hcju/isabelleps huggingface dataset:
class_les_trans: fixes cs :: "(String.literal \<times> String.literal) set"   and x :: "String.literal"   and y :: "String.literal"   and z :: "String.literal" assumes "wf_subclass_loc cs"   and "class_les cs x y"   and "class_les cs y z" shows "class_les cs x z"
RES_Sup_RETURN: fixes X :: "'a set" shows "Sup (RETURN ` X) = RES X"
mk_finally_def: fixes x :: "'a ltln" shows "mk_finally x \<equiv> case x of true\<^sub>n \<Rightarrow> true\<^sub>n | false\<^sub>n \<Rightarrow> false\<^sub>n | _ \<Rightarrow> F\<^sub>n remove_strong_ops x"
add_rel_commute: fixes X :: "('a set \<times> 'b) set"   and Y :: "('a set \<times> 'b) set" assumes "X \<in> carrier_direct_lim"   and "Y \<in> carrier_direct_lim" shows "add_rel X Y = add_rel Y X"
hom_1: fixes x :: "int" assumes "of_int x = (1::'d)" shows "x = 1"
bind_singleton_conv_image: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'b" shows "Set.bind A (\<lambda>x. {f x}) = f ` A"
count_pre_integrable: fixes N :: "'c measure"   and Y :: "'a \<Rightarrow> 'c"   and B :: "'c set"   and X :: "'a \<Rightarrow> real"   and n :: "nat" assumes "point_measurable M (space N) Y"   and "disc_fct Y"   and "B \<subseteq> space N"   and "countable B"   and "integrable M X"   and "\<forall>w\<in>space M. 0 \<le> X w" shows "integrable M (\<lambda>w. expl_cond_expect M Y X w * indicat_real (countable_preimages B Y n \<inter> space M) w)"
ring_simprules_15: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "ring R"   and "x \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = x"
elims_3: fixes x :: "('a, 'b) Combinators list" assumes "\<not> noDenyAll x" obtains xa :: "('a, 'b) Combinators" and xs :: "('a, 'b) Combinators list" where "x = xa # xs" and "\<not> (\<not> member DenyAll xa \<and> noDenyAll xs)"
mask_eqs_3: fixes a :: "'b word"   and n :: "nat"   and b :: "'b word" shows "(a AND mask n) - b AND mask n = a - b AND mask n"
lset_lnth_member: fixes xs :: "'c llist"   and A :: "'c set"   and n :: "nat" assumes "lset xs \<subseteq> A"   and "enat n < llength xs" shows "xs $ n \<in> A"
inv_impl: fixes l :: "char list \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" assumes "I l s"   and "bval b s" shows "pre C I l s"
P3_cong: fixes tms :: "trm list"   and P :: "trm \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> fmla"   and H :: "fmla set"   and x :: "trm"   and x' :: "trm"   and y :: "trm"   and y' :: "trm"   and z :: "trm"   and z' :: "trm" assumes "\<And>i t u v x. atom i \<sharp> tms \<Longrightarrow> (P t u v)(i::=x) = P (subst i x t) (subst i x u) (subst i x v)"   and "H \<turnstile> x EQ x'"   and "H \<turnstile> y EQ y'"   and "H \<turnstile> z EQ z'" shows "H \<turnstile> P x y z IFF P x' y' z'"
helper3: fixes x :: "'a word"   and y :: "'a word" shows "x || y = x || y && ~~ x"
Cons_path: fixes n'' :: "'node"   and as :: "'edge list"   and n' :: "'node"   and a :: "'edge"   and n :: "'node" assumes "n'' -as\<rightarrow>* n'"   and "valid_edge a"   and "sourcenode a = n"   and "targetnode a = n''" shows "n -a # as\<rightarrow>* n'"
