hcju/isabelleps huggingface dataset:
soc: shows "P,h,C\<^sub>0 \<turnstile>\<^sub>s sfs \<surd>"
IH_1: assumes "\<Gamma> \<turnstile> s" shows "\<exists>a. taval a1 s a"
finite_code: fixes A :: "'c set" assumes "class.finite TYPE('c)" shows "finite A = True"
length_butlast: fixes xs :: "'a list" shows "|butlast xs| = |xs| - 1"
loop_halts: fixes n :: "nat"   and l :: "cell list"   and r :: "cell list" assumes "0 < n"   and "inv_loop n (1, l, r)" shows "\<exists>stp. is_final (steps0 (1, l, r) tcopy_loop stp)"
stk_esl_def: fixes P :: "'b prog"   and mxs :: "nat" shows "stk_esl P mxs \<equiv> upto_esl mxs (SemiType.esl P)"
exhaust: fixes y :: "quadnat" obtains x :: "nat \<times> nat \<times> nat \<times> nat" where "y = Quad x"
pos: fixes i :: "nat" shows "AE x in M. 0 \<le> f i x"
deadlock_free_v2_implies_div_free: fixes P :: "'b process" assumes "deadlock_free_v2 P" shows "D P = {}"
face_cycle_succ_def: shows "face_cycle_succ \<equiv> edge_succ (to_map' (snd G_list) xss) \<circ> edge_rev (to_map' (snd G_list) xss)"
assms_1: shows "sympartp (\<sqsubseteq>) x y"
bi_unique_alt_def: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "bi_unique A = (left_unique A \<and> right_unique A)"
from_pre_digraph_def: fixes G :: "('b, 'c) pre_digraph" shows "from_pre_digraph G \<equiv> \<lparr>Graph.verts = pre_digraph.verts G, arcs = arcs_ends G\<rparr>"
simps_1: fixes D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" shows "tr [] D = [[]]"
Red_I_of_Inf_to_N: fixes \<iota> :: "'f inference"   and N :: "'f set" assumes "\<iota> \<in> Inf"   and "concl_of \<iota> \<in> N" shows "\<iota> \<in> {}"
input_sizes: fixes f :: "nat \<Rightarrow> real" shows "input_sizes (insert_weights s m f) = input_sizes m"
opp_half_apartmentI: fixes x :: "'a set"   and C :: "'a set" assumes "x \<in> X"   and "C \<in> \<C> - f \<turnstile> \<C>"   and "order.greater_eq C x" shows "x \<in> Y"
map_pair_id: fixes xs :: "('a \<times> 'b) list" assumes "tc.oalist_inv_raw xs" shows "map_pair id xs = xs"
invar_def: fixes g :: "'m" shows "invar g \<equiv> M_invar g \<and> finite (dom (M_lookup g)) \<and> (\<forall>u xs. M_lookup g u = Some xs \<longrightarrow> distinct (map fst xs) \<and> u \<notin> set (map fst xs) \<and> (\<forall>(v, d)\<in>set xs. (u, d) \<in> set (the_default [] (M_lookup g v))))"
igSubstIGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) model_scheme" shows "igSubstIGOp MOD \<equiv> \<forall>ys y Y delta inp binp. igWls MOD (asSort ys) Y \<and> igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igSubst MOD ys Y y (igOp MOD delta inp binp) = igOp MOD delta (igSubstInp MOD ys Y y inp) (igSubstBinp MOD ys Y y binp)"
rt_strictly_fresherI: fixes i :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "rt1 \<sqsubseteq>\<^bsub>i\<^esub> rt2"   and "\<not> rt1 \<approx>\<^bsub>i\<^esub> rt2" shows "rt1 \<sqsubset>\<^bsub>i\<^esub> rt2"
of_nat_simps_2: fixes m :: "nat" shows "of_nat (Suc m) = (1::'b) + of_nat m"
angle_actrans_sem: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "|~ \<langle>F\<rangle>_v = (F \<and> v$ \<noteq> $v)"
UP_domain_def: fixes R :: "('c, 'd) ring_scheme" shows "UP_domain R \<equiv> domain R"
