEC (Q X)" shows "FORWEAK XX d Inf_spec comb \<le> SPEC (\<lambda>i. \<forall>x\<in>XX. Q x i)"
1: shows "bprv (imp (subst \<chi> \<langle>neg \<phi>\<rangle> xx') (imp (eql \<langle>neg \<phi>\<rangle> (Var xx')) \<chi>))"
retraction_imp_surjective_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "retraction_map X Y f" shows "f ` topspace X = topspace Y"
A: shows "A \<in> sets (Pi\<^sub>F J M)"
f_expand_empty_conv: fixes xs :: "'b message_af list"   and k :: "nat" shows "(xs \<odot> k = []) = (xs = [] \<or> k = 0)"
numgcd0: fixes t :: "MIR.num"   and bs :: "real list" assumes "numgcd t = 0" shows "Inum bs t = 0"
add_neg_numeral_special_5: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "plus (uminus one) (uminus (numeral.numeral one plus n)) = uminus (numeral.numeral one plus (Num.inc n))"
splitFace_f21_oldF_neq: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and newVs :: "nat list"   and f12 :: "face"   and f21 :: "face"   and newGraph :: "graph" assumes "pre_splitFace g ram1 ram2 oldF newVs"   and "minGraphProps g"   and "(f12, f21, newGraph) = splitFace g ram1 ram2 oldF newVs" shows "oldF \<noteq> f21"
foldr_Cons_Nil: fixes xs :: "['b]" shows "foldr\<cdot>Cons\<cdot>[]\<cdot>xs = xs"