B_def: shows "B = {x \<in> {0<..(int p * int q - 1) div 2}. x mod int p \<in> {(int p - 1) div 2<..int p - 1} \<and> x mod int q \<in> {0<..(int q - 1) div 2}}"
mem_ball_imp_mem_cball: fixes x :: "'b"   and y :: "'b"   and e :: "real" assumes "x \<in> ball y e" shows "x \<in> cball y e"
PAR: fixes xs :: "'varSort" shows "(|vars xs| <o |UNIV| \<or> finite (vars xs)) \<and> (|terms| <o |UNIV| \<or> finite terms) \<and> Ball terms good \<and> (|abs| <o |UNIV| \<or> finite abs) \<and> Ball abs goodAbs \<and> (|envs| <o |UNIV| \<or> finite envs) \<and> Ball envs goodEnv"
obs_consistent_def: fixes R :: "('e \<times> 'f) set"   and pi :: "'g \<Rightarrow> 'h"   and Sa :: "('e, 'h) spec"   and Sc :: "('f, 'g) spec" shows "obs_consistent R pi Sa Sc \<equiv> \<forall>s t. (s, t) \<in> R \<longrightarrow> pi (obs Sc t) = obs Sa s"
nn_integral_Markov_inequality: fixes u :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and A :: "'b set"   and c :: "ennreal" assumes "u \<in> borel_measurable M"   and "A \<in> sets M" shows "emeasure M ({x \<in> space M. 1 \<le> c * u x} \<inter> A) \<le> c * \<integral>\<^sup>+ x. u x * indicator A x \<partial>M"
theNFTBoth_def: fixes cl :: "('g, 'h, 'i) cmd list"   and dl :: "('j, 'k, 'l) cmd list" shows "theNFTBoth cl dl \<equiv> theNFT cl \<inter> theNFT dl"
assms: shows "\<not> (liminf u = \<infinity> \<and> liminf v = - \<infinity> \<or> liminf u = - \<infinity> \<and> liminf v = \<infinity>)"
chine_assoc_def: shows "fgh.chine_assoc \<equiv> \<langle>fgh.Prj\<^sub>1\<^sub>1 \<lbrakk>fgh.\<mu>.leg0, fgh.\<nu>.leg1 \<cdot> fgh.\<nu>\<pi>.prj\<^sub>1\<rbrakk> \<langle>fgh.Prj\<^sub>0\<^sub>1 \<lbrakk>fgh.\<nu>.leg0, fgh.\<pi>.leg1\<rbrakk> fgh.Prj\<^sub>0\<rangle>\<rangle>"
are_endpoints_in_S_def: fixes p :: "proj2"   and q :: "proj2"   and a :: "proj2"   and b :: "proj2" shows "are_endpoints_in_S p q a b \<equiv> p \<noteq> q \<and> p \<in> S \<and> q \<in> S \<and> a \<in> hyp2 \<and> b \<in> hyp2 \<and> proj2_set_Col {p, q, a, b}"
lunit_naturality: fixes f :: "'d" assumes "arr f" shows "\<l>[cod f] \<cdot> (\<I> \<otimes> f) = f \<cdot> \<l>[local.dom f]"
cs: shows "closed_segment c b' \<subseteq> {x. k \<le> d \<bullet> x} \<and> closed_segment b' a' \<subseteq> {x. k \<le> d \<bullet> x}"
complete_imp_closed: fixes S :: "'a set" assumes "complete S" shows "closed S"
subst_btyping: fixes E :: "char list \<Rightarrow> Poly_Types.ty"   and b :: "bexp"   and S :: "nat \<Rightarrow> Poly_Types.ty" assumes "E \<turnstile>p b" shows "tsubst S \<circ> E \<turnstile>p b"
trivial_group_subset: shows "trivial_group (Z 1) = (carrier (Z 1) \<subseteq> {\<one>\<^bsub>Z 1\<^esub>})"
comp_arr_inv: fixes f :: "'b"   and g :: "'b" assumes "B.inverse_arrows f g" shows "f \<cdot>\<^sub>B g = B.dom g"
continuous_at_setdist: fixes x :: "'a"   and S :: "'a set" shows "isCont (\<lambda>y. setdist {y} S) x"