assms_1: shows "sound_mode_use (cms\<^sub>1, mem\<^sub>1)"
F1set1_bd': fixes x :: "('d, 'e, 'f) F1"   and r2 :: "(bd_type_F2 \<times> bd_type_F2) set" shows "|F1set1 x| \<le>o dir_image (bd_F1 +c r2) Abs_bd_type_F"
card_of_Well_order: fixes A :: "'b set" shows "Well_order |A|"
dims2_def: shows "ps_P.dims2 = nths ps2_P.dims0 (- ps2_P.vars1')"
cancels_to_1_def: fixes l1 :: "(bool \<times> 'd) list"   and l2 :: "(bool \<times> 'd) list" shows "cancels_to_1 l1 l2 = (\<exists>i. cancels_to_1_at i l1 l2)"
typeof_def: fixes v :: "Value" shows "\<tau> v = (case v of boolV b \<Rightarrow> BoolT | intgV i \<Rightarrow> IntgT | shortV sh \<Rightarrow> ShortT | byteV by \<Rightarrow> ByteT | objV C a \<Rightarrow> CClassT C | arrV T a \<Rightarrow> ArrT T | nullV \<Rightarrow> NullT)"
line_def: fixes t :: "real" shows "line t = X + t *\<^sub>R H"
t1_space_derived_set_of_finite: fixes X :: "'c topology" shows "t1_space X = (\<forall>S. finite S \<longrightarrow> X derived_set_of S = {})"
as_proj_eq_filter_action_proj: fixes as :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) list"   and vs :: "'d set" shows "as_proj as vs = filter (\<lambda>a. fmdom' (snd a) \<noteq> {}) (map (\<lambda>a. action_proj a vs) as)"
primary_ideal_def: fixes R :: "('c, 'd) Ring_scheme"   and q :: "'c set" shows "primary_ideal R q = (ideal R q \<and> 1\<^sub>r\<^bsub>R\<^esub> \<notin> q \<and> (\<forall>x\<in>carrier R. \<forall>y\<in>carrier R. x \<cdot>\<^sub>r\<^bsub>R\<^esub> y \<in> q \<longrightarrow> (\<exists>n. x^\<^bsup>R n\<^esup> \<in> q \<or> y \<in> q)))"
add_cancel_left_left: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and b :: "'g"   and a :: "'g" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(plus b a = a) = (b = zero)"
GuardsFlip_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"   and "\<Gamma>,\<Theta>\<Turnstile>\<^bsub>/- F\<^esub> P c UNIV,UNIV" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/{}\<^esub> P c Q,A"
mult_cancel_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.semiring_no_zero_divisors_cancel plus zero times" shows "(times a c = times b c) = (c = zero \<or> a = b)"
conv_backward_terminating: fixes x :: "'a" shows "backward_terminating x = forward_terminating (x\<^sup>T)"
prefix_code_3: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "prefix (x # xs) (y # ys) = (x = y \<and> prefix xs ys)"
sig_crit_alt_spp: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" assumes "0 \<notin> set bs" shows "sig_crit bs ss p = sig_crit_spp (map spp_of bs) ss (app_pair spp_of p)"
WPO_S_ctxt: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and f :: "'f"   and bef :: "('f, 'b) Term.term list"   and aft :: "('f, 'b) Term.term list" assumes "\<And>f k. set (\<sigma> (f, k)) = {0..<k}"   and "(s, t) \<in> WPO_S" shows "(Fun f (bef @ s # aft), Fun f (bef @ t # aft)) \<in> WPO_S"
wf_wcode_halt_case_le: shows "wf wcode_halt_case_le"
poly_div_minus_right: fixes x :: "'a poly"   and y :: "'a poly" shows "x div - y = - (x div y)"
cfun_arg_cong: fixes x :: "'b"   and y :: "'b"   and f :: "'b \<rightarrow> 'c" assumes "x = y" shows "f\<cdot>x = f\<cdot>y"