put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_put_ptrs: fixes document_ptr :: "(_) document_ptr"   and document :: "('c, 'a, 'b, 'j option) RDocument_scheme"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "put document_ptr document h = h'" shows "object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|cast document_ptr|}"
vars_subst_subseteq: fixes s :: "('b, 'a) tm"   and t :: "('b, 'a) tm"   and \<rho> :: "'a \<Rightarrow> ('b, 'a) tm" assumes "vars s \<subseteq> vars t" shows "vars (subst \<rho> s) \<subseteq> vars (subst \<rho> t)"
inject_1: fixes x1 :: "'d"   and y1 :: "'d" shows "(Le x1 = Le y1) = (x1 = y1)"
simps_1: fixes s :: "'b stream" shows "s !! 0 = shd s"
Extensionality: fixes H :: "fm set"   and x :: "tm"   and y :: "tm" shows "H \<turnstile> x EQ y IFF x SUBS y AND y SUBS x"
sse_invar_step: fixes S :: "'b set \<times> 'b set"   and \<Sigma>i :: "'b set"   and R :: "('b \<times> 'b) set"   and S' :: "'b set \<times> 'b set" assumes "S \<in> sse_invar \<Sigma>i R"   and "(S, S') \<in> sse_step R" shows "S' \<in> sse_invar \<Sigma>i R"
inv_solve_right: fixes a :: "'a set"   and b :: "'a set"   and c :: "'a set" assumes "a \<in> carrier (R Quot I)"   and "b \<in> carrier (R Quot I)"   and "c \<in> carrier (R Quot I)" shows "(a = b \<oplus>\<^bsub>R Quot I\<^esub> \<ominus>\<^bsub>R Quot I\<^esub> c) = (b = a \<oplus>\<^bsub>R Quot I\<^esub> c)"
LIMSEQ_neg_powr: fixes s :: "real" assumes "s < 0" shows "(\<lambda>x. real x powr s) \<longlonglongrightarrow> 0"
simps_7: fixes P :: "pi"   and Q :: "pi" shows "summands (P \<oplus> Q) = summands P \<union> summands Q"
filterlim_at_top_choose_lower: fixes s :: "nat \<Rightarrow> real"   and f :: "real \<Rightarrow> 'b"   and l :: "'b"   and b :: "real" assumes "s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>"   and "(f \<circ> s) \<longlonglongrightarrow> l" obtains t :: "nat \<Rightarrow> real" where "t \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>" and "(f \<circ> t) \<longlonglongrightarrow> l" and "\<forall>n. b \<le> t n"
gen_common_boundary_subdiv_exists_refl_twochain_boundary: fixes C :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "\<forall>(k, \<gamma>)\<in>C. valid_path \<gamma>"   and "boundary_chain C" shows "common_sudiv_exists C C"
h_minus_aux_source: fixes h' :: "'v \<times> 'v \<Rightarrow> ennreal"   and j :: "nat"   and y :: "'v" assumes "\<And>e. h' e \<noteq> \<top>"   and "cycles \<Delta> \<noteq> {}" shows "h_minus_aux h' j (source \<Delta>, y) = 0"
comp_MkArr: fixes A :: "'e set"   and B :: "'e set"   and F :: "'e \<Rightarrow> 'e"   and C :: "'e set"   and G :: "'e \<Rightarrow> 'e" assumes "arr (MkArr A B F)"   and "arr (MkArr B C G)" shows "MkArr B C G \<cdot> MkArr A B F = MkArr A C (compose A G F)"
rep_eq: fixes x :: "'c" shows "blinfun_apply (blinfun_inner_left x) = (\<lambda>y. y \<bullet> x)"
HEndPhase1_valueChosen: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and v :: "InputsOrNi" assumes "HEndPhase1 s s' q"   and "valueChosen s v"   and "Inv1 s"   and "Inv2a s"   and "Inv2b s"   and "v \<in> Inputs" shows "valueChosen s' v"