eq_upair_simps: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "eq_upair (a, b) (c, d) = (a = c \<and> b = d \<or> a = d \<and> b = c)"
elims: fixes x :: "d"   and y :: "(nat \<Rightarrow> contour option) set" assumes "benv_in_d x = y" obtains l :: "lambda" and \<beta> :: "nat \<Rightarrow> contour option" where "x = DC (l, \<beta>)" and "y = {\<beta>}"   | v :: "int" where "x = DI v__" and "y = {}"   | v :: "prim" where "x = DP v" and "y = {}"   | "x = Stop" and "y = {}"
continuous_at_imp_continuous_at_within: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'b"   and s :: "'a set" assumes "isCont f x" shows "continuous (at x within s) f"
Le_cases: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Le Bet Cong A B C D \<or> Tarski_neutral_dimensionless.Le Bet Cong B A C D \<or> Tarski_neutral_dimensionless.Le Bet Cong A B D C \<or> Tarski_neutral_dimensionless.Le Bet Cong B A D C" shows "Tarski_neutral_dimensionless.Le Bet Cong A B C D"
sum_nonneg: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> {#} \<subseteq># f x" shows "{#} \<subseteq># comm_monoid_add.sum (+) {#} f A"
o_apply: fixes f :: "'h \<Rightarrow> 'g"   and g :: "'i \<Rightarrow> 'h"   and x :: "'i" shows "(f \<circ> g) x = f (g x)"
\<F>\<G>_subfrmlsn: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<F>\<G> \<phi> w \<subseteq> subfrmlsn \<phi>"
Least_le_A: shows "(LEAST n. A $ ia $ n \<noteq> (0::'a)) < (LEAST n. A $ j $ n \<noteq> (0::'a))"
frechet_derivative_at: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "(f has_derivative f') (at x)" shows "f' = frechet_derivative f (at x)"
y_prop: shows "r < y \<and> (\<forall>(d, e, f)\<in>set leq. \<forall>x\<in>{r<..y}. d * x\<^sup>2 + e * x + f \<le> 0)"
adm_chfin: fixes P :: "'d \<Rightarrow> bool" shows "adm P"
IDirProds_def: fixes G :: "('c, 'd) monoid_scheme"   and S :: "'e \<Rightarrow> 'c set"   and I :: "'e set" shows "IDirProds G S I = generate G (\<Union> (S ` I))"
healthy_intros_19: fixes p :: "'b \<Rightarrow> 'a \<Rightarrow> real"   and f :: "'a \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>a s. a \<in> supp (p s) \<Longrightarrow> healthy (wp (f a))"   and "\<And>s. sound (p s)"   and "\<And>s. sum (p s) (supp (p s)) \<le> 1" shows "healthy (wp (SetPC f p))"
intro: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" assumes "\<And>a. f a z = a" shows "comm_monoid_axioms f z"
map_eq_conv: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and g :: "'d \<Rightarrow> 'c" shows "(map f xs = map g xs) = (\<forall>x\<in>set xs. f x = g x)"
cong: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x = y" shows "x * z = y * z"
pr_algo_len_bound_def: shows "pr_algo_len_bound \<equiv> 2 * (card V)\<^sup>2 + 22 * (card V)\<^sup>2 * card E"
env_delete_fun_upd: fixes x :: "'d"   and m :: "'d \<Rightarrow> 'e"   and v :: "'e" shows "env_delete x (m(x := v)) = env_delete x m"
tllist_ord_antisym: fixes xs :: "('c, 'b) tllist"   and ys :: "('c, 'b) tllist" assumes "local.tllist_ord xs ys"   and "local.tllist_ord ys xs" shows "xs = ys"
lookup_empty: fixes k :: "'b" shows "Mapping.lookup Mapping.empty k = None"
qtable1: shows "qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) ne \<phi>) rel1"
uncountable_cball: fixes r :: "real"   and a :: "'a" assumes "0 < r" shows "uncountable (cball a r)"
wp_cond_aux2: fixes p :: "'a"   and x :: "'a"   and y :: "'a"   and q :: "'a" shows "ad p \<cdot> wp (if p then x else y fi) q = ad p \<cdot> wp y q"
hom_decomp_shift: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat"   and m :: "nat" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "exact_decomp (Suc m) qs"   and "hom_decomp qs" shows "hom_decomp (local.shift k m qs)"
indep_subset_carrier: fixes X :: "'a set" assumes "indep X" shows "X \<subseteq> carrier"
distinct_merge_distinct: fixes xs :: "'c list"   and ys :: "'c list" assumes "sorted xs"   and "distinct xs"   and "sorted ys"   and "distinct ys" shows "distinct (merge_distinct xs ys)"
retract_of_singleton: fixes x :: "'c"   and S :: "'c set" shows "({x} retract_of S) = (x \<in> S)"
assms: shows "list_all tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p (unlabel S)"
user_reg_mod_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and w :: "word32"   and curr_win :: "'b word"   and rd :: "word5"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = user_reg_mod w curr_win rd s1 \<and> t2 = user_reg_mod w curr_win rd s2" shows "low_equal t1 t2"
m2_def: shows "m2 \<equiv> y$ = Suc<$y> \<and> x$ = $x"
ex_weakens: fixes x :: "'a \<Longrightarrow> 'b"   and P :: "'b upred" assumes "wb_lens x" shows "(\<exists> x \<bullet> P) \<sqsubseteq> P"
assms_2: shows "(\<lambda>x. \<Sum>\<^sub>ay\<in>Y. norm (f (x, y))) abs_summable_on X"
map_nth_append1: fixes xs :: "'c list"   and d :: "nat"   and ys :: "'c list" assumes "length xs = d" shows "map ((!) (xs @ ys)) [0..<d] = xs"
distinct_insort: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" shows "distinct (insort_key f x xs) = (x \<notin> set xs \<and> distinct xs)"
strong_reduction_bisimulations_impl_inverse_is_simulation: fixes Rel :: "('a \<times> 'a) set"   and Cal :: "'a processCalculus" assumes "strong_reduction_simulation Rel Cal \<and> (\<forall>P Q Q'. (P, Q) \<in> Rel \<and> Q \<longmapsto>Cal Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>Cal P' \<and> (P', Q') \<in> Rel))" shows "strong_reduction_simulation (Rel\<inverse>) Cal"
locals_gupd: fixes r :: "oref"   and obj :: "obj"   and s :: "st" shows "locals (gupd(r\<mapsto>obj) s) = locals s"
le_pp: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat_pp :: "'a \<Rightarrow> (nat, nat) pp"   and abs_nat_pp :: "(nat, nat) pp \<Rightarrow> 'a"   and lex_comp' :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and deg' :: "'a \<Rightarrow> nat"   and x :: "'a"   and y :: "'a" assumes "class.nat_pp_compare plus zero less_eq less rep_nat_pp abs_nat_pp lex_comp' deg'"   and "rep_nat_pp x \<le> rep_nat_pp y" shows "less_eq x y"
nat_of_fatom_def: shows "nat_of_fatom = (\<lambda>t. nat_of_hatom (hatom_of_fatom t))"
Proj_vimage_chart_basis_domainP: fixes b :: "'b" shows "Proj -` Collect (chart_basis_domainP b) = Collect (chart_basis_domain_nonzeroP b)"
case_eq_if: fixes f1 :: "'h \<Rightarrow> 'g"   and f2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'g"   and generat :: "('h, 'i, 'j) generat" shows "(case generat of Pure x \<Rightarrow> f1 x | IO x xa \<Rightarrow> f2 x xa) = (if is_Pure generat then f1 (generat.result generat) else f2 (generat.output generat) (continuation generat))"