dom_in_domains: fixes f :: "('o, 'a) arr" assumes "domains f \<noteq> {}" shows "local.dom f \<in> domains f"
shEx_lift_seq_2: fixes P :: "('b, 'd) urel"   and Q :: "'e \<Rightarrow> ('d, 'c) urel" shows "P ;; ushEx Q = (\<^bold>\<exists> x \<bullet> P ;; Q x)"
nn_comparable: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) \<le> a"   and "(0::'a) \<le> b" shows "a \<le> b \<or> b \<le> a"
paste_disj_domains: fixes P :: "('c \<times> 'd) set"   and Q :: "('c \<times> 'd) set" assumes "Domain P \<inter> Domain Q = {}" shows "P +< Q = P \<union> Q"
map_scale_one_left: fixes t :: "'c \<Rightarrow>\<^sub>0 'd" shows "(1::'d) \<cdot> t = t"
distrib_left: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "(b + c) \<cdot> a = b \<cdot> a + c \<cdot> a"
treeOf_seqOf: shows "fst \<circ> tree.root \<circ> treeOf pointsTo = seqOf"
llexord_coinduct: fixes X :: "'c llist \<Rightarrow> 'c llist \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'c llist"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "X xs ys"   and "\<And>xs ys. \<lbrakk>X xs ys; \<not> lnull xs\<rbrakk> \<Longrightarrow> \<not> lnull ys \<and> (\<not> lnull ys \<longrightarrow> r (lhd xs) (lhd ys) \<or> lhd xs = lhd ys \<and> (X (ltl xs) (ltl ys) \<or> llexord r (ltl xs) (ltl ys)))" shows "llexord r xs ys"
ex1_iff_Collect_singleton: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "P x" shows "(\<exists>!x. P x) = (Collect P = {x})"
freeIn_substInfinitesimalLinear: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and At :: "atom" assumes "var \<notin> vars a"   and "var \<notin> vars b" shows "freeIn var (substInfinitesimalLinear var a b At)"
crdt_ops_last: fixes xs :: "('c \<times> 'd) list"   and x :: "'c \<times> 'd"   and deps :: "'d \<Rightarrow> 'c set" assumes "crdt_ops (xs @ [x]) deps" obtains oid :: "'c" and oper :: "'d" where "x = (oid, oper)" and "crdt_ops xs deps" and "oid \<notin> fst ` set xs" and "\<forall>ref\<in>deps oper. ref \<in> fst ` set xs \<and> ref < oid"
assms_1: shows "g1 +++ g2 piecewise_C1_differentiable_on {0..1}"
quotients_butlast: assumes "1 < length \<HH>" shows "butlast comp\<HH>.quotients = normal_series.quotients (G\<lparr>carrier := \<HH> ! (length \<HH> - 1 - 1)\<rparr>) (take (length \<HH> - 1) \<HH>)"
contains_nreduced_imp_nreduced: fixes A :: "'a set"   and bs :: "'a list"   and as :: "'a list"   and cs :: "'a list" assumes "\<not> reduced_word A bs" shows "\<not> reduced_word A (as @ bs @ cs)"
InvariantImpliedLiteralsAfterBacktrack: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and M' :: "(Literal \<times> bool) list" assumes "InvariantImpliedLiterals F M"   and "InvariantUniq M"   and "InvariantConsistent M"   and "decisions M \<noteq> []"   and "formulaFalse F (elements M)"   and "M' = prefixBeforeLastDecision M @ [(opposite (lastDecision M), False)]" shows "InvariantImpliedLiterals F M'"