apex_def: shows "HH\<mu>\<nu>\<pi>.apex \<equiv> C.dom HH\<mu>\<nu>\<pi>.leg0"
real_alg_2''_def: fixes ri :: "root_info"   and p :: "int poly"   and l :: "rat"   and r :: "rat" shows "real_alg_2'' ri p l r = (case normalize_bounds_1 (case tighten_poly_bounds_for_x p 0 l r (sgn (ipoly p r)) of (l', r', sr') \<Rightarrow> (p, l', r')) of (p', l, r) \<Rightarrow> Irrational (root_info.number_root ri r) (p', l, r))"
cases: fixes a1 :: "char list"   and a2 :: "char list"   and P :: "'b prog" assumes "P \<turnstile> a1 \<prec>\<^sup>1 a2" obtains C :: "char list" and D :: "char list" and rest :: "fdecl list \<times> 'b mdecl list" where "a1 = C" and "a2 = D" and "class P C = \<lfloor>(D, rest)\<rfloor>" and "C \<noteq> Object"
assms_2: shows "\<BB> \<approx>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
IH: fixes \<sigma>1 :: "nat \<Rightarrow> val"   and \<sigma>2 :: "nat \<Rightarrow> val" assumes "\<And>x. x \<in> free_vars e \<Longrightarrow> \<sigma>1 x = \<sigma>2 x" shows "expr_sem_rf \<sigma>1 e = expr_sem_rf \<sigma>2 e"
not_eq_order_implies_strict: fixes a :: "'g"   and b :: "'g" assumes "a \<noteq> b"   and "a \<le> b" shows "a < b"
par_by_merge_def: fixes P :: "('d, 'f) urel"   and M :: "(('d, 'f, 'g) mrg, 'e) urel"   and Q :: "('d, 'g) urel" shows "P \<parallel>\<^bsub>M\<^esub> Q = (P ;; U0 \<and> Q ;; U1 \<and> $\<Sigma>\<^sub><\<acute> =\<^sub>u $\<^bold>v) ;; M"
curry_conv: fixes f :: "'b \<times> 'c \<Rightarrow> 'a"   and a :: "'b"   and b :: "'c" shows "curry f a b = f (a, b)"
zcf_ctrm: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "zcf (monom P (p 0) 0) = zcf p"
obtain_sorted_carrier_def: shows "obtain_sorted_carrier \<equiv> SPEC (\<lambda>L. wsorted L \<and> set L = E)"
attach_converter_of_resource_conv_parallel_resource2: fixes res :: "('i, 'k) resource"   and res' :: "('h, 'j) resource" shows "1\<^sub>C |\<^sub>\<propto> converter_of_resource res \<rhd> res' = res' \<parallel> res"
vimage_vunion_left: fixes r :: "V"   and s :: "V"   and A :: "V" shows "(r \<union>\<^sub>\<circ> s) `\<^sub>\<circ> A = r `\<^sub>\<circ> A \<union>\<^sub>\<circ> s `\<^sub>\<circ> A"
andalso_and: fixes t :: "tr"   and s :: "tr" assumes "t \<noteq> \<bottom>" shows "((t andalso s) \<noteq> FF) = (t \<noteq> FF \<and> s \<noteq> FF)"
g: shows "g = fpxs_compose_power (fpxs_of_fls g') (1 / rat_of_nat d)"
list_all2_map2: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and as :: "'g list"   and f :: "'i \<Rightarrow> 'h"   and bs :: "'i list" shows "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
fmeasurableI2: fixes A :: "'a set"   and M :: "'a measure"   and B :: "'a set" assumes "A \<in> fmeasurable M"   and "B \<subseteq> A"   and "B \<in> sets M" shows "B \<in> fmeasurable M"
subsubformulae: fixes G :: "'a formula"   and F :: "'a formula"   and H :: "'a formula" assumes "G \<in> set (subformulae F)"   and "H \<in> set (subformulae G)" shows "H \<in> set (subformulae F)"
simps: fixes a :: "C"   and b :: "nat" shows "(a \<in> hbase b) = (a = C [] \<or> (\<exists>i n ms. a = C (replicate i n @ ms) \<and> i \<noteq> 0 \<and> i < b \<and> n \<in> hbase b \<and> C ms \<in> hbase b \<and> (\<forall>x. x \<in> set ms \<longrightarrow> evalC b n < evalC b x)))"
fv_transform: fixes PropApp :: "'a \<Rightarrow> 'a"   and PropLam :: "'a \<Rightarrow> 'a"   and AnalLet :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> 'a \<Rightarrow> 'b"   and PropLetBody :: "'b \<Rightarrow> 'a"   and PropLetHeap :: "'b \<Rightarrow> Vars.var \<Rightarrow> 'a\<^sub>\<bottom>"   and PropIfScrut :: "'a \<Rightarrow> 'a"   and TransVar :: "'a \<Rightarrow> Vars.var \<Rightarrow> exp"   and TransApp :: "'a \<Rightarrow> exp \<Rightarrow> Vars.var \<Rightarrow> exp"   and TransLam :: "'a \<Rightarrow> Vars.var \<Rightarrow> exp \<Rightarrow> exp"   and TransLet :: "'b \<Rightarrow> (Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> exp"   and a :: "'a"   and e :: "exp" assumes "AbstractTransform PropApp PropLam AnalLet PropLetBody PropLetHeap PropIfScrut TransVar TransApp TransLam TransLet" shows "fv (AbstractTransform.transform PropApp PropLam AnalLet PropLetBody PropLetHeap PropIfScrut TransVar TransApp TransLam TransLet a e) \<subseteq> fv e"
card_Suc_eq: fixes A :: "'a set"   and k :: "nat" shows "(card A = Suc k) = (\<exists>b B. A = set_ins b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {}))"
Done_parametric': fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and C :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and R :: "'h \<Rightarrow> 'j \<Rightarrow> bool" shows "rel_fun A (rel_gpv'' A C R) Generative_Probabilistic_Value.Done Generative_Probabilistic_Value.Done"
qGoodInp_iff_goodInp_asInp: fixes qinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm option" shows "goodInp (asInp qinp) = qGoodInp qinp"
1: shows "state_succ s1 a \<subseteq>\<^sub>f state_succ s2 a"
PLemmas_2: fixes x :: "(int \<times> int) set set"   and y :: "(int \<times> int) set set" shows "C (DenyAllFromTo x y) = deny_all_from_to x y"
finsum_insert: fixes I :: "'c set"   and i :: "'c"   and f :: "'c \<Rightarrow> 'a" assumes "finite I"   and "i \<notin> I"   and "f i \<in> R"   and "\<And>j. j \<in> I \<Longrightarrow> f j \<in> R" shows "finsum (insert i I) f = f i + finsum I f"
Tree_wf_eqvt': shows "eqvt Tree_wf"
YMap_def: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and \<eta> :: "('d, 'g, 'h, ZF, 'i, 'j) NatTrans" shows "YMap C X \<eta> \<equiv> (\<eta> $$ X) |@| (m2z\<^bsub>C\<^esub>id\<^bsub>C\<^esub> X)"
extract_insert_weights_shared: fixes x :: "nat"   and m :: "(nat \<times> nat) convnet"   and w :: "nat \<Rightarrow> real" assumes "x < count_weights True m"   and "balanced_net m" shows "extract_weights True (insert_weights True m w) x = w x"
simps_6: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<alpha> :: "game" shows "usubstappp \<sigma> U \<alpha>** = (let V = fst (usubstappp \<sigma> U \<alpha>) in (V, Loopo (snd (usubstappp \<sigma> V \<alpha>))))"
rel_interior_eq_empty: fixes S :: "'a set" assumes "convex S" shows "(rel_interior S = {}) = (S = {})"
split0_nb0: fixes t :: "tm"   and c' :: "poly"   and t' :: "tm" assumes "split0 t = (c', t')" shows "tmbound 0 t'"