poly_deg_monom_mult_le: fixes c :: "'e"   and t :: "'d \<Rightarrow>\<^sub>0 'c"   and p :: "('d \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'e" shows "poly_deg (MPoly_Type_Class.punit.monom_mult c t p) \<le> deg_pm t + poly_deg p"
assms_2: shows "closedin (top_of_set (S \<union> T)) T"
simps: fixes h :: "'b" shows "the_state (Some h) = h"
spec_ops_add_last: fixes xs :: "('c \<times> 'd) list"   and deps :: "'d \<Rightarrow> 'c set"   and oid :: "'c"   and oper :: "'d" assumes "spec_ops xs deps"   and "\<forall>i\<in>set (map fst xs). i < oid"   and "\<forall>ref\<in>deps oper. ref < oid" shows "spec_ops (xs @ [(oid, oper)]) deps"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) \<nu>"
assms_7: shows "pf = (\<lambda>x. (a11 - 2 * a12 * k1 / k2 + a22 * k1\<^sup>2 / k2\<^sup>2) * x\<^sup>2 + (- 2 * a12 * n / k2 + b1 + a22 * 2 * n * k1 / k2\<^sup>2 - b2 * k1 / k2) * x + a22 * n\<^sup>2 / k2\<^sup>2 - b2 * n / k2 + c)"
circ_decompose_11: fixes x :: "'b"   and y :: "'b" shows "(x\<^sup>\<star> * y\<^sup>\<star>)\<^sup>\<star> = (x\<^sup>\<star> * y\<^sup>\<star>)\<^sup>\<star> * x\<^sup>\<star>"
fbox_whilei_break: fixes p :: "('b \<times> 'b) set"   and y :: "('b \<times> 'b) set"   and i :: "('b \<times> 'b) set"   and t :: "('b \<times> 'b) set"   and q :: "('b \<times> 'b) set"   and x :: "('b \<times> 'b) set" assumes "rdom p \<subseteq> wp y i"   and "rdom i ; rel_ad t \<subseteq> rdom q"   and "rdom i ; rdom t \<subseteq> wp x i" shows "rdom p \<subseteq> wp (y ; rel_antidomain_kleene_algebra.whilei t i x) q"
lookup_plus_pp: fixes s :: "('b, 'c) pp"   and t :: "('b, 'c) pp" shows "lookup_pp (s + t) = lookup_pp s + lookup_pp t"
paths: fixes xs :: "'a list" assumes "xs \<in> paths" shows "v0 \<leadsto>xs\<leadsto> v1"
integral_bigo: fixes f :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and a :: "real" assumes "f \<in> O(g')"   and "filterlim g at_top at_top"   and "\<And>a' x. \<lbrakk>a \<le> a'; a' \<le> x\<rbrakk> \<Longrightarrow> f integrable_on {a'..x}"   and "\<And>x. a \<le> x \<Longrightarrow> (g has_real_derivative g' x) (at x within {a..})"   and "continuous_on {a..} g'"   and "\<And>x. a \<le> x \<Longrightarrow> 0 \<le> g' x" shows "(\<lambda>x. integral {a..x} f) \<in> O(g)"
graph_successors: fixes v :: "nat \<times> 'a"   and A :: "('b, 'a) nba"   and w :: "'b stream"   and u :: "nat \<times> 'a"   and k :: "nat" assumes "v \<in> gunodes A w"   and "u \<in> gusuccessors A w v"   and "u \<in> graph A w k" shows "v \<in> graph A w k"
lift_cong: fixes inp :: "'a \<Rightarrow> 'b option"   and h :: "'b \<Rightarrow> 'c"   and h' :: "'b \<Rightarrow> 'c" assumes "\<And>i v. inp i = Some v \<Longrightarrow> h v = h' v" shows "lift h inp = lift h' inp"
simps_1: fixes h :: "'b \<Rightarrow> int"   and k :: "int" shows "tlookup h k Tip = None"
Pi_I': fixes A :: "'g set"   and f :: "'g \<Rightarrow> 'h"   and B :: "'g \<Rightarrow> 'h set" assumes "\<And>x. x \<in> A \<longrightarrow> f x \<in> B x" shows "f \<in> Pi A B"