le_mult_floor_Ints: fixes a :: "'c"   and b :: "'c" assumes "(0::'c) \<le> a"   and "a \<in> \<int>" shows "of_int (\<lfloor>a\<rfloor> * \<lfloor>b\<rfloor>) \<le> of_int \<lfloor>a * b\<rfloor>"
dsj_I2: fixes H :: "fmla set"   and B :: "fmla"   and A :: "fmla" assumes "H \<turnstile> B" shows "H \<turnstile> A OR B"
assms: shows "P \<Longrightarrow>\<tau> \<prec> P'"
square_eq_one: fixes x :: "'a" assumes "x \<in> carrier R"   and "x \<otimes> x = \<one>" shows "x = \<one> \<or> x = \<ominus> \<one>"
standard_uniqueness_pgallery_betw: fixes W :: "'c set set"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a"   and C :: "'c set"   and Cs :: "'c set list"   and D :: "'c set" assumes "ChamberComplexMorphism W X f"   and "ChamberComplexMorphism W X g"   and "fun_eq_on f g C"   and "SimplicialComplex.maxsimpchain W (C # Cs @ [D])"   and "pgallery (f \<Turnstile> (C # Cs @ [D]))"   and "pgallery (g \<Turnstile> (C # Cs @ [D]))" shows "fun_eq_on f g D"
all: shows "\<forall>i. (f i, g i) \<in> S\<^sup>* \<and> (g i, f (Suc i)) \<in> S\<^sup>+"
jozsa_transform_coeff_2: fixes i :: "nat"   and j :: "nat" assumes "i < dim_row U\<^sub>f \<and> j < dim_col U\<^sub>f" shows "i = j + 1 \<and> odd i \<longrightarrow> U\<^sub>f $$ (i, j) = of_nat (f (i div 2))"
HDomain_Incl_fresh_iff: fixes a :: "atom"   and r :: "tm"   and d :: "tm" shows "a \<sharp> HDomain_Incl r d = (a \<sharp> r \<and> a \<sharp> d)"
mat_of_rows_carrier_3: fixes n :: "nat"   and vs :: "'b Matrix.vec list" shows "dim_col (mat_of_rows n vs) = n"
pcr_ennreal_def: shows "pcr_ennreal \<equiv> (=) OO cr_ennreal"
f_Inf_distrib: fixes c :: "'a::type"   and D :: "'a::type set" shows "(f::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (c::'a::type) (\<Sqinter> (D::'a::type set)) = \<Sqinter> {f c d |d::'a::type. d \<in> D}"
group_by_keys: fixes f :: "'c \<Rightarrow> 'a \<times> 'b"   and M :: "'c set" shows "fst ` group_by f M = fst ` f ` M"
preserves_reflects_arr: fixes f :: "'c" shows "C.arr (C.map f) = C.arr f"
eqk: fixes \<rho> :: "complex mat"   and k :: "nat" assumes "\<rho> \<in> density_states" shows "trace (W k * \<rho>) = (\<Sum>k = 0..<k. trace (P * D0 k \<rho>)) + trace \<rho> - (\<Sum>k = 0..<k. trace (D0 k \<rho>))"
Skip_def: shows "Skip \<equiv> action_of (R (true \<turnstile> \<lambda>(A, A'). tr A' = tr A \<and> \<not> wait A' \<and> alpha_rp.more A = alpha_rp.more A'))"
leq_path_Subobjs: fixes C :: "char list"   and Cs :: "char list list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P,C \<turnstile> [C] \<sqsubseteq> Cs"   and "is_class P C"   and "wf_prog wf_md P" shows "Subobjs P C Cs"
isomorphic_def: fixes a :: "'b"   and a' :: "'b" shows "hhom\<^sub>D.isomorphic a a' = (\<exists>f. hhom\<^sub>D.in_hom f a a' \<and> hhom\<^sub>D.iso f)"
deg_q2: fixes t :: "'x \<Rightarrow>\<^sub>0 'a" assumes "t \<in> keys q2" shows "deg_pm t < poly_deg q"
fmeasurable_box: fixes a :: "'a"   and b :: "'a" shows "box a b \<in> fmeasurable lborel"
chain_subset_Total_Union: fixes R :: "('b \<times> 'b) set set" assumes "chain\<^sub>\<subseteq> R"   and "\<forall>r\<in>R. Total r" shows "Total (\<Union> R)"