H_eq_zero: fixes x :: "'c" assumes "x \<notin> ball (0::'c) 2" shows "H x = 0"
known_ptrs_subset: fixes h' :: "(_) heap"   and h :: "(_) heap" assumes "object_ptr_kinds h' |\<subseteq>| object_ptr_kinds h"   and "known_ptrs h" shows "known_ptrs h'"
SE2: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>F \<and> \<diamond>G \<longrightarrow> \<diamond>(F \<and> G)"
meet_3_in_def: fixes l :: "Lines"   and m :: "Lines"   and n :: "Lines"   and P :: "Points" shows "meet_3_in l m n P \<equiv> meet_in l m P \<and> meet_in l n P"
\<phi>_local_bij: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "bij_betw inC (C.hom b a) (HomC.set (b, a))"
PDG_wcd: shows "PDG sourcenode targetnode kind valid_edge (_Entry_) Def Use state_val (_Exit_) weak_control_dependence"
nullhomotopic_into_rel_frontier_extension: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set" assumes "closed S"   and "continuous_on S f"   and "convex T"   and "bounded T"   and "f ` S \<subseteq> rel_frontier T"   and "S \<noteq> {}" shows "(\<exists>c. homotopic_with_canon (\<lambda>x. True) S (rel_frontier T) f (\<lambda>x. c)) = (\<exists>g. continuous_on UNIV g \<and> range g \<subseteq> rel_frontier T \<and> (\<forall>x\<in>S. g x = f x))"
"0_2": shows "list_all (wf\<^sub>t\<^sub>r\<^sub>m' arity) (remdups (trms_list\<^sub>s\<^sub>s\<^sub>t (unlabel A) @ map (pair' Pair) (setops_list\<^sub>s\<^sub>s\<^sub>t (unlabel A))))"
bisimilar_symp: shows "symp (\<sim>\<cdot>\<^sub>S)"
InvariantImpliedLiteralsAndFormulaFalseThenFormulaAndDecisionsAreNotSatisfiable: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list" assumes "InvariantImpliedLiterals F M"   and "formulaFalse F (elements M)" shows "\<not> satisfiable (F @ val2form (decisions M))"
enn2real_mono: fixes a :: "ennreal"   and b :: "ennreal" assumes "a \<le> b"   and "b < top" shows "enn2real a \<le> enn2real b"
matrix_to_iarray_consistent: fixes A :: "(('d, 'e) vec, 'f) vec"   and b :: "('d, 'f) vec" shows "consistent A b = consistent_iarrays (matrix_to_iarray A) (vec_to_iarray b)"
lb_sqrt_lower_bound: fixes x :: "float"   and prec :: "nat" assumes "0 \<le> real_of_float x" shows "0 \<le> real_of_float (lb_sqrt prec x)"
Le_Le_dbm_lt_D: fixes a :: "'b"   and b :: "'b" assumes "Le a \<prec> Lt b" shows "a < b"
bigo_plus_self_subset: fixes f :: "'a \<Rightarrow> 'b" shows "O(f) + O(f) \<subseteq> O(f)"
simple_path_reversepath: fixes g :: "real \<Rightarrow> 'a" assumes "simple_path g" shows "simple_path (reversepath g)"
post: shows "post_del \<langle>rl, bl, rr\<rangle> r'"
2: shows "Y \<in> obj\<^bsub>Op C\<^esub>"
STEP': shows "(pc', s') \<in> set (exec (instrs ! pc) (pc, s))"
abs_sgn: fixes a :: "'a" shows "sgn \<bar>a\<bar> = of_bool (a \<noteq> (0::'a))"
induct: fixes P :: "'b rexp \<Rightarrow> 'b rexp \<Rightarrow> bool"   and a0 :: "'b rexp"   and a1 :: "'b rexp" assumes "\<And>r. P Zero r"   and "P One Zero"   and "\<And>v. P (Atom v) Zero"   and "\<And>v va. P (Plus v va) Zero"   and "\<And>v va. P (Times v va) Zero"   and "\<And>v. P (Star v) Zero"   and "\<And>r s. \<lbrakk>P s One; P r (nPlus s One)\<rbrakk> \<Longrightarrow> P (Plus r s) One"   and "\<And>r s v. \<lbrakk>P s (Atom v); P r (nPlus s (Atom v))\<rbrakk> \<Longrightarrow> P (Plus r s) (Atom v)"   and "\<And>r s v va. \<lbrakk>P s (Plus v va); P r (nPlus s (Plus v va))\<rbrakk> \<Longrightarrow> P (Plus r s) (Plus v va)"   and "\<And>r s v va. \<lbrakk>P s (Times v va); P r (nPlus s (Times v va))\<rbrakk> \<Longrightarrow> P (Plus r s) (Times v va)"   and "\<And>r s v. \<lbrakk>P s (Star v); P r (nPlus s (Star v))\<rbrakk> \<Longrightarrow> P (Plus r s) (Star v)"   and "\<And>s t. (\<lbrakk>One \<noteq> s; \<not> le_rexp One s\<rbrakk> \<Longrightarrow> P One t) \<Longrightarrow> P One (Plus s t)"   and "\<And>v s t. (\<lbrakk>Atom v \<noteq> s; \<not> le_rexp (Atom v) s\<rbrakk> \<Longrightarrow> P (Atom v) t) \<Longrightarrow> P (Atom v) (Plus s t)"   and "\<And>v va s t. (\<lbrakk>Times v va \<noteq> s; \<not> le_rexp (Times v va) s\<rbrakk> \<Longrightarrow> P (Times v va) t) \<Longrightarrow> P (Times v va) (Plus s t)"   and "\<And>v s t. (\<lbrakk>Star v \<noteq> s; \<not> le_rexp (Star v) s\<rbrakk> \<Longrightarrow> P (Star v) t) \<Longrightarrow> P (Star v) (Plus s t)"   and "P One One"   and "\<And>v. P One (Atom v)"   and "\<And>v va. P One (Times v va)"   and "\<And>v. P One (Star v)"   and "\<And>v. P (Atom v) One"   and "\<And>v va. P (Atom v) (Atom va)"   and "\<And>v va vb. P (Atom v) (Times va vb)"   and "\<And>v va. P (Atom v) (Star va)"   and "\<And>v va. P (Times v va) One"   and "\<And>v va vb. P (Times v va) (Atom vb)"   and "\<And>v va vb vc. P (Times v va) (Times vb vc)"   and "\<And>v va vb. P (Times v va) (Star vb)"   and "\<And>v. P (Star v) One"   and "\<And>v va. P (Star v) (Atom va)"   and "\<And>v va vb. P (Star v) (Times va vb)"   and "\<And>v va. P (Star v) (Star va)" shows "P a0 a1"
meet_def: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y = (THE inf. is_inf x y inf)"
S: fixes y :: "('c, 'a) vec" assumes "\<forall>i. y $ i \<in> A i" shows "y \<in> S"
elims_3: fixes x :: "'b queue" assumes "\<not> is_empty x" obtains q :: "'b queue" where "x = q" and "list q \<noteq> []"
assms_2: shows "\<forall>e\<in>pred c. finite (Bexp.vars e)"
max_min_same_3: fixes x :: "'c"   and y :: "'c" shows "max (min x y) y = y"
disj_conj_distr: fixes P :: "'a upred"   and Q :: "'a upred"   and R :: "'a upred" shows "(P \<or> Q \<and> R) = ((P \<or> Q) \<and> (P \<or> R))"
set_integral_fst': fixes A :: "'a set"   and B :: "'b set"   and f :: "'a \<times> 'b \<Rightarrow> 'c" assumes "set_integrable (lborel \<Otimes>\<^sub>M lborel) (A \<times> B) f" shows "set_lebesgue_integral (lborel \<Otimes>\<^sub>M lborel) (A \<times> B) f = LBINT x:A. (LBINT y:B. f (x, y))"
that: fixes h :: "nat \<Rightarrow> V" assumes "bij_betw h {..<card M} M"   and "strict_mono_on h {..<card M}" shows "thesis"
triangle_lemma: fixes x :: "real"   and y :: "real"   and z :: "real" assumes "0 \<le> x"   and "0 \<le> y"   and "0 \<le> z"   and "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2" shows "x \<le> y + z"
strict_is_neg_transitive: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "rational_preference carrier relation"   and "x \<in> carrier \<and> y \<in> carrier \<and> z \<in> carrier"   and "x \<succeq>[relation] y \<and> (y, x) \<notin> relation" shows "x \<succeq>[relation] z \<and> (z, x) \<notin> relation \<or> z \<succeq>[relation] y \<and> (y, z) \<notin> relation"
inject_3: fixes x31 :: "type_scheme"   and x32 :: "type_scheme"   and y31 :: "type_scheme"   and y32 :: "type_scheme" shows "(x31 =-> x32 = y31 =-> y32) = (x31 = y31 \<and> x32 = y32)"
boundedI: fixes b :: "'d"   and a :: "'d"   and c :: "'d" assumes "b \<le> a"   and "c \<le> a" shows "b \<squnion> c \<le> a"
isin_simps_2: fixes x :: "'b"   and xs :: "'b list" shows "sorted_less (x # xs) = ((\<forall>y\<in>set xs. x < y) \<and> sorted_less xs)"
IH: assumes "\<forall>u s. insert\<langle>u,s\<rangle> \<in> set B \<longrightarrow> Fun (Val n) [] \<noteq> u \<cdot> \<I>"   and "\<forall>u s. delete\<langle>u,s\<rangle> \<in> set B \<longrightarrow> Fun (Val n) [] \<noteq> u \<cdot> \<I>" shows "\<alpha>\<^sub>0 (db'\<^sub>s\<^sub>s\<^sub>t A \<I> D) n = \<alpha>\<^sub>0 (db'\<^sub>s\<^sub>s\<^sub>t (A @ B) \<I> D) n"
\<Z>_srngI: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z>_vfsequence \<alpha> \<SS>"   and "vcard \<SS> = 3\<^sub>\<nat>"   and "\<Z>_csgrp \<alpha> [\<SS>\<lparr>\<A>\<rparr>, \<SS>\<lparr>vplus\<rparr>]\<^sub>\<circ>"   and "\<Z>_sgrp \<alpha> [\<SS>\<lparr>\<A>\<rparr>, \<SS>\<lparr>vmult\<rparr>]\<^sub>\<circ>"   and "\<And>a b c. \<lbrakk>a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; c \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>\<rbrakk> \<Longrightarrow> (a +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b) *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c = a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c"   and "\<And>a b c. \<lbrakk>a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; c \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>\<rbrakk> \<Longrightarrow> a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> (b +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c) = a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c" shows "\<Z>_srng \<alpha> \<SS>"
Q_2: shows "pirreducible\<^bsub>M\<^esub> (carrier M) Q"
space_stream_space_M_ne: fixes x :: "real \<times> 'a" assumes "x \<in> space S" shows "space (stream_space S) \<noteq> {}"
simps_15: fixes x21 :: "'c"   and x22 :: "'c list" shows "set (x21 # x22) = Set.insert x21 (set x22)"
assms_1: shows "x \<in>\<^sub>\<circ> invimage D"
assms_3: shows "local.left_spec i\<^sub>0 \<le> Some i"
HInit_HInv3: fixes s :: "state" assumes "HInit s" shows "HInv3 s"
basis_ex1_supset_circuit: fixes B :: "'a set"   and x :: "'a" assumes "\<E>.basis B"   and "x \<in> \<E> - B" shows "\<exists>!C. \<E>.circuit C \<and> C \<subseteq> insert x B"